Group #23247 — subgroups

Subgroups — #23247

Order 256.

382 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1984	C₂	1	4	yes	maximal
3	#1939	C₂	1	4	yes	maximal
4	C₈×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈	C₂²	1	4	yes	—
13	#391	C₄	1	2	yes	—
14	#389	C₄	1	4	yes	—
15	#384	C₄	1	1	yes	characteristic
16	C₄×C₄⋊C₄	C₂²	1	4	yes	—
17	C₄³	C₂²	1	1	yes	characteristic
18	#336	C₂²	1	8	yes	—
19	#336	C₂²	1	4	yes	—
20	C₂²×Q₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂³	1	4	yes	—
22	C₂²×C₈	C₂³	1	4	yes	—
23	C₄⋊Q₈	C₈	1	8	yes	complemented
24	C₄×Q₈	—	4	2	no	—
25	C₄×Q₈	—	4	2	no	—
26	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₂×C₄²	C₂³	1	4	yes	—
35	C₂×C₄²	C₂³	1	2	yes	—
36	C₄⋊C₈	—	2	4	no	—
37	C₄⋊C₈	—	4	4	no	—
38	C₄⋊C₈	Q₈	1	8	yes	complemented
39	C₄×C₈	—	2	8	no	—
40	C₄×C₈	—	4	4	no	—
41	#95	C₂×C₄	1	8	yes	—
42	C₂×Q₈	—	4	2	no	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	C₂×C₈	1	8	yes	—
45	C₂²×C₄	C₂²×C₄	1	4	yes	—
46	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₄	1	2	yes	—
48	C₂²×C₄	C₂²×C₄	1	4	yes	—
49	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄	C₂²×C₄	1	2	yes	—
51	C₂²×C₄	C₂⁴	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	4	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	C₂×Q₈	1	8	yes	—
58	C₄⋊C₄	—	2	8	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	C₂×C₈	1	16	yes	—
62	C₄²	—	2	4	no	—
63	C₄²	—	2	4	no	—
64	C₄²	—	2	4	no	—
65	C₄²	—	2	1	no	—
66	C₄²	—	2	1	no	—
67	C₄²	—	4	1	no	—
68	C₄²	C₂×C₈	1	4	yes	—
69	C₄²	C₂×Q₈	1	4	yes	—
70	C₄²	#41	1	4	yes	—
71	C₂³	C₂³×C₄	1	1	yes	characteristic
72	Q₈	—	4	8	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	8	no	—
83	C₂×C₄	C₄×Q₈	1	4	yes	—
84	C₂×C₄	C₄×Q₈	1	4	yes	—
85	C₂×C₄	C₂²×C₈	1	2	yes	—
86	C₂×C₄	C₂²×C₈	1	8	yes	—
87	C₂×C₄	C₂²×C₈	1	4	yes	—
88	C₂×C₄	#131	1	4	yes	—
89	C₂×C₄	C₂²×Q₈	1	2	yes	—
90	C₂×C₄	C₂×#41	1	2	yes	—
91	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
93	C₈	—	2	8	no	—
94	C₈	—	4	4	no	—
95	C₂²	C₂×C₄×Q₈	1	2	yes	—
96	C₂²	#518	1	1	yes	characteristic
97	C₂²	C₂³×C₈	1	1	yes	characteristic
98	C₂²	C₂×#131	1	2	yes	—
99	C₂²	#568	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	8	no	—
102	C₄	—	2	2	no	—
103	C₄	—	4	1	no	—
104	C₄	—	4	4	no	—
105	C₄	C₈×Q₈	1	8	yes	—
106	C₄	#554	1	2	yes	—
107	C₄	#555	1	1	yes	characteristic
108	C₄	Q₈²	1	1	yes	characteristic
109	C₂	#2369	1	1	yes	characteristic
110	C₂	C₂×C₈×Q₈	1	2	yes	—
111	C₂	#2962	1	2	yes	—
112	C₂	#2972	1	1	yes	characteristic
113	C₂	#2979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

#1984

C₂

yes

maximal

#1939

C₂

yes

maximal

C₈×C₄⋊C₄

C₂

yes

maximal

C₄×C₄⋊C₈

C₂

yes

maximal

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

#391

C₄

yes

—

#389

C₄

yes

—

#384

C₄

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#336

C₂²

yes

—

#336

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

C₈

yes

complemented

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Q₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

#95

C₂×C₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂³×C₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂×#41

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₂²

C₂×C₄×Q₈

yes

—

C₂²

#518

yes

characteristic

C₂²

C₂³×C₈

yes

characteristic

C₂²

C₂×#131

yes

—

C₂²

#568

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₈×Q₈

yes

—

106

C₄

#554

yes

—

107

C₄

#555

yes

characteristic

108

C₄

Q₈²

yes

characteristic

109

C₂

#2369

yes

characteristic

110

C₂

C₂×C₈×Q₈

yes

—

111

C₂

#2962

yes

—

112

C₂

#2972

yes

characteristic

113

C₂

#2979

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂³×C₈
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₄

C₂⁴

commutator

C₂²

C₂³×C₈

socle, prime radical

C₂³

C₂³×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
23. C₄⋊Q₈	C₈	semidirect
38. C₄⋊C₈	Q₈	semidirect

H ≅ G/N

type

23. C₄⋊Q₈

C₈

semidirect

38. C₄⋊C₈

Q₈

semidirect

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

subgroup	index
C₄×C₄⋊Q₈	2
#1984	2
#1939	2
C₈×C₄⋊C₄	2
C₄×C₄⋊C₈	2

subgroup

index

C₄×C₄⋊Q₈

#1984

#1939

C₈×C₄⋊C₄

C₄×C₄⋊C₈

Subgroups — #23247

Conjugacy classes of subgroups

Complements

Maximal subgroups