Subgroups — #67441

Order 384.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4700	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	#4463	C₂	1	1	yes	maximal, characteristic, complemented
4	#1335	—	3	1	no	maximal, Sylow
5	#904	C₄	1	1	yes	characteristic
6	#901	—	2	1	no	—
7	C₃×C₄×C₈	—	2	1	no	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	#819	C₄	1	1	yes	characteristic
10	#812	—	2	1	no	—
11	#805	—	2	1	no	—
12	#443	—	3	1	no	—
13	#364	—	3	1	no	—
14	C₈²	D₃	1	1	yes	characteristic, complemented
15	#235	—	2	1	no	—
16	C₂×C₃×C₈	—	2	2	no	—
17	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄²	D₄	1	1	yes	characteristic
20	#208	—	2	1	no	—
21	Dic₁₂	C₈	1	1	yes	characteristic, complemented
22	D₂₄	C₈	1	1	yes	characteristic, complemented
23	#204	—	2	1	no	—
24	ΓL(1,𝔽₂₅)	—	4	1	no	—
25	D₃×C₈	—	4	1	no	—
26	C₃⋊C₁₆	—	4	1	no	—
27	#135	—	3	1	no	—
28	#131	—	6	1	no	—
29	M₃₂	—	6	1	no	—
30	#108	—	3	1	no	—
31	#104	—	6	1	no	—
32	C₄×C₈	—	2	1	no	—
33	C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
36	#67	—	4	1	no	—
37	D₁₂	—	2	1	no	—
38	C₄×D₃	—	4	1	no	—
39	Dic₆	—	2	1	no	—
40	C₃×C₈	—	2	4	no	—
41	C₃×C₈	C₂×C₈	1	1	yes	characteristic
42	C₃×C₈	M₁₆	1	1	yes	characteristic
43	C₃×C₈	D₈	1	1	yes	characteristic
44	C₃×C₈	SD₁₆	1	1	yes	characteristic
45	C₃⋊C₈	—	4	1	no	—
46	#41	—	6	1	no	—
47	Q₁₆	—	3	1	no	—
48	SD₁₆	—	6	1	no	—
49	D₈	—	3	1	no	—
50	M₁₆	—	6	1	no	—
51	M₁₆	—	12	1	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	12	1	no	—
54	C₂×C₈	C₄×D₃	1	1	yes	characteristic
55	C₂×C₈	D₁₂	1	1	yes	characteristic
56	C₄²	#67	1	1	yes	characteristic
57	C₁₆	—	12	1	no	—
58	C₂²×C₃	#104	1	1	yes	characteristic
59	C₂×D₃	—	4	1	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
62	C₃×C₄	#102	1	1	yes	characteristic
63	Dic₃	—	4	1	no	—
64	Q₈	—	6	1	no	—
65	D₄	—	6	1	no	—
66	D₄	—	12	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	12	1	no	—
69	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
70	C₈	—	2	4	no	—
71	C₈	—	12	1	no	—
72	C₈	D₃×C₈	1	1	yes	characteristic
73	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
74	C₈	#204	1	1	yes	characteristic
75	C₈	D₂₄	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
78	D₃	—	8	1	no	—
79	C₂²	—	12	1	no	—
80	C₂²	#805	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	12	1	no	—
83	C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
84	C₄	#821	1	1	yes	characteristic
85	C₃	#1335	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	24	1	no	—
88	C₂	#4459	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₄
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₈²	C₂
socle	C₂×C₃	D₄⋊C₈
prime radical	#904	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₈²	D₃
hypocenter	C₃	#1335
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. C₃×C₈²	C₂	semidirect
3. #4463	C₂	semidirect
14. C₈²	D₃	semidirect
21. Dic₁₂	C₈	semidirect
22. D₂₄	C₈	semidirect
85. C₃	#1335	semidirect

Maximal subgroups

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

subgroup	index
#4700	2
C₃×C₈²	2
#4463	2
#1335	3