Group #85604 — subgroups

Subgroups — #85604

Order 384.

292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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	Q₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(#41⋊C₃)	C₂	1	3	yes	maximal, complemented
3	C₄×#41⋊C₃	C₂	1	3	yes	maximal, complemented
4	#3483	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
5	C₂×#41⋊C₃	C₂²	1	3	yes	—
6	C₃×C₄×Q₈	—	4	1	no	maximal
7	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₄×SL(2,𝔽₃)	C₂²	1	3	yes	—
9	Q₈²	C₂×C₃	1	1	yes	characteristic
10	#555	—	3	3	no	—
11	#550	—	3	1	no	—
12	#550	C₂×C₃	1	3	yes	—
13	#532	—	3	3	no	—
14	C₄×#41	—	3	1	no	—
15	C₄×#41	C₂×C₃	1	3	yes	—
16	C₂×C₃×Q₈	—	4	1	no	—
17	#41⋊C₃	—	2	3	no	—
18	C₂×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
19	C₃×C₄⋊C₄	—	4	3	no	—
20	C₃×C₄²	—	4	3	no	—
21	C₂×#41	—	3	2	no	—
22	C₂×#41	—	3	1	no	—
23	C₂×#41	C₂²×C₃	1	3	yes	—
24	C₄⋊Q₈	—	3	3	no	—
25	#127	—	3	3	no	—
26	#124	—	3	6	no	—
27	#121	—	3	6	no	—
28	#121	—	3	6	no	—
29	C₄×Q₈	—	3	1	no	—
30	C₄×Q₈	A₄	1	1	yes	characteristic
31	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
32	C₄×Q₈	C₂²×C₃	1	3	yes	—
33	C₄×D₄	—	3	3	no	—
34	C₄×D₄	—	3	3	no	—
35	C₄×D₄	—	3	3	no	—
36	C₄×D₄	—	3	3	no	—
37	#117	—	3	3	no	—
38	#117	—	3	3	no	—
39	C₂×C₄²	—	3	3	no	—
40	C₃×Q₈	—	4	4	no	—
41	C₂×C₃×C₄	—	4	3	no	—
42	C₂×C₃×C₄	—	4	3	no	—
43	C₂×C₃×C₄	—	4	1	no	—
44	SL(2,𝔽₃)	#41	1	1	yes	characteristic, complemented
45	#41	—	2	3	no	—
46	#41	—	6	4	no	—
47	#41	—	6	3	no	—
48	#41	—	12	4	no	—
49	C₂×Q₈	—	3	2	no	—
50	C₂×Q₈	C₂×A₄	1	1	yes	characteristic
51	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
52	C₂×D₄	—	3	3	no	—
53	C₂×D₄	—	3	6	no	—
54	C₂×D₄	—	3	3	no	—
55	C₂²×C₄	—	3	3	no	—
56	C₂²×C₄	—	3	3	no	—
57	C₂²×C₄	—	3	3	no	—
58	C₄⋊C₄	—	3	1	no	—
59	C₄⋊C₄	—	3	3	no	—
60	C₄⋊C₄	—	3	3	no	—
61	C₄⋊C₄	C₂×A₄	1	3	yes	—
62	C₂²⋊C₄	—	3	6	no	—
63	C₂²⋊C₄	—	3	6	no	—
64	C₄²	—	3	3	no	—
65	C₄²	—	3	3	no	—
66	C₄²	—	3	1	no	—
67	C₄²	C₂×A₄	1	3	yes	—
68	C₂²×C₃	—	4	1	no	—
69	C₃×C₄	—	4	6	no	—
70	C₃×C₄	—	4	2	no	—
71	C₃×C₄	—	8	3	no	—
72	C₂³	—	3	3	no	—
73	Q₈	—	3	1	no	—
74	Q₈	—	12	2	no	—
75	Q₈	#41⋊C₃	1	4	yes	complemented
76	Q₈	C₃×#41	1	1	yes	characteristic
77	D₄	—	6	3	no	—
78	D₄	—	6	3	no	—
79	D₄	—	6	6	no	—
80	D₄	—	12	6	no	—
81	C₂×C₄	—	3	1	no	—
82	C₂×C₄	—	3	1	no	—
83	C₂×C₄	—	3	3	no	—
84	C₂×C₄	—	6	3	no	—
85	C₂×C₄	—	6	3	no	—
86	C₂×C₄	—	6	6	no	—
87	C₂×C₄	—	6	6	no	—
88	C₂×C₄	C₂²×A₄	1	3	yes	—
89	C₂×C₄	C₂²×A₄	1	3	yes	—
90	C₂×C₄	C₂²×A₄	1	1	yes	characteristic
91	C₂×C₃	—	4	1	no	—
92	C₂×C₃	—	4	1	no	—
93	C₂×C₃	—	4	1	no	—
94	C₂²	—	6	3	no	—
95	C₂²	—	6	3	no	—
96	C₂²	—	6	3	no	—
97	C₂²	C₂³×A₄	1	1	yes	characteristic
98	C₄	—	2	3	no	—
99	C₄	—	3	1	no	—
100	C₄	—	3	1	no	—
101	C₄	—	6	1	no	—
102	C₄	—	6	3	no	—
103	C₄	C₂×#41⋊C₃	1	6	yes	—
104	C₄	#994	1	2	yes	—
105	C₃	—	4	1	no	Sylow
106	C₂	—	12	3	no	—
107	C₂	C₂²×#41⋊C₃	1	1	yes	characteristic
108	C₂	A₄×#41	1	1	yes	characteristic
109	C₂	C₂×#994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×SL(2,𝔽₃)

C₂

yes

maximal, characteristic, complemented

C₄⋊(#41⋊C₃)

C₂

yes

maximal, complemented

C₄×#41⋊C₃

C₂

yes

maximal, complemented

#3483

C₃

yes

maximal, characteristic, complemented, Sylow

C₂×#41⋊C₃

C₂²

yes

—

C₃×C₄×Q₈

—

maximal

C₄×SL(2,𝔽₃)

C₂²

yes

characteristic

C₄×SL(2,𝔽₃)

C₂²

yes

—

Q₈²

C₂×C₃

yes

—

—

C₂×C₃

yes

—

#532

—

C₄×#41

—

C₄×#41

C₂×C₃

yes

—

C₂×C₃×Q₈

—

#41⋊C₃

—

C₂×SL(2,𝔽₃)

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂²×C₃

yes

—

C₄⋊Q₈

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

A₄

yes

characteristic

C₄×Q₈

C₂²×C₃

yes

characteristic

C₄×Q₈

C₂²×C₃

yes

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

SL(2,𝔽₃)

#41

yes

characteristic, complemented

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂×A₄

yes

characteristic

C₂×Q₈

C₂³×C₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×A₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×A₄

yes

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

Q₈

—

Q₈

—

Q₈

#41⋊C₃

yes

complemented

Q₈

C₃×#41

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×A₄

yes

—

C₂×C₄

C₂²×A₄

yes

—

C₂×C₄

C₂²×A₄

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×A₄

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#41⋊C₃

yes

—

104

C₄

#994

yes

—

105

C₃

—

Sylow

106

C₂

—

107

C₂

C₂²×#41⋊C₃

yes

characteristic

108

C₂

A₄×#41

yes

characteristic

109

C₂

C₂×#994

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃
Frattini, socle	C₂²	C₂³×A₄
Fitting	#3483	C₃
hypercenter	C₄×Q₈	A₄
hypocenter	Q₈	C₃×#41
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²×A₄

commutator, cosocle

C₂×Q₈

C₂³×C₃

Frattini, socle

C₂²

C₂³×A₄

Fitting

#3483

C₃

hypercenter

C₄×Q₈

A₄

hypocenter

Q₈

C₃×#41

LCS

G ▷ C₂×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₄ ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×SL(2,𝔽₃)	C₂	semidirect
2. C₄⋊(#41⋊C₃)	C₂	semidirect
3. C₄×#41⋊C₃	C₂	semidirect
4. #3483	C₃	semidirect
44. SL(2,𝔽₃)	#41	semidirect
75. Q₈	#41⋊C₃	semidirect

H ≅ G/N

type

1. Q₈×SL(2,𝔽₃)

C₂

semidirect

2. C₄⋊(#41⋊C₃)

C₂

semidirect

3. C₄×#41⋊C₃

C₂

semidirect

4. #3483

C₃

semidirect

44. SL(2,𝔽₃)

#41

semidirect

75. Q₈

#41⋊C₃

semidirect

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

subgroup	index
Q₈×SL(2,𝔽₃)	2
C₄⋊(#41⋊C₃)	2
C₄×#41⋊C₃	2
#3483	3
C₃×C₄×Q₈	4

subgroup

index

Q₈×SL(2,𝔽₃)

C₄⋊(#41⋊C₃)

C₄×#41⋊C₃

#3483

C₃×C₄×Q₈

Subgroups — #85604

Conjugacy classes of subgroups

Complements

Maximal subgroups