Subgroups — #67716

Order 384.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#423	C₂	1	1	yes	maximal, characteristic
2	#4926	C₂	1	2	yes	maximal
3	#1307	—	3	1	no	maximal, Sylow
4	C₂×C₃×M₁₆	C₄	1	2	yes	—
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₃×C₄⋊C₈	—	2	2	no	—
7	#830	C₄	1	4	yes	—
8	C₄⋊(C₃⋊C₈)	—	2	2	no	—
9	#803	—	2	2	no	—
10	#432	—	3	2	no	—
11	#423	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₃×M₁₆	—	4	2	no	—
15	C₂×C₃×C₈	—	2	2	no	—
16	C₃×C₄²	D₄	1	1	yes	characteristic
17	C₃×C₄²	D₄	1	2	yes	—
18	C₃×C₄²	Q₈	1	1	yes	characteristic
19	C₂×C₃⋊C₈	—	2	4	no	—
20	C₂×M₁₆	Dic₃	1	2	yes	—
21	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
22	C₄⋊C₈	—	2	2	no	—
23	C₄⋊C₈	—	6	2	no	—
24	C₂²⋊C₈	—	3	4	no	—
25	#97	—	6	2	no	—
26	C₂³×C₃	C₄²	1	1	yes	characteristic
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	4	2	no	—
30	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
33	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₃×C₈	—	4	2	no	—
35	C₃⋊C₈	—	4	4	no	—
36	C₂²×C₄	C₄×D₃	1	2	yes	—
37	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
38	M₁₆	—	4	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	6	4	no	—
41	C₄²	Dic₆	1	1	yes	characteristic
42	C₄²	D₁₂	1	1	yes	characteristic
43	C₄²	#67	1	2	yes	—
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	2	2	no	—
46	C₂²×C₃	#95	1	1	yes	characteristic
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	—	2	2	no	—
50	C₃×C₄	#100	1	1	yes	characteristic
51	C₃×C₄	#101	1	1	yes	characteristic
52	C₂³	C₄×Dic₃	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
58	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
59	C₂×C₄	#217	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	—	12	4	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	#343	1	1	yes	characteristic
64	C₂×C₃	#344	1	2	yes	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₃⋊#95	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	2	no	—
71	C₄	#833	1	1	yes	characteristic
72	C₄	#836	1	1	yes	characteristic
73	C₃	#1307	1	1	yes	characteristic, complemented, Sylow
74	C₂	—	4	1	no	—
75	C₂	#4530	1	1	yes	characteristic
76	C₂	#4559	1	2	yes	—

	subgroup	G/H
center	C₂²	C₃⋊#95
commutator, prime radical	C₂³×C₃	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	#95
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#423	D₃
hypocenter	C₃	#1307
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
73. C₃	#1307	semidirect

Maximal subgroups

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

subgroup	index
C₃×#423	2
#4926	2
#1307	3