Subgroups — #79704

Order 384.

332 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	#5568	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#5185	C₂	1	1	yes	maximal, characteristic
5	#5182	C₂	1	1	yes	maximal, characteristic, complemented
6	#5028	C₂	1	1	yes	maximal, characteristic
7	#5022	C₂	1	1	yes	maximal, characteristic, complemented
8	#4891	C₂	1	1	yes	maximal, characteristic
9	#4875	C₂	1	1	yes	maximal, characteristic, complemented
10	#4817	C₂	1	1	yes	maximal, characteristic
11	#4797	C₂	1	1	yes	maximal, characteristic, complemented
12	#4794	C₂	1	1	yes	maximal, characteristic
13	#4789	C₂	1	1	yes	maximal, characteristic, complemented
14	#4697	C₂	1	1	yes	maximal, characteristic
15	#4679	C₂	1	1	yes	maximal, characteristic, complemented
16	#3377	—	3	1	no	maximal, Sylow
17	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	#947	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃	C₂²	1	1	yes	characteristic
23	#944	C₂²	1	1	yes	characteristic
24	C₂×#216	C₂²	1	1	yes	characteristic
25	C₂×#215	C₂²	1	1	yes	characteristic
26	#898	—	2	1	no	—
27	#896	—	2	1	no	—
28	#893	C₂²	1	1	yes	characteristic
29	#890	—	2	1	no	—
30	#890	C₂²	1	1	yes	characteristic
31	#888	—	2	1	no	—
32	#888	C₂²	1	1	yes	characteristic
33	#887	C₂²	1	1	yes	characteristic
34	#876	—	2	1	no	—
35	#875	C₂²	1	1	yes	characteristic
36	#872	—	2	1	no	—
37	#870	—	2	1	no	—
38	#870	C₂²	1	1	yes	characteristic
39	C₄×Dic₆	—	2	1	no	—
40	C₄×Dic₆	C₂²	1	1	yes	characteristic
41	C₃×#107	C₂²	1	1	yes	characteristic
42	C₃×#103	C₂²	1	1	yes	characteristic
43	C₃×#103	C₂²	1	1	yes	characteristic
44	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
45	#835	C₂²	1	1	yes	characteristic
46	#835	C₂²	1	1	yes	characteristic
47	#821	C₂²	1	1	yes	characteristic
48	#821	C₂²	1	1	yes	characteristic
49	#816	C₂²	1	1	yes	characteristic
50	#816	C₂²	1	1	yes	characteristic
51	#815	C₂²	1	1	yes	characteristic
52	#815	C₂²	1	1	yes	characteristic
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	#810	C₂²	1	1	yes	characteristic
56	#809	C₂²	1	1	yes	characteristic
57	#808	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	#556	—	3	1	no	—
62	#491	—	3	1	no	—
63	#489	—	3	1	no	—
64	#488	—	3	1	no	—
65	#479	—	3	1	no	—
66	#478	—	3	1	no	—
67	#477	—	3	1	no	—
68	#464	—	3	1	no	—
69	#464	—	3	1	no	—
70	#462	—	3	1	no	—
71	#446	—	3	1	no	—
72	#441	—	3	1	no	—
73	#440	—	3	1	no	—
74	C₄×Q₁₆	D₃	1	1	yes	characteristic, complemented
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
77	C₂×D₁₂	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₃	—	2	1	no	—
79	C₂×Dic₆	—	2	1	no	—
80	C₂×Dic₆	D₄	1	1	yes	characteristic
81	C₂×Dic₆	C₂³	1	1	yes	characteristic
82	C₃×Q₁₆	—	2	2	no	—
83	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₄²	—	2	1	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	#216	—	2	2	no	—
93	#215	—	2	2	no	—
94	(C₂×D₃)⋊C₄	—	2	1	no	—
95	(C₂×D₃)⋊C₄	—	2	1	no	—
96	(C₂×D₃)⋊C₄	—	2	1	no	—
97	(C₂×D₃)⋊C₄	—	2	1	no	—
98	(C₂×D₃)⋊C₄	—	2	1	no	—
99	C₄⋊Dic₃	—	2	1	no	—
100	C₄⋊Dic₃	—	2	1	no	—
101	C₄⋊Dic₃	—	2	1	no	—
102	C₄⋊Dic₃	D₄	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	Dic₃⋊C₄	—	2	1	no	—
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	—	2	1	no	—
117	C₄×Dic₃	C₂³	1	1	yes	characteristic
118	C₄×Dic₃	C₂³	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×Q₁₆	—	3	1	no	—
122	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
123	C₂×SD₁₆	—	3	1	no	—
124	C₄⋊Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	6	1	no	—
127	#126	—	6	1	no	—
128	#126	—	6	1	no	—
129	#125	—	3	1	no	—
130	#125	—	3	1	no	—
131	#125	—	6	1	no	—
132	#125	—	6	1	no	—
133	#124	—	3	1	no	—
134	#124	—	3	1	no	—
135	C₂²⋊Q₈	—	6	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₄×Q₈	—	6	1	no	—
141	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
142	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
143	C₄×D₄	—	3	1	no	—
144	#117	—	6	1	no	—
145	#107	—	3	1	no	—
146	#107	C₂×D₃	1	1	yes	characteristic
147	#106	—	3	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	—	3	1	no	—
150	C₄⋊C₈	—	3	1	no	—
151	#103	—	3	1	no	—
152	#103	—	3	1	no	—
153	#103	—	3	1	no	—
154	#103	—	3	1	no	—
155	#103	—	3	1	no	—
156	#103	C₂×D₃	1	1	yes	characteristic
157	#103	C₂×D₃	1	1	yes	characteristic
158	#102	—	3	1	no	—
159	#102	—	3	1	no	—
160	#102	—	3	1	no	—
161	#97	—	3	1	no	—
162	#97	—	3	1	no	—
163	C₄×C₈	C₂×D₃	1	1	yes	characteristic
164	C₂²×D₃	—	2	1	no	—
165	C₃×Q₈	—	2	1	no	—
166	C₃×Q₈	—	2	1	no	—
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	#41	1	2	yes	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
180	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
181	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
183	D₁₂	—	2	1	no	—
184	D₁₂	—	2	1	no	—
185	C₄×D₃	—	4	2	no	—
186	Dic₆	—	2	1	no	—
187	Dic₆	—	2	1	no	—
188	Dic₆	—	4	1	no	—
189	Dic₆	—	4	2	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃⋊C₈	—	2	1	no	—
193	C₃⋊C₈	—	2	1	no	—
194	C₂×Q₈	—	3	1	no	—
195	C₂×Q₈	—	3	1	no	—
196	C₂×Q₈	—	6	1	no	—
197	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
198	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
199	C₂×D₄	—	3	1	no	—
200	C₂²×C₄	—	6	1	no	—
201	Q₁₆	—	2	2	no	—
202	Q₁₆	—	6	2	no	—
203	SD₁₆	—	6	2	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	3	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	—	6	1	no	—
216	C₄⋊C₄	—	6	1	no	—
217	C₄⋊C₄	—	6	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	C₂²×D₃	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
227	C₂²⋊C₄	—	6	1	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₂²⋊C₄	—	6	1	no	—
230	C₂²⋊C₄	—	6	1	no	—
231	C₂²⋊C₄	—	6	1	no	—
232	C₄²	—	2	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	—	6	1	no	—
239	C₄²	C₂²×D₃	1	1	yes	characteristic
240	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×D₃	—	4	1	no	—
242	C₂×D₃	—	4	1	no	—
243	C₂×D₃	—	4	1	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	4	1	no	—
248	C₃×C₄	—	4	1	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	C₂×#41	1	1	yes	characteristic
251	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
252	Dic₃	—	2	1	no	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	1	no	—
255	Dic₃	—	2	1	no	—
256	Dic₃	—	4	1	no	—
257	Dic₃	—	4	1	no	—
258	Dic₃	—	4	1	no	—
259	C₂³	—	6	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	2	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	6	1	no	—
264	Q₈	—	6	1	no	—
265	Q₈	—	12	1	no	—
266	Q₈	—	12	2	no	—
267	Q₈	#235	1	2	yes	—
268	D₄	—	6	1	no	—
269	D₄	—	6	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	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	—	2	1	no	—
287	C₈	—	6	1	no	—
288	C₈	—	6	1	no	—
289	C₂×C₃	#548	1	1	yes	characteristic
290	C₂×C₃	#575	1	1	yes	characteristic
291	C₂×C₃	#578	1	1	yes	characteristic
292	D₃	—	8	1	no	—
293	C₂²	—	12	1	no	—
294	C₂²	—	12	1	no	—
295	C₂²	—	12	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	—	4	1	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	—	6	1	no	—
307	C₄	—	12	1	no	—
308	C₄	—	12	1	no	—
309	C₄	—	12	1	no	—
310	C₄	C₂×#235	1	1	yes	characteristic
311	C₄	#1010	1	1	yes	characteristic
312	C₃	#3377	1	1	yes	characteristic, complemented, Sylow
313	C₂	—	24	1	no	—
314	C₂	#5552	1	1	yes	characteristic
315	C₂	#5766	1	1	yes	characteristic
316	C₂	#5768	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×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₁₂	C₂³
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#3377
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #5569	C₂	semidirect
3. C₃×C₄×Q₁₆	C₂	semidirect
5. #5182	C₂	semidirect
7. #5022	C₂	semidirect
9. #4875	C₂	semidirect
11. #4797	C₂	semidirect
13. #4789	C₂	semidirect
15. #4679	C₂	semidirect
74. C₄×Q₁₆	D₃	semidirect
312. C₃	#3377	semidirect

Maximal subgroups

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

subgroup	index
#5569	2
#5568	2
C₃×C₄×Q₁₆	2
#5185	2
#5182	2
#5028	2
#5022	2
#4891	2
#4875	2
#4817	2
#4797	2
#4794	2
#4789	2
#4697	2
#4679	2
#3377	3