Subgroups — #67547

Order 384.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5056	C₂	1	1	yes	maximal, characteristic
2	C₃×#355	C₂	1	1	yes	maximal, characteristic
3	#4541	C₂	1	1	yes	maximal, characteristic
4	#1414	—	3	1	no	maximal, Sylow
5	C₃×#124	C₂²	1	1	yes	characteristic
6	#870	C₄	1	1	yes	characteristic, complemented
7	C₃×#101	—	2	1	no	—
8	#835	—	2	1	no	—
9	#834	—	2	1	no	—
10	#832	—	2	1	no	—
11	C₄×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
12	#488	—	3	1	no	—
13	#355	D₃	1	1	yes	characteristic
14	#354	—	3	1	no	—
15	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
16	C₂×C₃×D₄	D₄	1	1	yes	characteristic
17	C₃×M₁₆	—	4	1	no	—
18	C₃×C₂²⋊C₄	—	4	1	no	—
19	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
20	#217	—	4	1	no	—
21	C₄⋊Dic₃	—	2	1	no	—
22	Dic₃⋊C₄	—	4	1	no	—
23	C₂×C₃⋊C₈	—	2	1	no	—
24	#125	—	3	1	no	—
25	#124	C₂×D₃	1	1	yes	characteristic
26	#103	—	6	1	no	—
27	#102	—	6	1	no	—
28	#101	—	2	1	no	—
29	#99	—	6	1	no	—
30	C₄×C₈	—	3	1	no	—
31	C₂³×C₃	—	2	1	no	—
32	C₃×Q₈	—	4	1	no	—
33	C₃×D₄	—	4	1	no	—
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₃	—	4	1	no	—
38	C₂×Dic₃	—	8	1	no	—
39	C₃×C₈	—	8	1	no	—
40	C₃⋊C₈	—	4	1	no	—
41	C₂×Q₈	#67	1	1	yes	characteristic
42	C₂×D₄	D₁₂	1	1	yes	characteristic
43	M₁₆	—	4	1	no	—
44	C₂×C₈	—	6	1	no	—
45	C₄⋊C₄	—	6	1	no	—
46	C₄⋊C₄	—	12	1	no	—
47	C₂²⋊C₄	—	4	1	no	—
48	C₂²⋊C₄	—	12	1	no	—
49	C₄²	C₄×D₃	1	1	yes	characteristic
50	C₂²×C₃	—	4	1	no	—
51	C₂²×C₃	—	8	1	no	—
52	C₂²×C₃	#99	1	1	yes	characteristic
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	—	4	1	no	—
55	C₃×C₄	—	4	1	no	—
56	Dic₃	—	8	1	no	—
57	Dic₃	—	16	1	no	—
58	C₂³	—	2	1	no	—
59	Q₈	—	4	1	no	—
60	D₄	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	12	1	no	—
64	C₂×C₄	—	24	1	no	—
65	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	12	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	8	1	no	—
70	C₂×C₃	#354	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	8	1	no	—
73	C₂²	#806	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	24	1	no	—
78	C₄	—	48	1	no	—
79	C₃	#1414	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#4473	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4473
commutator	C₃×C₄²	C₂×C₄
Frattini	#124	C₂×D₃
Fitting	C₃×#355	C₂
socle	C₂×C₃	#354
prime radical	C₂×C₃×D₄	D₄
cosocle	C₃×#124	C₂²
hypercenter	#355	D₃
hypocenter	C₃	#1414
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

Complements

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

N	H ≅ G/N	type
6. #870	C₄	semidirect
11. C₄×C₃⋊C₈	C₄	semidirect
79. C₃	#1414	semidirect

Maximal subgroups

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

subgroup	index
#5056	2
C₃×#355	2
#4541	2
#1414	3