Subgroups — #79760

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5569	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#440	C₂	1	1	yes	maximal, characteristic
4	#5164	C₂	1	1	yes	maximal, characteristic
5	Dic₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
6	#5021	C₂	1	1	yes	maximal, characteristic, complemented
7	#5009	C₂	1	1	yes	maximal, characteristic
8	#4876	C₂	1	1	yes	maximal, characteristic, complemented
9	#4874	C₂	1	1	yes	maximal, characteristic, complemented
10	#4793	C₂	1	1	yes	maximal, characteristic, complemented
11	#4791	C₂	1	1	yes	maximal, characteristic, complemented
12	#4763	C₂	1	1	yes	maximal, characteristic
13	#4759	C₂	1	1	yes	maximal, characteristic
14	#4702	C₂	1	1	yes	maximal, characteristic, complemented
15	#4701	C₂	1	1	yes	maximal, characteristic, complemented
16	#3386	—	3	1	no	maximal, Sylow
17	C₂²×Dic₆	—	2	1	no	—
18	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃	C₂²	1	1	yes	characteristic
23	D₄×Dic₃	C₂²	1	1	yes	characteristic
24	D₄×Dic₃	C₂²	1	1	yes	characteristic
25	#924	—	2	1	no	—
26	#890	—	2	1	no	—
27	#890	C₂²	1	1	yes	characteristic
28	#888	—	2	1	no	—
29	#888	C₂²	1	1	yes	characteristic
30	#888	C₂²	1	1	yes	characteristic
31	#888	C₂²	1	1	yes	characteristic
32	#878	—	2	1	no	—
33	#878	—	2	1	no	—
34	#870	—	2	1	no	—
35	#870	C₂²	1	1	yes	characteristic
36	#869	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	—	2	1	no	—
38	C₄×Dic₆	C₂²	1	1	yes	characteristic
39	C₃×#107	C₂²	1	1	yes	characteristic
40	C₃×#103	C₂²	1	1	yes	characteristic
41	C₃×#102	C₂²	1	1	yes	characteristic
42	C₃×#97	C₂²	1	1	yes	characteristic
43	#835	C₂²	1	1	yes	characteristic
44	#835	C₂²	1	1	yes	characteristic
45	#832	C₂²	1	1	yes	characteristic
46	#832	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#818	C₂²	1	1	yes	characteristic
49	#817	C₂²	1	1	yes	characteristic
50	#817	C₂²	1	1	yes	characteristic
51	#815	C₂²	1	1	yes	characteristic
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₃	C₂²	1	1	yes	characteristic
55	#808	C₂²	1	1	yes	characteristic
56	#807	C₂²	1	1	yes	characteristic
57	#807	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
60	#557	—	3	1	no	—
61	D₄×Q₈	—	3	1	no	—
62	#501	—	3	1	no	—
63	#501	—	3	1	no	—
64	#500	—	3	1	no	—
65	#478	—	3	1	no	—
66	#478	—	3	1	no	—
67	#477	—	3	1	no	—
68	#477	—	3	1	no	—
69	#475	—	3	1	no	—
70	#474	—	3	1	no	—
71	#446	—	3	1	no	—
72	#440	—	3	1	no	—
73	#440	D₃	1	1	yes	characteristic
74	C₄×SD₁₆	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×Dic₃	—	2	1	no	—
79	C₂²×Dic₃	—	2	1	no	—
80	C₂×Dic₆	—	2	1	no	—
81	C₂×Dic₆	—	2	1	no	—
82	C₂×Dic₆	—	2	1	no	—
83	C₂×Dic₆	—	2	1	no	—
84	C₂×Dic₆	—	4	4	no	—
85	C₂×Dic₆	D₄	1	1	yes	characteristic
86	C₃×SD₁₆	Q₈	1	4	yes	complemented
87	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₃×C₂²⋊C₄	—	2	1	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	#217	—	2	1	no	—
96	#217	—	2	1	no	—
97	C₄⋊Dic₃	—	2	1	no	—
98	C₄⋊Dic₃	—	2	1	no	—
99	C₄⋊Dic₃	—	2	1	no	—
100	C₄⋊Dic₃	—	2	1	no	—
101	C₄⋊Dic₃	D₄	1	1	yes	characteristic
102	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
106	Dic₃⋊C₄	—	2	1	no	—
107	Dic₃⋊C₄	—	2	1	no	—
108	Dic₃⋊C₄	—	2	1	no	—
109	Dic₃⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	Dic₃⋊C₄	—	2	1	no	—
112	Dic₃⋊C₄	D₄	1	1	yes	characteristic
113	Dic₃⋊C₄	D₄	1	1	yes	characteristic
114	C₄×Dic₃	—	2	1	no	—
115	C₄×Dic₃	—	2	1	no	—
116	C₄×Dic₃	C₂³	1	1	yes	characteristic
117	C₄×Dic₃	C₂³	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	C₂²×Q₈	—	6	1	no	—
121	C₂×SD₁₆	C₂×D₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	3	1	no	—
123	C₄⋊Q₈	—	3	1	no	—
124	C₄⋊Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	6	1	no	—
127	#125	—	3	1	no	—
128	#125	—	3	1	no	—
129	#125	—	6	1	no	—
130	#125	—	6	1	no	—
131	C₂²⋊Q₈	—	6	1	no	—
132	C₂²⋊Q₈	—	6	1	no	—
133	C₂²⋊Q₈	—	6	1	no	—
134	C₄×Q₈	—	3	1	no	—
135	C₄×Q₈	—	3	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	6	1	no	—
138	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
139	C₄×D₄	—	3	1	no	—
140	C₄×D₄	—	3	1	no	—
141	C₄×D₄	C₂×D₃	1	1	yes	characteristic
142	#107	—	3	1	no	—
143	#107	—	3	1	no	—
144	#107	—	3	1	no	—
145	#107	—	3	1	no	—
146	#107	—	3	1	no	—
147	#107	C₂×D₃	1	1	yes	characteristic
148	#106	—	3	1	no	—
149	#106	—	3	1	no	—
150	#106	—	3	1	no	—
151	C₄⋊C₈	—	3	1	no	—
152	C₄⋊C₈	—	3	1	no	—
153	C₄⋊C₈	—	3	1	no	—
154	#103	—	3	1	no	—
155	#103	—	3	1	no	—
156	#103	C₂×D₃	1	1	yes	characteristic
157	#102	—	3	1	no	—
158	#102	—	3	1	no	—
159	#102	C₂×D₃	1	1	yes	characteristic
160	#97	—	3	1	no	—
161	#97	C₂×D₃	1	1	yes	characteristic
162	C₄×C₈	—	3	1	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₃×Q₈	—	2	1	no	—
165	C₃×Q₈	C₂×Q₈	1	2	yes	complemented
166	C₃×D₄	—	2	1	no	—
167	C₃×D₄	C₂×Q₈	1	2	yes	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	4	2	no	—
180	C₂×Dic₃	—	4	2	no	—
181	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
185	Dic₆	—	2	2	no	—
186	Dic₆	—	4	2	no	—
187	Dic₆	—	4	2	no	—
188	Dic₆	—	4	2	no	—
189	Dic₆	—	4	2	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	C₂×Q₈	1	2	yes	complemented
192	C₃⋊C₈	—	2	1	no	—
193	C₃⋊C₈	—	2	1	no	—
194	C₂×Q₈	—	3	1	no	—
195	C₂×Q₈	—	6	1	no	—
196	C₂×Q₈	—	6	1	no	—
197	C₂×Q₈	—	6	1	no	—
198	C₂×Q₈	—	6	1	no	—
199	C₂×Q₈	—	12	4	no	—
200	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
201	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	6	1	no	—
204	C₂²×C₄	—	6	1	no	—
205	SD₁₆	Dic₆	1	4	yes	complemented
206	C₂×C₈	—	3	1	no	—
207	C₂×C₈	—	3	1	no	—
208	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	1	no	—
214	C₄⋊C₄	—	3	1	no	—
215	C₄⋊C₄	—	3	1	no	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₄⋊C₄	—	6	1	no	—
220	C₄⋊C₄	—	6	1	no	—
221	C₄⋊C₄	—	6	1	no	—
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	—	6	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄⋊C₄	—	6	1	no	—
228	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
229	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	6	1	no	—
232	C₂²⋊C₄	—	6	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	3	1	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	6	1	no	—
237	C₄²	—	6	1	no	—
238	C₄²	C₂²×D₃	1	1	yes	characteristic
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	4	1	no	—
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	4	1	no	—
247	C₃×C₄	—	4	1	no	—
248	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
249	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
250	Dic₃	—	2	1	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	2	1	no	—
253	Dic₃	—	4	1	no	—
254	Dic₃	—	4	1	no	—
255	Dic₃	—	4	1	no	—
256	Dic₃	—	4	1	no	—
257	Dic₃	AGL(1,ℤ/8)	1	2	yes	—
258	C₂³	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	6	2	no	—
261	Q₈	—	12	2	no	—
262	Q₈	—	12	2	no	—
263	Q₈	—	12	2	no	—
264	Q₈	—	12	2	no	—
265	Q₈	C₂×Dic₆	1	2	yes	complemented
266	D₄	—	2	1	no	—
267	D₄	C₂×Dic₆	1	2	yes	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	6	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	12	2	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	D₃×D₄	1	1	yes	characteristic
283	C₂×C₄	D₃×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	6	1	no	—
287	C₈	—	6	1	no	—
288	C₈	C₂×Dic₆	1	2	yes	complemented
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
291	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₂×C₃	#578	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	3	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	6	1	no	—
304	C₄	—	6	1	no	—
305	C₄	—	6	1	no	—
306	C₄	—	12	1	no	—
307	C₄	—	12	1	no	—
308	C₄	—	12	1	no	—
309	C₄	—	12	1	no	—
310	C₄	C₂²×Dic₆	1	1	yes	characteristic
311	C₄	#1010	1	1	yes	characteristic
312	C₃	#3386	1	1	yes	characteristic, complemented, Sylow
313	C₂	—	4	2	no	—
314	C₂	D₄×Dic₆	1	1	yes	characteristic
315	C₂	D₃×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂	#5779	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#440	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₄	C₂³
hypercenter	#440	D₃
hypocenter	C₃	#3386
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #5569	C₂	semidirect
6. #5021	C₂	semidirect
8. #4876	C₂	semidirect
9. #4874	C₂	semidirect
10. #4793	C₂	semidirect
11. #4791	C₂	semidirect
14. #4702	C₂	semidirect
15. #4701	C₂	semidirect
86. C₃×SD₁₆	Q₈	semidirect
165. C₃×Q₈	C₂×Q₈	semidirect
191. C₃×C₈	C₂×Q₈	semidirect
205. SD₁₆	Dic₆	semidirect
265. Q₈	C₂×Dic₆	semidirect
288. C₈	C₂×Dic₆	semidirect
312. C₃	#3386	semidirect

Maximal subgroups

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

subgroup	index
#5569	2
D₄×Dic₆	2
C₃×#440	2
#5164	2
Dic₃×SD₁₆	2
#5021	2
#5009	2
#4876	2
#4874	2
#4793	2
#4791	2
#4763	2
#4759	2
#4702	2
#4701	2
#3386	3