Subgroups — #69724

Order 384.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	2	yes	maximal
3	#4455	C₂	1	2	yes	maximal, complemented
4	C₈×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#1450	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#830	C₄	1	4	yes	—
12	C₄⋊(C₃⋊C₈)	C₄	1	4	yes	—
13	C₄×C₃⋊C₈	C₂²	1	4	yes	—
14	#431	—	3	2	no	—
15	C₂×C₄×C₈	D₃	1	1	yes	characteristic
16	#322	—	3	2	no	—
17	C₈²	—	3	2	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	C₈	1	8	yes	complemented
21	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₈	Dic₃	1	2	yes	—
28	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
29	C₄⋊C₈	—	3	4	no	—
30	C₂²⋊C₈	—	3	4	no	—
31	C₄×C₈	—	3	4	no	—
32	C₄×C₈	Dic₃	1	2	yes	—
33	C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₄×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂³×C₃	C₄²	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	4	no	—
37	C₂×C₃×C₄	C₄²	1	2	yes	—
38	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
41	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₈	C₂×C₈	1	4	yes	—
43	C₃×C₈	C₂×C₈	1	4	yes	—
44	C₃⋊C₈	—	2	8	no	—
45	C₂²×C₄	C₄×D₃	1	2	yes	—
46	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
47	C₂×C₈	—	3	8	no	—
48	C₂×C₈	C₃⋊C₈	1	8	yes	complemented
49	C₂×C₈	C₂×Dic₃	1	2	yes	—
50	C₂×C₈	C₂×Dic₃	1	2	yes	—
51	C₄²	C₄×D₃	1	2	yes	—
52	C₄²	C₂×Dic₃	1	1	yes	characteristic
53	C₄²	C₂²×D₃	1	1	yes	characteristic
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₄×C₈	1	2	yes	—
56	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
57	C₃×C₄	—	2	2	no	—
58	C₃×C₄	C₄×C₈	1	2	yes	—
59	C₃×C₄	C₂²×C₈	1	2	yes	—
60	C₃×C₄	#131	1	4	yes	—
61	C₂³	C₄×Dic₃	1	1	yes	characteristic
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
64	C₂×C₄	C₄×Dic₃	1	2	yes	—
65	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
66	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
67	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
68	C₈	—	6	8	no	—
69	C₈	C₂×C₃⋊C₈	1	4	yes	—
70	C₈	C₂×C₃⋊C₈	1	4	yes	—
71	C₂×C₃	—	2	2	no	—
72	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂×C₃	#405	1	2	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₄×C₃⋊C₈	1	2	yes	—
76	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	C₄×C₃⋊C₈	1	2	yes	—
79	C₄	#901	1	4	yes	—
80	C₄	C₂²×C₃⋊C₈	1	2	yes	—
81	C₃	#1450	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
84	C₂	#5101	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1450
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #4455	C₂	semidirect
4. C₈×C₃⋊C₈	C₂	semidirect
20. C₂×C₃×C₈	C₈	semidirect
48. C₂×C₈	C₃⋊C₈	semidirect
81. C₃	#1450	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
#4925	2
#4455	2
C₈×C₃⋊C₈	2
#1450	3