Subgroups — #78980

Order 384.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#5819	C₂	1	1	yes	maximal, characteristic, complemented
3	#5745	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	#5124	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#208	C₂	1	1	yes	maximal, characteristic, complemented
8	#4824	C₂	1	4	yes	maximal, complemented
9	#4823	C₂	1	4	yes	maximal, complemented
10	#2956	—	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	C₂×C₃×C₄²	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	2	no	—
19	#948	C₂²	1	4	yes	complemented
20	#942	C₄	1	2	yes	complemented
21	#932	C₄	1	2	yes	complemented
22	C₂×#208	—	2	2	no	—
23	C₂×#208	C₂²	1	1	yes	characteristic
24	C₂×#208	C₂²	1	1	yes	characteristic
25	C₂×#208	C₂²	1	1	yes	characteristic
26	#907	C₂²	1	4	yes	complemented
27	D₃×M₁₆	—	2	2	no	—
28	#901	—	2	2	no	—
29	C₃×#104	C₂²	1	2	yes	complemented
30	C₃×#104	C₂²	1	2	yes	complemented
31	#822	C₂²	1	4	yes	complemented
32	#805	C₂²	1	2	yes	complemented
33	#805	C₂²	1	2	yes	complemented
34	#803	C₂²	1	2	yes	complemented
35	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
36	#575	—	3	1	no	—
37	#568	—	3	1	no	—
38	#568	—	3	1	no	—
39	#444	—	3	4	no	—
40	#443	—	3	4	no	—
41	#430	—	3	1	no	—
42	C₂×#104	—	3	1	no	—
43	C₂×#104	D₃	1	1	yes	characteristic, complemented
44	C₄×M₁₆	—	3	1	no	—
45	C₃×#41	—	2	1	no	—
46	C₃×#41	—	2	1	no	—
47	C₃×#41	C₂×C₄	1	2	yes	—
48	C₃×#41	C₂³	1	2	yes	—
49	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	—	2	1	no	—
53	C₂²×C₃×C₄	—	2	1	no	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂×#67	—	2	1	no	—
56	#235	—	2	1	no	—
57	#235	—	2	1	no	—
58	#235	C₂×C₄	1	2	yes	—
59	#235	C₂³	1	2	yes	—
60	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃	—	2	1	no	—
62	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
63	C₃×M₁₆	—	2	1	no	—
64	C₃×M₁₆	C₂³	1	2	yes	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₃×C₄²	—	2	1	no	—
67	C₃×C₄²	C₂³	1	1	yes	characteristic
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	#216	C₂×C₄	1	4	yes	complemented
70	#215	C₂×C₄	1	4	yes	complemented
71	#214	C₂×C₄	1	4	yes	complemented
72	#213	C₂×C₄	1	4	yes	complemented
73	#208	—	2	1	no	—
74	#208	—	2	4	no	—
75	#208	C₂×C₄	1	2	yes	—
76	#208	C₂×C₄	1	2	yes	—
77	#208	D₄	1	2	yes	—
78	#208	D₄	1	2	yes	—
79	#208	C₂³	1	2	yes	—
80	C₂×C₃⋊C₈	—	2	1	no	—
81	C₂×C₃⋊C₈	—	2	4	no	—
82	C₂×C₃⋊C₈	C₂³	1	2	yes	—
83	C₂×C₃⋊C₈	C₂³	1	2	yes	—
84	ΓL(1,𝔽₂₅)	—	2	4	no	—
85	D₃×C₈	—	2	4	no	—
86	C₂×#41	—	3	1	no	—
87	C₂×#41	C₂×D₃	1	1	yes	characteristic
88	#137	—	3	2	no	—
89	#137	—	3	2	no	—
90	AGL(1,ℤ/8)	—	3	2	no	—
91	AGL(1,ℤ/8)	—	3	2	no	—
92	#135	—	3	4	no	—
93	#131	—	3	4	no	—
94	#131	—	3	4	no	—
95	#131	—	6	2	no	—
96	#131	—	6	2	no	—
97	C₂×M₁₆	—	3	1	no	—
98	C₂×M₁₆	—	3	1	no	—
99	C₂×M₁₆	—	3	1	no	—
100	C₂×M₁₆	—	6	2	no	—
101	C₂×M₁₆	—	6	2	no	—
102	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
104	#108	—	3	4	no	—
105	#104	—	3	2	no	—
106	#104	—	3	2	no	—
107	#104	C₂×D₃	1	2	yes	complemented
108	#104	C₂×D₃	1	2	yes	complemented
109	#97	—	3	2	no	—
110	C₄×C₈	—	3	2	no	—
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	#41	1	1	yes	characteristic
113	C₂²×D₃	—	2	1	no	—
114	C₃×Q₈	—	2	1	no	—
115	C₃×Q₈	C₂²×C₄	1	2	yes	—
116	C₃×D₄	—	2	2	no	—
117	C₃×D₄	—	2	2	no	—
118	C₃×D₄	—	2	1	no	—
119	C₃×D₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	4	2	no	—
126	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	#41	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
132	#67	—	2	2	no	—
133	#67	—	2	2	no	—
134	C₂×Dic₃	—	2	1	no	—
135	D₁₂	—	2	1	no	—
136	D₁₂	C₂²×C₄	1	2	yes	—
137	C₄×D₃	—	2	2	no	—
138	C₄×D₃	—	2	2	no	—
139	Dic₆	—	2	1	no	—
140	Dic₆	C₂²×C₄	1	2	yes	—
141	C₃×C₈	—	2	2	no	—
142	C₃⋊C₈	—	2	2	no	—
143	C₃⋊C₈	C₂²×C₄	1	4	yes	—
144	C₃⋊C₈	C₂×D₄	1	4	yes	—
145	#41	—	2	1	no	—
146	#41	—	2	1	no	—
147	#41	—	3	2	no	—
148	#41	—	3	2	no	—
149	#41	—	6	1	no	—
150	#41	—	6	1	no	—
151	#41	C₄×D₃	1	2	yes	—
152	#41	C₂²×D₃	1	2	yes	—
153	C₂×Q₈	—	3	1	no	—
154	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	3	1	no	—
157	C₂×D₄	—	6	1	no	—
158	C₂×D₄	C₄×D₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	6	1	no	—
162	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
163	Q₁₆	—	3	4	no	—
164	SD₁₆	—	3	4	no	—
165	SD₁₆	—	3	4	no	—
166	D₈	—	3	4	no	—
167	M₁₆	—	2	1	no	—
168	M₁₆	—	3	2	no	—
169	M₁₆	—	3	2	no	—
170	M₁₆	—	3	2	no	—
171	M₁₆	—	3	2	no	—
172	M₁₆	—	3	2	no	—
173	M₁₆	—	6	4	no	—
174	M₁₆	—	6	4	no	—
175	M₁₆	—	6	1	no	—
176	M₁₆	C₂²×D₃	1	2	yes	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂×C₈	—	6	4	no	—
181	C₂×C₈	—	6	4	no	—
182	C₂×C₈	—	6	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	C₂²×D₃	1	1	yes	characteristic
185	C₄²	C₂²×D₃	1	1	yes	characteristic
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	4	1	no	—
188	C₂²×C₃	—	8	1	no	—
189	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
190	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
191	C₂²×C₃	C₂×#41	1	1	yes	characteristic
192	C₂×D₃	—	2	2	no	—
193	C₂×D₃	—	8	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	C₄×D₄	1	1	yes	characteristic
198	C₃×C₄	C₄×D₄	1	1	yes	characteristic
199	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
200	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
201	Dic₃	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	6	1	no	—
204	C₂³	#237	1	1	yes	characteristic
205	Q₈	—	2	1	no	—
206	Q₈	—	3	2	no	—
207	Q₈	—	6	1	no	—
208	Q₈	C₂×C₄×D₃	1	2	yes	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	3	2	no	—
213	D₄	—	6	2	no	—
214	D₄	—	6	2	no	—
215	D₄	—	6	1	no	—
216	D₄	C₂×C₄×D₃	1	2	yes	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	6	1	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
228	C₂×C₄	D₃×D₄	1	1	yes	characteristic
229	C₂×C₄	D₃×D₄	1	1	yes	characteristic
230	C₂×C₄	#237	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	3	4	no	—
234	C₈	—	3	4	no	—
235	C₈	—	6	2	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
241	D₃	—	4	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	6	2	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	24	1	no	—
247	C₂²	#67⋊C₄	1	1	yes	characteristic
248	C₂²	#67⋊C₄	1	1	yes	characteristic
249	C₂²	C₂×#237	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	6	2	no	—
254	C₄	#67⋊C₄	1	1	yes	characteristic
255	C₄	#67⋊C₄	1	1	yes	characteristic
256	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
257	C₄	C₂×D₃×D₄	1	1	yes	characteristic
258	C₃	#2956	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	12	2	no	—
264	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	C₂×C₃×#104	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	#5822	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#104	D₃
hypocenter	C₃	#2956
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #5819	C₂	semidirect
3. #5745	C₂	semidirect
4. C₂×C₃×#104	C₂	semidirect
5. #5124	C₂	semidirect
6. C₂×#805	C₂	semidirect
7. C₄×#208	C₂	semidirect
8. #4824	C₂	semidirect
9. #4823	C₂	semidirect
15. #951	C₄	semidirect
17. #949	C₄	semidirect
19. #948	C₂²	semidirect
20. #942	C₄	semidirect
21. #932	C₄	semidirect
26. #907	C₂²	semidirect
29. C₃×#104	C₂²	semidirect
30. C₃×#104	C₂²	semidirect
31. #822	C₂²	semidirect
32. #805	C₂²	semidirect
33. #805	C₂²	semidirect
34. #803	C₂²	semidirect
35. C₄×C₃⋊C₈	C₂²	semidirect
43. C₂×#104	D₃	semidirect
69. #216	C₂×C₄	semidirect
70. #215	C₂×C₄	semidirect
71. #214	C₂×C₄	semidirect
72. #213	C₂×C₄	semidirect
107. #104	C₂×D₃	semidirect
108. #104	C₂×D₃	semidirect
258. C₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#5822	2
#5819	2
#5745	2
C₂×C₃×#104	2
#5124	2
C₂×#805	2
C₄×#208	2
#4824	2
#4823	2
#2956	3