Subgroups — #67571

Order 384.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4695	C₂	1	1	yes	maximal, characteristic
2	C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#4463	C₂	1	1	yes	maximal, characteristic, complemented
4	#1357	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	—	2	1	no	—
6	C₂×D₂₄	—	2	1	no	—
7	#874	C₄	1	2	yes	—
8	C₃×M₃₂	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#812	—	2	1	no	—
11	#493	—	3	1	no	—
12	#364	—	3	1	no	—
13	#364	D₃	1	1	yes	characteristic, complemented
14	C₂×D₁₂	—	2	2	no	—
15	C₂×D₁₂	—	4	2	no	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
19	D₂₄	—	4	1	no	—
20	D₂₄	—	4	1	no	—
21	D₂₄	—	4	1	no	—
22	D₂₄	—	4	1	no	—
23	C₃×C₁₆	—	4	1	no	—
24	C₃⋊C₁₆	—	4	1	no	—
25	C₂×D₈	—	6	1	no	—
26	C₂×D₈	—	6	1	no	—
27	#127	—	3	2	no	—
28	M₃₂	—	2	1	no	—
29	M₃₂	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂²×D₃	—	4	2	no	—
32	C₂×C₃×C₄	—	2	1	no	—
33	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	D₁₂	—	4	2	no	—
35	D₁₂	—	4	2	no	—
36	D₁₂	—	8	4	no	—
37	C₃×C₈	D₈	1	1	yes	characteristic
38	C₃×C₈	D₈	1	1	yes	characteristic
39	C₃×C₈	SD₁₆	1	1	yes	characteristic
40	C₃×C₈	SD₁₆	1	1	yes	characteristic
41	C₂×D₄	—	6	2	no	—
42	C₂×D₄	—	12	2	no	—
43	D₈	—	12	1	no	—
44	D₈	—	12	1	no	—
45	D₈	—	12	1	no	—
46	D₈	—	12	1	no	—
47	C₂×C₈	D₁₂	1	1	yes	characteristic
48	C₂×C₈	#67	1	1	yes	characteristic
49	C₄²	C₄×D₃	1	1	yes	characteristic
50	C₁₆	—	4	1	no	—
51	C₁₆	—	12	1	no	—
52	C₂²×C₃	#100	1	1	yes	characteristic
53	C₂×D₃	—	8	4	no	—
54	C₂×D₃	—	8	2	no	—
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	#102	1	1	yes	characteristic
57	C₃×C₄	#102	1	1	yes	characteristic
58	C₂³	—	12	2	no	—
59	D₄	—	12	2	no	—
60	D₄	—	12	2	no	—
61	D₄	—	24	4	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
64	C₈	#204	1	1	yes	characteristic
65	C₈	D₂₄	1	1	yes	characteristic
66	C₈	#213	1	1	yes	characteristic
67	C₈	#215	1	1	yes	characteristic
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	#331	1	1	yes	characteristic
70	D₃	—	16	2	no	—
71	C₂²	—	24	4	no	—
72	C₂²	—	24	2	no	—
73	C₂²	#823	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#809	1	1	yes	characteristic
76	C₄	#821	1	1	yes	characteristic
77	C₃	#1357	1	1	yes	characteristic, complemented, Sylow
78	C₂	—	2	1	no	—
79	C₂	—	48	2	no	—
80	C₂	#4485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4485
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#364	C₂
socle	C₂×C₃	#331
prime radical	#4695	C₂
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#364	D₃
hypocenter	C₃	#1357
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₃×#364	C₂	semidirect
3. #4463	C₂	semidirect
13. #364	D₃	semidirect
77. C₃	#1357	semidirect

Maximal subgroups

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

subgroup	index
#4695	2
C₃×#364	2
#4463	2
#1357	3