Subgroups — #16857569

Order 3968.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521699	C₂	1	1	yes	maximal, characteristic
3	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521380	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521378	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521332	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521276	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521275	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521252	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521186	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#11758423	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#448	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
18	C₂×#92497	—	2	1	no	—
19	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#121	C₂²	1	1	yes	characteristic
22	#11758523	—	2	1	no	—
23	#11758523	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
25	#11758521	—	2	1	no	—
26	#11758521	C₂²	1	1	yes	characteristic
27	#11758515	C₂²	1	1	yes	characteristic
28	#11758512	—	2	1	no	—
29	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	C₂×#92480	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₃₁	—	2	1	no	—
32	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
34	#11758501	—	2	1	no	—
35	#11758501	C₂²	1	1	yes	characteristic
36	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
39	#11758465	C₂²	1	1	yes	characteristic
40	#11758460	—	2	1	no	—
41	#11758455	—	2	1	no	—
42	C₃₁×#103	C₂²	1	1	yes	characteristic
43	C₃₁×#103	C₂²	1	1	yes	characteristic
44	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758423	C₂²	1	1	yes	characteristic
47	#11758423	C₂²	1	1	yes	characteristic
48	#11758423	C₂²	1	1	yes	characteristic
49	#11758420	C₂²	1	1	yes	characteristic
50	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#11758405	C₂²	1	1	yes	characteristic
52	#11758405	C₂²	1	1	yes	characteristic
53	#11758404	C₂²	1	1	yes	characteristic
54	#11758404	C₂²	1	1	yes	characteristic
55	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
57	#11758398	C₂²	1	1	yes	characteristic
58	#11758396	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₃₁	—	2	1	no	—
69	C₂×C₂²⋊D₃₁	D₄	1	1	yes	characteristic
70	C₂²×Dic₃₁	—	2	1	no	—
71	C₂²×Dic₃₁	—	2	1	no	—
72	Q₈×D₃₁	—	2	2	no	—
73	Q₈×D₃₁	—	4	1	no	—
74	#92497	—	4	4	no	—
75	C₂×C₄×D₃₁	—	2	1	no	—
76	C₂×C₄×D₃₁	—	2	1	no	—
77	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
78	C₂×Dic₆₂	—	2	1	no	—
79	C₂×Dic₆₂	—	2	1	no	—
80	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
81	SD₁₆×C₃₁	—	2	2	no	—
82	SD₁₆×C₃₁	—	4	1	no	—
83	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂²⋊Dic₃₁	—	2	1	no	—
88	C₂²⋊Dic₃₁	—	2	1	no	—
89	C₂²⋊Dic₃₁	—	2	1	no	—
90	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
91	#92480	—	4	1	no	—
92	(C₂×D₃₁)⋊C₄	—	2	1	no	—
93	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
94	C₄⋊Dic₃₁	—	2	1	no	—
95	C₄⋊Dic₃₁	—	2	1	no	—
96	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
99	Dic₃₁⋊C₄	—	2	1	no	—
100	Dic₃₁⋊C₄	—	2	1	no	—
101	Dic₃₁⋊C₄	—	2	1	no	—
102	Dic₃₁⋊C₄	—	2	1	no	—
103	Dic₃₁⋊C₄	—	2	1	no	—
104	Dic₃₁⋊C₄	—	2	1	no	—
105	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
106	C₄×Dic₃₁	—	2	1	no	—
107	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₃₁⋊C₈	—	4	1	no	—
109	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
110	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
111	C₈×D₃₁	—	4	1	no	—
112	C₂³×C₃₁	—	2	1	no	—
113	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	—	2	1	no	—
117	Q₈×C₃₁	—	2	1	no	—
118	Q₈×C₃₁	—	4	2	no	—
119	Q₈×C₃₁	#41	1	2	yes	complemented
120	D₄×C₃₁	—	2	1	no	—
121	D₄×C₃₁	—	2	1	no	—
122	D₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₃₁	—	4	2	no	—
132	C₂²⋊D₃₁	SD₁₆	1	4	yes	complemented
133	C₂×Dic₃₁	—	2	1	no	—
134	C₂×Dic₃₁	—	2	1	no	—
135	C₂×Dic₃₁	—	2	1	no	—
136	C₂×Dic₃₁	—	2	1	no	—
137	C₂×Dic₃₁	—	4	2	no	—
138	C₂×Dic₃₁	—	4	2	no	—
139	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄×D₃₁	—	2	1	no	—
143	C₄×D₃₁	—	2	1	no	—
144	C₄×D₃₁	—	4	2	no	—
145	C₄×D₃₁	—	4	2	no	—
146	Dic₆₂	—	2	1	no	—
147	Dic₆₂	—	2	1	no	—
148	Dic₆₂	—	4	2	no	—
149	Dic₆₂	—	4	2	no	—
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	—	2	1	no	—
152	C₃₁⋊C₈	—	2	1	no	—
153	C₃₁⋊C₈	—	2	1	no	—
154	#3298	—	31	1	no	maximal, Sylow
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	—	4	1	no	—
159	C₂²×C₃₁	—	4	1	no	—
160	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
161	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₃₁	—	2	1	no	—
163	C₂×D₃₁	—	2	1	no	—
164	C₂×D₃₁	C₂×SD₁₆	1	2	yes	—
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	4	1	no	—
168	C₄×C₃₁	—	4	1	no	—
169	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₃₁	—	2	1	no	—
172	Dic₃₁	—	2	1	no	—
173	Dic₃₁	—	4	1	no	—
174	Dic₃₁	—	4	1	no	—
175	Dic₃₁	—	4	1	no	—
176	Dic₃₁	—	4	1	no	—
177	Dic₃₁	C₂×SD₁₆	1	2	yes	—
178	D₄×Q₈	—	31	1	no	—
179	#547	—	31	1	no	—
180	#487	—	31	1	no	—
181	#483	—	31	1	no	—
182	#483	—	31	1	no	—
183	#475	—	31	1	no	—
184	#465	—	31	1	no	—
185	#465	—	31	1	no	—
186	#461	—	31	1	no	—
187	#448	—	31	1	no	—
188	#448	D₃₁	1	1	yes	characteristic, complemented
189	C₄×SD₁₆	—	31	1	no	—
190	C₈×D₄	—	31	1	no	—
191	C₂×#103	—	31	1	no	—
192	C₂×#103	—	31	1	no	—
193	C₂×C₃₁	—	2	2	no	—
194	C₂×C₃₁	—	8	1	no	—
195	C₂×C₃₁	#546	1	1	yes	characteristic
196	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
197	C₂×C₃₁	#578	1	1	yes	characteristic
198	D₃₁	—	2	2	no	—
199	C₂×#41	—	62	1	no	—
200	C₂²×Q₈	—	31	1	no	—
201	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
202	C₂×SD₁₆	—	31	1	no	—
203	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
205	C₂²×C₈	—	31	1	no	—
206	C₂²×C₈	—	31	1	no	—
207	C₄⋊Q₈	—	31	1	no	—
208	C₄⋊Q₈	—	31	1	no	—
209	C₄⋊Q₈	—	62	1	no	—
210	#123	—	62	1	no	—
211	#123	—	62	1	no	—
212	C₂²⋊Q₈	—	31	1	no	—
213	C₂²⋊Q₈	—	31	1	no	—
214	C₂²⋊Q₈	—	62	1	no	—
215	C₂²⋊Q₈	—	62	1	no	—
216	C₂²⋊Q₈	—	62	1	no	—
217	#121	—	31	1	no	—
218	#121	C₂×D₃₁	1	1	yes	characteristic
219	C₄×Q₈	—	31	1	no	—
220	C₄×D₄	—	31	1	no	—
221	C₄×D₄	—	31	1	no	—
222	C₄×D₄	—	62	1	no	—
223	C₂×C₄⋊C₄	—	31	1	no	—
224	C₂×C₄⋊C₄	—	31	1	no	—
225	#106	—	31	1	no	—
226	#106	—	31	1	no	—
227	#106	—	31	1	no	—
228	C₄⋊C₈	—	31	1	no	—
229	#103	—	31	1	no	—
230	#103	—	31	1	no	—
231	#103	—	31	1	no	—
232	#103	—	31	1	no	—
233	#103	—	31	1	no	—
234	#103	—	31	1	no	—
235	#103	—	31	1	no	—
236	#103	C₂×D₃₁	1	1	yes	characteristic
237	#103	C₂×D₃₁	1	1	yes	characteristic
238	#102	—	31	1	no	—
239	C₂²⋊C₈	—	31	1	no	—
240	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
241	C₄×C₈	—	31	1	no	—
242	C₃₁	#3298	1	1	yes	characteristic, complemented, Sylow
243	#41	—	124	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	31	1	no	—
248	C₂×Q₈	—	62	1	no	—
249	C₂×Q₈	—	62	1	no	—
250	C₂×Q₈	—	62	2	no	—
251	C₂×Q₈	—	124	1	no	—
252	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	31	1	no	—
256	C₂×D₄	—	62	1	no	—
257	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	31	1	no	—
260	C₂²×C₄	—	62	1	no	—
261	C₂²×C₄	—	62	1	no	—
262	C₂²×C₄	—	62	1	no	—
263	C₂²×C₄	—	62	1	no	—
264	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
265	SD₁₆	—	2	2	no	—
266	SD₁₆	—	4	1	no	—
267	SD₁₆	—	124	1	no	—
268	C₂×C₈	—	31	1	no	—
269	C₂×C₈	—	31	1	no	—
270	C₂×C₈	—	124	1	no	—
271	C₂×C₈	—	124	1	no	—
272	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
274	C₄⋊C₄	—	31	1	no	—
275	C₄⋊C₄	—	31	1	no	—
276	C₄⋊C₄	—	31	1	no	—
277	C₄⋊C₄	—	31	1	no	—
278	C₄⋊C₄	—	62	1	no	—
279	C₄⋊C₄	—	62	1	no	—
280	C₄⋊C₄	—	62	1	no	—
281	C₄⋊C₄	—	62	1	no	—
282	C₄⋊C₄	—	62	1	no	—
283	C₄⋊C₄	—	62	1	no	—
284	C₄⋊C₄	—	62	1	no	—
285	C₄⋊C₄	—	62	1	no	—
286	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	31	1	no	—
289	C₂²⋊C₄	—	31	1	no	—
290	C₂²⋊C₄	—	62	1	no	—
291	C₂²⋊C₄	—	62	1	no	—
292	C₂²⋊C₄	—	62	1	no	—
293	C₂²⋊C₄	—	62	1	no	—
294	C₄²	—	31	1	no	—
295	C₄²	—	62	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	31	1	no	—
298	C₂³	D₄×D₃₁	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	62	1	no	—
304	Q₈	—	62	1	no	—
305	Q₈	—	124	2	no	—
306	Q₈	—	124	2	no	—
307	Q₈	#92497	1	2	yes	complemented
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	31	4	no	—
312	D₄	—	124	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	31	1	no	—
320	C₂×C₄	—	31	1	no	—
321	C₂×C₄	—	31	1	no	—
322	C₂×C₄	—	62	1	no	—
323	C₂×C₄	—	62	1	no	—
324	C₂×C₄	—	62	1	no	—
325	C₂×C₄	—	62	1	no	—
326	C₂×C₄	—	62	1	no	—
327	C₂×C₄	—	62	1	no	—
328	C₂×C₄	—	124	2	no	—
329	C₂×C₄	—	124	2	no	—
330	C₂×C₄	—	124	2	no	—
331	C₂×C₄	—	124	2	no	—
332	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	62	1	no	—
337	C₈	—	62	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	31	2	no	—
344	C₂²	—	62	1	no	—
345	C₂²	—	62	1	no	—
346	C₂²	SD₁₆×D₃₁	1	2	yes	—
347	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	31	2	no	—
353	C₄	—	62	1	no	—
354	C₄	—	62	1	no	—
355	C₄	—	124	1	no	—
356	C₄	—	124	1	no	—
357	C₄	—	124	1	no	—
358	C₄	—	124	1	no	—
359	C₄	C₂×#92497	1	1	yes	characteristic
360	C₄	#11758561	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	62	2	no	—
364	C₂	#14521686	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₃₁	1	1	yes	characteristic
366	C₂	#14521865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#448	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#14521699	C₂
hypercenter	#448	D₃₁
hypocenter	C₃₁	#3298
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₃₁	C₂	semidirect
3. #14521398	C₂	semidirect
4. #14521380	C₂	semidirect
5. #14521378	C₂	semidirect
6. #14521332	C₂	semidirect
7. #14521276	C₂	semidirect
8. #14521275	C₂	semidirect
9. #14521252	C₂	semidirect
10. #14521186	C₂	semidirect
11. SD₁₆×Dic₃₁	C₂	semidirect
12. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
13. C₂×#11758423	C₂	semidirect
14. #103×D₃₁	C₂	semidirect
15. C₃₁×#448	C₂	semidirect
119. Q₈×C₃₁	#41	semidirect
132. C₂²⋊D₃₁	SD₁₆	semidirect
188. #448	D₃₁	semidirect
242. C₃₁	#3298	semidirect
307. Q₈	#92497	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₃₁	2
#14521699	2
#14521398	2
#14521380	2
#14521378	2
#14521332	2
#14521276	2
#14521275	2
#14521252	2
#14521186	2
SD₁₆×Dic₃₁	2
(C₂²⋊D₃₁)⋊C₈	2
C₂×#11758423	2
#103×D₃₁	2
C₃₁×#448	2
#3298	31