Subgroups — #67577

Order 384.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#1415	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	—	2	1	no	—
6	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₃×#107	—	2	1	no	—
8	D₄₈	—	2	1	no	—
9	D₄₈	C₄	1	2	yes	complemented
10	C₃⋊C₃₂	—	2	1	no	—
11	C₂×D₁₆	—	3	1	no	—
12	C₂×C₃₂	—	3	1	no	—
13	#366	D₃	1	1	yes	characteristic, complemented
14	C₂×D₁₂	—	4	1	no	—
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	C₃×C₄⋊C₄	—	4	1	no	—
17	D₂₄	—	2	2	no	—
18	D₂₄	—	4	1	no	—
19	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×D₈	—	6	1	no	—
22	D₁₆	—	3	2	no	—
23	D₁₆	—	6	1	no	—
24	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
25	#107	—	2	1	no	—
26	C₃₂	—	6	1	no	—
27	C₂²×D₃	—	8	1	no	—
28	C₂×C₃×C₄	—	8	1	no	—
29	C₂×C₃×C₄	D₈	1	1	yes	characteristic
30	D₁₂	—	4	2	no	—
31	D₁₂	—	8	1	no	—
32	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₈	SD₁₆	1	1	yes	characteristic
34	C₂×D₄	—	12	1	no	—
35	D₈	—	6	2	no	—
36	D₈	—	12	1	no	—
37	C₂×C₈	#67	1	1	yes	characteristic
38	C₄⋊C₄	—	4	1	no	—
39	C₁₆	C₄×D₃	1	1	yes	characteristic
40	C₁₆	D₁₂	1	1	yes	characteristic
41	C₂²×C₃	D₁₆	1	1	yes	characteristic
42	C₂×D₃	—	8	2	no	—
43	C₂×D₃	—	16	1	no	—
44	C₂×D₃	—	16	1	no	—
45	C₃×C₄	—	16	1	no	—
46	C₃×C₄	#102	1	1	yes	characteristic
47	C₃×C₄	SD₃₂	1	1	yes	characteristic
48	C₂³	—	24	1	no	—
49	D₄	—	12	2	no	—
50	D₄	—	24	1	no	—
51	C₂×C₄	—	8	1	no	—
52	C₂×C₄	#213	1	1	yes	characteristic
53	C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
54	C₈	#215	1	1	yes	characteristic
55	C₂×C₃	#357	1	1	yes	characteristic
56	C₂×C₃	D₃₂	1	1	yes	characteristic
57	C₂×C₃	SD₆₄	1	1	yes	characteristic
58	D₃	—	16	2	no	—
59	C₂²	—	24	2	no	—
60	C₂²	—	48	1	no	—
61	C₂²	—	48	1	no	—
62	C₂²	#826	1	1	yes	characteristic
63	C₄	—	16	1	no	—
64	C₄	#809	1	1	yes	characteristic
65	C₄	#828	1	1	yes	characteristic
66	C₃	#1415	1	1	yes	characteristic, complemented, Sylow
67	C₂	—	48	2	no	—
68	C₂	#4491	1	1	yes	characteristic
69	C₂	#4519	1	1	yes	characteristic
70	C₂	#4521	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#826
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×#366	C₂
socle	C₂²×C₃	D₁₆
prime radical	C₂×D₄₈	C₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	#366	D₃
hypocenter	C₃	#1415
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₃×#366	C₂	semidirect
3. C₂×C₃⋊C₃₂	C₂	semidirect
9. D₄₈	C₄	semidirect
13. #366	D₃	semidirect
66. C₃	#1415	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
C₃×#366	2
C₂×C₃⋊C₃₂	2
#1415	3