Group #68505 — subgroups

Subgroups — #68505

Order 384.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5722	C₂	1	1	yes	maximal, characteristic
2	#5092	C₂	1	1	yes	maximal, characteristic
3	#4850	C₂	1	2	yes	maximal
4	C₃×#333	C₂	1	1	yes	maximal, characteristic
5	#4484	C₂	1	2	yes	maximal
6	#1633	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#944	—	2	2	no	—
9	C₂×#216	—	2	2	no	—
10	#888	—	2	2	no	—
11	#887	—	2	2	no	—
12	#869	C₂²	1	1	yes	characteristic
13	C₄×Dic₆	C₂²	1	2	yes	—
14	C₃×C₄⋊C₈	C₂²	1	2	yes	—
15	#818	—	2	2	no	—
16	#816	—	2	2	no	—
17	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	Q₈²	—	3	1	no	—
19	#494	—	3	1	no	—
20	#477	—	3	2	no	—
21	#333	D₃	1	1	yes	characteristic
22	Q₈⋊C₈	—	3	2	no	—
23	C₂×C₃×Q₈	D₄	1	2	yes	—
24	C₂×Dic₆	—	2	1	no	—
25	C₂×Dic₆	—	2	2	no	—
26	C₂×Dic₆	D₄	1	2	yes	—
27	C₂×C₃×C₈	—	2	2	no	—
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₄²	C₂³	1	1	yes	characteristic
30	#216	—	4	4	no	—
31	C₄⋊Dic₃	—	2	2	no	—
32	C₄⋊Dic₃	D₄	1	2	yes	—
33	Dic₃⋊C₄	—	2	4	no	—
34	Dic₃⋊C₄	—	2	2	no	—
35	C₄×Dic₃	—	2	2	no	—
36	C₄×Dic₃	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	1	no	—
38	C₂×Q₁₆	—	6	2	no	—
39	C₄⋊Q₈	—	3	1	no	—
40	C₄⋊Q₈	—	6	2	no	—
41	C₄⋊Q₈	—	6	2	no	—
42	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
43	C₄×Q₈	—	3	2	no	—
44	C₄×Q₈	—	6	2	no	—
45	#107	—	6	2	no	—
46	C₄⋊C₈	C₂×D₃	1	2	yes	—
47	#103	—	6	2	no	—
48	C₄×C₈	—	3	1	no	—
49	C₃×Q₈	—	4	2	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
53	C₂×Dic₃	—	2	2	no	—
54	C₂×Dic₃	—	2	2	no	—
55	C₂×Dic₃	—	4	1	no	—
56	Dic₆	—	2	2	no	—
57	Dic₆	—	4	2	no	—
58	Dic₆	—	8	2	no	—
59	Dic₆	Q₁₆	1	4	yes	complemented
60	C₃×C₈	—	4	2	no	—
61	C₃⋊C₈	—	2	2	no	—
62	C₂×Q₈	—	3	2	no	—
63	C₂×Q₈	—	6	2	no	—
64	C₂×Q₈	—	6	1	no	—
65	C₂×Q₈	D₁₂	1	2	yes	—
66	Q₁₆	—	12	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	6	1	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	3	2	no	—
71	C₄⋊C₄	—	6	4	no	—
72	C₄⋊C₄	—	6	2	no	—
73	C₄⋊C₄	—	6	2	no	—
74	C₄²	—	6	2	no	—
75	C₄²	—	6	2	no	—
76	C₄²	C₂²×D₃	1	1	yes	characteristic
77	C₂²×C₃	#120	1	1	yes	characteristic
78	C₃×C₄	—	2	1	no	—
79	C₃×C₄	—	4	2	no	—
80	C₃×C₄	C₂×Q₁₆	1	2	yes	—
81	C₃×C₄	#137	1	2	yes	—
82	Dic₃	—	2	4	no	—
83	Dic₃	—	4	2	no	—
84	Dic₃	—	8	1	no	—
85	Q₈	—	3	4	no	—
86	Q₈	—	4	2	no	—
87	Q₈	—	6	2	no	—
88	Q₈	—	12	2	no	—
89	Q₈	—	24	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	6	2	no	—
92	C₂×C₄	—	6	2	no	—
93	C₂×C₄	—	12	1	no	—
94	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
95	C₂×C₄	D₃×D₄	1	2	yes	—
96	C₈	—	4	2	no	—
97	C₈	—	6	2	no	—
98	C₂×C₃	#451	1	2	yes	—
99	C₂×C₃	#453	1	1	yes	characteristic
100	C₂²	#882	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	4	2	no	—
103	C₄	—	6	4	no	—
104	C₄	—	12	2	no	—
105	C₄	—	24	1	no	—
106	C₄	#915	1	2	yes	—
107	C₄	D₃×Q₁₆	1	2	yes	—
108	C₃	#1633	1	1	yes	characteristic, complemented, Sylow
109	C₂	#4747	1	1	yes	characteristic
110	C₂	#4809	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5722

C₂

yes

maximal, characteristic

#5092

C₂

yes

maximal, characteristic

#4850

C₂

yes

maximal

C₃×#333

C₂

yes

maximal, characteristic

#4484

C₂

yes

maximal

#1633

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₆

C₂²

yes

—

C₃×C₄⋊C₈

C₂²

yes

—

#818

—

#816

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

Q₈²

—

#494

—

#477

—

#333

D₃

yes

characteristic

Q₈⋊C₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

D₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

C₂×D₃

yes

—

#103

—

C₄×C₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

Q₁₆

yes

complemented

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#120

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×Q₁₆

yes

—

C₃×C₄

#137

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

D₃×D₄

yes

—

C₈

—

C₈

—

C₂×C₃

#451

yes

—

C₂×C₃

#453

yes

characteristic

100

C₂²

#882

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#915

yes

—

107

C₄

D₃×Q₁₆

yes

—

108

C₃

#1633

yes

characteristic, complemented, Sylow

109

C₂

#4747

yes

characteristic

110

C₂

#4809

yes

—

	subgroup	G/H
center	C₂²	#882
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#333	C₂
socle, prime radical	C₂²×C₃	#120
hypercenter	#333	D₃
hypocenter	C₃	#1633
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#882

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
59. Dic₆	Q₁₆	semidirect
108. C₃	#1633	semidirect

H ≅ G/N

type

59. Dic₆

Q₁₆

semidirect

108. C₃

#1633

semidirect

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

subgroup	index
#5722	2
#5092	2
#4850	2
C₃×#333	2
#4484	2
#1633	3

subgroup

index

Subgroups — #68505

Conjugacy classes of subgroups

Complements

Maximal subgroups