Subgroups — #16858446

Order 3968.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521768	C₂	1	1	yes	maximal, characteristic
3	#14521408	C₂	1	1	yes	maximal, characteristic
4	#14521407	C₂	1	1	yes	maximal, characteristic
5	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521337	C₂	1	1	yes	maximal, characteristic
7	#14521335	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521332	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521269	C₂	1	1	yes	maximal, characteristic
10	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521156	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521154	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#464	C₂	1	1	yes	maximal, characteristic
16	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#124	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
23	#11758521	C₂²	1	1	yes	characteristic
24	C₂×#92482	C₂²	1	1	yes	characteristic
25	#11758513	C₂²	1	1	yes	characteristic
26	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
27	#11758465	—	2	1	no	—
28	#11758465	—	2	1	no	—
29	#11758465	C₂²	1	1	yes	characteristic
30	#11758465	C₂²	1	1	yes	characteristic
31	#11758456	—	2	1	no	—
32	#11758456	—	2	1	no	—
33	#11758455	—	2	1	no	—
34	#11758454	—	2	1	no	—
35	#11758446	—	2	1	no	—
36	#11758446	C₂²	1	1	yes	characteristic
37	C₄×Dic₆₂	—	2	1	no	—
38	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	#11758423	C₂²	1	1	yes	characteristic
44	#11758423	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758420	C₂²	1	1	yes	characteristic
48	#11758405	C₂²	1	1	yes	characteristic
49	#11758405	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758403	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
56	#11758398	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758391	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₁	—	2	1	no	—
63	C₂×Dic₆₂	—	2	1	no	—
64	C₂×Dic₆₂	—	2	1	no	—
65	C₂×Dic₆₂	—	2	1	no	—
66	C₂×Dic₆₂	D₄	1	1	yes	characteristic
67	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	2	no	—
69	SD₁₆×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₃₁	—	2	1	no	—
79	C₂²⋊Dic₃₁	—	2	1	no	—
80	C₂²⋊Dic₃₁	—	2	1	no	—
81	#92482	—	2	2	no	—
82	C₄⋊Dic₃₁	—	2	1	no	—
83	C₄⋊Dic₃₁	—	2	1	no	—
84	C₄⋊Dic₃₁	—	2	1	no	—
85	C₄⋊Dic₃₁	—	2	1	no	—
86	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
87	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	Dic₃₁⋊C₄	—	2	1	no	—
91	Dic₃₁⋊C₄	—	2	1	no	—
92	Dic₃₁⋊C₄	—	2	1	no	—
93	Dic₃₁⋊C₄	—	2	1	no	—
94	Dic₃₁⋊C₄	—	2	1	no	—
95	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
96	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₃₁	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂³×C₃₁	—	2	1	no	—
106	Q₈×C₃₁	—	2	1	no	—
107	Q₈×C₃₁	—	2	1	no	—
108	Q₈×C₃₁	—	2	1	no	—
109	Q₈×C₃₁	#41	1	2	yes	—
110	D₄×C₃₁	—	2	1	no	—
111	D₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₃₁	—	2	1	no	—
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	—	2	1	no	—
121	C₂×Dic₃₁	—	2	1	no	—
122	C₂×Dic₃₁	—	4	2	no	—
123	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
127	Dic₆₂	—	2	2	no	—
128	Dic₆₂	—	2	1	no	—
129	Dic₆₂	—	2	1	no	—
130	Dic₆₂	—	8	1	no	—
131	Dic₆₂	—	8	1	no	—
132	Dic₆₂	—	8	1	no	—
133	C₈×C₃₁	—	2	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₃₁⋊C₈	—	2	1	no	—
136	C₃₁⋊C₈	—	2	1	no	—
137	#3369	—	31	1	no	maximal, Sylow
138	C₂²×C₃₁	—	4	1	no	—
139	C₂²×C₃₁	—	4	1	no	—
140	C₂²×C₃₁	—	4	1	no	—
141	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	4	1	no	—
146	C₄×C₃₁	—	4	1	no	—
147	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₃₁	—	2	1	no	—
150	Dic₃₁	—	2	1	no	—
151	Dic₃₁	—	2	1	no	—
152	Dic₃₁	—	4	1	no	—
153	Dic₃₁	—	4	1	no	—
154	Dic₃₁	—	4	1	no	—
155	Dic₃₁	—	4	1	no	—
156	Dic₃₁	#137	1	2	yes	—
157	Q₈²	—	31	1	no	—
158	#556	—	31	1	no	—
159	#489	—	31	1	no	—
160	#489	—	31	1	no	—
161	#487	—	31	1	no	—
162	#476	—	31	1	no	—
163	#475	—	31	1	no	—
164	#475	—	31	1	no	—
165	#464	—	31	1	no	—
166	#464	D₃₁	1	1	yes	characteristic
167	#462	—	31	1	no	—
168	#446	—	31	1	no	—
169	#441	—	31	1	no	—
170	#441	—	31	1	no	—
171	C₄×SD₁₆	—	31	1	no	—
172	C₂×C₃₁	—	8	1	no	—
173	C₂×C₃₁	#548	1	1	yes	characteristic
174	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
175	C₂×C₃₁	#577	1	1	yes	characteristic
176	C₂×Q₁₆	—	31	1	no	—
177	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
178	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
179	C₄⋊Q₈	—	31	1	no	—
180	C₄⋊Q₈	—	31	1	no	—
181	C₄⋊Q₈	—	31	1	no	—
182	C₄⋊Q₈	—	31	1	no	—
183	C₄⋊Q₈	—	62	1	no	—
184	C₄⋊Q₈	—	62	1	no	—
185	C₄⋊Q₈	—	62	1	no	—
186	#126	—	62	1	no	—
187	#126	—	62	1	no	—
188	#124	—	31	1	no	—
189	#124	C₂×D₃₁	1	1	yes	characteristic
190	C₂²⋊Q₈	—	62	1	no	—
191	C₄×Q₈	—	31	1	no	—
192	C₄×Q₈	—	31	1	no	—
193	C₄×Q₈	—	31	1	no	—
194	C₄×Q₈	—	31	1	no	—
195	C₄×Q₈	—	31	1	no	—
196	C₄×Q₈	—	62	1	no	—
197	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
198	C₄×D₄	—	31	1	no	—
199	#117	—	62	1	no	—
200	#106	—	31	1	no	—
201	#106	—	31	1	no	—
202	#106	—	31	1	no	—
203	C₄⋊C₈	—	31	1	no	—
204	C₄⋊C₈	—	31	1	no	—
205	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
206	#103	—	31	1	no	—
207	#103	—	31	1	no	—
208	#103	—	31	1	no	—
209	#103	—	31	1	no	—
210	#103	—	31	1	no	—
211	#103	—	31	1	no	—
212	#103	C₂×D₃₁	1	1	yes	characteristic
213	#102	—	31	1	no	—
214	#102	—	31	1	no	—
215	#102	C₂×D₃₁	1	1	yes	characteristic
216	#97	—	31	1	no	—
217	#97	—	31	1	no	—
218	C₄×C₈	—	31	1	no	—
219	C₃₁	#3369	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	31	1	no	—
221	C₂×Q₈	—	31	1	no	—
222	C₂×Q₈	—	62	1	no	—
223	C₂×Q₈	—	62	1	no	—
224	C₂×Q₈	—	62	1	no	—
225	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
228	C₂²×C₄	—	62	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	62	2	no	—
231	SD₁₆	—	2	2	no	—
232	C₂×C₈	—	31	1	no	—
233	C₂×C₈	—	31	1	no	—
234	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	31	1	no	—
238	C₄⋊C₄	—	31	1	no	—
239	C₄⋊C₄	—	31	1	no	—
240	C₄⋊C₄	—	31	1	no	—
241	C₄⋊C₄	—	31	1	no	—
242	C₄⋊C₄	—	31	1	no	—
243	C₄⋊C₄	—	62	1	no	—
244	C₄⋊C₄	—	62	1	no	—
245	C₄⋊C₄	—	62	1	no	—
246	C₄⋊C₄	—	62	1	no	—
247	C₄⋊C₄	—	62	1	no	—
248	C₄⋊C₄	—	62	1	no	—
249	C₄⋊C₄	—	62	1	no	—
250	C₄⋊C₄	—	62	1	no	—
251	C₄⋊C₄	—	62	1	no	—
252	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	62	1	no	—
256	C₂²⋊C₄	—	62	1	no	—
257	C₂²⋊C₄	—	62	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	31	1	no	—
260	C₄²	—	31	1	no	—
261	C₄²	—	62	1	no	—
262	C₄²	—	62	1	no	—
263	C₄²	—	62	1	no	—
264	C₄²	—	62	1	no	—
265	C₄²	C₂²×D₃₁	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	62	2	no	—
271	Q₈	—	62	1	no	—
272	Q₈	—	62	1	no	—
273	Q₈	—	248	1	no	—
274	Q₈	—	248	1	no	—
275	Q₈	—	248	1	no	—
276	Q₈	#92497	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	31	1	no	—
283	C₂×C₄	—	31	1	no	—
284	C₂×C₄	—	31	1	no	—
285	C₂×C₄	—	31	1	no	—
286	C₂×C₄	—	62	1	no	—
287	C₂×C₄	—	62	1	no	—
288	C₂×C₄	—	62	1	no	—
289	C₂×C₄	—	62	1	no	—
290	C₂×C₄	—	124	2	no	—
291	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	62	1	no	—
297	C₈	—	62	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	31	2	no	—
308	C₄	—	62	1	no	—
309	C₄	—	62	1	no	—
310	C₄	—	62	1	no	—
311	C₄	—	124	1	no	—
312	C₄	—	124	1	no	—
313	C₄	—	124	1	no	—
314	C₄	—	124	1	no	—
315	C₄	C₂×#92497	1	1	yes	characteristic
316	C₄	#11758567	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#14521705	1	1	yes	characteristic
319	C₂	#137×D₃₁	1	1	yes	characteristic
320	C₂	#14521856	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×#464	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₄×C₃₁	C₂³
hypercenter	#464	D₃₁
hypocenter	C₃₁	#3369
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₆₂	C₂	semidirect
5. #14521398	C₂	semidirect
7. #14521335	C₂	semidirect
8. #14521332	C₂	semidirect
10. #14521257	C₂	semidirect
11. #14521178	C₂	semidirect
12. #14521156	C₂	semidirect
13. #14521154	C₂	semidirect
219. C₃₁	#3369	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₆₂	2
#14521768	2
#14521408	2
#14521407	2
#14521398	2
#14521337	2
#14521335	2
#14521332	2
#14521269	2
#14521257	2
#14521178	2
#14521156	2
#14521154	2
SD₁₆×Dic₃₁	2
C₃₁×#464	2
#3369	31