Subgroups — #71313

Order 384.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#828	C₂	1	4	yes	maximal, complemented
3	#4695	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#2248	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₁₆	—	2	1	no	—
7	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
8	C₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×D₂₄	—	2	1	no	—
10	C₂×D₂₄	C₂²	1	2	yes	—
11	#874	—	2	1	no	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#828	—	2	8	no	—
14	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×SD₃₂	—	3	4	no	—
16	#494	D₃	1	1	yes	characteristic, complemented
17	#493	—	3	1	no	—
18	C₄×C₁₆	—	3	1	no	—
19	C₂×C₃×Q₈	—	2	2	no	—
20	C₂×D₁₂	—	2	2	no	—
21	C₂×D₁₂	—	4	2	no	—
22	C₃×Q₁₆	—	2	4	no	—
23	C₃×Q₁₆	—	4	2	no	—
24	C₂×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄⋊C₄	—	4	2	no	—
27	C₃×C₄²	D₄	1	1	yes	characteristic
28	D₂₄	—	2	4	no	—
29	D₂₄	—	4	2	no	—
30	C₃⋊C₁₆	D₄	1	4	yes	complemented
31	C₂×Q₁₆	—	2	1	no	—
32	C₂×Q₁₆	C₂×D₃	1	2	yes	—
33	C₂×D₈	—	3	2	no	—
34	C₂×D₈	—	6	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	#127	—	6	1	no	—
37	SD₃₂	—	6	8	no	—
38	C₂×C₁₆	—	3	2	no	—
39	C₄×C₈	C₂×D₃	1	1	yes	characteristic
40	C₂²×D₃	—	4	2	no	—
41	C₃×Q₈	—	4	2	no	—
42	C₃×Q₈	—	4	2	no	—
43	C₂×C₃×C₄	—	4	2	no	—
44	C₂×C₃×C₄	D₈	1	2	yes	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	D₁₂	—	4	2	no	—
47	D₁₂	—	4	2	no	—
48	D₁₂	—	8	4	no	—
49	C₃×C₈	D₈	1	2	yes	—
50	C₃×C₈	C₂×D₄	1	2	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×D₄	—	6	2	no	—
53	C₂×D₄	—	12	2	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	2	no	—
56	D₈	—	6	4	no	—
57	D₈	—	12	2	no	—
58	C₂×C₈	#67	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
60	C₄⋊C₄	—	4	2	no	—
61	C₄²	#67	1	1	yes	characteristic
62	C₁₆	—	3	4	no	—
63	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
64	C₂×D₃	—	8	2	no	—
65	C₂×D₃	—	8	4	no	—
66	C₃×C₄	—	8	2	no	—
67	C₃×C₄	SD₃₂	1	4	yes	complemented
68	C₃×C₄	#127	1	1	yes	characteristic
69	C₃×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂³	—	12	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	D₄	—	12	2	no	—
74	D₄	—	12	2	no	—
75	D₄	—	24	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	#213	1	2	yes	—
78	C₂×C₄	C₂×#67	1	1	yes	characteristic
79	C₈	#213	1	2	yes	—
80	C₈	D₃×D₄	1	2	yes	—
81	C₂×C₃	#493	1	1	yes	characteristic
82	C₂×C₃	C₂×SD₃₂	1	2	yes	—
83	D₃	—	16	2	no	—
84	C₂²	—	24	4	no	—
85	C₂²	—	24	2	no	—
86	C₂²	C₂×#213	1	1	yes	characteristic
87	C₄	—	8	2	no	—
88	C₄	#828	1	4	yes	complemented
89	C₄	C₂×#213	1	1	yes	characteristic
90	C₄	#940	1	1	yes	characteristic
91	C₃	#2248	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	48	2	no	—
93	C₂	#5073	1	1	yes	characteristic
94	C₂	C₂×#828	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#494	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	#4695	C₂
hypercenter	#494	D₃
hypocenter	C₃	#2248
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₃×#494	C₂	semidirect
2. C₂×#828	C₂	semidirect
4. C₄×C₃⋊C₁₆	C₂	semidirect
16. #494	D₃	semidirect
30. C₃⋊C₁₆	D₄	semidirect
67. C₃×C₄	SD₃₂	semidirect
88. C₄	#828	semidirect
91. C₃	#2248	semidirect

Maximal subgroups

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

subgroup	index
C₃×#494	2
C₂×#828	2
#4695	2
C₄×C₃⋊C₁₆	2
#2248	3