Subgroups — #69882

Order 384.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₃⋊C₁₆)	C₂	1	4	yes	maximal, complemented
5	#2151	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₃×#117	C₄	1	1	yes	characteristic
9	C₃×#97	C₄	1	2	yes	—
10	C₃×C₄×C₈	C₂²	1	2	yes	—
11	#812	—	2	2	no	—
12	C₂×C₃⋊C₁₆	—	2	2	no	—
13	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×M₃₂	—	3	1	no	—
16	C₂²×C₁₆	—	3	1	no	—
17	#405	D₃	1	1	yes	characteristic
18	C₄⋊C₁₆	—	3	4	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃×M₁₆	—	2	2	no	—
21	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	D₄	1	2	yes	—
24	C₂×C₃×C₈	Q₈	1	2	yes	—
25	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄⋊C₄	C₈	1	2	yes	—
27	C₃×C₂²⋊C₄	C₈	1	2	yes	—
28	C₃×C₄²	C₂×C₄	1	2	yes	—
29	C₃⋊C₁₆	—	2	2	no	—
30	C₃⋊C₁₆	—	2	2	no	—
31	C₂×M₁₆	Dic₃	1	1	yes	characteristic
32	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
33	#117	Dic₃	1	1	yes	characteristic
34	M₃₂	—	6	2	no	—
35	C₂×C₁₆	—	3	2	no	—
36	C₂×C₁₆	—	3	2	no	—
37	C₂×C₁₆	—	6	2	no	—
38	#97	Dic₃	1	2	yes	—
39	C₄×C₈	C₂×D₃	1	2	yes	—
40	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
43	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃×C₄	M₁₆	1	2	yes	—
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₈	—	2	2	no	—
47	C₃×C₈	C₄⋊C₄	1	2	yes	—
48	C₃×C₈	C₂×D₄	1	1	yes	characteristic
49	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	Dic₆	1	2	yes	—
53	C₂×C₈	D₁₂	1	2	yes	—
54	C₂×C₈	C₂×Dic₃	1	2	yes	—
55	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
57	C₄⋊C₄	C₃⋊C₈	1	2	yes	—
58	C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
59	C₄²	C₂×Dic₃	1	2	yes	—
60	C₁₆	—	6	2	no	—
61	C₁₆	—	6	2	no	—
62	C₂²×C₃	—	2	2	no	—
63	C₂²×C₃	C₄⋊C₈	1	2	yes	—
64	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
65	C₃×C₄	—	2	4	no	—
66	C₃×C₄	C₄⋊C₈	1	2	yes	—
67	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
69	C₂³	C₂×C₃⋊C₈	1	1	yes	characteristic
70	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
71	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
72	C₂×C₄	#208	1	2	yes	—
73	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
74	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	C₄⋊Dic₃	1	2	yes	—
77	C₈	C₂×Dic₆	1	1	yes	characteristic
78	C₈	C₂×D₁₂	1	1	yes	characteristic
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₃	#504	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊(C₃⋊C₈)	1	2	yes	—
84	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
87	C₄	C₂×#208	1	1	yes	characteristic
88	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
89	C₃	#2151	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
92	C₂	#5140	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₃×#405	C₂
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#405	D₃
hypocenter	C₃	#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. C₄⋊(C₃⋊C₁₆)	C₂	semidirect
89. C₃	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₃×#405	2
C₂×#812	2
C₂²×C₃⋊C₁₆	2
C₄⋊(C₃⋊C₁₆)	2
#2151	3