Subgroups — Q₈×C₂₃×AGL(1,ℤ/8) #19208992

Order 5888.

2010 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3386	C₂	1	6	yes	maximal, complemented
4	C₂₃×#3384	C₂	1	6	yes	maximal, complemented
5	Q₈×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
6	Q₈×D₈×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3165	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3079	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3077	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	3	yes	maximal
12	C₂³×Q₈×C₂₃	—	2	1	no	—
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	Q₈²×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#557	C₂²	1	3	yes	complemented
16	C₂₃×#554	—	2	3	no	—
17	C₂₃×#554	C₂²	1	3	yes	complemented
18	D₄×Q₈×C₂₃	—	2	1	no	—
19	D₄×Q₈×C₂₃	—	2	1	no	—
20	D₄×Q₈×C₂₃	—	2	3	no	—
21	D₄×Q₈×C₂₃	C₂²	1	2	yes	—
22	D₄×Q₈×C₂₃	C₂²	1	6	yes	—
23	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#533	—	2	3	no	—
25	C₂₃×#533	C₂²	1	3	yes	—
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
27	C₂×C₂₃×C₂²⋊Q₈	—	2	3	no	—
28	C₄×#41×C₂₃	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
32	C₂₃×#501	C₂²	1	12	yes	complemented
33	C₂₃×#500	C₂²	1	6	yes	complemented
34	C₂₃×#498	C₂²	1	6	yes	complemented
35	C₂₃×#478	C₂²	1	12	yes	—
36	C₂₃×#477	C₂²	1	6	yes	complemented
37	C₂₃×#476	C₂²	1	6	yes	—
38	C₂₃×#476	C₂²	1	6	yes	complemented
39	C₂₃×#475	C₂²	1	6	yes	complemented
40	C₂₃×#474	C₂²	1	6	yes	—
41	C₂₃×#474	C₂²	1	6	yes	complemented
42	C₂₃×#446	C₂²	1	6	yes	complemented
43	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
44	C₂₃×#442	C₂²	1	6	yes	—
45	C₂₃×#440	C₂²	1	6	yes	—
46	C₄×SD₁₆×C₂₃	C₂²	1	6	yes	—
47	C₄×D₈×C₂₃	C₂²	1	6	yes	—
48	C₂₃×#428	C₂²	1	3	yes	—
49	C₂₃×#428	C₂²	1	6	yes	—
50	C₂₃×#423	C₂²	1	3	yes	—
51	C₂₃×#418	C₂²	1	3	yes	—
52	C₂₃×#417	C₂²	1	3	yes	—
53	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
54	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	6	no	—
56	C₂²×Q₈×C₂₃	—	2	2	no	—
57	C₂²×Q₈×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	—	2	3	no	—
59	C₂²×Q₈×C₂₃	—	4	8	no	—
60	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	—	2	3	no	—
63	C₂₃×AGL(1,ℤ/8)	Q₈	1	8	yes	direct product factor
64	C₂×SD₁₆×C₂₃	C₂³	1	2	yes	—
65	C₂×D₈×C₂₃	C₂³	1	2	yes	—
66	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
67	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	—	2	3	no	—
69	C₂₃×C₄⋊Q₈	—	2	3	no	—
70	C₂₃×C₄⋊Q₈	—	2	3	no	—
71	C₂₃×C₄⋊Q₈	—	2	3	no	—
72	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
76	C₂₃×#125	—	2	6	no	—
77	C₂₃×#125	C₂³	1	6	yes	—
78	C₂₃×C₂²⋊Q₈	—	2	3	no	—
79	C₂₃×C₂²⋊Q₈	—	2	3	no	—
80	C₂₃×C₂²⋊Q₈	—	2	3	no	—
81	C₂₃×C₂²⋊Q₈	—	2	6	no	—
82	C₂₃×C₂²⋊Q₈	—	2	12	no	—
83	C₂₃×C₂²⋊Q₈	—	2	6	no	—
84	C₂₃×C₂²⋊Q₈	—	2	3	no	—
85	C₂₃×C₂²⋊Q₈	—	2	6	no	—
86	C₄×Q₈×C₂₃	—	2	1	no	—
87	C₄×Q₈×C₂₃	—	2	3	no	—
88	C₄×Q₈×C₂₃	—	2	3	no	—
89	C₄×Q₈×C₂₃	—	2	3	no	—
90	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₂₃	D₄	1	3	yes	—
92	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
94	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
96	C₄×D₄×C₂₃	—	2	3	no	—
97	C₄×D₄×C₂₃	—	2	3	no	—
98	C₄×D₄×C₂₃	—	2	3	no	—
99	C₄×D₄×C₂₃	—	2	3	no	—
100	C₄×D₄×C₂₃	C₂³	1	6	yes	—
101	C₄×D₄×C₂₃	C₂³	1	6	yes	—
102	C₄×D₄×C₂₃	C₂³	1	3	yes	—
103	C₄×D₄×C₂₃	C₂³	1	3	yes	—
104	C₂₃×#117	—	2	3	no	—
105	C₂₃×#117	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
111	C₂×C₄²×C₂₃	—	2	3	no	—
112	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
113	C₂₃×#107	C₂³	1	6	yes	—
114	C₂₃×#107	C₂³	1	12	yes	—
115	C₂₃×#106	C₂³	1	6	yes	—
116	C₂₃×#106	C₂³	1	12	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
119	C₂₃×#103	C₂³	1	6	yes	—
120	C₂₃×#102	C₂³	1	6	yes	—
121	C₂₃×#102	C₂³	1	6	yes	—
122	C₂₃×#102	C₂³	1	6	yes	—
123	C₂₃×#97	C₂³	1	6	yes	—
124	C₄×C₈×C₂₃	C₂³	1	6	yes	—
125	C₂⁴×C₂₃	—	2	1	no	—
126	#41×C₂₃	—	2	2	no	—
127	#41×C₂₃	C₂×Q₈	1	4	yes	complemented
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	—	2	3	no	—
130	C₂×Q₈×C₂₃	—	2	6	no	—
131	C₂×Q₈×C₂₃	—	2	6	no	—
132	C₂×Q₈×C₂₃	—	2	3	no	—
133	C₂×Q₈×C₂₃	—	4	12	no	—
134	C₂×Q₈×C₂₃	—	4	24	no	—
135	C₂×Q₈×C₂₃	—	4	4	no	—
136	C₂×Q₈×C₂₃	—	4	8	no	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
138	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	2	2	no	—
143	C₂×D₄×C₂₃	C₂×Q₈	1	4	yes	complemented
144	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
145	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	3	no	—
148	C₂²×C₄×C₂₃	—	2	3	no	—
149	C₂²×C₄×C₂₃	—	2	6	no	—
150	C₂²×C₄×C₂₃	—	2	6	no	—
151	C₂²×C₄×C₂₃	—	2	3	no	—
152	C₂²×C₄×C₂₃	—	4	12	no	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₂₃	C₂×Q₈	1	8	yes	complemented
157	D₈×C₂₃	C₂×Q₈	1	8	yes	complemented
158	M₁₆×C₂₃	—	2	6	no	—
159	M₁₆×C₂₃	C₂×Q₈	1	4	yes	complemented
160	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₂₃	—	2	6	no	—
163	C₄⋊C₄×C₂₃	—	2	3	no	—
164	C₄⋊C₄×C₂₃	—	2	6	no	—
165	C₄⋊C₄×C₂₃	—	2	6	no	—
166	C₄⋊C₄×C₂₃	—	2	3	no	—
167	C₄⋊C₄×C₂₃	—	2	3	no	—
168	C₄⋊C₄×C₂₃	—	2	3	no	—
169	C₄⋊C₄×C₂₃	—	2	3	no	—
170	C₄⋊C₄×C₂₃	—	2	6	no	—
171	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
175	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
177	C₂²⋊C₄×C₂₃	—	2	3	no	—
178	C₂²⋊C₄×C₂₃	—	2	6	no	—
179	C₂²⋊C₄×C₂₃	—	2	6	no	—
180	C₂²⋊C₄×C₂₃	—	2	3	no	—
181	C₂²⋊C₄×C₂₃	—	2	3	no	—
182	C₄²×C₂₃	—	2	3	no	—
183	C₄²×C₂₃	—	2	3	no	—
184	C₄²×C₂₃	—	2	3	no	—
185	C₄²×C₂₃	C₂×D₄	1	3	yes	—
186	C₄²×C₂₃	C₂×D₄	1	3	yes	—
187	C₄²×C₂₃	C₂⁴	1	3	yes	—
188	C₄²×C₂₃	C₂⁴	1	3	yes	—
189	Q₈×AGL(1,ℤ/8)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	2	1	no	—
192	C₂³×C₂₃	—	2	2	no	—
193	C₂³×C₂₃	—	4	2	no	—
194	C₂³×C₂₃	—	4	2	no	—
195	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₂₃	—	2	6	no	—
197	Q₈×C₂₃	—	2	1	no	—
198	Q₈×C₂₃	—	4	12	no	—
199	Q₈×C₂₃	—	4	6	no	—
200	Q₈×C₂₃	—	4	12	no	—
201	Q₈×C₂₃	—	8	3	no	—
202	Q₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	direct product factor
203	Q₈×C₂₃	C₂²×Q₈	1	2	yes	complemented
204	D₄×C₂₃	—	2	4	no	—
205	D₄×C₂₃	—	2	4	no	—
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	2	1	no	—
208	D₄×C₂₃	C₂²×Q₈	1	4	yes	—
209	D₄×C₂₃	C₂²×Q₈	1	2	yes	complemented
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	6	no	—
213	C₂×C₄×C₂₃	—	2	3	no	—
214	C₂×C₄×C₂₃	—	2	3	no	—
215	C₂×C₄×C₂₃	—	2	6	no	—
216	C₂×C₄×C₂₃	—	4	3	no	—
217	C₂×C₄×C₂₃	—	4	12	no	—
218	C₂×C₄×C₂₃	—	4	6	no	—
219	C₂×C₄×C₂₃	—	4	12	no	—
220	C₂×C₄×C₂₃	—	4	6	no	—
221	C₂×C₄×C₂₃	—	8	3	no	—
222	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
228	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
229	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
230	C₈×C₂₃	—	2	6	no	—
231	C₈×C₂₃	C₂²×Q₈	1	4	yes	complemented
232	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#3386	C₂×C₂₃	1	6	yes	complemented
235	#3384	C₂×C₂₃	1	6	yes	complemented
236	Q₈×SD₁₆	C₂×C₂₃	1	2	yes	complemented
237	Q₈×D₈	C₂×C₂₃	1	2	yes	complemented
238	#3165	C₂×C₂₃	1	3	yes	complemented
239	#3079	C₂×C₂₃	1	3	yes	complemented
240	#3077	C₂×C₂₃	1	3	yes	complemented
241	Q₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	3	yes	—
243	C₂²×C₂₃	—	2	1	no	—
244	C₂²×C₂₃	—	2	1	no	—
245	C₂²×C₂₃	—	2	2	no	—
246	C₂²×C₂₃	—	2	4	no	—
247	C₂²×C₂₃	—	4	8	no	—
248	C₂²×C₂₃	—	4	1	no	—
249	C₂²×C₂₃	—	4	2	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	1	no	—
252	C₂²×C₂₃	D₄×Q₈	1	2	yes	—
253	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
254	C₄×C₂₃	—	2	3	no	—
255	C₄×C₂₃	—	2	3	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	2	3	no	—
258	C₄×C₂₃	—	4	3	no	—
259	C₄×C₂₃	—	4	6	no	—
260	C₄×C₂₃	—	4	3	no	—
261	C₄×C₂₃	D₄×Q₈	1	2	yes	—
262	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
263	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
264	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	C₂³×Q₈	—	2	1	no	—
266	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
267	Q₈²	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	#557	C₂²×C₂₃	1	3	yes	complemented
269	#554	—	2	3	no	—
270	#554	C₂²×C₂₃	1	3	yes	complemented
271	D₄×Q₈	—	2	1	no	—
272	D₄×Q₈	—	2	1	no	—
273	D₄×Q₈	—	2	3	no	—
274	D₄×Q₈	C₂²×C₂₃	1	2	yes	—
275	D₄×Q₈	C₂²×C₂₃	1	6	yes	—
276	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
277	#533	—	2	3	no	—
278	#533	C₂²×C₂₃	1	3	yes	—
279	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
280	C₂×C₂²⋊Q₈	—	2	3	no	—
281	C₄×#41	C₂²×C₂₃	1	3	yes	—
282	C₂×C₄×Q₈	—	2	1	no	—
283	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
284	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
285	#501	C₂²×C₂₃	1	12	yes	complemented
286	#500	C₂²×C₂₃	1	6	yes	complemented
287	#498	C₂²×C₂₃	1	6	yes	complemented
288	#478	C₂²×C₂₃	1	12	yes	—
289	#477	C₂²×C₂₃	1	6	yes	complemented
290	#476	C₂²×C₂₃	1	6	yes	—
291	#476	C₂²×C₂₃	1	6	yes	complemented
292	#475	C₂²×C₂₃	1	6	yes	complemented
293	#474	C₂²×C₂₃	1	6	yes	—
294	#474	C₂²×C₂₃	1	6	yes	complemented
295	#446	C₂²×C₂₃	1	6	yes	complemented
296	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
297	#442	C₂²×C₂₃	1	6	yes	—
298	#440	C₂²×C₂₃	1	6	yes	—
299	C₄×SD₁₆	C₂²×C₂₃	1	6	yes	—
300	C₄×D₈	C₂²×C₂₃	1	6	yes	—
301	#428	C₂²×C₂₃	1	6	yes	—
302	#428	C₂²×C₂₃	1	3	yes	—
303	#423	C₂²×C₂₃	1	3	yes	—
304	#418	C₂²×C₂₃	1	3	yes	—
305	#417	C₂²×C₂₃	1	3	yes	—
306	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
307	C₂×C₂₃	—	2	2	no	—
308	C₂×C₂₃	—	4	2	no	—
309	C₂×C₂₃	—	4	4	no	—
310	C₂×C₂₃	C₂×D₄×Q₈	1	1	yes	characteristic
311	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂×C₂₃	#3586	1	1	yes	characteristic
313	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	—	2	6	no	—
316	C₂²×Q₈	—	2	2	no	—
317	C₂²×Q₈	—	2	3	no	—
318	C₂²×Q₈	—	4	8	no	—
319	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
320	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
321	C₂³×C₄	—	2	3	no	—
322	AGL(1,ℤ/8)	Q₈×C₂₃	1	8	yes	direct product factor
323	C₂×SD₁₆	C₂³×C₂₃	1	2	yes	—
324	C₂×D₈	C₂³×C₂₃	1	2	yes	—
325	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
326	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	3	no	—
328	C₄⋊Q₈	—	2	3	no	—
329	C₄⋊Q₈	—	2	3	no	—
330	C₄⋊Q₈	—	2	3	no	—
331	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
332	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
333	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
334	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
335	#125	—	2	6	no	—
336	#125	C₂³×C₂₃	1	6	yes	—
337	C₂²⋊Q₈	—	2	3	no	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	—	2	6	no	—
341	C₂²⋊Q₈	—	2	12	no	—
342	C₂²⋊Q₈	—	2	6	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	6	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×Q₈	—	2	3	no	—
347	C₄×Q₈	—	2	3	no	—
348	C₄×Q₈	—	2	3	no	—
349	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
350	C₄×Q₈	D₄×C₂₃	1	3	yes	—
351	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
352	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
353	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
354	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
355	C₄×D₄	—	2	3	no	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	C₂³×C₂₃	1	6	yes	—
360	C₄×D₄	C₂³×C₂₃	1	6	yes	—
361	C₄×D₄	C₂³×C₂₃	1	3	yes	—
362	C₄×D₄	C₂³×C₂₃	1	3	yes	—
363	#117	—	2	3	no	—
364	#117	C₂³×C₂₃	1	3	yes	—
365	C₂×C₄⋊C₄	—	2	3	no	—
366	C₂×C₄⋊C₄	—	2	3	no	—
367	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
368	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
369	C₂×C₂²⋊C₄	—	2	3	no	—
370	C₂×C₄²	—	2	3	no	—
371	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
372	#107	C₂³×C₂₃	1	6	yes	—
373	#107	C₂³×C₂₃	1	12	yes	—
374	#106	C₂³×C₂₃	1	6	yes	—
375	#106	C₂³×C₂₃	1	12	yes	—
376	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
377	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
378	#103	C₂³×C₂₃	1	6	yes	—
379	#102	C₂³×C₂₃	1	6	yes	—
380	#102	C₂³×C₂₃	1	6	yes	—
381	#102	C₂³×C₂₃	1	6	yes	—
382	#97	C₂³×C₂₃	1	6	yes	—
383	C₄×C₈	C₂³×C₂₃	1	6	yes	—
384	C₂₃	Q₈×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
385	C₂⁴	—	2	1	no	—
386	#41	—	2	2	no	—
387	#41	C₂×Q₈×C₂₃	1	4	yes	complemented
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	6	no	—
391	C₂×Q₈	—	2	6	no	—
392	C₂×Q₈	—	2	3	no	—
393	C₂×Q₈	—	4	12	no	—
394	C₂×Q₈	—	4	24	no	—
395	C₂×Q₈	—	4	8	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
398	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	C₂×Q₈×C₂₃	1	4	yes	complemented
404	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
405	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	6	no	—
411	C₂²×C₄	—	2	3	no	—
412	C₂²×C₄	—	4	12	no	—
413	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
414	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
415	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
416	SD₁₆	C₂×Q₈×C₂₃	1	8	yes	complemented
417	D₈	C₂×Q₈×C₂₃	1	8	yes	complemented
418	M₁₆	—	2	6	no	—
419	M₁₆	C₂×Q₈×C₂₃	1	4	yes	complemented
420	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
421	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
422	C₄⋊C₄	—	2	6	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	6	no	—
425	C₄⋊C₄	—	2	6	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	3	no	—
429	C₄⋊C₄	—	2	3	no	—
430	C₄⋊C₄	—	2	6	no	—
431	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
432	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
433	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
434	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
435	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
436	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
437	C₂²⋊C₄	—	2	3	no	—
438	C₂²⋊C₄	—	2	6	no	—
439	C₂²⋊C₄	—	2	6	no	—
440	C₂²⋊C₄	—	2	3	no	—
441	C₂²⋊C₄	—	2	3	no	—
442	C₄²	—	2	3	no	—
443	C₄²	—	2	3	no	—
444	C₄²	—	2	3	no	—
445	C₄²	C₂×D₄×C₂₃	1	3	yes	—
446	C₄²	C₂×D₄×C₂₃	1	3	yes	—
447	C₄²	C₂⁴×C₂₃	1	3	yes	—
448	C₄²	C₂⁴×C₂₃	1	3	yes	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
455	Q₈	—	2	6	no	—
456	Q₈	—	2	1	no	—
457	Q₈	—	4	12	no	—
458	Q₈	—	4	12	no	—
459	Q₈	—	4	6	no	—
460	Q₈	—	8	3	no	—
461	Q₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	direct product factor
462	Q₈	C₂²×Q₈×C₂₃	1	2	yes	complemented
463	D₄	—	2	4	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	2	no	—
466	D₄	—	2	1	no	—
467	D₄	C₂²×Q₈×C₂₃	1	4	yes	—
468	D₄	C₂²×Q₈×C₂₃	1	2	yes	complemented
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	6	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	6	no	—
475	C₂×C₄	—	4	3	no	—
476	C₂×C₄	—	4	12	no	—
477	C₂×C₄	—	4	6	no	—
478	C₂×C₄	—	4	12	no	—
479	C₂×C₄	—	4	6	no	—
480	C₂×C₄	—	8	3	no	—
481	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
482	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
486	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
487	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
488	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
489	C₈	—	2	6	no	—
490	C₈	C₂²×Q₈×C₂₃	1	4	yes	complemented
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	D₄×Q₈×C₂₃	1	2	yes	—
501	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
502	C₄	—	2	2	no	—
503	C₄	—	2	3	no	—
504	C₄	—	2	3	no	—
505	C₄	—	2	3	no	—
506	C₄	—	4	6	no	—
507	C₄	—	4	3	no	—
508	C₄	—	4	3	no	—
509	C₄	D₄×Q₈×C₂₃	1	2	yes	—
510	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	6	yes	—
511	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
512	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	4	2	no	—
516	C₂	C₂×D₄×Q₈×C₂₃	1	1	yes	characteristic
517	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
518	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×#41×C₂₃	C₂	semidirect
2. C₂×D₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#3386	C₂	semidirect
4. C₂₃×#3384	C₂	semidirect
5. Q₈×SD₁₆×C₂₃	C₂	semidirect
6. Q₈×D₈×C₂₃	C₂	semidirect
7. C₂₃×#3165	C₂	semidirect
8. C₂₃×#3079	C₂	semidirect
9. C₂₃×#3077	C₂	semidirect
10. Q₈×M₁₆×C₂₃	C₂	semidirect
14. Q₈²×C₂₃	C₂²	semidirect
15. C₂₃×#557	C₂²	semidirect
17. C₂₃×#554	C₂²	semidirect
23. D₄×Q₈×C₂₃	C₂²	semidirect
32. C₂₃×#501	C₂²	semidirect
33. C₂₃×#500	C₂²	semidirect
34. C₂₃×#498	C₂²	semidirect
36. C₂₃×#477	C₂²	semidirect
38. C₂₃×#476	C₂²	semidirect
39. C₂₃×#475	C₂²	semidirect
41. C₂₃×#474	C₂²	semidirect
42. C₂₃×#446	C₂²	semidirect
43. C₈×Q₈×C₂₃	C₂²	semidirect
63. C₂₃×AGL(1,ℤ/8)	Q₈	direct (N × H)
127. #41×C₂₃	C₂×Q₈	semidirect
143. C₂×D₄×C₂₃	C₂×Q₈	semidirect
156. SD₁₆×C₂₃	C₂×Q₈	semidirect
157. D₈×C₂₃	C₂×Q₈	semidirect
159. M₁₆×C₂₃	C₂×Q₈	semidirect
189. Q₈×AGL(1,ℤ/8)	C₂₃	direct (N × H)
202. Q₈×C₂₃	AGL(1,ℤ/8)	direct (N × H)
203. Q₈×C₂₃	C₂²×Q₈	semidirect
209. D₄×C₂₃	C₂²×Q₈	semidirect
231. C₈×C₂₃	C₂²×Q₈	semidirect
232. Q₈×#41	C₂×C₂₃	semidirect
233. C₂×D₄×Q₈	C₂×C₂₃	semidirect
234. #3386	C₂×C₂₃	semidirect
235. #3384	C₂×C₂₃	semidirect
236. Q₈×SD₁₆	C₂×C₂₃	semidirect
237. Q₈×D₈	C₂×C₂₃	semidirect
238. #3165	C₂×C₂₃	semidirect
239. #3079	C₂×C₂₃	semidirect
240. #3077	C₂×C₂₃	semidirect
241. Q₈×M₁₆	C₂×C₂₃	semidirect
267. Q₈²	C₂²×C₂₃	semidirect
268. #557	C₂²×C₂₃	semidirect
270. #554	C₂²×C₂₃	semidirect
276. D₄×Q₈	C₂²×C₂₃	semidirect
285. #501	C₂²×C₂₃	semidirect
286. #500	C₂²×C₂₃	semidirect
287. #498	C₂²×C₂₃	semidirect
289. #477	C₂²×C₂₃	semidirect
291. #476	C₂²×C₂₃	semidirect
292. #475	C₂²×C₂₃	semidirect
294. #474	C₂²×C₂₃	semidirect
295. #446	C₂²×C₂₃	semidirect
296. C₈×Q₈	C₂²×C₂₃	semidirect
322. AGL(1,ℤ/8)	Q₈×C₂₃	direct (N × H)
384. C₂₃	Q₈×AGL(1,ℤ/8)	direct (N × H)
387. #41	C₂×Q₈×C₂₃	semidirect
403. C₂×D₄	C₂×Q₈×C₂₃	semidirect
416. SD₁₆	C₂×Q₈×C₂₃	semidirect
417. D₈	C₂×Q₈×C₂₃	semidirect
419. M₁₆	C₂×Q₈×C₂₃	semidirect
461. Q₈	C₂₃×AGL(1,ℤ/8)	direct (N × H)
462. Q₈	C₂²×Q₈×C₂₃	semidirect
468. D₄	C₂²×Q₈×C₂₃	semidirect
490. C₈	C₂²×Q₈×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×#41×C₂₃	2
C₂×D₄×Q₈×C₂₃	2
C₂₃×#3386	2
C₂₃×#3384	2
Q₈×SD₁₆×C₂₃	2
Q₈×D₈×C₂₃	2
C₂₃×#3165	2
C₂₃×#3079	2
C₂₃×#3077	2
Q₈×M₁₆×C₂₃	2
C₄×C₂₃×AGL(1,ℤ/8)	2
Q₈×AGL(1,ℤ/8)	23

Subgroups — Q8×C23×AGL(1,ℤ/8) #19208992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₃×AGL(1,ℤ/8) #19208992