Subgroups — #67808

Order 384.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#819	C₂	1	2	yes	maximal
3	#1381	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
5	C₂×#208	—	2	2	no	—
6	C₂×C₃×C₁₆	—	2	2	no	—
7	C₂×C₃×C₁₆	C₄	1	2	yes	—
8	#819	—	2	2	no	—
9	#819	C₄	1	4	yes	—
10	C₂²×C₁₆	D₃	1	1	yes	characteristic
11	C₂×#108	—	3	2	no	—
12	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
13	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₈	D₄	1	2	yes	—
16	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
17	#208	—	2	4	no	—
18	#208	—	4	2	no	—
19	C₂×C₃⋊C₈	—	4	2	no	—
20	C₃×C₁₆	—	2	2	no	—
21	C₂×M₁₆	—	6	2	no	—
22	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	Dic₃	1	2	yes	—
25	#108	—	3	4	no	—
26	#108	—	6	2	no	—
27	C₂³×C₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
29	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	D₈	1	1	yes	characteristic
31	C₂×C₃×C₄	SD₁₆	1	2	yes	—
32	C₃×C₈	C₄²	1	1	yes	characteristic
33	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×C₈	C₄⋊C₄	1	2	yes	—
35	C₃⋊C₈	—	4	4	no	—
36	C₂²×C₄	D₁₂	1	1	yes	characteristic
37	M₁₆	—	6	4	no	—
38	M₁₆	—	12	2	no	—
39	C₂×C₈	—	12	2	no	—
40	C₂×C₈	Dic₆	1	1	yes	characteristic
41	C₂×C₈	C₄×D₃	1	2	yes	—
42	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₂×C₈	#67	1	2	yes	—
44	C₁₆	—	2	2	no	—
45	C₂²×C₃	—	2	2	no	—
46	C₂²×C₃	#103	1	2	yes	—
47	C₂²×C₃	#107	1	1	yes	characteristic
48	C₃×C₄	#95	1	1	yes	characteristic
49	C₃×C₄	#102	1	2	yes	—
50	C₃×C₄	#106	1	1	yes	characteristic
51	C₂³	Dic₁₂	1	1	yes	characteristic
52	C₂×C₄	#204	1	2	yes	—
53	C₂×C₄	D₂₄	1	1	yes	characteristic
54	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
55	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
56	C₈	—	12	4	no	—
57	C₈	C₄×Dic₃	1	1	yes	characteristic
58	C₈	Dic₃⋊C₄	1	2	yes	—
59	C₈	#217	1	1	yes	characteristic
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	#340	1	1	yes	characteristic
62	C₂×C₃	#368	1	2	yes	—
63	C₂²	—	2	2	no	—
64	C₂²	#816	1	2	yes	—
65	C₂²	#818	1	1	yes	characteristic
66	C₄	#817	1	1	yes	characteristic
67	C₄	#821	1	2	yes	—
68	C₄	C₃⋊#95	1	1	yes	characteristic
69	C₃	#1381	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	2	no	—
71	C₂	#4506	1	2	yes	—
72	C₂	#4551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₄
commutator	C₃×C₈	C₄²
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂²×C₃	#107
prime radical	C₂³×C₃	Q₁₆
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	#1381
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
69. C₃	#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₁₆	2
C₂×#819	2
#1381	3