Subgroups — #78596

Order 384.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5791	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#901	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#5110	C₂	1	1	yes	maximal, characteristic, complemented
6	#5102	C₂	1	1	yes	maximal, characteristic, complemented
7	#5101	C₂	1	1	yes	maximal, characteristic, complemented
8	#4730	C₂	1	2	yes	maximal, complemented
9	#4729	C₂	1	2	yes	maximal, complemented
10	#4728	C₂	1	2	yes	maximal, complemented
11	#67⋊C₈	C₂	1	2	yes	maximal, complemented
12	#2933	—	3	1	no	maximal, Sylow
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂×#235	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	#953	C₄	1	2	yes	—
18	#930	C₄	1	1	yes	characteristic
19	#930	C₄	1	1	yes	characteristic
20	#929	C₄	1	2	yes	—
21	C₄×#67	C₂²	1	2	yes	—
22	C₄×#67	C₂²	1	2	yes	—
23	#926	C₂²	1	1	yes	characteristic
24	#924	C₄	1	1	yes	characteristic
25	#924	C₄	1	1	yes	characteristic
26	C₂×#208	—	2	4	no	—
27	C₂×#208	C₂²	1	2	yes	—
28	C₂×#208	C₂²	1	1	yes	characteristic
29	C₂×#208	C₂²	1	1	yes	characteristic
30	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
31	#901	—	2	4	no	—
32	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
33	C₂×D₃×C₈	C₂²	1	2	yes	complemented
34	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
39	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
40	#815	C₂²	1	2	yes	complemented
41	Dic₃⋊C₈	C₂²	1	2	yes	complemented
42	Dic₃⋊C₈	C₂²	1	2	yes	complemented
43	C₈×Dic₃	C₂²	1	2	yes	complemented
44	C₂×#131	—	3	1	no	—
45	C₂²×M₁₆	—	3	1	no	—
46	#525	—	3	1	no	—
47	#436	—	3	2	no	—
48	#435	—	3	2	no	—
49	#435	—	3	2	no	—
50	C₈×D₄	—	3	2	no	—
51	#424	—	3	1	no	—
52	#408	—	3	1	no	—
53	C₂×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
54	#405	—	3	1	no	—
55	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂×#67	C₂×C₄	1	2	yes	—
65	C₂×#67	C₂×C₄	1	2	yes	—
66	C₂×#67	C₂×C₄	1	2	yes	—
67	#235	—	2	2	no	—
68	#235	—	2	2	no	—
69	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₃	C₂³	1	2	yes	—
71	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
78	#217	C₂×C₄	1	2	yes	—
79	#217	C₂×C₄	1	2	yes	—
80	#217	C₂×C₄	1	2	yes	—
81	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
82	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
83	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
84	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
85	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
86	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
87	C₄×Dic₃	C₂³	1	2	yes	—
88	#208	—	2	2	no	—
89	#208	—	2	4	no	—
90	#208	D₄	1	4	yes	complemented
91	C₂×C₃⋊C₈	—	2	4	no	—
92	C₂×C₃⋊C₈	C₂³	1	2	yes	—
93	C₂×C₃⋊C₈	C₂³	1	2	yes	—
94	ΓL(1,𝔽₂₅)	—	2	4	no	—
95	D₃×C₈	—	2	4	no	—
96	C₂×#41	—	3	1	no	—
97	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
98	#131	—	6	4	no	—
99	C₂×M₁₆	—	3	1	no	—
100	C₂×M₁₆	—	3	1	no	—
101	C₂×M₁₆	—	3	2	no	—
102	C₂×M₁₆	—	3	2	no	—
103	C₂×M₁₆	—	6	4	no	—
104	C₂²×C₈	—	3	2	no	—
105	C₂²×C₈	—	3	2	no	—
106	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
108	#123	—	3	2	no	—
109	C₂²⋊Q₈	—	3	1	no	—
110	C₂²⋊Q₈	—	3	1	no	—
111	#121	—	3	1	no	—
112	#121	—	3	1	no	—
113	#120	—	3	2	no	—
114	C₄×D₄	—	3	2	no	—
115	C₄×D₄	—	3	2	no	—
116	#117	—	3	1	no	—
117	C₄⋊C₈	—	3	2	no	—
118	C₄⋊C₈	—	3	2	no	—
119	C₂²⋊C₈	—	3	2	no	—
120	C₂²⋊C₈	—	3	2	no	—
121	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
122	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
123	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
124	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
125	#97	—	3	2	no	—
126	C₄×C₈	—	3	2	no	—
127	C₂³×C₃	—	2	2	no	—
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	—	4	1	no	—
130	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
131	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
132	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×D₃	C₂²×C₄	1	2	yes	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
145	C₂×C₃×C₄	#41	1	2	yes	—
146	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
147	#67	—	2	4	no	—
148	#67	—	2	4	no	—
149	#67	—	4	2	no	—
150	C₂×Dic₃	C₂²×C₄	1	2	yes	—
151	C₂×Dic₃	C₂²×C₄	1	2	yes	—
152	C₂×Dic₃	C₂²×C₄	1	2	yes	—
153	D₁₂	—	2	2	no	—
154	C₄×D₃	—	2	2	no	—
155	C₄×D₃	—	2	2	no	—
156	Dic₆	—	2	2	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃⋊C₈	—	2	2	no	—
162	C₃⋊C₈	C₂×D₄	1	4	yes	—
163	C₂⁴	C₄×D₃	1	1	yes	characteristic
164	#41	—	6	2	no	—
165	#41	—	6	2	no	—
166	C₂×Q₈	—	3	1	no	—
167	C₂×D₄	—	3	2	no	—
168	C₂×D₄	—	3	2	no	—
169	C₂×D₄	—	3	2	no	—
170	C₂×D₄	—	3	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	3	2	no	—
174	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
175	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
176	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
180	M₁₆	—	3	4	no	—
181	M₁₆	—	6	4	no	—
182	M₁₆	—	6	4	no	—
183	M₁₆	—	6	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	3	2	no	—
187	C₂×C₈	—	3	2	no	—
188	C₂×C₈	—	6	4	no	—
189	C₂×C₈	—	6	4	no	—
190	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
192	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
194	C₄⋊C₄	—	3	1	no	—
195	C₄⋊C₄	—	3	1	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₂²⋊C₄	—	3	2	no	—
199	C₂²⋊C₄	—	3	2	no	—
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	—	3	2	no	—
203	C₄²	—	3	2	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	4	4	no	—
212	C₂²×C₃	C₄×D₄	1	2	yes	—
213	C₂²×C₃	#131	1	2	yes	complemented
214	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
215	C₂×D₃	—	2	2	no	—
216	C₂×D₃	—	2	2	no	—
217	C₂×D₃	—	2	2	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	C₄×D₄	1	2	yes	—
221	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₃×C₄	C₂×#41	1	1	yes	characteristic
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	3	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
231	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
232	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
233	Q₈	—	6	2	no	—
234	D₄	—	6	4	no	—
235	D₄	—	6	4	no	—
236	D₄	—	6	2	no	—
237	D₄	—	12	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	6	2	no	—
249	C₂×C₄	—	6	2	no	—
250	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
253	C₂×C₄	D₃×D₄	1	2	yes	—
254	C₂×C₄	#237	1	2	yes	—
255	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	3	4	no	—
261	C₈	—	6	2	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂×C₃	C₂×#131	1	1	yes	characteristic
267	C₂×C₃	#568	1	1	yes	characteristic
268	D₃	—	4	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	6	2	no	—
278	C₂²	—	6	2	no	—
279	C₂²	—	6	2	no	—
280	C₂²	#67⋊C₄	1	2	yes	—
281	C₂²	#901	1	2	yes	complemented
282	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	6	2	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	2	no	—
288	C₄	#67⋊C₄	1	2	yes	—
289	C₄	C₂×D₃×D₄	1	1	yes	characteristic
290	C₄	C₂×#237	1	1	yes	characteristic
291	C₃	#2933	1	1	yes	characteristic, complemented, Sylow
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	12	2	no	—
296	C₂	C₂×#67⋊C₄	1	1	yes	characteristic
297	C₂	C₂×#901	1	1	yes	characteristic
298	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₂²⋊C₈	C₂
prime radical	#5791	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	#2933
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#208	C₂	semidirect
3. C₂×#901	C₂	semidirect
4. C₂×C₃×C₂²⋊C₈	C₂	semidirect
5. #5110	C₂	semidirect
6. #5102	C₂	semidirect
7. #5101	C₂	semidirect
8. #4730	C₂	semidirect
9. #4729	C₂	semidirect
10. #4728	C₂	semidirect
11. #67⋊C₈	C₂	semidirect
32. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
33. C₂×D₃×C₈	C₂²	semidirect
38. (C₂×D₃)⋊C₈	C₂²	semidirect
39. (C₂×D₃)⋊C₈	C₂²	semidirect
40. #815	C₂²	semidirect
41. Dic₃⋊C₈	C₂²	semidirect
42. Dic₃⋊C₈	C₂²	semidirect
43. C₈×Dic₃	C₂²	semidirect
53. C₂×C₂²⋊C₈	D₃	semidirect
90. #208	D₄	semidirect
213. C₂²×C₃	#131	semidirect
281. C₂²	#901	semidirect
291. C₃	#2933	semidirect

Maximal subgroups

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

subgroup	index
#5791	2
C₂²×#208	2
C₂×#901	2
C₂×C₃×C₂²⋊C₈	2
#5110	2
#5102	2
#5101	2
#4730	2
#4729	2
#4728	2
#67⋊C₈	2
#2933	3