Subgroups — #78554

Order 384.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#901	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	#5524	C₂	1	1	yes	maximal, characteristic
4	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	#5129	C₂	1	1	yes	maximal, characteristic, complemented
6	#5126	C₂	1	1	yes	maximal, characteristic, complemented
7	#4999	C₂	1	1	yes	maximal, characteristic, complemented
8	#4706	C₂	1	4	yes	maximal, complemented
9	#4688	C₂	1	4	yes	maximal, complemented
10	#2930	—	3	1	no	maximal, Sylow
11	C₂×#235	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×#117	C₂²	1	1	yes	characteristic
16	C₂×#208	C₂²	1	1	yes	characteristic
17	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	#901	—	2	4	no	—
19	C₂×ΓL(1,𝔽₂₅)	—	2	4	no	—
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
22	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
23	C₂×D₃×C₈	C₂²	1	2	yes	complemented
24	#897	C₄	1	2	yes	—
25	#895	C₄	1	2	yes	—
26	#886	C₄	1	2	yes	—
27	#882	C₄	1	2	yes	—
28	C₄×D₁₂	C₂²	1	4	yes	—
29	C₃×#97	C₂²	1	2	yes	complemented
30	C₃×C₄×C₈	C₂²	1	2	yes	complemented
31	(C₂×D₃)⋊C₈	C₂²	1	4	yes	—
32	(C₂×D₃)⋊C₈	C₂²	1	4	yes	complemented
33	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	complemented
34	C₂×#131	—	3	1	no	—
35	C₂²×M₁₆	—	3	1	no	—
36	#525	—	3	1	no	—
37	#436	—	3	4	no	—
38	#435	—	3	4	no	—
39	#424	—	3	1	no	—
40	#408	—	3	1	no	—
41	#407	—	3	1	no	—
42	#405	D₃	1	1	yes	characteristic, complemented
43	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂×#67	C₂×C₄	1	2	yes	—
46	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
47	#235	—	2	2	no	—
48	#235	—	2	2	no	—
49	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₁₂	C₂×C₄	1	4	yes	—
51	C₂×C₄×D₃	—	2	2	no	—
52	C₂×C₄×D₃	—	2	2	no	—
53	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₃	C₂³	1	2	yes	—
55	C₂×C₄×D₃	C₂³	1	2	yes	—
56	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
57	C₃×M₁₆	—	2	2	no	—
58	C₂×C₃×C₈	D₄	1	4	yes	complemented
59	C₂×C₃×C₈	C₂³	1	2	yes	—
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
64	C₃×C₄²	C₂³	1	2	yes	—
65	(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
66	(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
67	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
68	#208	—	2	2	no	—
69	C₂×C₃⋊C₈	—	2	2	no	—
70	C₂×C₃⋊C₈	C₂³	1	2	yes	—
71	C₂×C₃⋊C₈	C₂³	1	2	yes	—
72	ΓL(1,𝔽₂₅)	—	2	4	no	—
73	ΓL(1,𝔽₂₅)	—	2	4	no	—
74	ΓL(1,𝔽₂₅)	—	2	4	no	—
75	D₃×C₈	—	2	4	no	—
76	C₂×#41	—	3	1	no	—
77	C₂³×C₄	—	3	1	no	—
78	#131	—	6	4	no	—
79	C₂×M₁₆	—	3	1	no	—
80	C₂×M₁₆	—	3	2	no	—
81	C₂×M₁₆	—	3	2	no	—
82	C₂×M₁₆	—	3	2	no	—
83	C₂×M₁₆	—	6	4	no	—
84	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₈	—	3	1	no	—
86	C₂²×C₈	—	3	2	no	—
87	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
88	#123	—	3	2	no	—
89	C₂²⋊Q₈	—	3	2	no	—
90	#121	—	3	2	no	—
91	#120	—	3	2	no	—
92	C₄×D₄	—	3	4	no	—
93	#117	C₂×D₃	1	1	yes	characteristic
94	C₄⋊C₈	—	3	4	no	—
95	C₂²⋊C₈	—	3	4	no	—
96	C₂²⋊C₈	—	3	4	no	—
97	#97	C₂×D₃	1	2	yes	complemented
98	C₄×C₈	C₂×D₃	1	2	yes	complemented
99	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₃	—	2	2	no	—
101	C₂²×D₃	—	4	2	no	—
102	C₂²×D₃	C₂²×C₄	1	2	yes	—
103	C₂²×D₃	C₂²×C₄	1	2	yes	—
104	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
105	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
107	C₂×C₃×C₄	#41	1	2	yes	—
108	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
109	#67	—	2	4	no	—
110	C₂×Dic₃	—	2	2	no	—
111	C₂×Dic₃	C₂²×C₄	1	2	yes	—
112	C₂×Dic₃	C₂²×C₄	1	2	yes	—
113	D₁₂	—	2	2	no	—
114	D₁₂	—	4	4	no	—
115	C₄×D₃	—	2	4	no	—
116	C₄×D₃	—	2	2	no	—
117	C₄×D₃	—	2	2	no	—
118	C₄×D₃	—	2	2	no	—
119	C₄×D₃	—	2	2	no	—
120	Dic₆	—	2	2	no	—
121	C₃×C₈	—	2	2	no	—
122	C₃×C₈	C₂×D₄	1	2	yes	—
123	C₃×C₈	C₂×D₄	1	2	yes	—
124	C₃⋊C₈	—	2	2	no	—
125	C₃⋊C₈	—	2	2	no	—
126	C₂⁴	—	3	1	no	—
127	#41	—	6	2	no	—
128	#41	—	6	2	no	—
129	C₂×Q₈	—	3	1	no	—
130	C₂×D₄	—	3	2	no	—
131	C₂×D₄	—	3	1	no	—
132	C₂×D₄	—	3	4	no	—
133	C₂²×C₄	—	3	1	no	—
134	C₂²×C₄	—	3	2	no	—
135	C₂²×C₄	—	3	2	no	—
136	C₂²×C₄	—	3	2	no	—
137	C₂²×C₄	—	6	2	no	—
138	C₂²×C₄	—	6	2	no	—
139	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
140	M₁₆	—	2	2	no	—
141	M₁₆	—	6	2	no	—
142	M₁₆	—	6	4	no	—
143	M₁₆	—	6	4	no	—
144	M₁₆	—	6	4	no	—
145	C₂×C₈	—	3	2	no	—
146	C₂×C₈	—	3	2	no	—
147	C₂×C₈	—	6	2	no	—
148	C₂×C₈	—	6	4	no	—
149	C₂×C₈	D₁₂	1	4	yes	complemented
150	C₂×C₈	C₂²×D₃	1	2	yes	—
151	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₄⋊C₄	—	3	4	no	—
154	C₄⋊C₄	C₄×D₃	1	2	yes	—
155	C₂²⋊C₄	—	3	4	no	—
156	C₂²⋊C₄	—	3	4	no	—
157	C₂²⋊C₄	C₄×D₃	1	2	yes	—
158	C₄²	C₂²×D₃	1	2	yes	—
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	C₄×D₄	1	2	yes	—
161	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
162	C₂×D₃	—	2	2	no	—
163	C₂×D₃	—	2	2	no	—
164	C₂×D₃	—	2	2	no	—
165	C₂×D₃	—	2	4	no	—
166	C₂×D₃	—	2	2	no	—
167	C₂×D₃	—	4	4	no	—
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	C₄×D₄	1	2	yes	—
170	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₃×C₄	C₂×#41	1	1	yes	characteristic
172	Dic₃	—	2	2	no	—
173	Dic₃	—	2	2	no	—
174	C₂³	—	3	2	no	—
175	C₂³	—	3	2	no	—
176	C₂³	—	6	2	no	—
177	C₂³	—	12	2	no	—
178	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
179	Q₈	—	6	2	no	—
180	D₄	—	6	4	no	—
181	D₄	—	6	2	no	—
182	D₄	—	12	4	no	—
183	C₂×C₄	—	3	2	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	6	2	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	—	6	2	no	—
188	C₂×C₄	—	6	2	no	—
189	C₂×C₄	—	6	2	no	—
190	C₂×C₄	—	6	2	no	—
191	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
192	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₄	C₂×D₁₂	1	2	yes	—
194	C₂×C₄	#235	1	2	yes	—
195	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	6	2	no	—
198	C₈	—	6	2	no	—
199	C₈	C₂×D₁₂	1	2	yes	—
200	C₈	C₂×D₁₂	1	2	yes	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂×C₃	#568	1	2	yes	—
204	D₃	—	4	2	no	—
205	D₃	—	4	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	6	2	no	—
208	C₂²	—	6	2	no	—
209	C₂²	—	6	2	no	—
210	C₂²	—	6	4	no	—
211	C₂²	—	6	2	no	—
212	C₂²	—	12	4	no	—
213	C₂²	C₄×D₁₂	1	2	yes	—
214	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	6	2	no	—
217	C₄	—	6	2	no	—
218	C₄	C₄×D₁₂	1	2	yes	—
219	C₄	C₂²×D₁₂	1	1	yes	characteristic
220	C₄	C₂×#235	1	1	yes	characteristic
221	C₃	#2930	1	1	yes	characteristic, complemented, Sylow
222	C₂	—	2	2	no	—
223	C₂	—	12	2	no	—
224	C₂	—	12	2	no	—
225	C₂	C₂×C₄×D₁₂	1	1	yes	characteristic
226	C₂	#5750	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#405	C₂
prime radical	#5524	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2930
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#901	C₂	semidirect
2. C₂²×ΓL(1,𝔽₂₅)	C₂	semidirect
4. C₃×#405	C₂	semidirect
5. #5129	C₂	semidirect
6. #5126	C₂	semidirect
7. #4999	C₂	semidirect
8. #4706	C₂	semidirect
9. #4688	C₂	semidirect
22. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
23. C₂×D₃×C₈	C₂²	semidirect
29. C₃×#97	C₂²	semidirect
30. C₃×C₄×C₈	C₂²	semidirect
32. (C₂×D₃)⋊C₈	C₂²	semidirect
33. C₄⋊(C₃⋊C₈)	C₂²	semidirect
42. #405	D₃	semidirect
58. C₂×C₃×C₈	D₄	semidirect
97. #97	C₂×D₃	semidirect
98. C₄×C₈	C₂×D₃	semidirect
149. C₂×C₈	D₁₂	semidirect
221. C₃	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×#901	2
C₂²×ΓL(1,𝔽₂₅)	2
#5524	2
C₃×#405	2
#5129	2
#5126	2
#4999	2
#4706	2
#4688	2
#2930	3