Subgroups — #67561

Order 384.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4458	C₂	1	1	yes	maximal, characteristic
4	#1362	—	3	1	no	maximal, Sylow
5	C₂×Dic₁₂	—	2	1	no	—
6	C₂×Dic₁₂	C₄	1	2	yes	—
7	#869	—	2	1	no	—
8	C₂×C₃×C₁₆	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊C₈)	—	2	1	no	—
11	#494	—	3	1	no	—
12	C₄⋊C₁₆	D₃	1	1	yes	characteristic
13	#335	—	3	1	no	—
14	C₂×Dic₆	—	2	2	no	—
15	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₃×C₄²	D₄	1	1	yes	characteristic
18	C₄⋊Dic₃	—	4	1	no	—
19	C₄⋊Dic₃	—	4	1	no	—
20	C₂×C₃⋊C₈	—	4	1	no	—
21	Dic₁₂	—	4	2	no	—
22	Dic₁₂	—	4	1	no	—
23	Dic₁₂	—	4	1	no	—
24	C₃×C₁₆	—	4	1	no	—
25	C₂×Q₁₆	—	3	2	no	—
26	C₂×Q₁₆	—	6	1	no	—
27	C₄⋊Q₈	—	6	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	C₄⋊C₈	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	D₈	1	1	yes	characteristic
33	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
34	C₂×Dic₃	—	4	2	no	—
35	Dic₆	—	4	2	no	—
36	Dic₆	—	4	2	no	—
37	C₃×C₈	—	2	1	no	—
38	C₃×C₈	D₈	1	1	yes	characteristic
39	C₃×C₈	SD₁₆	1	1	yes	characteristic
40	C₃⋊C₈	—	8	1	no	—
41	C₂×Q₈	—	6	2	no	—
42	Q₁₆	—	12	2	no	—
43	Q₁₆	—	12	1	no	—
44	Q₁₆	—	12	1	no	—
45	C₂×C₈	—	12	1	no	—
46	C₂×C₈	C₄×D₃	1	1	yes	characteristic
47	C₂×C₈	#67	1	1	yes	characteristic
48	C₄⋊C₄	—	12	1	no	—
49	C₄⋊C₄	—	12	1	no	—
50	C₄²	D₁₂	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₂²×C₃	#102	1	1	yes	characteristic
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	#100	1	1	yes	characteristic
55	C₃×C₄	#102	1	1	yes	characteristic
56	C₃×C₄	SD₃₂	1	1	yes	characteristic
57	C₃×C₄	Q₃₂	1	1	yes	characteristic
58	Dic₃	—	8	2	no	—
59	Q₈	—	12	2	no	—
60	Q₈	—	12	2	no	—
61	C₂×C₄	—	12	2	no	—
62	C₂×C₄	#204	1	1	yes	characteristic
63	C₂×C₄	D₂₄	1	1	yes	characteristic
64	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	24	1	no	—
67	C₈	#213	1	1	yes	characteristic
68	C₈	#215	1	1	yes	characteristic
69	C₂×C₃	#331	1	1	yes	characteristic
70	C₂×C₃	#358	1	1	yes	characteristic
71	C₂×C₃	#362	1	1	yes	characteristic
72	C₂²	#821	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	24	2	no	—
75	C₄	#800	1	1	yes	characteristic
76	C₄	Dic₂₄	1	1	yes	characteristic
77	C₄	#809	1	1	yes	characteristic
78	C₄	#823	1	1	yes	characteristic
79	C₃	#1362	1	1	yes	characteristic, complemented, Sylow
80	C₂	#4485	1	1	yes	characteristic
81	C₂	#4503	1	1	yes	characteristic
82	C₂	#4517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#821
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	#1362
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
79. C₃	#1362	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
C₃×C₄⋊C₁₆	2
#4458	2
#1362	3