Subgroups — #67614

Order 384.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#376	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	2	yes	maximal
3	#1274	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄²	C₄	1	2	yes	—
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₃×#95	—	2	2	no	—
7	#830	—	2	2	no	—
8	C₄⋊(C₃⋊C₈)	—	2	2	no	—
9	C₄×C₃⋊C₈	C₄	1	4	yes	complemented
10	#431	—	3	2	no	—
11	#376	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	—	2	2	no	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	D₄	1	2	yes	—
15	C₃×C₄²	—	2	2	no	—
16	C₃×C₄²	C₈	1	4	yes	complemented
17	C₃×C₄²	C₂×C₄	1	2	yes	—
18	C₃×C₄²	D₄	1	1	yes	characteristic
19	C₃×C₄²	Q₈	1	1	yes	characteristic
20	C₂×C₃⋊C₈	—	2	4	no	—
21	C₂×C₄²	Dic₃	1	2	yes	—
22	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
23	C₄⋊C₈	—	6	2	no	—
24	C₂²⋊C₈	—	6	2	no	—
25	C₄×C₈	—	3	4	no	—
26	#95	—	2	2	no	—
27	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	—	2	4	no	—
29	C₂×C₃×C₄	—	4	2	no	—
30	C₂×C₃×C₄	—	4	4	no	—
31	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
33	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
34	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
36	C₂×C₃×C₄	M₁₆	1	2	yes	—
37	C₃⋊C₈	—	4	4	no	—
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
40	C₂²×C₄	#67	1	2	yes	—
41	C₂×C₈	—	6	4	no	—
42	C₄²	—	2	2	no	—
43	C₄²	C₃⋊C₈	1	4	yes	complemented
44	C₄²	Dic₆	1	1	yes	characteristic
45	C₄²	C₄×D₃	1	2	yes	—
46	C₄²	D₁₂	1	1	yes	characteristic
47	C₂²×C₃	—	2	2	no	—
48	C₂²×C₃	#95	1	1	yes	characteristic
49	C₂²×C₃	C₂²⋊C₈	1	2	yes	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	—	4	4	no	—
53	C₃×C₄	C₄×C₈	1	1	yes	characteristic
54	C₃×C₄	#97	1	1	yes	characteristic
55	C₃×C₄	C₄⋊C₈	1	2	yes	—
56	C₂³	#217	1	1	yes	characteristic
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	4	4	no	—
60	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
61	C₂×C₄	#208	1	2	yes	—
62	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
63	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
64	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
66	C₈	—	12	4	no	—
67	C₂×C₃	—	2	2	no	—
68	C₂×C₃	#336	1	1	yes	characteristic
69	C₂×C₃	H(3,ℤ/4)	1	2	yes	—
70	C₂²	—	2	2	no	—
71	C₂²	#830	1	2	yes	—
72	C₂²	C₃⋊#95	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	4	no	—
76	C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
77	C₄	#803	1	1	yes	characteristic
78	C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
79	C₃	#1274	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	2	2	no	—
81	C₂	#4524	1	1	yes	characteristic
82	C₂	#4531	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₃
commutator	C₃×C₄	C₄×C₈
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#376	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#376	D₃
hypocenter	C₃	#1274
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
9. C₄×C₃⋊C₈	C₄	semidirect
16. C₃×C₄²	C₈	semidirect
43. C₄²	C₃⋊C₈	semidirect
79. C₃	#1274	semidirect

Maximal subgroups

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

subgroup	index
C₃×#376	2
#4925	2
#1274	3