Group #73055 — subgroups

Subgroups — #73055

Order 384.

130 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 34 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	C₂×C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	SL(2,𝔽₃)⋊C₈	C₂	1	2	yes	maximal, complemented
3	#1487	—	3	1	no	maximal, Sylow
4	C₂²×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
5	C₂×#208	—	4	1	no	maximal
6	C₄×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
7	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈	D₃	1	1	yes	characteristic
9	#431	—	3	1	no	—
10	#423	—	3	1	no	—
11	Q₈⋊C₈	—	3	2	no	—
12	Q₈⋊C₈	—	3	2	no	—
13	C₂²×C₃×C₄	—	4	1	no	—
14	C₂×SL(2,𝔽₃)	—	2	1	no	—
15	C₂×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	#208	—	4	4	no	—
17	C₂×C₃⋊C₈	—	4	2	no	—
18	C₂²×Q₈	Dic₃	1	1	yes	characteristic
19	C₂×M₁₆	—	6	1	no	—
20	C₄×Q₈	—	3	1	no	—
21	C₄×Q₈	—	3	1	no	—
22	C₄×Q₈	Dic₃	1	1	yes	characteristic
23	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	3	1	no	—
25	C₂×C₄²	—	3	1	no	—
26	C₄⋊C₈	—	3	2	no	—
27	C₄⋊C₈	—	6	1	no	—
28	C₄⋊C₈	—	6	1	no	—
29	C₂²⋊C₈	—	6	1	no	—
30	C₄×C₈	—	3	2	no	—
31	C₂³×C₃	—	4	1	no	—
32	C₂×C₃×C₄	—	4	2	no	—
33	C₂×C₃×C₄	—	4	2	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	C₂×C₃×C₄	—	4	1	no	—
36	SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic, complemented
37	C₃⋊C₈	—	4	4	no	—
38	C₂×Q₈	—	2	1	no	—
39	C₂×Q₈	—	3	1	no	—
40	C₂×Q₈	—	6	1	no	—
41	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic
42	C₂²×C₄	—	3	1	no	—
43	C₂²×C₄	—	3	1	no	—
44	C₂²×C₄	S₄	1	1	yes	characteristic
45	M₁₆	—	12	2	no	—
46	C₂×C₈	—	6	2	no	—
47	C₂×C₈	—	6	2	no	—
48	C₄⋊C₄	—	3	1	no	—
49	C₄⋊C₄	—	3	1	no	—
50	C₄⋊C₄	—	6	1	no	—
51	C₄²	—	3	1	no	—
52	C₄²	—	3	1	no	—
53	C₄²	—	3	1	no	—
54	C₄²	—	3	1	no	—
55	C₂²×C₃	—	4	2	no	—
56	C₂²×C₃	—	4	1	no	—
57	C₂²×C₃	—	8	1	no	—
58	C₂²×C₃	—	8	1	no	—
59	C₃×C₄	—	4	2	no	—
60	C₃×C₄	—	4	1	no	—
61	C₃×C₄	—	4	1	no	—
62	C₂³	SL(2,ℤ/4)	1	1	yes	characteristic
63	Q₈	—	3	1	no	—
64	Q₈	—	3	2	no	—
65	Q₈	—	6	1	no	—
66	Q₈	#208	1	1	yes	characteristic, complemented
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	6	1	no	—
72	C₂×C₄	—	6	1	no	—
73	C₂×C₄	—	12	1	no	—
74	C₂×C₄	#226	1	2	yes	—
75	C₂×C₄	GL(2,𝔽₃)	1	2	yes	—
76	C₂×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
77	C₂×C₄	C₂×S₄	1	1	yes	characteristic
78	C₈	—	12	2	no	—
79	C₈	—	12	2	no	—
80	C₂×C₃	—	4	1	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂×C₃	—	8	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	#859	1	2	yes	—
87	C₂²	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
88	C₄	—	3	1	no	—
89	C₄	—	3	1	no	—
90	C₄	—	6	1	no	—
91	C₄	—	6	1	no	—
92	C₄	—	6	1	no	—
93	C₄	#859	1	2	yes	—
94	C₄	C₂×#226	1	1	yes	characteristic
95	C₄	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
96	C₃	—	4	1	no	Sylow
97	C₂	—	2	2	no	—
98	C₂	#5409	1	1	yes	characteristic
99	C₂	C₂×#859	1	1	yes	characteristic
100	C₂	#5423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×SL(2,𝔽₃)

C₂

yes

maximal, characteristic

SL(2,𝔽₃)⋊C₈

C₂

yes

maximal, complemented

#1487

—

maximal, Sylow

C₂²×SL(2,𝔽₃)

C₄

yes

characteristic

C₂×#208

—

maximal

C₄×SL(2,𝔽₃)

C₄

yes

characteristic

C₄×SL(2,𝔽₃)

C₂²

yes

characteristic

C₂×C₄×Q₈

D₃

yes

characteristic

#431

—

#423

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂²×C₃×C₄

—

C₂×SL(2,𝔽₃)

—

C₂×SL(2,𝔽₃)

C₂×C₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂²×Q₈

Dic₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₃

yes

characteristic

C₄×Q₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

SL(2,𝔽₃)

M₁₆

yes

characteristic, complemented

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

S₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

SL(2,ℤ/4)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

#208

yes

characteristic, complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#226

yes

—

C₂×C₄

GL(2,𝔽₃)

yes

—

C₂×C₄

SL(2,ℤ/4)

yes

characteristic

C₂×C₄

C₂×S₄

yes

characteristic

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂²

—

C₂²

—

C₂²

#859

yes

—

C₂²

C₂×SL(2,ℤ/4)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#859

yes

—

C₄

C₂×#226

yes

characteristic

C₄

C₂×GL(2,𝔽₃)

yes

characteristic

C₃

—

Sylow

C₂

—

C₂

yes

C₂

yes

100

C₂

#5423

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×S₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄
Fitting	C₂×C₄×Q₈	D₃
socle	C₂²	C₂×SL(2,ℤ/4)
prime radical	C₂²×SL(2,𝔽₃)	C₄
cosocle	C₄×SL(2,𝔽₃)	C₂²
hypercenter	C₂²×C₄	S₄
hypocenter	SL(2,𝔽₃)	M₁₆
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂×S₄

commutator

C₂×SL(2,𝔽₃)

C₂×C₄

Fitting

C₂×C₄×Q₈

D₃

socle

C₂²

C₂×SL(2,ℤ/4)

prime radical

C₂²×SL(2,𝔽₃)

C₄

cosocle

C₄×SL(2,𝔽₃)

C₂²

hypercenter

C₂²×C₄

S₄

hypocenter

SL(2,𝔽₃)

M₁₆

LCS

G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
2. SL(2,𝔽₃)⋊C₈	C₂	semidirect
36. SL(2,𝔽₃)	M₁₆	semidirect
66. Q₈	#208	semidirect

H ≅ G/N

type

2. SL(2,𝔽₃)⋊C₈

C₂

semidirect

36. SL(2,𝔽₃)

M₁₆

semidirect

66. Q₈

#208

semidirect

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

subgroup	index
C₂×C₄×SL(2,𝔽₃)	2
SL(2,𝔽₃)⋊C₈	2
#1487	3
C₂×#208	4

subgroup

index

C₂×C₄×SL(2,𝔽₃)

SL(2,𝔽₃)⋊C₈

#1487

C₂×#208

Subgroups — #73055

Conjugacy classes of subgroups

Complements

Maximal subgroups