Subgroups — #67762

Order 384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	#4461	C₂	1	1	yes	maximal, characteristic, complemented
3	#4457	C₂	1	1	yes	maximal, characteristic, complemented
4	#1333	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₈	C₄	1	1	yes	characteristic
6	C₃×C₄×D₄	—	2	1	no	—
7	C₃×#107	C₄	1	1	yes	characteristic
8	C₃×#102	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₄×D₈	D₃	1	1	yes	characteristic
13	#346	—	3	1	no	—
14	#334	—	3	1	no	—
15	C₂×C₃×D₄	—	2	1	no	—
16	C₂²×C₃×C₄	—	4	1	no	—
17	C₃×D₈	—	2	1	no	—
18	C₃×D₈	C₈	1	2	yes	complemented
19	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	C₃×C₄⋊C₄	—	2	1	no	—
22	C₃×C₂²⋊C₄	—	4	1	no	—
23	C₃×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₃⋊C₈	—	4	1	no	—
25	C₃⋊C₁₆	—	2	2	no	—
26	C₂×D₈	Dic₃	1	1	yes	characteristic
27	C₄×D₄	—	2	1	no	—
28	C₂×C₁₆	—	6	1	no	—
29	#107	Dic₃	1	1	yes	characteristic
30	C₄⋊C₈	—	6	1	no	—
31	#102	—	2	1	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂³×C₃	—	4	1	no	—
34	C₃×D₄	—	2	2	no	—
35	C₃×D₄	—	4	1	no	—
36	C₂×C₃×C₄	—	4	1	no	—
37	C₂×C₃×C₄	—	8	2	no	—
38	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	D₈	1	1	yes	characteristic
40	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
41	C₃×C₈	—	2	1	no	—
42	C₃×C₈	C₂×C₈	1	1	yes	characteristic
43	C₃×C₈	M₁₆	1	1	yes	characteristic
44	C₃⋊C₈	—	8	1	no	—
45	C₂×D₄	—	2	1	no	—
46	C₂²×C₄	—	4	1	no	—
47	D₈	—	2	1	no	—
48	D₈	C₃⋊C₈	1	2	yes	complemented
49	C₂×C₈	—	12	1	no	—
50	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
51	C₂×C₈	#67	1	1	yes	characteristic
52	C₄⋊C₄	—	2	1	no	—
53	C₂²⋊C₄	—	4	1	no	—
54	C₄²	#67	1	1	yes	characteristic
55	C₁₆	—	6	2	no	—
56	C₂²×C₃	—	4	2	no	—
57	C₂²×C₃	—	8	1	no	—
58	C₂²×C₃	—	8	1	no	—
59	C₂²×C₃	#102	1	1	yes	characteristic
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	8	1	no	—
63	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
64	C₃×C₄	#104	1	1	yes	characteristic
65	C₂³	—	4	1	no	—
66	D₄	—	2	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	8	2	no	—
70	C₂×C₄	#213	1	1	yes	characteristic
71	C₂×C₄	#214	1	1	yes	characteristic
72	C₂×C₄	#217	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	24	1	no	—
75	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
76	C₈	#208	1	1	yes	characteristic
77	C₂×C₃	—	8	2	no	—
78	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
79	C₂×C₃	#360	1	1	yes	characteristic
80	C₂×C₃	#361	1	1	yes	characteristic
81	C₂²	—	4	2	no	—
82	C₂²	—	8	1	no	—
83	C₂²	—	8	1	no	—
84	C₂²	#832	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	8	1	no	—
88	C₄	#830	1	1	yes	characteristic
89	C₄	#837	1	1	yes	characteristic
90	C₃	#1333	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	8	2	no	—
92	C₂	D₄⋊(C₃⋊C₈)	1	1	yes	characteristic
93	C₂	#4563	1	1	yes	characteristic
94	C₂	#4566	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#832
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄×D₈	C₂
socle	C₂²×C₃	#102
prime radical	C₂×C₃×D₈	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄×D₈	D₃
hypocenter	C₃	#1333
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #4461	C₂	semidirect
3. #4457	C₂	semidirect
18. C₃×D₈	C₈	semidirect
48. D₈	C₃⋊C₈	semidirect
90. C₃	#1333	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₈	2
#4461	2
#4457	2
#1333	3