Subgroups — #83185

Order 384.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#940	C₂	1	12	yes	maximal, complemented
3	C₂×#874	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#2867	—	3	1	no	maximal, Sylow
6	C₂²×C₃×D₄	C₂²	1	6	yes	—
7	C₂²×#67	C₂²	1	16	yes	—
8	C₂²×D₁₂	C₂²	1	6	yes	—
9	C₃×#127	—	2	4	no	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	#940	—	2	48	no	—
12	C₂×C₄×Dic₃	C₂²	1	6	yes	—
13	#874	—	2	4	no	—
14	C₂×#127	—	3	12	no	—
15	C₂×#127	—	3	1	no	—
16	C₂×#127	D₃	1	1	yes	characteristic, complemented
17	C₄³	—	3	1	no	—
18	C₂⁴×C₃	C₂³	1	4	yes	—
19	C₂³×D₃	C₂³	1	4	yes	—
20	C₂×C₃×D₄	—	2	24	no	—
21	C₂×C₃×D₄	—	2	12	no	—
22	C₂²×C₃×C₄	C₂³	1	3	yes	—
23	C₂×#67	—	2	96	no	—
24	C₂²×Dic₃	C₂³	1	4	yes	—
25	C₂×D₁₂	—	2	12	no	—
26	C₂×D₁₂	—	2	24	no	—
27	C₃×C₄²	D₄	1	4	yes	complemented
28	C₄×Dic₃	D₄	1	24	yes	complemented
29	C₂²×D₄	—	3	16	no	—
30	C₂²×D₄	—	3	6	no	—
31	C₂²×D₄	C₂×D₃	1	6	yes	—
32	#127	—	2	4	no	—
33	#127	—	6	48	no	—
34	#127	—	6	4	no	—
35	C₂×C₄²	—	3	6	no	—
36	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂³×C₃	—	2	24	no	—
38	C₂³×C₃	—	2	4	no	—
39	C₂³×C₃	C₂⁴	1	1	yes	characteristic
40	C₂²×D₃	—	2	4	no	—
41	C₂²×D₃	—	2	24	no	—
42	C₃×D₄	—	4	24	no	—
43	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
44	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
45	#67	—	4	64	no	—
46	C₂×Dic₃	C₂×D₄	1	24	yes	—
47	D₁₂	—	4	24	no	—
48	C₂⁴	—	3	4	no	—
49	C₂⁴	C₂²×D₃	1	4	yes	—
50	C₂×D₄	—	2	24	no	—
51	C₂×D₄	—	2	12	no	—
52	C₂×D₄	—	6	96	no	—
53	C₂×D₄	—	6	24	no	—
54	C₂×D₄	—	6	12	no	—
55	C₂²×C₄	—	3	4	no	—
56	C₂²×C₄	C₂²×D₃	1	3	yes	—
57	C₄²	—	3	24	no	—
58	C₄²	#67	1	4	yes	complemented
59	C₂²×C₃	—	4	16	no	—
60	C₂²×C₃	—	4	12	no	—
61	C₂²×C₃	C₂²×D₄	1	6	yes	—
62	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₃	—	4	12	no	—
64	C₂×D₃	—	4	16	no	—
65	C₃×C₄	#127	1	12	yes	complemented
66	Dic₃	#127	1	16	yes	complemented
67	C₂³	—	2	24	no	—
68	C₂³	—	2	4	no	—
69	C₂³	—	6	24	no	—
70	C₂³	—	6	4	no	—
71	C₂³	C₂³×D₃	1	1	yes	characteristic
72	D₄	—	4	24	no	—
73	D₄	—	12	64	no	—
74	D₄	—	12	24	no	—
75	C₂×C₄	—	3	24	no	—
76	C₂×C₄	D₃×D₄	1	12	yes	—
77	C₂×C₄	C₂×#67	1	6	yes	—
78	C₂×C₃	—	8	4	no	—
79	C₂×C₃	C₂×#127	1	4	yes	—
80	C₂×C₃	C₂×#127	1	3	yes	—
81	D₃	—	8	4	no	—
82	C₂²	—	4	16	no	—
83	C₂²	—	4	12	no	—
84	C₂²	—	12	16	no	—
85	C₂²	—	12	12	no	—
86	C₂²	C₂×D₃×D₄	1	6	yes	—
87	C₂²	C₂²×#67	1	1	yes	characteristic
88	C₄	—	3	16	no	—
89	C₄	#940	1	12	yes	complemented
90	C₃	#2867	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	8	4	no	—
92	C₂	—	24	4	no	—
93	C₂	D₃×#127	1	4	yes	—
94	C₂	C₂×#940	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂×C₃×#127	C₂
hypercenter	C₂×#127	D₃
hypocenter	C₃	#2867
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#127	C₂	semidirect
2. C₂×#940	C₂	semidirect
3. C₂×#874	C₂	semidirect
4. C₄²×Dic₃	C₂	semidirect
16. C₂×#127	D₃	semidirect
27. C₃×C₄²	D₄	semidirect
28. C₄×Dic₃	D₄	semidirect
58. C₄²	#67	semidirect
65. C₃×C₄	#127	semidirect
66. Dic₃	#127	semidirect
89. C₄	#940	semidirect
90. C₃	#2867	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#127	2
C₂×#940	2
C₂×#874	2
C₄²×Dic₃	2
#2867	3