Subgroups — #80576

Order 384.

354 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5680	C₂	1	1	yes	maximal, characteristic
2	#5578	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
4	#5188	C₂	1	1	yes	maximal, characteristic, complemented
5	#5182	C₂	1	1	yes	maximal, characteristic, complemented
6	#5171	C₂	1	1	yes	maximal, characteristic
7	#5162	C₂	1	1	yes	maximal, characteristic, complemented
8	#5054	C₂	1	1	yes	maximal, characteristic, complemented
9	#5022	C₂	1	1	yes	maximal, characteristic, complemented
10	D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#4829	C₂	1	1	yes	maximal, characteristic, complemented
12	#4813	C₂	1	1	yes	maximal, characteristic, complemented
13	#4791	C₂	1	1	yes	maximal, characteristic, complemented
14	#4778	C₂	1	1	yes	maximal, characteristic
15	#4759	C₂	1	1	yes	maximal, characteristic, complemented
16	#3248	—	3	1	no	maximal, Sylow
17	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₃×#124	C₂²	1	1	yes	characteristic
20	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
21	#946	C₂²	1	1	yes	characteristic
22	#946	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃	C₂²	1	1	yes	characteristic
24	#944	C₂²	1	1	yes	characteristic
25	C₂×#216	C₂²	1	1	yes	characteristic
26	#938	C₂²	1	1	yes	characteristic
27	D₄×Dic₃	C₂²	1	1	yes	characteristic
28	C₂×#214	C₂²	1	1	yes	characteristic
29	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
30	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
31	#897	C₂²	1	1	yes	characteristic
32	#896	—	2	1	no	—
33	#889	—	2	1	no	—
34	#888	C₂²	1	1	yes	characteristic
35	#883	—	2	1	no	—
36	#67⋊C₄	—	2	1	no	—
37	#879	—	2	1	no	—
38	#877	—	2	1	no	—
39	#872	—	2	1	no	—
40	C₄²×D₃	C₂²	1	1	yes	characteristic
41	#870	C₂²	1	1	yes	characteristic
42	C₄×Dic₆	—	2	1	no	—
43	C₄×Dic₆	C₂²	1	1	yes	characteristic
44	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃×#103	C₂²	1	1	yes	characteristic
46	C₃×#102	C₂²	1	1	yes	characteristic
47	#835	C₂²	1	1	yes	characteristic
48	#835	C₂²	1	1	yes	characteristic
49	#835	C₂²	1	1	yes	characteristic
50	#832	C₂²	1	1	yes	characteristic
51	#818	C₂²	1	1	yes	characteristic
52	#817	C₂²	1	1	yes	characteristic
53	#816	C₂²	1	1	yes	characteristic
54	#816	C₂²	1	1	yes	characteristic
55	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₃⋊C₈	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	#533	—	3	1	no	—
61	#529	—	3	1	no	—
62	#479	—	3	1	no	—
63	#479	—	3	1	no	—
64	#475	—	3	1	no	—
65	#474	—	3	1	no	—
66	#467	—	3	1	no	—
67	#467	—	3	1	no	—
68	#465	—	3	1	no	—
69	#465	—	3	1	no	—
70	#464	—	3	1	no	—
71	#464	D₃	1	1	yes	characteristic, complemented
72	#462	—	3	1	no	—
73	#461	—	3	1	no	—
74	C₂×C₄⋊C₈	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
78	C₂×#67	—	2	1	no	—
79	C₂²×Dic₃	—	2	1	no	—
80	C₂×C₄×D₃	D₄	1	1	yes	characteristic
81	C₂×C₄×D₃	D₄	1	1	yes	characteristic
82	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
83	C₂×Dic₆	—	2	1	no	—
84	C₂×Dic₆	C₂³	1	1	yes	characteristic
85	C₃×Q₁₆	—	4	1	no	—
86	C₃×SD₁₆	—	4	1	no	—
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×C₂²⋊C₄	—	2	1	no	—
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	#216	—	4	1	no	—
98	#214	—	4	1	no	—
99	(C₂×D₃)⋊C₄	—	2	1	no	—
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	(C₂×D₃)⋊C₄	—	2	1	no	—
102	C₄⋊Dic₃	—	2	1	no	—
103	C₄⋊Dic₃	—	2	1	no	—
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
113	Dic₃⋊C₄	—	2	1	no	—
114	Dic₃⋊C₄	—	2	1	no	—
115	C₄×Dic₃	—	2	1	no	—
116	C₄×Dic₃	—	2	1	no	—
117	C₄×Dic₃	D₄	1	1	yes	characteristic
118	C₄×Dic₃	D₄	1	1	yes	characteristic
119	C₄×Dic₃	C₂³	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	D₃×C₈	—	4	1	no	—
123	D₃×C₈	—	4	1	no	—
124	C₂×Q₁₆	—	3	1	no	—
125	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
126	C₂×SD₁₆	—	3	1	no	—
127	C₂×SD₁₆	C₂×D₃	1	1	yes	characteristic
128	C₂²×C₈	—	3	1	no	—
129	C₂²×C₈	—	3	1	no	—
130	C₄⋊Q₈	—	3	1	no	—
131	C₄⋊Q₈	—	3	1	no	—
132	#126	—	6	1	no	—
133	#125	—	3	1	no	—
134	#125	—	6	1	no	—
135	#124	C₂×D₃	1	1	yes	characteristic
136	#123	—	6	1	no	—
137	C₂²⋊Q₈	—	3	1	no	—
138	C₂²⋊Q₈	—	3	1	no	—
139	C₂²⋊Q₈	—	3	1	no	—
140	C₂²⋊Q₈	—	6	1	no	—
141	#121	—	3	1	no	—
142	C₄×Q₈	—	3	1	no	—
143	C₄×Q₈	—	3	1	no	—
144	C₄×Q₈	—	6	1	no	—
145	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
146	C₄×D₄	—	3	1	no	—
147	C₄×D₄	—	6	1	no	—
148	#117	—	6	1	no	—
149	#117	—	6	1	no	—
150	C₂×C₄²	—	3	1	no	—
151	#107	—	3	1	no	—
152	#107	—	3	1	no	—
153	#106	—	3	1	no	—
154	#106	—	3	1	no	—
155	C₄⋊C₈	—	3	1	no	—
156	C₄⋊C₈	—	3	1	no	—
157	C₄⋊C₈	—	3	1	no	—
158	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	#103	—	3	1	no	—
162	#103	—	3	1	no	—
163	#103	—	3	1	no	—
164	#103	C₂×D₃	1	1	yes	characteristic
165	#102	—	3	1	no	—
166	#102	C₂×D₃	1	1	yes	characteristic
167	C₂³×C₃	—	2	1	no	—
168	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
169	C₃×Q₈	—	2	1	no	—
170	C₃×Q₈	—	2	1	no	—
171	C₃×Q₈	—	2	1	no	—
172	C₃×Q₈	—	2	1	no	—
173	C₃×D₄	—	2	1	no	—
174	C₃×D₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
181	#67	—	4	2	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	4	2	no	—
187	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
188	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
190	C₄×D₃	—	2	2	no	—
191	C₄×D₃	—	2	1	no	—
192	C₄×D₃	—	2	1	no	—
193	C₄×D₃	—	2	2	no	—
194	Dic₆	—	2	1	no	—
195	Dic₆	—	2	1	no	—
196	Dic₆	—	4	2	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃⋊C₈	—	2	1	no	—
200	C₃⋊C₈	—	2	1	no	—
201	C₂×Q₈	—	3	1	no	—
202	C₂×Q₈	—	6	1	no	—
203	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
204	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
205	C₂×D₄	—	6	1	no	—
206	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
207	C₂²×C₄	—	3	1	no	—
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	Q₁₆	—	4	1	no	—
212	Q₁₆	—	12	1	no	—
213	SD₁₆	—	4	1	no	—
214	SD₁₆	—	12	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	12	1	no	—
218	C₂×C₈	—	12	1	no	—
219	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄⋊C₄	—	6	1	no	—
228	C₄⋊C₄	—	6	1	no	—
229	C₄⋊C₄	—	6	1	no	—
230	C₄⋊C₄	—	6	1	no	—
231	C₄⋊C₄	—	6	1	no	—
232	C₄⋊C₄	—	6	1	no	—
233	C₄⋊C₄	—	6	1	no	—
234	C₄⋊C₄	—	6	1	no	—
235	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	6	1	no	—
239	C₂²⋊C₄	—	6	1	no	—
240	C₂²⋊C₄	—	6	1	no	—
241	C₂²⋊C₄	—	6	1	no	—
242	C₂²⋊C₄	—	6	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	3	1	no	—
245	C₄²	—	3	1	no	—
246	C₄²	—	3	1	no	—
247	C₄²	—	6	1	no	—
248	C₄²	—	6	1	no	—
249	C₄²	C₂²×D₃	1	1	yes	characteristic
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	—	4	1	no	—
253	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	#137	1	2	yes	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
263	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
264	Dic₃	—	2	1	no	—
265	Dic₃	—	2	1	no	—
266	Dic₃	—	4	1	no	—
267	Dic₃	—	4	1	no	—
268	Dic₃	—	4	1	no	—
269	Dic₃	—	4	1	no	—
270	Dic₃	#135	1	2	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	3	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	6	1	no	—
278	Q₈	—	6	1	no	—
279	Q₈	—	12	2	no	—
280	D₄	—	2	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	12	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	12	2	no	—
298	C₂×C₄	D₃×D₄	1	1	yes	characteristic
299	C₂×C₄	D₃×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	6	1	no	—
304	C₈	—	6	1	no	—
305	C₂×C₃	—	8	1	no	—
306	C₂×C₃	#537	1	1	yes	characteristic
307	C₂×C₃	C₂×#135	1	1	yes	characteristic
308	C₂×C₃	C₂×#137	1	1	yes	characteristic
309	D₃	—	2	2	no	—
310	C₂²	—	3	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	6	1	no	—
315	C₂²	—	6	1	no	—
316	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	3	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	—	12	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	—	12	1	no	—
329	C₄	#1004	1	1	yes	characteristic
330	C₄	#1010	1	1	yes	characteristic
331	C₃	#3248	1	1	yes	characteristic, complemented, Sylow
332	C₂	—	6	2	no	—
333	C₂	—	8	1	no	—
334	C₂	#5608	1	1	yes	characteristic
335	C₂	D₃×#135	1	1	yes	characteristic
336	C₂	D₃×#137	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×#464	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#938	C₂²
hypercenter	#464	D₃
hypocenter	C₃	#3248
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. #5578	C₂	semidirect
3. C₃×#464	C₂	semidirect
4. #5188	C₂	semidirect
5. #5182	C₂	semidirect
7. #5162	C₂	semidirect
8. #5054	C₂	semidirect
9. #5022	C₂	semidirect
10. D₃×C₄⋊C₈	C₂	semidirect
11. #4829	C₂	semidirect
12. #4813	C₂	semidirect
13. #4791	C₂	semidirect
15. #4759	C₂	semidirect
71. #464	D₃	semidirect
331. C₃	#3248	semidirect

Maximal subgroups

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

subgroup	index
#5680	2
#5578	2
C₃×#464	2
#5188	2
#5182	2
#5171	2
#5162	2
#5054	2
#5022	2
D₃×C₄⋊C₈	2
#4829	2
#4813	2
#4791	2
#4778	2
#4759	2
#3248	3