Subgroups — #69743

Order 384.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	#2150	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₃×C₄×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	2	yes	—
10	#812	—	2	4	no	—
11	C₂×C₃⋊C₁₆	C₂²	1	4	yes	—
12	C₂×M₃₂	—	3	2	no	—
13	C₂×C₄×C₈	D₃	1	1	yes	characteristic
14	C₄⋊C₁₆	—	3	4	no	—
15	C₂²×C₃×C₄	C₈	1	2	yes	—
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₈	—	2	2	no	—
18	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	D₄	1	2	yes	—
21	C₂×C₃×C₈	Q₈	1	2	yes	—
22	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄²	C₈	1	1	yes	characteristic
24	C₃×C₄²	C₈	1	1	yes	characteristic
25	C₃×C₄²	C₂×C₄	1	2	yes	—
26	C₃⋊C₁₆	—	2	4	no	—
27	C₂²×C₈	Dic₃	1	1	yes	characteristic
28	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂×C₄²	Dic₃	1	1	yes	characteristic
30	M₃₂	—	6	4	no	—
31	C₂×C₁₆	—	3	4	no	—
32	C₄×C₈	Dic₃	1	2	yes	—
33	C₄×C₈	C₂×D₃	1	2	yes	—
34	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₃×C₄	—	2	4	no	—
36	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
37	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
38	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
40	C₂×C₃×C₄	M₁₆	1	2	yes	—
41	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₈	—	2	2	no	—
43	C₃×C₈	C₄⋊C₄	1	2	yes	—
44	C₃×C₈	C₂×D₄	1	1	yes	characteristic
45	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄	C₃⋊C₈	1	2	yes	—
47	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	Dic₆	1	2	yes	—
50	C₂×C₈	D₁₂	1	2	yes	—
51	C₂×C₈	C₂×Dic₃	1	2	yes	—
52	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
53	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
54	C₄²	C₃⋊C₈	1	1	yes	characteristic
55	C₄²	C₃⋊C₈	1	1	yes	characteristic
56	C₄²	C₂×Dic₃	1	2	yes	—
57	C₁₆	—	6	4	no	—
58	C₂²×C₃	—	2	2	no	—
59	C₂²×C₃	C₄⋊C₈	1	2	yes	—
60	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	C₄⋊C₈	1	2	yes	—
63	C₃×C₄	M₃₂	1	4	yes	complemented
64	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
66	C₂³	C₂×C₃⋊C₈	1	1	yes	characteristic
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
69	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
70	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
71	C₂×C₄	#208	1	2	yes	—
72	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
73	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	C₄⋊Dic₃	1	2	yes	—
76	C₈	C₂×Dic₆	1	1	yes	characteristic
77	C₈	C₂×D₁₂	1	1	yes	characteristic
78	C₂×C₃	—	2	2	no	—
79	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₃	C₂×M₃₂	1	2	yes	—
81	C₂²	—	2	2	no	—
82	C₂²	C₄⋊(C₃⋊C₈)	1	2	yes	—
83	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
86	C₄	#812	1	4	yes	complemented
87	C₄	C₂×#208	1	1	yes	characteristic
88	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
89	C₃	#2150	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
92	C₂	C₂×#812	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂×C₃×C₄×C₈	C₂
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#2150
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊C₁₆)	C₂	semidirect
63. C₃×C₄	M₃₂	semidirect
86. C₄	#812	semidirect
89. C₃	#2150	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
C₂×#812	2
C₄⋊(C₃⋊C₁₆)	2
#2150	3