Subgroups — #81230

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5688	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
4	#5063	C₂	1	1	yes	maximal, characteristic, complemented
5	#5022	C₂	1	1	yes	maximal, characteristic, complemented
6	#4881	C₂	1	1	yes	maximal, characteristic
7	#4875	C₂	1	1	yes	maximal, characteristic, complemented
8	#4861	C₂	1	1	yes	maximal, characteristic
9	#4854	C₂	1	1	yes	maximal, characteristic, complemented
10	#4837	C₂	1	1	yes	maximal, characteristic
11	#4829	C₂	1	1	yes	maximal, characteristic, complemented
12	#4802	C₂	1	1	yes	maximal, characteristic
13	#4802	C₂	1	1	yes	maximal, characteristic
14	#4793	C₂	1	1	yes	maximal, characteristic, complemented
15	#4791	C₂	1	1	yes	maximal, characteristic, complemented
16	#3078	—	3	1	no	maximal, Sylow
17	C₂×D₃×Q₈	C₂²	1	1	yes	characteristic
18	C₃×#125	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
22	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
23	#896	—	2	1	no	—
24	#896	—	2	1	no	—
25	#892	C₂²	1	1	yes	characteristic
26	#892	C₂²	1	1	yes	characteristic
27	D₃×C₄⋊C₄	—	2	1	no	—
28	D₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#888	C₂²	1	1	yes	characteristic
30	#888	C₂²	1	1	yes	characteristic
31	#887	—	2	1	no	—
32	#887	—	2	1	no	—
33	#887	—	2	1	no	—
34	#887	C₂²	1	1	yes	characteristic
35	C₄²×D₃	—	2	1	no	—
36	C₄²×D₃	C₂²	1	1	yes	characteristic
37	#870	C₂²	1	1	yes	characteristic
38	C₄×Dic₆	—	2	1	no	—
39	C₄×Dic₆	C₂²	1	1	yes	characteristic
40	C₃×#107	C₂²	1	1	yes	characteristic
41	C₃×#106	C₂²	1	1	yes	characteristic
42	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×#103	C₂²	1	1	yes	characteristic
44	C₃×#103	C₂²	1	1	yes	characteristic
45	#835	C₂²	1	1	yes	characteristic
46	#835	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#817	C₂²	1	1	yes	characteristic
49	#816	C₂²	1	1	yes	characteristic
50	#816	C₂²	1	1	yes	characteristic
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	#810	C₂²	1	1	yes	characteristic
54	#810	C₂²	1	1	yes	characteristic
55	#808	C₂²	1	1	yes	characteristic
56	#808	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	#533	—	3	1	no	—
61	C₂×C₄×Q₈	—	3	1	no	—
62	#479	—	3	1	no	—
63	#479	—	3	1	no	—
64	#479	—	3	1	no	—
65	#479	D₃	1	1	yes	characteristic, complemented
66	#477	—	3	1	no	—
67	#477	—	3	1	no	—
68	#475	—	3	1	no	—
69	#475	—	3	1	no	—
70	#428	—	3	1	no	—
71	#428	—	3	1	no	—
72	#423	—	3	1	no	—
73	#419	—	3	1	no	—
74	#419	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	D₃×Q₈	—	2	2	no	—
77	D₃×Q₈	Q₈	1	4	yes	complemented
78	C₂×C₄×D₃	—	2	1	no	—
79	C₂×C₄×D₃	—	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	—	2	1	no	—
85	C₂×Dic₆	—	2	1	no	—
86	C₂×Dic₆	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
91	C₃×C₄⋊C₄	—	2	1	no	—
92	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	(C₂×D₃)⋊C₄	—	2	1	no	—
98	(C₂×D₃)⋊C₄	—	2	1	no	—
99	C₄⋊Dic₃	—	2	1	no	—
100	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
102	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	Dic₃⋊C₄	—	2	1	no	—
110	C₄×Dic₃	—	2	1	no	—
111	C₄×Dic₃	—	2	1	no	—
112	C₄×Dic₃	—	2	1	no	—
113	C₄×Dic₃	—	2	1	no	—
114	C₄×Dic₃	—	2	1	no	—
115	C₄×Dic₃	D₄	1	1	yes	characteristic
116	C₄×Dic₃	D₄	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	ΓL(1,𝔽₂₅)	—	2	2	no	—
121	ΓL(1,𝔽₂₅)	—	2	2	no	—
122	C₂²×Q₈	—	3	1	no	—
123	C₂×M₁₆	—	3	1	no	—
124	C₂×M₁₆	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	#125	—	3	1	no	—
128	#125	C₂×D₃	1	1	yes	characteristic
129	C₂²⋊Q₈	—	6	1	no	—
130	C₂²⋊Q₈	—	6	1	no	—
131	C₄×Q₈	—	3	1	no	—
132	C₄×Q₈	—	3	1	no	—
133	C₄×Q₈	—	3	1	no	—
134	C₄×Q₈	—	6	1	no	—
135	C₄×Q₈	—	6	1	no	—
136	C₄×Q₈	—	6	1	no	—
137	C₄×Q₈	—	6	1	no	—
138	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
139	#117	—	3	1	no	—
140	#117	—	3	1	no	—
141	C₂×C₄⋊C₄	—	3	1	no	—
142	C₂×C₄⋊C₄	—	6	1	no	—
143	C₂×C₄²	—	3	1	no	—
144	C₂×C₄²	—	6	1	no	—
145	#107	—	3	1	no	—
146	#107	—	3	1	no	—
147	#107	—	3	1	no	—
148	#107	C₂×D₃	1	1	yes	characteristic
149	#106	—	3	1	no	—
150	#106	—	3	1	no	—
151	#106	—	3	1	no	—
152	#106	C₂×D₃	1	1	yes	characteristic
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	—	3	1	no	—
155	C₄⋊C₈	—	3	1	no	—
156	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	#103	—	3	1	no	—
162	#103	—	3	1	no	—
163	#103	C₂×D₃	1	1	yes	characteristic
164	#103	C₂×D₃	1	1	yes	characteristic
165	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
166	C₃×Q₈	—	2	1	no	—
167	C₃×Q₈	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	—	2	1	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	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	C₄×D₃	—	2	4	no	—
183	C₄×D₃	—	2	2	no	—
184	C₄×D₃	—	4	1	no	—
185	C₄×D₃	—	4	2	no	—
186	C₄×D₃	C₂×Q₈	1	2	yes	—
187	C₄×D₃	#41	1	2	yes	—
188	Dic₆	—	2	1	no	—
189	Dic₆	—	2	4	no	—
190	Dic₆	—	4	2	no	—
191	Dic₆	—	4	2	no	—
192	Dic₆	C₂×Q₈	1	2	yes	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃⋊C₈	—	2	1	no	—
196	C₃⋊C₈	—	2	1	no	—
197	C₂×Q₈	—	3	4	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	—	6	1	no	—
200	C₂×Q₈	—	6	1	no	—
201	C₂×Q₈	—	6	1	no	—
202	C₂×Q₈	—	6	2	no	—
203	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	6	1	no	—
208	C₂²×C₄	—	6	1	no	—
209	M₁₆	—	6	2	no	—
210	M₁₆	—	6	2	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
214	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	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	—	6	1	no	—
229	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
230	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
231	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
232	C₂²⋊C₄	—	6	1	no	—
233	C₂²⋊C₄	—	6	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	6	1	no	—
239	C₄²	—	6	1	no	—
240	C₄²	—	6	1	no	—
241	C₄²	—	6	1	no	—
242	C₄²	—	6	1	no	—
243	C₄²	C₂²×D₃	1	1	yes	characteristic
244	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	C₂²⋊Q₈	1	2	yes	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	—	4	1	no	—
254	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
255	C₃×C₄	C₂×#41	1	1	yes	characteristic
256	Dic₃	—	2	2	no	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	4	1	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	4	1	no	—
261	Dic₃	C₂²⋊Q₈	1	2	yes	—
262	Dic₃	#137	1	2	yes	—
263	C₂³	—	3	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	3	2	no	—
266	Q₈	—	6	4	no	—
267	Q₈	—	6	1	no	—
268	Q₈	—	12	2	no	—
269	Q₈	—	12	2	no	—
270	Q₈	D₃×Q₈	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	6	1	no	—
286	C₂×C₄	—	12	1	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	D₃×D₄	1	1	yes	characteristic
289	C₂×C₄	D₃×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	6	1	no	—
294	C₈	—	6	1	no	—
295	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
296	C₂×C₃	C₂×#137	1	1	yes	characteristic
297	C₂×C₃	#575	1	1	yes	characteristic
298	D₃	—	2	2	no	—
299	C₂²	—	3	2	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	3	2	no	—
307	C₄	—	3	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	6	1	no	—
312	C₄	—	6	2	no	—
313	C₄	—	12	1	no	—
314	C₄	—	12	1	no	—
315	C₄	—	12	1	no	—
316	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
317	C₄	D₃×#41	1	1	yes	characteristic
318	C₃	#3078	1	1	yes	characteristic, complemented, Sylow
319	C₂	—	6	2	no	—
320	C₂	D₃×C₂²⋊Q₈	1	1	yes	characteristic
321	C₂	#5768	1	1	yes	characteristic
322	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×#479	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#479	D₃
hypocenter	C₃	#3078
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. C₃×#479	C₂	semidirect
4. #5063	C₂	semidirect
5. #5022	C₂	semidirect
7. #4875	C₂	semidirect
9. #4854	C₂	semidirect
11. #4829	C₂	semidirect
14. #4793	C₂	semidirect
15. #4791	C₂	semidirect
65. #479	D₃	semidirect
77. D₃×Q₈	Q₈	semidirect
318. C₃	#3078	semidirect

Maximal subgroups

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

subgroup	index
#5688	2
C₄×D₃×Q₈	2
C₃×#479	2
#5063	2
#5022	2
#4881	2
#4875	2
#4861	2
#4854	2
#4837	2
#4829	2
#4802	2
#4793	2
#4791	2
#3078	3