Subgroups — #71276

Order 384.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	#5073	C₂	1	4	yes	maximal, complemented
3	#4695	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#1712	—	3	1	no	maximal, Sylow
6	C₂×C₃×D₈	—	2	2	no	—
7	C₃×#127	C₂²	1	2	yes	—
8	C₂×#213	—	2	8	no	—
9	C₂×D₂₄	—	2	2	no	—
10	#874	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊C₈	C₂²	1	2	yes	—
13	#493	—	3	4	no	—
14	#493	—	3	1	no	—
15	#493	D₃	1	1	yes	characteristic, complemented
16	C₈²	—	3	1	no	—
17	C₂×C₃×D₄	—	2	4	no	—
18	C₂×C₃×D₄	—	2	2	no	—
19	C₂×D₁₂	—	2	2	no	—
20	C₂×D₁₂	—	2	4	no	—
21	C₃×D₈	—	4	4	no	—
22	C₂×C₃×C₈	D₄	1	2	yes	—
23	C₃×C₄²	C₂³	1	1	yes	characteristic
24	#213	—	4	16	no	—
25	C₂×C₃⋊C₈	D₄	1	4	yes	—
26	D₂₄	—	4	4	no	—
27	C₂×D₈	—	2	2	no	—
28	C₂×D₈	—	6	8	no	—
29	C₂×D₈	—	6	2	no	—
30	#127	—	3	2	no	—
31	#127	C₂×D₃	1	2	yes	—
32	C₄×C₈	—	3	2	no	—
33	C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂³×C₃	—	4	2	no	—
35	C₂²×D₃	—	4	2	no	—
36	C₃×D₄	—	4	8	no	—
37	C₃×D₄	—	4	4	no	—
38	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
39	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
40	D₁₂	—	4	4	no	—
41	D₁₂	—	4	8	no	—
42	C₃×C₈	D₈	1	4	yes	complemented
43	C₃⋊C₈	D₈	1	8	yes	complemented
44	C₂×D₄	—	2	4	no	—
45	C₂×D₄	—	2	2	no	—
46	C₂×D₄	—	6	4	no	—
47	C₂×D₄	—	6	2	no	—
48	D₈	—	4	4	no	—
49	D₈	—	12	16	no	—
50	D₈	—	12	4	no	—
51	C₂×C₈	—	3	4	no	—
52	C₂×C₈	#67	1	2	yes	—
53	C₄²	C₂²×D₃	1	1	yes	characteristic
54	C₂²×C₃	—	8	4	no	—
55	C₂²×C₃	—	8	2	no	—
56	C₂²×C₃	#127	1	1	yes	characteristic
57	C₂×D₃	—	8	4	no	—
58	C₂×D₃	—	8	2	no	—
59	C₃×C₄	C₂×D₈	1	4	yes	—
60	C₃×C₄	C₂×D₈	1	2	yes	—
61	C₂³	—	4	2	no	—
62	C₂³	—	12	2	no	—
63	D₄	—	4	8	no	—
64	D₄	—	4	4	no	—
65	D₄	—	12	8	no	—
66	D₄	—	12	4	no	—
67	C₂×C₄	D₃×D₄	1	2	yes	—
68	C₂×C₄	C₂×#67	1	1	yes	characteristic
69	C₈	—	3	8	no	—
70	C₈	#213	1	4	yes	complemented
71	C₂×C₃	—	16	2	no	—
72	C₂×C₃	#493	1	2	yes	—
73	C₂×C₃	#493	1	1	yes	characteristic
74	D₃	—	16	2	no	—
75	C₂²	—	8	4	no	—
76	C₂²	—	8	2	no	—
77	C₂²	—	24	4	no	—
78	C₂²	—	24	2	no	—
79	C₂²	#940	1	1	yes	characteristic
80	C₄	D₃×D₈	1	4	yes	—
81	C₄	C₂×#213	1	2	yes	—
82	C₃	#1712	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	16	2	no	—
84	C₂	—	48	2	no	—
85	C₂	#5073	1	1	yes	characteristic
86	C₂	#5151	1	2	yes	—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#493	C₂
socle	C₂²×C₃	#127
hypercenter	#493	D₃
hypocenter	C₃	#1712
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. C₃×#493	C₂	semidirect
2. #5073	C₂	semidirect
3. #4695	C₂	semidirect
4. C₈×C₃⋊C₈	C₂	semidirect
15. #493	D₃	semidirect
42. C₃×C₈	D₈	semidirect
43. C₃⋊C₈	D₈	semidirect
70. C₈	#213	semidirect
82. C₃	#1712	semidirect

Maximal subgroups

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

subgroup	index
C₃×#493	2
#5073	2
#4695	2
C₈×C₃⋊C₈	2
#1712	3