Subgroups — #78989

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5822	C₂	1	1	yes	maximal, characteristic
2	C₂×#948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
5	#5124	C₂	1	1	yes	maximal, characteristic, complemented
6	#5005	C₂	1	1	yes	maximal, characteristic, complemented
7	#4997	C₂	1	1	yes	maximal, characteristic, complemented
8	#4824	C₂	1	4	yes	maximal, complemented
9	#4823	C₂	1	4	yes	maximal, complemented
10	#2955	—	3	1	no	maximal, Sylow
11	C₂×C₃×#41	C₂²	1	1	yes	characteristic
12	C₂×#235	C₂²	1	1	yes	characteristic
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₃×#117	C₂²	1	1	yes	characteristic
15	#951	C₄	1	2	yes	complemented
16	#950	C₂²	1	4	yes	—
17	#949	C₄	1	2	yes	complemented
18	#948	—	2	1	no	—
19	#948	—	2	1	no	—
20	#948	C₂²	1	2	yes	complemented
21	#948	C₂²	1	2	yes	complemented
22	#942	C₄	1	2	yes	complemented
23	#932	C₄	1	2	yes	complemented
24	C₂×#208	—	2	1	no	—
25	C₂×#208	C₂²	1	1	yes	characteristic
26	C₂×#208	C₂²	1	1	yes	characteristic
27	C₂²×C₃⋊C₈	—	2	1	no	—
28	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
29	#907	—	2	1	no	—
30	#907	—	2	1	no	—
31	#907	C₂²	1	2	yes	complemented
32	#907	C₂²	1	2	yes	complemented
33	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
34	C₂×D₃×C₈	—	2	1	no	—
35	C₃×#104	C₂²	1	4	yes	complemented
36	#822	C₂²	1	4	yes	complemented
37	#805	C₂²	1	4	yes	complemented
38	#803	C₂²	1	2	yes	complemented
39	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
40	#575	—	3	1	no	—
41	C₂×#131	—	3	1	no	—
42	C₂×#131	—	3	1	no	—
43	#444	—	3	4	no	—
44	#443	—	3	4	no	—
45	#430	—	3	1	no	—
46	#421	—	3	1	no	—
47	#421	D₃	1	1	yes	characteristic, complemented
48	#405	—	3	1	no	—
49	C₃×#41	—	2	1	no	—
50	C₃×#41	—	2	1	no	—
51	C₃×#41	C₂×C₄	1	2	yes	—
52	C₃×#41	C₂³	1	2	yes	—
53	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×D₄	—	2	1	no	—
55	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	—	2	1	no	—
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂×#67	—	2	1	no	—
59	#235	—	2	1	no	—
60	#235	—	2	1	no	—
61	#235	C₂×C₄	1	2	yes	—
62	#235	C₂³	1	2	yes	—
63	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃	—	2	1	no	—
65	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
66	C₃×M₁₆	—	2	1	no	—
67	C₃×M₁₆	C₂³	1	2	yes	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₂²⋊C₄	—	2	1	no	—
71	C₃×C₄²	C₂³	1	2	yes	—
72	#216	C₂×C₄	1	4	yes	—
73	#215	C₂×C₄	1	4	yes	—
74	#214	C₂×C₄	1	4	yes	—
75	#213	C₂×C₄	1	4	yes	—
76	#208	—	2	1	no	—
77	#208	—	2	2	no	—
78	#208	—	2	2	no	—
79	#208	C₂×C₄	1	2	yes	—
80	#208	C₂×C₄	1	2	yes	—
81	#208	C₂³	1	2	yes	—
82	C₂×C₃⋊C₈	—	2	1	no	—
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
86	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
87	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
88	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
89	C₂×C₃⋊C₈	C₂³	1	2	yes	—
90	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
92	ΓL(1,𝔽₂₅)	—	2	2	no	—
93	ΓL(1,𝔽₂₅)	—	2	2	no	—
94	D₃×C₈	—	2	2	no	—
95	D₃×C₈	—	2	2	no	—
96	C₂×#41	—	3	1	no	—
97	C₂×#41	C₂×D₃	1	1	yes	characteristic
98	#137	—	3	2	no	—
99	#137	—	3	2	no	—
100	AGL(1,ℤ/8)	—	3	2	no	—
101	AGL(1,ℤ/8)	—	3	2	no	—
102	#135	—	3	4	no	—
103	#131	—	3	2	no	—
104	#131	—	3	2	no	—
105	#131	—	3	2	no	—
106	#131	—	3	2	no	—
107	#131	—	6	1	no	—
108	#131	—	6	1	no	—
109	#131	—	6	1	no	—
110	#131	—	6	1	no	—
111	C₂×M₁₆	—	3	1	no	—
112	C₂×M₁₆	—	3	1	no	—
113	C₂×M₁₆	—	6	1	no	—
114	C₂×M₁₆	—	6	1	no	—
115	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
116	C₂²×C₈	—	3	1	no	—
117	C₂²×C₈	—	6	1	no	—
118	C₂²×C₈	—	6	1	no	—
119	#117	C₂×D₃	1	1	yes	characteristic
120	#108	—	3	4	no	—
121	#104	—	3	4	no	—
122	#104	C₂×D₃	1	4	yes	complemented
123	#97	—	3	2	no	—
124	C₄×C₈	—	3	2	no	—
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	#41	1	1	yes	characteristic
127	C₂²×D₃	—	2	1	no	—
128	C₃×Q₈	—	2	1	no	—
129	C₃×Q₈	C₂²×C₄	1	2	yes	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	2	1	no	—
133	C₃×D₄	C₂²×C₄	1	2	yes	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	#41	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
144	#67	—	2	2	no	—
145	#67	—	2	2	no	—
146	C₂×Dic₃	—	2	1	no	—
147	D₁₂	—	2	1	no	—
148	D₁₂	C₂²×C₄	1	2	yes	—
149	C₄×D₃	—	2	2	no	—
150	C₄×D₃	—	2	2	no	—
151	Dic₆	—	2	1	no	—
152	Dic₆	C₂²×C₄	1	2	yes	—
153	C₃×C₈	—	2	2	no	—
154	C₃⋊C₈	—	2	2	no	—
155	C₃⋊C₈	C₂²×C₄	1	4	yes	—
156	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
157	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
158	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
159	C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
160	#41	—	2	1	no	—
161	#41	—	2	1	no	—
162	#41	—	3	2	no	—
163	#41	—	3	2	no	—
164	#41	—	6	1	no	—
165	#41	—	6	1	no	—
166	#41	C₄×D₃	1	2	yes	—
167	#41	C₂²×D₃	1	2	yes	—
168	C₂×Q₈	—	3	1	no	—
169	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	3	1	no	—
172	C₂×D₄	—	6	1	no	—
173	C₂×D₄	C₄×D₃	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	6	1	no	—
176	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
177	Q₁₆	—	3	4	no	—
178	SD₁₆	—	3	4	no	—
179	SD₁₆	—	3	4	no	—
180	D₈	—	3	4	no	—
181	M₁₆	—	2	1	no	—
182	M₁₆	—	3	2	no	—
183	M₁₆	—	3	2	no	—
184	M₁₆	—	3	2	no	—
185	M₁₆	—	6	2	no	—
186	M₁₆	—	6	2	no	—
187	M₁₆	—	6	2	no	—
188	M₁₆	—	6	2	no	—
189	M₁₆	—	6	1	no	—
190	M₁₆	C₂²×D₃	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	3	1	no	—
193	C₂×C₈	—	3	1	no	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	3	1	no	—
196	C₂×C₈	—	3	1	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	3	1	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	—	6	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	—	6	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	C₂²×D₃	1	2	yes	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
211	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
212	C₂²×C₃	C₂×#41	1	1	yes	characteristic
213	C₂×D₃	—	2	2	no	—
214	C₂×D₃	—	4	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	C₄×D₄	1	1	yes	characteristic
218	C₃×C₄	C₄×D₄	1	1	yes	characteristic
219	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
220	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
221	Dic₃	—	2	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	6	1	no	—
224	C₂³	#237	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	3	2	no	—
227	Q₈	—	6	1	no	—
228	Q₈	C₂×C₄×D₃	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	1	no	—
232	D₄	—	3	2	no	—
233	D₄	—	6	2	no	—
234	D₄	—	6	2	no	—
235	D₄	—	6	1	no	—
236	D₄	C₂×C₄×D₃	1	2	yes	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	6	1	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
246	C₂×C₄	D₃×D₄	1	1	yes	characteristic
247	C₂×C₄	D₃×D₄	1	1	yes	characteristic
248	C₂×C₄	#237	1	1	yes	characteristic
249	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	3	4	no	—
252	C₈	—	3	1	no	—
253	C₈	—	3	1	no	—
254	C₈	—	3	1	no	—
255	C₈	—	3	1	no	—
256	C₈	—	6	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
262	D₃	—	4	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	6	2	no	—
267	C₂²	—	12	2	no	—
268	C₂²	#67⋊C₄	1	1	yes	characteristic
269	C₂²	#67⋊C₄	1	1	yes	characteristic
270	C₂²	C₂×#237	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	6	2	no	—
274	C₄	#67⋊C₄	1	1	yes	characteristic
275	C₄	#67⋊C₄	1	1	yes	characteristic
276	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
277	C₄	C₂×D₃×D₄	1	1	yes	characteristic
278	C₃	#2955	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	12	2	no	—
284	C₂	C₂×#67⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#421	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	#5822	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#421	D₃
hypocenter	C₃	#2955
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. C₂×#948	C₂	semidirect
3. C₂×#907	C₂	semidirect
4. C₃×#421	C₂	semidirect
5. #5124	C₂	semidirect
6. #5005	C₂	semidirect
7. #4997	C₂	semidirect
8. #4824	C₂	semidirect
9. #4823	C₂	semidirect
15. #951	C₄	semidirect
17. #949	C₄	semidirect
20. #948	C₂²	semidirect
21. #948	C₂²	semidirect
22. #942	C₄	semidirect
23. #932	C₄	semidirect
31. #907	C₂²	semidirect
32. #907	C₂²	semidirect
35. C₃×#104	C₂²	semidirect
36. #822	C₂²	semidirect
37. #805	C₂²	semidirect
38. #803	C₂²	semidirect
39. C₄×C₃⋊C₈	C₂²	semidirect
47. #421	D₃	semidirect
122. #104	C₂×D₃	semidirect
278. C₃	#2955	semidirect

Maximal subgroups

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

subgroup	index
#5822	2
C₂×#948	2
C₂×#907	2
C₃×#421	2
#5124	2
#5005	2
#4997	2
#4824	2
#4823	2
#2955	3