Subgroups — #81500

Order 384.

326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5622	C₂	1	1	yes	maximal, characteristic
2	#5517	C₂	1	1	yes	maximal, characteristic
3	C₃×#485	C₂	1	1	yes	maximal, characteristic
4	#5045	C₂	1	1	yes	maximal, characteristic
5	#4996	C₂	1	1	yes	maximal, characteristic
6	#4877	C₂	1	1	yes	maximal, characteristic, complemented
7	#4874	C₂	1	1	yes	maximal, characteristic, complemented
8	#4857	C₂	1	1	yes	maximal, characteristic, complemented
9	#4855	C₂	1	1	yes	maximal, characteristic, complemented
10	#4799	C₂	1	1	yes	maximal, characteristic, complemented
11	#4796	C₂	1	1	yes	maximal, characteristic, complemented
12	#4792	C₂	1	1	yes	maximal, characteristic, complemented
13	#4792	C₂	1	1	yes	maximal, characteristic, complemented
14	#4723	C₂	1	1	yes	maximal, characteristic
15	#4718	C₂	1	1	yes	maximal, characteristic
16	#3231	—	3	1	no	maximal, Sylow
17	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#117	C₂²	1	1	yes	characteristic
19	Q₈×Dic₃	C₂²	1	1	yes	characteristic
20	#944	C₂²	1	1	yes	characteristic
21	#926	C₂²	1	1	yes	characteristic
22	#924	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
24	#890	C₂²	1	1	yes	characteristic
25	#890	C₂²	1	1	yes	characteristic
26	#889	—	2	1	no	—
27	#888	C₂²	1	1	yes	characteristic
28	#888	C₂²	1	1	yes	characteristic
29	#888	C₂²	1	1	yes	characteristic
30	#887	—	2	1	no	—
31	#887	C₂²	1	1	yes	characteristic
32	#878	—	2	1	no	—
33	#878	—	2	1	no	—
34	#878	—	2	1	no	—
35	#877	—	2	1	no	—
36	C₄×Dic₆	—	2	1	no	—
37	C₄×Dic₆	—	2	1	no	—
38	C₃×#107	C₂²	1	1	yes	characteristic
39	C₃×#106	C₂²	1	1	yes	characteristic
40	C₃×#103	C₂²	1	1	yes	characteristic
41	C₃×#103	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	#835	C₂²	1	1	yes	characteristic
44	#835	C₂²	1	1	yes	characteristic
45	#830	C₂²	1	1	yes	characteristic
46	#818	C₂²	1	1	yes	characteristic
47	#817	C₂²	1	1	yes	characteristic
48	#816	C₂²	1	1	yes	characteristic
49	#816	C₂²	1	1	yes	characteristic
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₃	C₂²	1	1	yes	characteristic
54	#810	C₂²	1	1	yes	characteristic
55	#810	C₂²	1	1	yes	characteristic
56	#808	C₂²	1	1	yes	characteristic
57	#808	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	#807	C₂²	1	1	yes	characteristic
60	#533	—	3	1	no	—
61	#533	—	3	1	no	—
62	#500	—	3	1	no	—
63	#498	—	3	1	no	—
64	#487	—	3	1	no	—
65	#487	—	3	1	no	—
66	#485	—	3	1	no	—
67	#485	—	3	1	no	—
68	#485	—	3	1	no	—
69	#485	D₃	1	1	yes	characteristic
70	#479	—	3	1	no	—
71	#479	—	3	1	no	—
72	#479	—	3	1	no	—
73	#479	—	3	1	no	—
74	#431	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×Dic₃	D₄	1	1	yes	characteristic
78	C₂²×Dic₃	D₄	1	1	yes	characteristic
79	C₂×Dic₆	—	2	1	no	—
80	C₂×Dic₆	—	2	1	no	—
81	C₂×Dic₆	—	2	1	no	—
82	C₂×Dic₆	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	#217	—	2	1	no	—
93	#217	—	2	1	no	—
94	C₄⋊Dic₃	—	2	1	no	—
95	C₄⋊Dic₃	—	2	1	no	—
96	C₄⋊Dic₃	—	2	1	no	—
97	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
100	Dic₃⋊C₄	—	2	1	no	—
101	Dic₃⋊C₄	—	2	1	no	—
102	Dic₃⋊C₄	—	2	1	no	—
103	Dic₃⋊C₄	—	2	1	no	—
104	Dic₃⋊C₄	—	2	1	no	—
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	C₄×Dic₃	—	2	1	no	—
110	C₄×Dic₃	—	2	1	no	—
111	C₄×Dic₃	—	2	1	no	—
112	C₄×Dic₃	D₄	1	1	yes	characteristic
113	C₄×Dic₃	D₄	1	1	yes	characteristic
114	C₄×Dic₃	C₂³	1	1	yes	characteristic
115	C₄×Dic₃	C₂³	1	1	yes	characteristic
116	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈	—	3	1	no	—
119	C₄⋊Q₈	—	3	1	no	—
120	C₄⋊Q₈	—	3	1	no	—
121	C₄⋊Q₈	—	3	1	no	—
122	#125	—	3	1	no	—
123	#125	—	3	1	no	—
124	#125	—	6	1	no	—
125	C₂²⋊Q₈	—	3	1	no	—
126	C₂²⋊Q₈	—	6	1	no	—
127	C₂²⋊Q₈	—	6	1	no	—
128	C₂²⋊Q₈	—	6	1	no	—
129	C₂²⋊Q₈	C₂×D₃	1	1	yes	characteristic
130	C₄×Q₈	—	3	1	no	—
131	C₄×Q₈	—	3	1	no	—
132	C₄×Q₈	—	6	1	no	—
133	C₄×Q₈	—	6	1	no	—
134	C₄×Q₈	—	6	1	no	—
135	#117	—	3	1	no	—
136	#117	—	6	1	no	—
137	#117	C₂×D₃	1	1	yes	characteristic
138	C₂×C₄²	—	3	1	no	—
139	#107	—	3	1	no	—
140	#107	—	3	1	no	—
141	#107	—	3	1	no	—
142	#107	C₂×D₃	1	1	yes	characteristic
143	#106	—	3	1	no	—
144	#106	—	3	1	no	—
145	#106	—	3	1	no	—
146	#106	C₂×D₃	1	1	yes	characteristic
147	C₄⋊C₈	—	3	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	#103	—	3	1	no	—
150	#103	—	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	C₂×D₃	1	1	yes	characteristic
156	#103	C₂×D₃	1	1	yes	characteristic
157	C₂²⋊C₈	—	3	1	no	—
158	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
162	C₃×Q₈	—	2	1	no	—
163	C₃×Q₈	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
172	C₂×Dic₃	—	2	2	no	—
173	C₂×Dic₃	—	2	2	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
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	Dic₆	—	2	1	no	—
183	Dic₆	—	2	1	no	—
184	Dic₆	—	4	2	no	—
185	Dic₆	—	4	2	no	—
186	Dic₆	—	4	2	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃⋊C₈	—	2	1	no	—
190	C₃⋊C₈	—	2	1	no	—
191	C₂×Q₈	—	3	1	no	—
192	C₂×Q₈	—	6	1	no	—
193	C₂×Q₈	—	6	1	no	—
194	C₂×Q₈	—	6	1	no	—
195	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
199	C₂×C₈	—	3	1	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	3	1	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	6	1	no	—
208	C₄⋊C₄	—	6	1	no	—
209	C₄⋊C₄	—	6	1	no	—
210	C₄⋊C₄	—	6	1	no	—
211	C₄⋊C₄	—	6	1	no	—
212	C₄⋊C₄	—	6	1	no	—
213	C₄⋊C₄	—	6	1	no	—
214	C₄⋊C₄	—	6	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	C₂²×D₃	1	1	yes	characteristic
220	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
221	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	6	1	no	—
225	C₂²⋊C₄	—	6	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	6	1	no	—
232	C₄²	—	6	1	no	—
233	C₄²	—	6	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	4	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	4	1	no	—
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
244	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	—	4	1	no	—
248	Dic₃	—	4	1	no	—
249	Dic₃	—	4	1	no	—
250	Dic₃	—	4	1	no	—
251	Dic₃	#135	1	2	yes	—
252	Dic₃	#137	1	2	yes	—
253	C₂³	D₃×D₄	1	1	yes	characteristic
254	Q₈	—	2	1	no	—
255	Q₈	—	2	1	no	—
256	Q₈	—	6	1	no	—
257	Q₈	—	6	1	no	—
258	Q₈	—	12	2	no	—
259	Q₈	—	12	2	no	—
260	Q₈	—	12	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	6	2	no	—
272	C₂×C₄	—	6	2	no	—
273	C₂×C₄	—	6	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	D₃×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	6	1	no	—
282	C₈	—	6	1	no	—
283	C₂×C₃	—	4	1	no	—
284	C₂×C₃	#536	1	1	yes	characteristic
285	C₂×C₃	C₂×#135	1	1	yes	characteristic
286	C₂×C₃	C₂×#137	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	3	2	no	—
293	C₄	—	3	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	6	1	no	—
299	C₄	—	6	1	no	—
300	C₄	—	12	1	no	—
301	C₄	—	12	1	no	—
302	C₄	—	12	1	no	—
303	C₄	—	12	1	no	—
304	C₄	#1010	1	1	yes	characteristic
305	C₄	#1010	1	1	yes	characteristic
306	C₃	#3231	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	4	1	no	—
308	C₂	#5648	1	1	yes	characteristic
309	C₂	D₃×#135	1	1	yes	characteristic
310	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×#485	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	#485	D₃
hypocenter	C₃	#3231
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
6. #4877	C₂	semidirect
7. #4874	C₂	semidirect
8. #4857	C₂	semidirect
9. #4855	C₂	semidirect
10. #4799	C₂	semidirect
11. #4796	C₂	semidirect
12. #4792	C₂	semidirect
13. #4792	C₂	semidirect
306. C₃	#3231	semidirect

Maximal subgroups

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

subgroup	index
#5622	2
#5517	2
C₃×#485	2
#5045	2
#4996	2
#4877	2
#4874	2
#4857	2
#4855	2
#4799	2
#4796	2
#4792	2
#4723	2
#4718	2
#3231	3