Subgroups — #69060

Order 384.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₈	C₂	1	4	yes	maximal, complemented
2	#4695	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#2246	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	—	2	2	no	—
6	C₂×D₂₄	C₂²	1	4	yes	—
7	#874	—	2	2	no	—
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	D₄₈	—	2	8	no	—
11	C₂×D₁₆	—	3	4	no	—
12	#493	—	3	2	no	—
13	C₄×C₁₆	D₃	1	1	yes	characteristic, complemented
14	C₂×D₁₂	—	2	4	no	—
15	C₂×D₁₂	—	4	4	no	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₃×C₄²	D₄	1	1	yes	characteristic
19	D₂₄	—	2	8	no	—
20	D₂₄	—	4	4	no	—
21	C₃×C₁₆	D₄	1	4	yes	complemented
22	C₂×D₈	—	3	4	no	—
23	C₂×D₈	—	6	2	no	—
24	#127	—	6	2	no	—
25	D₁₆	—	6	8	no	—
26	C₂×C₁₆	C₂×D₃	1	2	yes	—
27	C₄×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂²×D₃	—	4	4	no	—
29	C₂×C₃×C₄	D₈	1	2	yes	—
30	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
31	D₁₂	—	4	4	no	—
32	D₁₂	—	4	4	no	—
33	D₁₂	—	8	8	no	—
34	C₃×C₈	D₈	1	2	yes	—
35	C₃×C₈	C₂×D₄	1	2	yes	—
36	C₂×D₄	—	6	4	no	—
37	C₂×D₄	—	12	4	no	—
38	D₈	—	6	8	no	—
39	D₈	—	12	4	no	—
40	C₂×C₈	D₁₂	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
42	C₄²	D₁₂	1	1	yes	characteristic
43	C₁₆	D₁₂	1	4	yes	complemented
44	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
45	C₂×D₃	—	8	4	no	—
46	C₂×D₃	—	8	8	no	—
47	C₃×C₄	D₁₆	1	4	yes	complemented
48	C₃×C₄	#127	1	1	yes	characteristic
49	C₃×C₄	C₂×D₈	1	1	yes	characteristic
50	C₂³	—	12	4	no	—
51	D₄	—	12	4	no	—
52	D₄	—	12	4	no	—
53	D₄	—	24	8	no	—
54	C₂×C₄	D₂₄	1	2	yes	—
55	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
56	C₈	D₂₄	1	2	yes	—
57	C₈	C₂×D₁₂	1	2	yes	—
58	C₂×C₃	#493	1	1	yes	characteristic
59	C₂×C₃	C₂×D₁₆	1	2	yes	—
60	D₃	—	16	4	no	—
61	C₂²	—	24	8	no	—
62	C₂²	—	24	4	no	—
63	C₂²	C₂×D₂₄	1	1	yes	characteristic
64	C₄	D₄₈	1	4	yes	complemented
65	C₄	#874	1	1	yes	characteristic
66	C₄	C₂×D₂₄	1	1	yes	characteristic
67	C₃	#2246	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	48	4	no	—
69	C₂	#4695	1	1	yes	characteristic
70	C₂	C₂×D₄₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle	C₂²×C₃	C₂×D₈
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#2246
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₈	C₂	semidirect
2. #4695	C₂	semidirect
3. C₃×C₄×C₁₆	C₂	semidirect
13. C₄×C₁₆	D₃	semidirect
21. C₃×C₁₆	D₄	semidirect
43. C₁₆	D₁₂	semidirect
47. C₃×C₄	D₁₆	semidirect
64. C₄	D₄₈	semidirect
67. C₃	#2246	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₈	2
#4695	2
C₃×C₄×C₁₆	2
#2246	3