Subgroups — C₇×Q₈×AGL(1,ℤ/8) #13238582

Order 1792.

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	C₇×Q₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3386	C₂	1	6	yes	maximal, complemented
4	C₇×#3384	C₂	1	6	yes	maximal, complemented
5	C₇×Q₈×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₇×Q₈×D₈	C₂	1	2	yes	maximal, complemented
7	C₇×#3165	C₂	1	3	yes	maximal, complemented
8	C₇×#3079	C₂	1	3	yes	maximal, complemented
9	C₇×#3077	C₂	1	3	yes	maximal, complemented
10	C₇×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×AGL(1,ℤ/8)	C₂	1	3	yes	maximal
12	C₂³×C₇×Q₈	—	2	1	no	—
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	C₇×Q₈²	C₂²	1	1	yes	characteristic, complemented
15	C₇×#557	C₂²	1	3	yes	complemented
16	C₇×#554	—	2	3	no	—
17	C₇×#554	C₂²	1	3	yes	complemented
18	C₇×D₄×Q₈	—	2	1	no	—
19	C₇×D₄×Q₈	—	2	1	no	—
20	C₇×D₄×Q₈	—	2	3	no	—
21	C₇×D₄×Q₈	C₂²	1	2	yes	—
22	C₇×D₄×Q₈	C₂²	1	6	yes	—
23	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₇×#533	—	2	3	no	—
25	C₇×#533	C₂²	1	3	yes	—
26	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
27	C₂×C₇×C₂²⋊Q₈	—	2	3	no	—
28	C₄×C₇×#41	C₂²	1	3	yes	—
29	C₂×C₄×C₇×Q₈	—	2	1	no	—
30	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
32	C₇×#501	C₂²	1	12	yes	complemented
33	C₇×#500	C₂²	1	6	yes	complemented
34	C₇×#498	C₂²	1	6	yes	complemented
35	C₇×#478	C₂²	1	12	yes	—
36	C₇×#477	C₂²	1	6	yes	complemented
37	C₇×#476	C₂²	1	6	yes	—
38	C₇×#476	C₂²	1	6	yes	complemented
39	C₇×#475	C₂²	1	6	yes	complemented
40	C₇×#474	C₂²	1	6	yes	—
41	C₇×#474	C₂²	1	6	yes	complemented
42	C₇×#446	C₂²	1	6	yes	complemented
43	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
44	C₇×#442	C₂²	1	6	yes	—
45	C₇×#440	C₂²	1	6	yes	—
46	C₄×C₇×SD₁₆	C₂²	1	6	yes	—
47	C₄×C₇×D₈	C₂²	1	6	yes	—
48	C₇×#428	C₂²	1	3	yes	—
49	C₇×#428	C₂²	1	6	yes	—
50	C₇×#423	C₂²	1	3	yes	—
51	C₇×#418	C₂²	1	3	yes	—
52	C₇×#417	C₂²	1	3	yes	—
53	C₄×C₇×M₁₆	C₂²	1	3	yes	—
54	Q₈×AGL(1,ℤ/8)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₇×#41	C₂³	1	1	yes	characteristic
56	C₂²×C₇×Q₈	—	2	6	no	—
57	C₂²×C₇×Q₈	—	2	2	no	—
58	C₂²×C₇×Q₈	—	2	1	no	—
59	C₂²×C₇×Q₈	—	2	3	no	—
60	C₂²×C₇×Q₈	—	4	8	no	—
61	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
62	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₇	—	2	3	no	—
64	C₇×AGL(1,ℤ/8)	Q₈	1	8	yes	direct product factor
65	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
66	C₂×C₇×D₈	C₂³	1	2	yes	—
67	C₂×C₇×M₁₆	C₂³	1	3	yes	—
68	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊Q₈	—	2	3	no	—
70	C₇×C₄⋊Q₈	—	2	3	no	—
71	C₇×C₄⋊Q₈	—	2	3	no	—
72	C₇×C₄⋊Q₈	—	2	3	no	—
73	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
74	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
75	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
76	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
77	C₇×#125	—	2	6	no	—
78	C₇×#125	C₂³	1	6	yes	—
79	C₇×C₂²⋊Q₈	—	2	3	no	—
80	C₇×C₂²⋊Q₈	—	2	3	no	—
81	C₇×C₂²⋊Q₈	—	2	3	no	—
82	C₇×C₂²⋊Q₈	—	2	6	no	—
83	C₇×C₂²⋊Q₈	—	2	12	no	—
84	C₇×C₂²⋊Q₈	—	2	6	no	—
85	C₇×C₂²⋊Q₈	—	2	3	no	—
86	C₇×C₂²⋊Q₈	—	2	6	no	—
87	C₄×C₇×Q₈	—	2	1	no	—
88	C₄×C₇×Q₈	—	2	3	no	—
89	C₄×C₇×Q₈	—	2	3	no	—
90	C₄×C₇×Q₈	—	2	3	no	—
91	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
92	C₄×C₇×Q₈	D₄	1	3	yes	—
93	C₄×C₇×Q₈	C₂³	1	3	yes	—
94	C₄×C₇×Q₈	C₂³	1	3	yes	—
95	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
96	C₄×C₇×Q₈	C₂³	1	3	yes	—
97	C₄×C₇×D₄	—	2	3	no	—
98	C₄×C₇×D₄	—	2	3	no	—
99	C₄×C₇×D₄	—	2	3	no	—
100	C₄×C₇×D₄	—	2	3	no	—
101	C₄×C₇×D₄	C₂³	1	6	yes	—
102	C₄×C₇×D₄	C₂³	1	3	yes	—
103	C₄×C₇×D₄	C₂³	1	3	yes	—
104	C₄×C₇×D₄	C₂³	1	6	yes	—
105	C₇×#117	—	2	3	no	—
106	C₇×#117	C₂³	1	3	yes	—
107	C₂×C₇×C₄⋊C₄	—	2	3	no	—
108	C₂×C₇×C₄⋊C₄	—	2	3	no	—
109	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
111	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
112	C₂×C₄²×C₇	—	2	3	no	—
113	C₂×C₄²×C₇	C₂³	1	3	yes	—
114	C₇×#107	C₂³	1	12	yes	—
115	C₇×#107	C₂³	1	6	yes	—
116	C₇×#106	C₂³	1	12	yes	—
117	C₇×#106	C₂³	1	6	yes	—
118	C₇×C₄⋊C₈	C₂³	1	6	yes	—
119	C₇×C₄⋊C₈	C₂³	1	6	yes	—
120	C₇×#103	C₂³	1	6	yes	—
121	C₇×#102	C₂³	1	6	yes	—
122	C₇×#102	C₂³	1	6	yes	—
123	C₇×#102	C₂³	1	6	yes	—
124	C₇×#97	C₂³	1	6	yes	—
125	C₄×C₇×C₈	C₂³	1	6	yes	—
126	Q₈×#41	C₂×C₇	1	1	yes	characteristic, complemented
127	C₂×D₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
128	#3386	C₂×C₇	1	6	yes	complemented
129	#3384	C₂×C₇	1	6	yes	complemented
130	Q₈×SD₁₆	C₂×C₇	1	2	yes	complemented
131	Q₈×D₈	C₂×C₇	1	2	yes	complemented
132	#3165	C₂×C₇	1	3	yes	complemented
133	#3079	C₂×C₇	1	3	yes	complemented
134	#3077	C₂×C₇	1	3	yes	complemented
135	Q₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
136	C₄×AGL(1,ℤ/8)	C₂×C₇	1	3	yes	—
137	C₂⁴×C₇	—	2	1	no	—
138	C₇×#41	—	2	2	no	—
139	C₇×#41	C₂×Q₈	1	4	yes	complemented
140	C₂×C₇×Q₈	—	2	4	no	—
141	C₂×C₇×Q₈	—	2	3	no	—
142	C₂×C₇×Q₈	—	2	6	no	—
143	C₂×C₇×Q₈	—	2	6	no	—
144	C₂×C₇×Q₈	—	2	3	no	—
145	C₂×C₇×Q₈	—	4	12	no	—
146	C₂×C₇×Q₈	—	4	24	no	—
147	C₂×C₇×Q₈	—	4	4	no	—
148	C₂×C₇×Q₈	—	4	8	no	—
149	C₂×C₇×Q₈	C₂×D₄	1	3	yes	—
150	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
152	C₂×C₇×D₄	—	2	1	no	—
153	C₂×C₇×D₄	—	2	1	no	—
154	C₂×C₇×D₄	—	2	2	no	—
155	C₂×C₇×D₄	C₂×Q₈	1	4	yes	complemented
156	C₂×C₇×D₄	C₂⁴	1	2	yes	—
157	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₇	—	2	1	no	—
159	C₂²×C₄×C₇	—	2	3	no	—
160	C₂²×C₄×C₇	—	2	3	no	—
161	C₂²×C₄×C₇	—	2	6	no	—
162	C₂²×C₄×C₇	—	2	6	no	—
163	C₂²×C₄×C₇	—	2	3	no	—
164	C₂²×C₄×C₇	—	4	12	no	—
165	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
166	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
167	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
168	C₇×SD₁₆	C₂×Q₈	1	8	yes	complemented
169	C₇×D₈	C₂×Q₈	1	8	yes	complemented
170	C₇×M₁₆	—	2	6	no	—
171	C₇×M₁₆	C₂×Q₈	1	4	yes	complemented
172	C₂×C₇×C₈	C₂⁴	1	6	yes	—
173	C₂×C₇×C₈	C₂⁴	1	2	yes	—
174	C₇×C₄⋊C₄	—	2	6	no	—
175	C₇×C₄⋊C₄	—	2	3	no	—
176	C₇×C₄⋊C₄	—	2	6	no	—
177	C₇×C₄⋊C₄	—	2	6	no	—
178	C₇×C₄⋊C₄	—	2	3	no	—
179	C₇×C₄⋊C₄	—	2	3	no	—
180	C₇×C₄⋊C₄	—	2	3	no	—
181	C₇×C₄⋊C₄	—	2	3	no	—
182	C₇×C₄⋊C₄	—	2	6	no	—
183	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
184	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
185	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
186	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
187	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
188	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
189	C₇×C₂²⋊C₄	—	2	3	no	—
190	C₇×C₂²⋊C₄	—	2	6	no	—
191	C₇×C₂²⋊C₄	—	2	6	no	—
192	C₇×C₂²⋊C₄	—	2	3	no	—
193	C₇×C₂²⋊C₄	—	2	3	no	—
194	C₄²×C₇	—	2	3	no	—
195	C₄²×C₇	—	2	3	no	—
196	C₄²×C₇	—	2	3	no	—
197	C₄²×C₇	C₂×D₄	1	3	yes	—
198	C₄²×C₇	C₂×D₄	1	3	yes	—
199	C₄²×C₇	C₂⁴	1	3	yes	—
200	C₄²×C₇	C₂⁴	1	3	yes	—
201	C₂³×Q₈	—	2	1	no	—
202	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	1	yes	characteristic
203	Q₈²	C₂²×C₇	1	1	yes	characteristic, complemented
204	#557	C₂²×C₇	1	3	yes	complemented
205	#554	—	2	3	no	—
206	#554	C₂²×C₇	1	3	yes	complemented
207	D₄×Q₈	—	2	1	no	—
208	D₄×Q₈	—	2	1	no	—
209	D₄×Q₈	—	2	3	no	—
210	D₄×Q₈	C₂²×C₇	1	2	yes	—
211	D₄×Q₈	C₂²×C₇	1	6	yes	—
212	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
213	#533	—	2	3	no	—
214	#533	C₂²×C₇	1	3	yes	—
215	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
216	C₂×C₂²⋊Q₈	—	2	3	no	—
217	C₄×#41	C₂²×C₇	1	3	yes	—
218	C₂×C₄×Q₈	—	2	1	no	—
219	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
220	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
221	#501	C₂²×C₇	1	12	yes	complemented
222	#500	C₂²×C₇	1	6	yes	complemented
223	#498	C₂²×C₇	1	6	yes	complemented
224	#478	C₂²×C₇	1	12	yes	—
225	#477	C₂²×C₇	1	6	yes	complemented
226	#476	C₂²×C₇	1	6	yes	—
227	#476	C₂²×C₇	1	6	yes	complemented
228	#475	C₂²×C₇	1	6	yes	complemented
229	#474	C₂²×C₇	1	6	yes	—
230	#474	C₂²×C₇	1	6	yes	complemented
231	#446	C₂²×C₇	1	6	yes	complemented
232	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
233	#442	C₂²×C₇	1	6	yes	—
234	#440	C₂²×C₇	1	6	yes	—
235	C₄×SD₁₆	C₂²×C₇	1	6	yes	—
236	C₄×D₈	C₂²×C₇	1	6	yes	—
237	#428	C₂²×C₇	1	6	yes	—
238	#428	C₂²×C₇	1	3	yes	—
239	#423	C₂²×C₇	1	3	yes	—
240	#418	C₂²×C₇	1	3	yes	—
241	#417	C₂²×C₇	1	3	yes	—
242	C₄×M₁₆	C₂²×C₇	1	3	yes	—
243	C₂³×C₇	—	2	4	no	—
244	C₂³×C₇	—	2	1	no	—
245	C₂³×C₇	—	2	2	no	—
246	C₂³×C₇	—	4	2	no	—
247	C₂³×C₇	—	4	2	no	—
248	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
249	C₇×Q₈	—	2	6	no	—
250	C₇×Q₈	—	2	1	no	—
251	C₇×Q₈	—	4	12	no	—
252	C₇×Q₈	—	4	6	no	—
253	C₇×Q₈	—	4	12	no	—
254	C₇×Q₈	—	8	3	no	—
255	C₇×Q₈	AGL(1,ℤ/8)	1	4	yes	direct product factor
256	C₇×Q₈	C₂²×Q₈	1	2	yes	complemented
257	C₇×D₄	—	2	4	no	—
258	C₇×D₄	—	2	4	no	—
259	C₇×D₄	—	2	2	no	—
260	C₇×D₄	—	2	1	no	—
261	C₇×D₄	C₂²×Q₈	1	4	yes	—
262	C₇×D₄	C₂²×Q₈	1	2	yes	complemented
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	4	no	—
265	C₂×C₄×C₇	—	2	6	no	—
266	C₂×C₄×C₇	—	2	3	no	—
267	C₂×C₄×C₇	—	2	3	no	—
268	C₂×C₄×C₇	—	2	6	no	—
269	C₂×C₄×C₇	—	4	3	no	—
270	C₂×C₄×C₇	—	4	12	no	—
271	C₂×C₄×C₇	—	4	6	no	—
272	C₂×C₄×C₇	—	4	12	no	—
273	C₂×C₄×C₇	—	4	6	no	—
274	C₂×C₄×C₇	—	8	3	no	—
275	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
276	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
277	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
278	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
280	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
281	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
282	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
283	C₇×C₈	—	2	6	no	—
284	C₇×C₈	C₂²×Q₈	1	4	yes	complemented
285	C₂×#41	C₂³×C₇	1	1	yes	characteristic
286	C₂²×Q₈	—	2	1	no	—
287	C₂²×Q₈	—	2	6	no	—
288	C₂²×Q₈	—	2	2	no	—
289	C₂²×Q₈	—	2	3	no	—
290	C₂²×Q₈	—	4	8	no	—
291	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
292	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
293	C₂³×C₄	—	2	3	no	—
294	AGL(1,ℤ/8)	C₇×Q₈	1	8	yes	direct product factor
295	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
296	C₂×D₈	C₂³×C₇	1	2	yes	—
297	C₂×M₁₆	C₂³×C₇	1	3	yes	—
298	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
299	C₄⋊Q₈	—	2	3	no	—
300	C₄⋊Q₈	—	2	3	no	—
301	C₄⋊Q₈	—	2	3	no	—
302	C₄⋊Q₈	—	2	3	no	—
303	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
304	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
305	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
306	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
307	#125	—	2	6	no	—
308	#125	C₂³×C₇	1	6	yes	—
309	C₂²⋊Q₈	—	2	3	no	—
310	C₂²⋊Q₈	—	2	3	no	—
311	C₂²⋊Q₈	—	2	3	no	—
312	C₂²⋊Q₈	—	2	6	no	—
313	C₂²⋊Q₈	—	2	12	no	—
314	C₂²⋊Q₈	—	2	6	no	—
315	C₂²⋊Q₈	—	2	3	no	—
316	C₂²⋊Q₈	—	2	6	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	3	no	—
319	C₄×Q₈	—	2	3	no	—
320	C₄×Q₈	—	2	3	no	—
321	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
322	C₄×Q₈	C₇×D₄	1	3	yes	—
323	C₄×Q₈	C₂³×C₇	1	3	yes	—
324	C₄×Q₈	C₂³×C₇	1	3	yes	—
325	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
326	C₄×Q₈	C₂³×C₇	1	3	yes	—
327	C₄×D₄	—	2	3	no	—
328	C₄×D₄	—	2	3	no	—
329	C₄×D₄	—	2	3	no	—
330	C₄×D₄	—	2	3	no	—
331	C₄×D₄	C₂³×C₇	1	6	yes	—
332	C₄×D₄	C₂³×C₇	1	6	yes	—
333	C₄×D₄	C₂³×C₇	1	3	yes	—
334	C₄×D₄	C₂³×C₇	1	3	yes	—
335	#117	—	2	3	no	—
336	#117	C₂³×C₇	1	3	yes	—
337	C₂×C₄⋊C₄	—	2	3	no	—
338	C₂×C₄⋊C₄	—	2	3	no	—
339	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
340	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
341	C₂×C₂²⋊C₄	—	2	3	no	—
342	C₂×C₄²	—	2	3	no	—
343	C₂×C₄²	C₂³×C₇	1	3	yes	—
344	#107	C₂³×C₇	1	6	yes	—
345	#107	C₂³×C₇	1	12	yes	—
346	#106	C₂³×C₇	1	6	yes	—
347	#106	C₂³×C₇	1	12	yes	—
348	C₄⋊C₈	C₂³×C₇	1	6	yes	—
349	C₄⋊C₈	C₂³×C₇	1	6	yes	—
350	#103	C₂³×C₇	1	6	yes	—
351	#102	C₂³×C₇	1	6	yes	—
352	#102	C₂³×C₇	1	6	yes	—
353	#102	C₂³×C₇	1	6	yes	—
354	#97	C₂³×C₇	1	6	yes	—
355	C₄×C₈	C₂³×C₇	1	6	yes	—
356	C₂²×C₇	—	2	1	no	—
357	C₂²×C₇	—	2	1	no	—
358	C₂²×C₇	—	2	2	no	—
359	C₂²×C₇	—	2	4	no	—
360	C₂²×C₇	—	4	8	no	—
361	C₂²×C₇	—	4	1	no	—
362	C₂²×C₇	—	4	2	no	—
363	C₂²×C₇	—	4	2	no	—
364	C₂²×C₇	—	4	1	no	—
365	C₂²×C₇	D₄×Q₈	1	2	yes	—
366	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
367	C₄×C₇	—	2	3	no	—
368	C₄×C₇	—	2	3	no	—
369	C₄×C₇	—	2	2	no	—
370	C₄×C₇	—	2	3	no	—
371	C₄×C₇	—	4	3	no	—
372	C₄×C₇	—	4	6	no	—
373	C₄×C₇	—	4	3	no	—
374	C₄×C₇	D₄×Q₈	1	2	yes	—
375	C₄×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
376	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
377	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
378	C₂⁴	—	2	1	no	—
379	#41	—	2	2	no	—
380	#41	C₂×C₇×Q₈	1	4	yes	complemented
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	3	no	—
383	C₂×Q₈	—	2	6	no	—
384	C₂×Q₈	—	2	6	no	—
385	C₂×Q₈	—	2	3	no	—
386	C₂×Q₈	—	4	12	no	—
387	C₂×Q₈	—	4	24	no	—
388	C₂×Q₈	—	4	8	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	C₂×C₇×D₄	1	3	yes	—
391	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
392	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	C₂×C₇×Q₈	1	4	yes	complemented
397	C₂×D₄	C₂⁴×C₇	1	2	yes	—
398	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	3	no	—
401	C₂²×C₄	—	2	3	no	—
402	C₂²×C₄	—	2	6	no	—
403	C₂²×C₄	—	2	6	no	—
404	C₂²×C₄	—	2	3	no	—
405	C₂²×C₄	—	4	12	no	—
406	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
407	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
408	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
409	SD₁₆	C₂×C₇×Q₈	1	8	yes	complemented
410	D₈	C₂×C₇×Q₈	1	8	yes	complemented
411	M₁₆	—	2	6	no	—
412	M₁₆	C₂×C₇×Q₈	1	4	yes	complemented
413	C₂×C₈	C₂⁴×C₇	1	6	yes	—
414	C₂×C₈	C₂⁴×C₇	1	2	yes	—
415	C₄⋊C₄	—	2	6	no	—
416	C₄⋊C₄	—	2	3	no	—
417	C₄⋊C₄	—	2	6	no	—
418	C₄⋊C₄	—	2	6	no	—
419	C₄⋊C₄	—	2	3	no	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	6	no	—
424	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
425	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
426	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
427	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
428	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
429	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
430	C₂²⋊C₄	—	2	3	no	—
431	C₂²⋊C₄	—	2	6	no	—
432	C₂²⋊C₄	—	2	6	no	—
433	C₂²⋊C₄	—	2	3	no	—
434	C₂²⋊C₄	—	2	3	no	—
435	C₄²	—	2	3	no	—
436	C₄²	—	2	3	no	—
437	C₄²	—	2	3	no	—
438	C₄²	C₂×C₇×D₄	1	3	yes	—
439	C₄²	C₂×C₇×D₄	1	3	yes	—
440	C₄²	C₂⁴×C₇	1	3	yes	—
441	C₄²	C₂⁴×C₇	1	3	yes	—
442	C₂×C₇	—	2	2	no	—
443	C₂×C₇	—	4	2	no	—
444	C₂×C₇	—	4	4	no	—
445	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
446	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
447	C₂×C₇	#3586	1	1	yes	characteristic
448	C₂³	—	2	1	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
454	Q₈	—	2	1	no	—
455	Q₈	—	2	6	no	—
456	Q₈	—	4	12	no	—
457	Q₈	—	4	6	no	—
458	Q₈	—	4	12	no	—
459	Q₈	—	8	3	no	—
460	Q₈	C₇×AGL(1,ℤ/8)	1	4	yes	direct product factor
461	Q₈	C₂²×C₇×Q₈	1	2	yes	complemented
462	D₄	—	2	4	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	1	no	—
465	D₄	—	2	4	no	—
466	D₄	C₂²×C₇×Q₈	1	4	yes	—
467	D₄	C₂²×C₇×Q₈	1	2	yes	complemented
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	3	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	6	no	—
474	C₂×C₄	—	4	3	no	—
475	C₂×C₄	—	4	12	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	12	no	—
478	C₂×C₄	—	4	6	no	—
479	C₂×C₄	—	8	3	no	—
480	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
481	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
482	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
483	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
484	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
485	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
486	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
487	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
488	C₈	—	2	6	no	—
489	C₈	C₂²×C₇×Q₈	1	4	yes	complemented
490	C₇	Q₈×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
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	C₇×D₄×Q₈	1	2	yes	—
501	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
502	C₄	—	2	3	no	—
503	C₄	—	2	3	no	—
504	C₄	—	2	3	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	6	no	—
507	C₄	—	4	3	no	—
508	C₄	—	4	3	no	—
509	C₄	C₇×D₄×Q₈	1	2	yes	—
510	C₄	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
511	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
512	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	4	2	no	—
516	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
517	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
518	C₂	C₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×Q₈×#41	C₂	semidirect
2. C₂×C₇×D₄×Q₈	C₂	semidirect
3. C₇×#3386	C₂	semidirect
4. C₇×#3384	C₂	semidirect
5. C₇×Q₈×SD₁₆	C₂	semidirect
6. C₇×Q₈×D₈	C₂	semidirect
7. C₇×#3165	C₂	semidirect
8. C₇×#3079	C₂	semidirect
9. C₇×#3077	C₂	semidirect
10. C₇×Q₈×M₁₆	C₂	semidirect
14. C₇×Q₈²	C₂²	semidirect
15. C₇×#557	C₂²	semidirect
17. C₇×#554	C₂²	semidirect
23. C₇×D₄×Q₈	C₂²	semidirect
32. C₇×#501	C₂²	semidirect
33. C₇×#500	C₂²	semidirect
34. C₇×#498	C₂²	semidirect
36. C₇×#477	C₂²	semidirect
38. C₇×#476	C₂²	semidirect
39. C₇×#475	C₂²	semidirect
41. C₇×#474	C₂²	semidirect
42. C₇×#446	C₂²	semidirect
43. C₇×C₈×Q₈	C₂²	semidirect
54. Q₈×AGL(1,ℤ/8)	C₇	direct (N × H)
64. C₇×AGL(1,ℤ/8)	Q₈	direct (N × H)
126. Q₈×#41	C₂×C₇	semidirect
127. C₂×D₄×Q₈	C₂×C₇	semidirect
128. #3386	C₂×C₇	semidirect
129. #3384	C₂×C₇	semidirect
130. Q₈×SD₁₆	C₂×C₇	semidirect
131. Q₈×D₈	C₂×C₇	semidirect
132. #3165	C₂×C₇	semidirect
133. #3079	C₂×C₇	semidirect
134. #3077	C₂×C₇	semidirect
135. Q₈×M₁₆	C₂×C₇	semidirect
139. C₇×#41	C₂×Q₈	semidirect
155. C₂×C₇×D₄	C₂×Q₈	semidirect
168. C₇×SD₁₆	C₂×Q₈	semidirect
169. C₇×D₈	C₂×Q₈	semidirect
171. C₇×M₁₆	C₂×Q₈	semidirect
203. Q₈²	C₂²×C₇	semidirect
204. #557	C₂²×C₇	semidirect
206. #554	C₂²×C₇	semidirect
212. D₄×Q₈	C₂²×C₇	semidirect
221. #501	C₂²×C₇	semidirect
222. #500	C₂²×C₇	semidirect
223. #498	C₂²×C₇	semidirect
225. #477	C₂²×C₇	semidirect
227. #476	C₂²×C₇	semidirect
228. #475	C₂²×C₇	semidirect
230. #474	C₂²×C₇	semidirect
231. #446	C₂²×C₇	semidirect
232. C₈×Q₈	C₂²×C₇	semidirect
255. C₇×Q₈	AGL(1,ℤ/8)	direct (N × H)
256. C₇×Q₈	C₂²×Q₈	semidirect
262. C₇×D₄	C₂²×Q₈	semidirect
284. C₇×C₈	C₂²×Q₈	semidirect
294. AGL(1,ℤ/8)	C₇×Q₈	direct (N × H)
380. #41	C₂×C₇×Q₈	semidirect
396. C₂×D₄	C₂×C₇×Q₈	semidirect
409. SD₁₆	C₂×C₇×Q₈	semidirect
410. D₈	C₂×C₇×Q₈	semidirect
412. M₁₆	C₂×C₇×Q₈	semidirect
460. Q₈	C₇×AGL(1,ℤ/8)	direct (N × H)
461. Q₈	C₂²×C₇×Q₈	semidirect
467. D₄	C₂²×C₇×Q₈	semidirect
489. C₈	C₂²×C₇×Q₈	semidirect
490. C₇	Q₈×AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×Q₈×#41	2
C₂×C₇×D₄×Q₈	2
C₇×#3386	2
C₇×#3384	2
C₇×Q₈×SD₁₆	2
C₇×Q₈×D₈	2
C₇×#3165	2
C₇×#3079	2
C₇×#3077	2
C₇×Q₈×M₁₆	2
C₄×C₇×AGL(1,ℤ/8)	2
Q₈×AGL(1,ℤ/8)	7

Subgroups — C7×Q8×AGL(1,ℤ/8) #13238582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×AGL(1,ℤ/8) #13238582