Subgroups — #69726

Order 384.

196 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	#4454	C₂	1	4	yes	maximal, complemented
4	#1452	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₈	C₄	1	2	yes	—
8	C₃×C₄×C₈	C₂²	1	2	yes	—
9	#830	C₄	1	4	yes	—
10	C₄⋊(C₃⋊C₈)	C₄	1	4	yes	—
11	C₄×C₃⋊C₈	C₂²	1	4	yes	—
12	#431	—	3	2	no	—
13	C₂×C₄×C₈	D₃	1	1	yes	characteristic
14	#322	—	3	4	no	—
15	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₈	—	2	4	no	—
18	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
19	C₃×C₄²	C₂×C₄	1	2	yes	—
20	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄²	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
23	C₂²×C₈	Dic₃	1	2	yes	—
24	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
25	C₄⋊C₈	—	3	4	no	—
26	C₂²⋊C₈	—	3	4	no	—
27	C₄×C₈	—	3	4	no	—
28	C₄×C₈	Dic₃	1	2	yes	—
29	C₄×C₈	C₂×D₃	1	2	yes	—
30	C₂³×C₃	C₄²	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	4	no	—
32	C₂×C₃×C₄	C₄²	1	2	yes	—
33	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
34	C₂×C₃×C₄	M₁₆	1	4	yes	—
35	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
36	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₃×C₈	—	2	4	no	—
38	C₃⋊C₈	—	2	8	no	—
39	C₂²×C₄	C₄×D₃	1	2	yes	—
40	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
41	C₂×C₈	—	2	4	no	—
42	C₂×C₈	—	3	8	no	—
43	C₂×C₈	C₂×Dic₃	1	4	yes	—
44	C₄²	C₄×D₃	1	2	yes	—
45	C₄²	C₂×Dic₃	1	1	yes	characteristic
46	C₄²	C₂²×D₃	1	1	yes	characteristic
47	C₂²×C₃	—	2	2	no	—
48	C₂²×C₃	#97	1	2	yes	—
49	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	#97	1	2	yes	—
52	C₃×C₄	C₂×M₁₆	1	2	yes	—
53	C₃×C₄	#131	1	4	yes	—
54	C₂³	C₄×Dic₃	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	#208	1	4	yes	—
57	C₂×C₄	C₄×Dic₃	1	2	yes	—
58	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
59	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
60	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
61	C₈	—	2	4	no	—
62	C₈	—	6	8	no	—
63	C₂×C₃	—	2	2	no	—
64	C₂×C₃	C₂×#97	1	1	yes	characteristic
65	C₂×C₃	#405	1	2	yes	—
66	C₂²	—	2	2	no	—
67	C₂²	#803	1	2	yes	—
68	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	#803	1	2	yes	—
71	C₄	#901	1	4	yes	—
72	C₄	C₂×#208	1	2	yes	—
73	C₃	#1452	1	1	yes	characteristic, complemented, Sylow
74	C₂	—	2	2	no	—
75	C₂	C₂×#803	1	1	yes	characteristic
76	C₂	#5101	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₂×C₃×C₄×C₈	C₂
prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1452
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #4454	C₂	semidirect
73. C₃	#1452	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
#4454	2
#1452	3