Subgroups — Q₈×#13063347 #25977066

Order 11008.

1786 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10621898	C₂	1	2	yes	maximal, complemented
2	#19147487	C₂	1	3	yes	maximal, complemented
3	Q₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19147472	C₂	1	6	yes	maximal, complemented
5	#19147463	C₂	1	3	yes	maximal, complemented
6	Q₈×#10621873	C₂	1	2	yes	maximal, complemented
7	Q₈×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#19145109	C₂	1	3	yes	maximal, complemented
9	#19144385	C₂	1	6	yes	maximal, complemented
10	C₄×#13063347	C₂	1	3	yes	maximal
11	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	Q₈×Dic₈₆	C₂²	1	2	yes	—
14	#14583627	C₂²	1	6	yes	—
15	#14583609	C₂²	1	6	yes	—
16	#14583604	—	2	6	no	—
17	Q₈×D₁₇₂	C₂²	1	2	yes	—
18	Q₈×C₂²⋊D₄₃	—	2	2	no	—
19	#14583456	C₂²	1	6	yes	—
20	#14583453	—	2	6	no	—
21	C₄×#10621898	C₂²	1	6	yes	—
22	C₄×Q₈×D₄₃	—	2	2	no	—
23	#14583304	C₂²	1	3	yes	—
24	#14583304	C₂²	1	3	yes	—
25	#14583301	C₂²	1	6	yes	—
26	#14583301	C₂²	1	6	yes	—
27	#14583297	C₂²	1	6	yes	—
28	#14583207	C₂²	1	12	yes	—
29	#14583200	C₂²	1	12	yes	—
30	#14583195	C₂²	1	6	yes	—
31	#14583190	C₂²	1	6	yes	—
32	#14583185	C₂²	1	6	yes	—
33	#14583180	C₂²	1	6	yes	—
34	C₄×Dic₁₇₂	C₂²	1	3	yes	—
35	C₄×Dic₁₇₂	C₂²	1	3	yes	—
36	C₄×#10621873	C₂²	1	6	yes	—
37	C₄×#10621873	C₂²	1	6	yes	—
38	C₄×D₃₄₄	C₂²	1	3	yes	—
39	C₄×D₃₄₄	C₂²	1	3	yes	—
40	#14582937	C₂²	1	3	yes	—
41	#14582937	C₂²	1	6	yes	—
42	#14582896	C₂²	1	6	yes	—
43	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
44	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
45	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
46	C₈×Q₈×C₄₃	C₂²	1	3	yes	—
47	C₈×Q₈×C₄₃	C₂²	1	3	yes	—
48	C₂×C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
49	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
50	C₂×#10621898	C₂³	1	2	yes	—
51	C₂×Q₈×D₄₃	—	2	2	no	—
52	#13063347	Q₈	1	8	yes	direct product factor
53	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
54	C₂×#10621873	C₂³	1	2	yes	—
55	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
56	#13063325	C₂³	1	6	yes	—
57	#13063325	C₂³	1	6	yes	—
58	#13063324	—	2	6	no	—
59	#13063323	—	2	6	no	—
60	#13063317	C₂³	1	12	yes	—
61	#13063315	—	2	12	no	—
62	#13063305	—	2	6	no	—
63	#13063304	—	2	6	no	—
64	#13063304	—	2	6	no	—
65	#13063303	—	2	12	no	—
66	#13063301	—	2	6	no	—
67	Q₈×Dic₄₃	—	2	2	no	—
68	C₄×Dic₈₆	—	2	6	no	—
69	C₄×Dic₈₆	—	2	6	no	—
70	C₄×Dic₈₆	C₂³	1	6	yes	—
71	C₄×Dic₈₆	C₂³	1	6	yes	—
72	#13063290	—	2	6	no	—
73	C₄×D₁₇₂	C₂³	1	6	yes	—
74	C₄×D₁₇₂	C₂³	1	6	yes	—
75	C₄×C₂²⋊D₄₃	—	2	6	no	—
76	C₄×C₂²⋊D₄₃	—	2	6	no	—
77	#13063284	C₂³	1	6	yes	—
78	#13063282	—	2	6	no	—
79	C₄⋊C₄×D₄₃	—	2	6	no	—
80	C₄²×D₄₃	—	2	6	no	—
81	#13063259	C₂³	1	3	yes	—
82	#13063259	C₂³	1	3	yes	—
83	#13063259	C₂³	1	6	yes	—
84	#13063259	C₂³	1	6	yes	—
85	#13063257	C₂³	1	6	yes	—
86	#13063257	C₂³	1	12	yes	—
87	#13063250	C₂³	1	6	yes	—
88	#13063250	C₂³	1	6	yes	—
89	#13063247	C₂³	1	6	yes	—
90	#13063247	C₂³	1	6	yes	—
91	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
92	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
93	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₄₃	D₄	1	3	yes	—
96	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₄₃	C₂³	1	3	yes	—
98	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
99	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
100	C₄⋊C₈×C₄₃	C₂³	1	3	yes	—
101	C₄⋊C₈×C₄₃	C₂³	1	3	yes	—
102	C₄⋊C₈×C₄₃	C₂³	1	6	yes	—
103	C₄×C₈×C₄₃	C₂³	1	3	yes	—
104	C₄×C₈×C₄₃	C₂³	1	3	yes	—
105	C₄×C₈×C₄₃	C₂³	1	3	yes	—
106	C₄×C₈×C₄₃	C₂³	1	3	yes	—
107	C₂×Q₈×C₄₃	—	2	4	no	—
108	C₂×Q₈×C₄₃	C₂×D₄	1	3	yes	—
109	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
111	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
112	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₂²⋊D₄₃	—	2	2	no	—
114	Q₈×D₄₃	—	4	8	no	—
115	#10621898	—	2	4	no	—
116	#10621898	C₂×Q₈	1	8	yes	complemented
117	C₂×D₁₇₂	C₂⁴	1	2	yes	—
118	C₂×C₄×D₄₃	—	2	2	no	—
119	C₂×C₄×D₄₃	—	2	6	no	—
120	C₂×C₄×D₄₃	—	2	6	no	—
121	C₂×Dic₈₆	—	2	6	no	—
122	C₂×Dic₈₆	—	2	6	no	—
123	C₂×Dic₈₆	C₂⁴	1	2	yes	—
124	C₂×C₈×C₄₃	—	2	6	no	—
125	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	complemented
126	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
127	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
128	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₄₃	C₂×D₄	1	3	yes	—
131	C₄⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
132	C₄⋊C₄×C₄₃	C₂×D₄	1	3	yes	—
133	C₄²×C₄₃	C₂×D₄	1	3	yes	—
134	C₄²×C₄₃	C₂×D₄	1	3	yes	—
135	C₄²×C₄₃	C₂⁴	1	3	yes	—
136	C₄²×C₄₃	C₂⁴	1	3	yes	—
137	C₂²⋊Dic₄₃	—	2	6	no	—
138	(C₂×D₄₃)⋊C₄	—	2	6	no	—
139	(C₂×D₄₃)⋊C₄	—	2	6	no	—
140	C₄⋊Dic₄₃	—	2	6	no	—
141	C₄⋊Dic₄₃	—	2	6	no	—
142	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
143	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
144	Dic₄₃⋊C₄	—	2	12	no	—
145	Dic₄₃⋊C₄	—	2	6	no	—
146	Dic₄₃⋊C₄	—	2	6	no	—
147	Dic₄₃⋊C₄	—	2	12	no	—
148	C₄×Dic₄₃	—	2	6	no	—
149	C₄×Dic₄₃	—	2	6	no	—
150	C₄×Dic₄₃	—	2	6	no	—
151	Dic₁₇₂	C₂×Q₈	1	4	yes	complemented
152	D₃₄₄	C₂×Q₈	1	4	yes	complemented
153	#10621873	C₂×Q₈	1	8	yes	complemented
154	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
155	C₂²×D₄₃	—	2	2	no	—
156	Q₈×C₄₃	—	2	6	no	—
157	Q₈×C₄₃	#135	1	4	yes	complemented
158	C₂×C₄×C₄₃	—	2	6	no	—
159	C₂×C₄×C₄₃	—	4	3	no	—
160	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
161	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
162	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
163	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
164	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
166	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
167	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
168	C₂²⋊D₄₃	—	2	8	no	—
169	C₂×Dic₄₃	—	2	2	no	—
170	C₂×Dic₄₃	—	2	6	no	—
171	C₂×Dic₄₃	—	2	6	no	—
172	D₁₇₂	—	2	2	no	—
173	D₁₇₂	C₂²×Q₈	1	4	yes	—
174	C₄×D₄₃	—	2	8	no	—
175	C₄×D₄₃	—	4	12	no	—
176	C₄×D₄₃	—	8	6	no	—
177	Dic₈₆	—	2	2	no	—
178	Dic₈₆	—	4	12	no	—
179	Dic₈₆	—	8	6	no	—
180	Dic₈₆	C₂²×Q₈	1	4	yes	—
181	C₈×C₄₃	—	2	3	no	—
182	C₈×C₄₃	—	2	3	no	—
183	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
184	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
185	Q₈×#135	—	43	1	no	maximal, Sylow
186	C₂²×C₄₃	—	2	1	no	—
187	C₂²×C₄₃	—	2	1	no	—
188	C₂²×C₄₃	D₄×Q₈	1	2	yes	—
189	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
190	C₂×D₄₃	—	2	4	no	—
191	C₂×D₄₃	—	4	2	no	—
192	C₂×D₄₃	—	4	2	no	—
193	C₄×C₄₃	—	2	3	no	—
194	C₄×C₄₃	—	2	3	no	—
195	C₄×C₄₃	—	2	3	no	—
196	C₄×C₄₃	D₄×Q₈	1	2	yes	—
197	C₄×C₄₃	C₂×#135	1	6	yes	—
198	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
199	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
200	Dic₄₃	—	2	4	no	—
201	Dic₄₃	—	4	6	no	—
202	Dic₄₃	—	4	6	no	—
203	Q₈×#41	—	43	2	no	—
204	#3383	—	43	3	no	—
205	Q₈×Q₁₆	—	43	1	no	—
206	#3381	—	43	6	no	—
207	#3380	—	43	3	no	—
208	Q₈×SD₁₆	—	43	2	no	—
209	Q₈×D₈	—	43	1	no	—
210	#3160	—	43	3	no	—
211	#3076	—	43	6	no	—
212	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic, complemented
213	C₄×#135	—	43	3	no	—
214	C₂×C₄₃	—	2	2	no	—
215	C₂×C₄₃	C₂×D₄×Q₈	1	1	yes	characteristic
216	C₂×C₄₃	C₂²×#135	1	1	yes	characteristic
217	C₂×C₄₃	#3584	1	1	yes	characteristic
218	D₄₃	—	4	4	no	—
219	C₂×#135	—	43	1	no	—
220	Q₈²	—	43	2	no	—
221	#557	—	43	6	no	—
222	#554	—	43	6	no	—
223	#554	—	86	6	no	—
224	D₄×Q₈	—	43	2	no	—
225	D₄×Q₈	—	86	2	no	—
226	#533	—	43	6	no	—
227	#533	—	86	6	no	—
228	C₄×#41	—	43	6	no	—
229	C₂×C₄×Q₈	—	86	2	no	—
230	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
231	#500	—	43	3	no	—
232	#500	—	43	3	no	—
233	#499	—	43	6	no	—
234	#499	—	43	6	no	—
235	#498	—	43	6	no	—
236	#479	—	43	12	no	—
237	#478	—	43	12	no	—
238	#477	—	43	6	no	—
239	#476	—	43	6	no	—
240	#475	—	43	6	no	—
241	#474	—	43	6	no	—
242	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
243	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
244	C₈×Q₈	C₂×D₄₃	1	3	yes	—
245	C₈×Q₈	C₂×D₄₃	1	3	yes	—
246	C₄×Q₁₆	—	43	3	no	—
247	C₄×Q₁₆	—	43	3	no	—
248	C₄×SD₁₆	—	43	6	no	—
249	C₄×SD₁₆	—	43	6	no	—
250	C₄×D₈	—	43	3	no	—
251	C₄×D₈	—	43	3	no	—
252	#427	—	43	6	no	—
253	#427	—	43	3	no	—
254	C₂×C₄⋊C₈	C₂×D₄₃	1	3	yes	—
255	#416	—	43	6	no	—
256	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
257	C₄₃	Q₈×#135	1	1	yes	characteristic, complemented, Sylow
258	C₂×#41	—	43	2	no	—
259	C₂²×Q₈	—	86	2	no	—
260	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
261	#135	—	43	8	no	—
262	C₂×Q₁₆	—	43	1	no	—
263	C₂×SD₁₆	—	43	2	no	—
264	C₂×D₈	—	43	1	no	—
265	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
266	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
267	C₄⋊Q₈	—	43	6	no	—
268	C₄⋊Q₈	—	43	6	no	—
269	C₄⋊Q₈	—	86	6	no	—
270	C₄⋊Q₈	—	86	6	no	—
271	#125	—	43	12	no	—
272	#125	—	86	12	no	—
273	C₂²⋊Q₈	—	86	6	no	—
274	C₂²⋊Q₈	—	86	6	no	—
275	C₂²⋊Q₈	—	86	6	no	—
276	C₂²⋊Q₈	—	86	6	no	—
277	C₂²⋊Q₈	—	86	12	no	—
278	C₄×Q₈	—	43	6	no	—
279	C₄×Q₈	—	43	6	no	—
280	C₄×Q₈	—	86	2	no	—
281	C₄×Q₈	—	86	6	no	—
282	C₄×Q₈	—	86	6	no	—
283	C₄×Q₈	—	86	6	no	—
284	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
285	C₄×Q₈	D₁₇₂	1	3	yes	—
286	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
287	C₄×Q₈	C₂²×D₄₃	1	3	yes	—
288	C₄×D₄	—	43	6	no	—
289	C₄×D₄	—	43	6	no	—
290	C₄×D₄	—	86	6	no	—
291	C₄×D₄	—	86	6	no	—
292	#117	—	43	6	no	—
293	#117	—	86	6	no	—
294	C₂×C₄⋊C₄	—	86	6	no	—
295	C₂×C₄⋊C₄	D₁₇₂	1	3	yes	—
296	C₂×C₄²	—	86	6	no	—
297	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
298	#107	—	43	3	no	—
299	#107	—	43	3	no	—
300	#107	—	43	6	no	—
301	#107	—	43	6	no	—
302	#106	—	43	6	no	—
303	#106	—	43	12	no	—
304	C₄⋊C₈	C₂²×D₄₃	1	3	yes	—
305	C₄⋊C₈	C₂²×D₄₃	1	3	yes	—
306	C₄⋊C₈	C₂²×D₄₃	1	6	yes	—
307	#103	—	43	6	no	—
308	#103	—	43	6	no	—
309	#102	—	43	6	no	—
310	#102	—	43	6	no	—
311	C₄×C₈	C₂²×D₄₃	1	3	yes	—
312	C₄×C₈	C₂²×D₄₃	1	3	yes	—
313	C₄×C₈	C₂²×D₄₃	1	3	yes	—
314	C₄×C₈	C₂²×D₄₃	1	3	yes	—
315	#41	—	43	8	no	—
316	#41	—	86	4	no	—
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	43	2	no	—
319	C₂×Q₈	—	86	6	no	—
320	C₂×Q₈	—	86	6	no	—
321	C₂×Q₈	—	172	8	no	—
322	C₂×Q₈	C₂×D₁₇₂	1	3	yes	—
323	C₂×Q₈	C₂×D₁₇₂	1	1	yes	characteristic
324	C₂×D₄	—	43	2	no	—
325	C₂×D₄	—	86	2	no	—
326	C₂²×C₄	—	86	2	no	—
327	C₂²×C₄	—	86	6	no	—
328	C₂²×C₄	—	86	6	no	—
329	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
330	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
331	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
332	Q₁₆	—	43	4	no	—
333	SD₁₆	—	43	8	no	—
334	D₈	—	43	4	no	—
335	C₂×C₈	—	2	6	no	—
336	C₂×C₈	Q₈×D₄₃	1	4	yes	complemented
337	C₂×C₈	C₂³×D₄₃	1	3	yes	—
338	C₂×C₈	C₂³×D₄₃	1	3	yes	—
339	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
340	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
341	C₄⋊C₄	—	43	6	no	—
342	C₄⋊C₄	—	43	6	no	—
343	C₄⋊C₄	—	86	12	no	—
344	C₄⋊C₄	—	86	6	no	—
345	C₄⋊C₄	—	86	6	no	—
346	C₄⋊C₄	—	86	6	no	—
347	C₄⋊C₄	—	86	12	no	—
348	C₄⋊C₄	—	86	6	no	—
349	C₄⋊C₄	C₂×D₁₇₂	1	3	yes	—
350	C₄⋊C₄	C₂×D₁₇₂	1	6	yes	—
351	C₄⋊C₄	C₂×D₁₇₂	1	3	yes	—
352	C₂²⋊C₄	—	86	6	no	—
353	C₂²⋊C₄	—	86	6	no	—
354	C₂²⋊C₄	—	86	6	no	—
355	C₄²	—	86	6	no	—
356	C₄²	—	86	6	no	—
357	C₄²	—	86	6	no	—
358	C₄²	C₂×D₁₇₂	1	3	yes	—
359	C₄²	C₂×D₁₇₂	1	3	yes	—
360	C₄²	C₂³×D₄₃	1	3	yes	—
361	C₄²	C₂³×D₄₃	1	3	yes	—
362	C₂³	—	86	2	no	—
363	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
364	Q₈	—	2	6	no	—
365	Q₈	—	43	4	no	—
366	Q₈	—	86	2	no	—
367	Q₈	—	344	6	no	—
368	Q₈	—	172	12	no	—
369	Q₈	#13063347	1	4	yes	direct product factor
370	D₄	—	43	4	no	—
371	D₄	—	86	8	no	—
372	D₄	—	86	2	no	—
373	C₂×C₄	—	2	6	no	—
374	C₂×C₄	—	4	3	no	—
375	C₂×C₄	—	86	2	no	—
376	C₂×C₄	—	86	8	no	—
377	C₂×C₄	—	86	6	no	—
378	C₂×C₄	—	86	6	no	—
379	C₂×C₄	—	344	6	no	—
380	C₂×C₄	—	172	12	no	—
381	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
382	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
383	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
384	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
385	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
386	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
387	C₂×C₄	#13063372	1	2	yes	—
388	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
389	C₈	—	2	3	no	—
390	C₈	—	2	3	no	—
391	C₈	C₂×Q₈×D₄₃	1	2	yes	—
392	C₈	C₂×Q₈×D₄₃	1	2	yes	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	86	4	no	—
396	C₂²	—	172	2	no	—
397	C₂²	—	172	2	no	—
398	C₂²	Q₈×D₁₇₂	1	2	yes	—
399	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
400	C₄	—	2	3	no	—
401	C₄	—	2	3	no	—
402	C₄	—	2	3	no	—
403	C₄	—	86	4	no	—
404	C₄	—	172	6	no	—
405	C₄	—	172	6	no	—
406	C₄	Q₈×D₁₇₂	1	2	yes	—
407	C₄	C₂×#13063347	1	6	yes	—
408	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
409	C₄	C₂×#13063372	1	1	yes	characteristic
410	C₂	—	2	2	no	—
411	C₂	—	172	4	no	—
412	C₂	C₂×Q₈×D₁₇₂	1	1	yes	characteristic
413	C₂	C₂²×#13063347	1	1	yes	characteristic
414	C₂	#19149369	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂³×D₄
prime radical	C₂×#13063347	C₂²
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	Q₈×#135
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#10621898	C₂	semidirect
2. #19147487	C₂	semidirect
3. Q₈×Dic₁₇₂	C₂	semidirect
4. #19147472	C₂	semidirect
5. #19147463	C₂	semidirect
6. Q₈×#10621873	C₂	semidirect
7. Q₈×D₃₄₄	C₂	semidirect
8. #19145109	C₂	semidirect
9. #19144385	C₂	semidirect
11. C₂×C₈×Q₈×C₄₃	C₂	semidirect
52. #13063347	Q₈	direct (N × H)
116. #10621898	C₂×Q₈	semidirect
125. C₂×C₈×C₄₃	C₂×Q₈	semidirect
151. Dic₁₇₂	C₂×Q₈	semidirect
152. D₃₄₄	C₂×Q₈	semidirect
153. #10621873	C₂×Q₈	semidirect
157. Q₈×C₄₃	#135	semidirect
212. C₂×C₈×Q₈	D₄₃	semidirect
257. C₄₃	Q₈×#135	semidirect
336. C₂×C₈	Q₈×D₄₃	semidirect
369. Q₈	#13063347	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#10621898	2
#19147487	2
Q₈×Dic₁₇₂	2
#19147472	2
#19147463	2
Q₈×#10621873	2
Q₈×D₃₄₄	2
#19145109	2
#19144385	2
C₄×#13063347	2
C₂×C₈×Q₈×C₄₃	2
Q₈×#135	43

Subgroups — Q8×#13063347 #25977066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13063347 #25977066