Subgroups — #69374

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#4904	C₂	1	1	yes	maximal, characteristic
3	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₄₈	C₂	1	2	yes	maximal, complemented
5	#4449	C₂	1	2	yes	maximal, complemented
6	#2264	—	3	1	no	maximal, Sylow
7	C₂×Dic₁₂	—	2	1	no	—
8	#904	—	2	1	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	Dic₂₄	—	2	1	no	—
11	Dic₂₄	C₂²	1	1	yes	characteristic
12	Dic₂₄	C₂²	1	2	yes	—
13	#800	—	2	1	no	—
14	D₄₈	C₂²	1	1	yes	characteristic
15	C₃×C₃₂	C₂²	1	2	yes	—
16	#508	—	3	1	no	—
17	C₂×Q₃₂	—	3	1	no	—
18	Q₆₄	—	3	2	no	—
19	SD₆₄	—	3	2	no	—
20	M₆₄	D₃	1	1	yes	characteristic, complemented
21	#235	—	4	1	no	—
22	C₂×Dic₆	—	4	1	no	—
23	C₂×C₃×C₈	D₄	1	1	yes	characteristic
24	Dic₁₂	—	2	1	no	—
25	Dic₁₂	—	2	2	no	—
26	Dic₁₂	—	4	1	no	—
27	D₂₄	—	2	1	no	—
28	#204	—	4	1	no	—
29	C₃×C₁₆	D₄	1	1	yes	characteristic
30	C₃×C₁₆	C₂³	1	1	yes	characteristic
31	#135	—	6	1	no	—
32	C₂×Q₁₆	—	6	1	no	—
33	Q₃₂	—	3	2	no	—
34	Q₃₂	—	3	1	no	—
35	Q₃₂	—	6	1	no	—
36	SD₃₂	—	6	1	no	—
37	D₁₆	—	3	1	no	—
38	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
39	C₃₂	C₂×D₃	1	2	yes	—
40	C₂×C₃×C₄	D₈	1	1	yes	characteristic
41	#67	—	8	1	no	—
42	C₂×Dic₃	—	8	1	no	—
43	D₁₂	—	4	1	no	—
44	C₄×D₃	—	8	1	no	—
45	Dic₆	—	4	1	no	—
46	Dic₆	—	4	2	no	—
47	Dic₆	—	8	1	no	—
48	C₃×C₈	D₈	1	1	yes	characteristic
49	C₃×C₈	C₂×D₄	1	1	yes	characteristic
50	#41	—	12	1	no	—
51	C₂×Q₈	—	12	1	no	—
52	Q₁₆	—	6	1	no	—
53	Q₁₆	—	6	2	no	—
54	Q₁₆	—	12	1	no	—
55	SD₁₆	—	12	1	no	—
56	D₈	—	6	1	no	—
57	C₂×C₈	D₁₂	1	1	yes	characteristic
58	C₁₆	D₁₂	1	1	yes	characteristic
59	C₁₆	C₂²×D₃	1	1	yes	characteristic
60	C₂²×C₃	D₁₆	1	1	yes	characteristic
61	C₂×D₃	—	8	1	no	—
62	C₃×C₄	D₁₆	1	1	yes	characteristic
63	C₃×C₄	C₂×D₈	1	1	yes	characteristic
64	Dic₃	—	8	2	no	—
65	Dic₃	—	8	1	no	—
66	Q₈	—	12	1	no	—
67	Q₈	—	12	2	no	—
68	Q₈	—	24	1	no	—
69	D₄	—	12	1	no	—
70	D₄	—	24	1	no	—
71	C₂×C₄	—	24	1	no	—
72	C₂×C₄	—	24	1	no	—
73	C₂×C₄	D₂₄	1	1	yes	characteristic
74	C₈	D₂₄	1	1	yes	characteristic
75	C₈	C₂×D₁₂	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
78	D₃	—	16	1	no	—
79	C₂²	—	24	1	no	—
80	C₂²	D₄₈	1	1	yes	characteristic
81	C₄	—	24	1	no	—
82	C₄	—	24	2	no	—
83	C₄	D₄₈	1	1	yes	characteristic
84	C₄	C₂×D₂₄	1	1	yes	characteristic
85	C₃	#2264	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	48	1	no	—
88	C₂	C₂×D₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₈
commutator, cosocle	C₃×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	C₂×D₁₆
prime radical	#4904	C₂
hypercenter	M₆₄	D₃
hypocenter	C₃	#2264
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₄	C₂	semidirect
3. C₃×M₆₄	C₂	semidirect
4. Dic₄₈	C₂	semidirect
5. #4449	C₂	semidirect
20. M₆₄	D₃	semidirect
85. C₃	#2264	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄	2
#4904	2
C₃×M₆₄	2
Dic₄₈	2
#4449	2
#2264	3