Subgroups — Q₈×#13094049 #27096851

Order 12032.

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	C₂×Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20256305	C₂	1	6	yes	maximal, complemented
3	#20256287	C₂	1	6	yes	maximal, complemented
4	Q₈×#10625336	C₂	1	2	yes	maximal, complemented
5	Q₈×#10625334	C₂	1	2	yes	maximal, complemented
6	#20253935	C₂	1	3	yes	maximal, complemented
7	#20253203	C₂	1	3	yes	maximal, complemented
8	#20253181	C₂	1	3	yes	maximal, complemented
9	Q₈×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13094049	C₂	1	3	yes	maximal
11	Q₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×D₄₇	—	2	1	no	—
13	C₂×#13094049	C₂²	1	1	yes	characteristic
14	Q₈×D₁₈₈	—	2	1	no	—
15	Q₈×D₁₈₈	—	2	3	no	—
16	Q₈×D₁₈₈	C₂²	1	6	yes	—
17	Q₈×D₁₈₈	C₂²	1	2	yes	—
18	C₂×#13094024	C₂²	1	3	yes	—
19	C₂×#13094004	—	2	3	no	—
20	C₂×C₄×D₁₈₈	C₂²	1	3	yes	—
21	#15689533	C₂²	1	12	yes	complemented
22	#15689526	C₂²	1	6	yes	complemented
23	#15689519	C₂²	1	6	yes	complemented
24	#15689420	C₂²	1	12	yes	—
25	#15689418	C₂²	1	6	yes	complemented
26	#15689413	C₂²	1	6	yes	complemented
27	#15689410	C₂²	1	6	yes	—
28	#15689408	C₂²	1	6	yes	complemented
29	#15689403	C₂²	1	6	yes	complemented
30	#15689400	C₂²	1	6	yes	—
31	#15689254	C₂²	1	6	yes	complemented
32	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
33	#15689233	C₂²	1	6	yes	—
34	#15689221	C₂²	1	6	yes	—
35	C₄×#10625336	C₂²	1	6	yes	—
36	C₄×#10625334	C₂²	1	6	yes	—
37	#15689166	C₂²	1	6	yes	—
38	#15689166	C₂²	1	3	yes	—
39	#15689147	C₂²	1	3	yes	—
40	#15689127	C₂²	1	3	yes	—
41	#15689124	C₂²	1	3	yes	—
42	C₄×#10625329	C₂²	1	3	yes	—
43	Q₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
44	C₄₇×#557	C₂²	1	3	yes	complemented
45	C₄₇×#554	—	2	3	no	—
46	C₄₇×#554	C₂²	1	3	yes	complemented
47	D₄×Q₈×C₄₇	—	2	1	no	—
48	D₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
49	C₄₇×#533	—	2	3	no	—
50	C₄₇×#533	C₂²	1	3	yes	—
51	C₄×#41×C₄₇	C₂²	1	3	yes	—
52	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
53	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
54	C₂²×Dic₉₄	—	2	3	no	—
55	C₂×Q₈×D₄₇	—	2	6	no	—
56	C₂×Q₈×D₄₇	—	2	2	no	—
57	C₂×Q₈×D₄₇	—	4	8	no	—
58	C₂²×D₁₈₈	C₂³	1	1	yes	characteristic
59	C₂²×C₄×D₄₇	—	2	3	no	—
60	#13094049	Q₈	1	8	yes	direct product factor
61	C₂×#10625336	C₂³	1	2	yes	—
62	C₂×#10625334	C₂³	1	2	yes	—
63	C₂×#10625329	C₂³	1	3	yes	—
64	C₂×#10625329	C₂³	1	1	yes	characteristic
65	#13094024	—	2	3	no	—
66	#13094024	—	2	3	no	—
67	#13094024	C₂³	1	6	yes	—
68	#13094024	C₂³	1	6	yes	—
69	#13094004	—	2	6	no	—
70	#13094004	—	2	12	no	—
71	#13094004	—	2	6	no	—
72	C₄×D₁₈₈	—	2	3	no	—
73	C₄×D₁₈₈	—	2	3	no	—
74	C₄×D₁₈₈	C₂³	1	6	yes	—
75	C₄×D₁₈₈	C₂³	1	6	yes	—
76	C₂×C₄⋊Dic₄₇	—	2	3	no	—
77	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
78	C₂×(C₂×D₄₇)⋊C₄	—	2	3	no	—
79	#13093957	C₂³	1	6	yes	—
80	#13093957	C₂³	1	12	yes	—
81	#13093955	C₂³	1	6	yes	—
82	#13093955	C₂³	1	12	yes	—
83	C₄⋊(C₄₇⋊C₈)	C₂³	1	6	yes	—
84	C₄⋊(C₄₇⋊C₈)	C₂³	1	6	yes	—
85	#13093948	C₂³	1	6	yes	—
86	#13093945	C₂³	1	6	yes	—
87	#13093944	C₂³	1	6	yes	—
88	#13093944	C₂³	1	6	yes	—
89	#13093935	C₂³	1	6	yes	—
90	C₄×C₄₇⋊C₈	C₂³	1	6	yes	—
91	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
92	C₂²×Q₈×C₄₇	—	2	1	no	—
93	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
94	C₄⋊Q₈×C₄₇	—	2	3	no	—
95	C₄⋊Q₈×C₄₇	—	2	3	no	—
96	C₄⋊Q₈×C₄₇	C₂³	1	3	yes	—
97	C₄⋊Q₈×C₄₇	C₂³	1	3	yes	—
98	#125×C₄₇	—	2	6	no	—
99	#125×C₄₇	C₂³	1	6	yes	—
100	C₂²⋊Q₈×C₄₇	—	2	3	no	—
101	C₂²⋊Q₈×C₄₇	—	2	3	no	—
102	C₂²⋊Q₈×C₄₇	—	2	3	no	—
103	C₂²⋊Q₈×C₄₇	—	2	3	no	—
104	C₂²⋊Q₈×C₄₇	—	2	6	no	—
105	C₄×Q₈×C₄₇	—	2	1	no	—
106	C₄×Q₈×C₄₇	—	2	3	no	—
107	C₄×Q₈×C₄₇	—	2	3	no	—
108	C₄×Q₈×C₄₇	—	2	3	no	—
109	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₄₇	D₄	1	3	yes	—
111	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
112	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
113	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
114	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
115	C₄×D₄×C₄₇	—	2	3	no	—
116	C₄×D₄×C₄₇	—	2	3	no	—
117	C₄×D₄×C₄₇	C₂³	1	3	yes	—
118	C₄×D₄×C₄₇	C₂³	1	3	yes	—
119	#117×C₄₇	—	2	3	no	—
120	#117×C₄₇	C₂³	1	3	yes	—
121	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
122	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
123	C₂×C₄²×C₄₇	—	2	3	no	—
124	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
125	C₂³×D₄₇	—	2	1	no	—
126	#41×C₄₇	—	2	2	no	—
127	#41×C₄₇	C₂×Q₈	1	4	yes	complemented
128	C₂×Q₈×C₄₇	—	2	4	no	—
129	C₂×Q₈×C₄₇	—	2	3	no	—
130	C₂×Q₈×C₄₇	—	2	3	no	—
131	C₂×Q₈×C₄₇	—	4	4	no	—
132	C₂×Q₈×C₄₇	C₂×D₄	1	3	yes	—
133	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Q₈×C₄₇	C₂⁴	1	1	yes	characteristic
135	C₂×D₄×C₄₇	—	2	1	no	—
136	C₂×D₄×C₄₇	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₄₇	—	2	1	no	—
138	C₂²×C₄×C₄₇	—	2	3	no	—
139	C₂²×C₄×C₄₇	—	2	3	no	—
140	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
141	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
142	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
143	C₂²×Dic₄₇	—	2	3	no	—
144	Q₈×D₄₇	—	4	24	no	—
145	Q₈×D₄₇	—	4	8	no	—
146	C₂×D₁₈₈	—	2	1	no	—
147	C₂×D₁₈₈	—	2	2	no	—
148	C₂×D₁₈₈	C₂×Q₈	1	4	yes	complemented
149	C₂×D₁₈₈	C₂⁴	1	2	yes	—
150	C₂×C₄×D₄₇	—	2	6	no	—
151	C₂×C₄×D₄₇	—	2	6	no	—
152	C₂×C₄×D₄₇	—	4	12	no	—
153	C₂×Dic₉₄	—	2	6	no	—
154	C₂×Dic₉₄	—	2	6	no	—
155	C₂×Dic₉₄	—	4	12	no	—
156	C₄⋊C₄×C₄₇	—	2	6	no	—
157	C₄⋊C₄×C₄₇	—	2	3	no	—
158	C₄⋊C₄×C₄₇	—	2	3	no	—
159	C₄⋊C₄×C₄₇	—	2	3	no	—
160	C₄⋊C₄×C₄₇	—	2	3	no	—
161	C₄⋊C₄×C₄₇	—	2	6	no	—
162	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
163	C₄⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
164	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
165	C₄⋊C₄×C₄₇	C₂⁴	1	3	yes	—
166	C₄⋊C₄×C₄₇	C₂⁴	1	3	yes	—
167	C₂²⋊C₄×C₄₇	—	2	3	no	—
168	C₂²⋊C₄×C₄₇	—	2	3	no	—
169	C₂²⋊C₄×C₄₇	—	2	3	no	—
170	C₄²×C₄₇	—	2	3	no	—
171	C₄²×C₄₇	—	2	3	no	—
172	C₄²×C₄₇	—	2	3	no	—
173	C₄²×C₄₇	C₂×D₄	1	3	yes	—
174	C₄²×C₄₇	C₂×D₄	1	3	yes	—
175	C₄²×C₄₇	C₂⁴	1	3	yes	—
176	C₄²×C₄₇	C₂⁴	1	3	yes	—
177	#10625336	C₂×Q₈	1	8	yes	complemented
178	#10625334	C₂×Q₈	1	8	yes	complemented
179	(C₂×D₄₇)⋊C₄	—	2	6	no	—
180	(C₂×D₄₇)⋊C₄	—	2	6	no	—
181	C₄⋊Dic₄₇	—	2	6	no	—
182	C₄⋊Dic₄₇	—	2	6	no	—
183	C₄⋊Dic₄₇	—	2	3	no	—
184	C₄⋊Dic₄₇	C₂⁴	1	6	yes	—
185	#10625329	—	2	6	no	—
186	#10625329	C₂×Q₈	1	4	yes	complemented
187	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
188	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
189	C₂³×C₄₇	—	2	1	no	—
190	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
191	C₂²×D₄₇	—	2	4	no	—
192	C₂²×D₄₇	—	2	2	no	—
193	C₂²×D₄₇	—	4	2	no	—
194	C₂²×D₄₇	—	4	2	no	—
195	Q₈×C₄₇	—	2	6	no	—
196	Q₈×C₄₇	—	2	1	no	—
197	Q₈×C₄₇	—	4	6	no	—
198	Q₈×C₄₇	—	8	3	no	—
199	Q₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	complemented
200	Q₈×C₄₇	C₂²×Q₈	1	2	yes	complemented
201	D₄×C₄₇	—	2	4	no	—
202	D₄×C₄₇	—	2	1	no	—
203	D₄×C₄₇	C₂²×Q₈	1	2	yes	complemented
204	C₂×C₄×C₄₇	—	2	1	no	—
205	C₂×C₄×C₄₇	—	2	4	no	—
206	C₂×C₄×C₄₇	—	2	6	no	—
207	C₂×C₄×C₄₇	—	2	3	no	—
208	C₂×C₄×C₄₇	—	2	3	no	—
209	C₂×C₄×C₄₇	—	4	3	no	—
210	C₂×C₄×C₄₇	—	4	6	no	—
211	C₂×C₄×C₄₇	—	8	3	no	—
212	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
213	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
214	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
218	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
219	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
220	C₂×Dic₄₇	—	2	6	no	—
221	C₂×Dic₄₇	—	4	6	no	—
222	D₁₈₈	—	2	4	no	—
223	D₁₈₈	—	2	2	no	—
224	D₁₈₈	C₂²×Q₈	1	4	yes	—
225	C₄×D₄₇	—	4	12	no	—
226	C₄×D₄₇	—	4	12	no	—
227	Dic₉₄	—	4	12	no	—
228	Dic₉₄	—	4	12	no	—
229	C₄₇⋊C₈	—	2	6	no	—
230	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	complemented
231	Q₈×AGL(1,ℤ/8)	—	47	1	no	maximal, Sylow
232	C₂²×C₄₇	—	2	1	no	—
233	C₂²×C₄₇	—	2	1	no	—
234	C₂²×C₄₇	—	2	2	no	—
235	C₂²×C₄₇	—	4	1	no	—
236	C₂²×C₄₇	—	4	1	no	—
237	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
238	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
239	C₂×D₄₇	—	2	4	no	—
240	C₂×D₄₇	—	4	8	no	—
241	C₂×D₄₇	—	4	2	no	—
242	C₂×D₄₇	—	4	2	no	—
243	C₄×C₄₇	—	2	3	no	—
244	C₄×C₄₇	—	2	3	no	—
245	C₄×C₄₇	—	2	2	no	—
246	C₄×C₄₇	—	2	3	no	—
247	C₄×C₄₇	—	4	3	no	—
248	C₄×C₄₇	—	4	3	no	—
249	C₄×C₄₇	D₄×Q₈	1	2	yes	—
250	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
251	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
252	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
253	Dic₄₇	—	4	6	no	—
254	Q₈×#41	D₄₇	1	1	yes	characteristic, complemented
255	C₂×D₄×Q₈	—	47	1	no	—
256	#3386	—	47	6	no	—
257	#3384	—	47	6	no	—
258	Q₈×SD₁₆	—	47	2	no	—
259	Q₈×D₈	—	47	2	no	—
260	#3165	—	47	3	no	—
261	#3079	—	47	3	no	—
262	#3077	—	47	3	no	—
263	Q₈×M₁₆	—	47	1	no	—
264	C₄×AGL(1,ℤ/8)	—	47	3	no	—
265	C₂×C₄₇	—	2	2	no	—
266	C₂×C₄₇	—	4	2	no	—
267	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
268	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₂×C₄₇	#3586	1	1	yes	characteristic
270	D₄₇	—	4	4	no	—
271	C₂³×Q₈	—	94	1	no	—
272	C₂×AGL(1,ℤ/8)	—	47	1	no	—
273	Q₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
274	#557	C₂×D₄₇	1	3	yes	complemented
275	#554	—	2	3	no	—
276	#554	C₂×D₄₇	1	3	yes	complemented
277	D₄×Q₈	—	2	1	no	—
278	D₄×Q₈	—	47	2	no	—
279	D₄×Q₈	—	47	6	no	—
280	D₄×Q₈	—	94	1	no	—
281	D₄×Q₈	—	94	3	no	—
282	D₄×Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
283	#533	—	2	3	no	—
284	#533	C₂×D₄₇	1	3	yes	—
285	C₂×C₄⋊Q₈	—	47	3	no	—
286	C₂×C₂²⋊Q₈	—	94	3	no	—
287	C₄×#41	C₂×D₄₇	1	3	yes	—
288	C₂×C₄×Q₈	—	2	1	no	—
289	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
290	C₂×C₄×D₄	—	47	3	no	—
291	#501	—	47	12	no	—
292	#500	—	47	6	no	—
293	#498	—	47	6	no	—
294	#478	—	47	12	no	—
295	#477	—	47	6	no	—
296	#476	—	47	6	no	—
297	#476	—	47	6	no	—
298	#475	—	47	6	no	—
299	#474	—	47	6	no	—
300	#474	—	47	6	no	—
301	#446	—	47	6	no	—
302	C₈×Q₈	—	47	2	no	—
303	#442	—	47	6	no	—
304	#440	—	47	6	no	—
305	C₄×SD₁₆	—	47	6	no	—
306	C₄×D₈	—	47	6	no	—
307	#428	—	47	6	no	—
308	#428	—	47	3	no	—
309	#423	—	47	3	no	—
310	#418	—	47	3	no	—
311	#417	—	47	3	no	—
312	C₄×M₁₆	—	47	3	no	—
313	C₄₇	Q₈×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
314	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	—	94	6	no	—
317	C₂²×Q₈	—	94	2	no	—
318	C₂²×Q₈	—	94	3	no	—
319	C₂²×Q₈	—	188	8	no	—
320	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
321	C₂²×D₄	—	47	1	no	—
322	C₂³×C₄	—	94	3	no	—
323	AGL(1,ℤ/8)	—	47	8	no	—
324	C₂×SD₁₆	—	47	2	no	—
325	C₂×D₈	—	47	2	no	—
326	C₂×M₁₆	—	47	1	no	—
327	C₂×M₁₆	—	47	3	no	—
328	C₄⋊Q₈	—	2	3	no	—
329	C₄⋊Q₈	—	2	3	no	—
330	C₄⋊Q₈	—	47	6	no	—
331	C₄⋊Q₈	—	47	6	no	—
332	C₄⋊Q₈	—	94	3	no	—
333	C₄⋊Q₈	—	94	3	no	—
334	C₄⋊Q₈	C₂²×D₄₇	1	3	yes	—
335	C₄⋊Q₈	C₂²×D₄₇	1	3	yes	—
336	#125	—	2	6	no	—
337	#125	C₂²×D₄₇	1	6	yes	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	—	2	3	no	—
341	C₂²⋊Q₈	—	2	3	no	—
342	C₂²⋊Q₈	—	2	6	no	—
343	C₂²⋊Q₈	—	94	6	no	—
344	C₂²⋊Q₈	—	94	12	no	—
345	C₂²⋊Q₈	—	94	6	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	—	2	3	no	—
348	C₄×Q₈	—	2	3	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
351	C₄×Q₈	C₂²⋊D₄₇	1	3	yes	—
352	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
353	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
354	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
355	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	47	6	no	—
359	C₄×D₄	—	47	6	no	—
360	C₄×D₄	—	94	3	no	—
361	C₄×D₄	—	94	3	no	—
362	C₄×D₄	C₂²×D₄₇	1	3	yes	—
363	C₄×D₄	C₂²×D₄₇	1	3	yes	—
364	#117	—	2	3	no	—
365	#117	C₂²×D₄₇	1	3	yes	—
366	C₂×C₄⋊C₄	—	2	3	no	—
367	C₂×C₄⋊C₄	—	47	3	no	—
368	C₂×C₄⋊C₄	—	94	3	no	—
369	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
370	C₂×C₂²⋊C₄	—	94	3	no	—
371	C₂×C₄²	—	2	3	no	—
372	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
373	#107	—	47	6	no	—
374	#107	—	47	12	no	—
375	#106	—	47	6	no	—
376	#106	—	47	12	no	—
377	C₄⋊C₈	—	47	6	no	—
378	C₄⋊C₈	—	47	6	no	—
379	#103	—	47	6	no	—
380	#102	—	47	6	no	—
381	#102	—	47	6	no	—
382	#102	—	47	6	no	—
383	#97	—	47	6	no	—
384	C₄×C₈	—	47	6	no	—
385	C₂⁴	—	94	1	no	—
386	#41	—	2	2	no	—
387	#41	Q₈×D₄₇	1	4	yes	complemented
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	3	no	—
391	C₂×Q₈	—	4	4	no	—
392	C₂×Q₈	—	94	6	no	—
393	C₂×Q₈	—	94	6	no	—
394	C₂×Q₈	—	188	12	no	—
395	C₂×Q₈	—	188	24	no	—
396	C₂×Q₈	—	188	8	no	—
397	C₂×Q₈	C₂×C₂²⋊D₄₇	1	3	yes	—
398	C₂×Q₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
399	C₂×Q₈	C₂³×D₄₇	1	1	yes	characteristic
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	47	4	no	—
402	C₂×D₄	—	47	2	no	—
403	C₂×D₄	—	94	1	no	—
404	C₂×D₄	—	94	2	no	—
405	C₂×D₄	C₂³×D₄₇	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	—	94	3	no	—
410	C₂²×C₄	—	94	6	no	—
411	C₂²×C₄	—	94	6	no	—
412	C₂²×C₄	—	188	12	no	—
413	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
414	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
415	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
416	SD₁₆	—	47	8	no	—
417	D₈	—	47	8	no	—
418	M₁₆	—	47	4	no	—
419	M₁₆	—	94	6	no	—
420	C₂×C₈	—	47	2	no	—
421	C₂×C₈	—	47	6	no	—
422	C₄⋊C₄	—	2	6	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	—	47	6	no	—
429	C₄⋊C₄	—	94	3	no	—
430	C₄⋊C₄	—	94	6	no	—
431	C₄⋊C₄	—	94	6	no	—
432	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
433	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
434	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
435	C₄⋊C₄	C₂³×D₄₇	1	3	yes	—
436	C₄⋊C₄	C₂³×D₄₇	1	3	yes	—
437	C₂²⋊C₄	—	2	3	no	—
438	C₂²⋊C₄	—	2	3	no	—
439	C₂²⋊C₄	—	2	3	no	—
440	C₂²⋊C₄	—	94	6	no	—
441	C₂²⋊C₄	—	94	6	no	—
442	C₄²	—	2	3	no	—
443	C₄²	—	2	3	no	—
444	C₄²	—	2	3	no	—
445	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
446	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
447	C₄²	C₂³×D₄₇	1	3	yes	—
448	C₄²	C₂³×D₄₇	1	3	yes	—
449	C₂³	—	2	1	no	—
450	C₂³	—	94	4	no	—
451	C₂³	—	94	2	no	—
452	C₂³	—	188	2	no	—
453	C₂³	—	188	2	no	—
454	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
455	Q₈	—	2	6	no	—
456	Q₈	—	2	1	no	—
457	Q₈	—	4	6	no	—
458	Q₈	—	8	3	no	—
459	Q₈	—	188	12	no	—
460	Q₈	—	188	12	no	—
461	Q₈	#13094049	1	4	yes	direct product factor
462	Q₈	C₂×Q₈×D₄₇	1	2	yes	complemented
463	D₄	—	2	4	no	—
464	D₄	—	2	1	no	—
465	D₄	—	47	4	no	—
466	D₄	—	94	4	no	—
467	D₄	—	94	2	no	—
468	D₄	C₂×Q₈×D₄₇	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	—	4	3	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	—	8	3	no	—
477	C₂×C₄	—	94	6	no	—
478	C₂×C₄	—	188	12	no	—
479	C₂×C₄	—	188	6	no	—
480	C₂×C₄	—	188	12	no	—
481	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
482	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
483	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
484	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
486	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
487	C₂×C₄	#13094072	1	2	yes	—
488	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
489	C₈	—	47	4	no	—
490	C₈	—	94	6	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	94	4	no	—
497	C₂²	—	188	8	no	—
498	C₂²	—	188	2	no	—
499	C₂²	—	188	2	no	—
500	C₂²	Q₈×C₂²⋊D₄₇	1	2	yes	—
501	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
502	C₄	—	2	3	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	3	no	—
505	C₄	—	2	3	no	—
506	C₄	—	4	3	no	—
507	C₄	—	4	3	no	—
508	C₄	—	188	6	no	—
509	C₄	Q₈×C₂²⋊D₄₇	1	2	yes	—
510	C₄	C₂×#13094049	1	6	yes	—
511	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
512	C₄	C₂×#13094072	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	4	2	no	—
515	C₂	—	188	4	no	—
516	C₂	C₂×Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
517	C₂	C₂²×#13094049	1	1	yes	characteristic
518	C₂	#20258174	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	Q₈×#41×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#13094049	C₂²
hypercenter	Q₈×#41	D₄₇
hypocenter	C₄₇	Q₈×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ Q₈×#41

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₁₈₈	C₂	semidirect
2. #20256305	C₂	semidirect
3. #20256287	C₂	semidirect
4. Q₈×#10625336	C₂	semidirect
5. Q₈×#10625334	C₂	semidirect
6. #20253935	C₂	semidirect
7. #20253203	C₂	semidirect
8. #20253181	C₂	semidirect
9. Q₈×#10625329	C₂	semidirect
11. Q₈×#41×C₄₇	C₂	semidirect
21. #15689533	C₂²	semidirect
22. #15689526	C₂²	semidirect
23. #15689519	C₂²	semidirect
25. #15689418	C₂²	semidirect
26. #15689413	C₂²	semidirect
28. #15689408	C₂²	semidirect
29. #15689403	C₂²	semidirect
31. #15689254	C₂²	semidirect
32. Q₈×C₄₇⋊C₈	C₂²	semidirect
43. Q₈²×C₄₇	C₂²	semidirect
44. C₄₇×#557	C₂²	semidirect
46. C₄₇×#554	C₂²	semidirect
48. D₄×Q₈×C₄₇	C₂²	semidirect
60. #13094049	Q₈	direct (N × H)
127. #41×C₄₇	C₂×Q₈	semidirect
148. C₂×D₁₈₈	C₂×Q₈	semidirect
177. #10625336	C₂×Q₈	semidirect
178. #10625334	C₂×Q₈	semidirect
186. #10625329	C₂×Q₈	semidirect
199. Q₈×C₄₇	AGL(1,ℤ/8)	semidirect
200. Q₈×C₄₇	C₂²×Q₈	semidirect
203. D₄×C₄₇	C₂²×Q₈	semidirect
230. C₄₇⋊C₈	C₂²×Q₈	semidirect
254. Q₈×#41	D₄₇	semidirect
273. Q₈²	C₂×D₄₇	semidirect
274. #557	C₂×D₄₇	semidirect
276. #554	C₂×D₄₇	semidirect
282. D₄×Q₈	C₂×D₄₇	semidirect
313. C₄₇	Q₈×AGL(1,ℤ/8)	semidirect
387. #41	Q₈×D₄₇	semidirect
461. Q₈	#13094049	direct (N × H)
462. Q₈	C₂×Q₈×D₄₇	semidirect
468. D₄	C₂×Q₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₁₈₈	2
#20256305	2
#20256287	2
Q₈×#10625336	2
Q₈×#10625334	2
#20253935	2
#20253203	2
#20253181	2
Q₈×#10625329	2
C₄×#13094049	2
Q₈×#41×C₄₇	2
Q₈×AGL(1,ℤ/8)	47

Subgroups — Q8×#13094049 #27096851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13094049 #27096851