Group #21716 — subgroups

Subgroups — #21716

Order 256.

373 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 195 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×#446	C₂	1	2	yes	maximal
2	#2399	C₂	1	1	yes	maximal, characteristic
3	#1985	C₂	1	2	yes	maximal
4	#1940	C₂	1	4	yes	maximal
5	#1843	C₂	1	1	yes	maximal, characteristic
6	#1770	C₂	1	4	yes	maximal
7	#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈	C₂²	1	2	yes	—
9	#446	—	2	8	no	—
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	C₂×#97	C₂²	1	4	yes	—
14	C₂×#97	C₂²	1	2	yes	—
15	C₂×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	2	yes	—
18	#398	C₄	1	4	yes	—
19	#395	C₄	1	4	yes	—
20	C₄×C₄⋊C₄	C₂²	1	4	yes	—
21	#376	C₂²	1	1	yes	characteristic
22	#336	C₂²	1	8	yes	—
23	C₂²×Q₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈	C₂³	1	4	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	2	yes	—
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	C₄×Q₈	—	2	4	no	—
29	C₄×Q₈	—	2	4	no	—
30	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₂×C₄²	C₂³	1	4	yes	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	C₂×C₄²	C₂³	1	2	yes	—
36	C₄⋊C₈	—	2	8	no	—
37	C₄⋊C₈	—	2	4	no	—
38	C₄⋊C₈	—	2	8	no	—
39	#97	—	2	8	no	—
40	#97	—	2	2	no	—
41	#97	—	2	4	no	—
42	C₄×C₈	D₄	1	4	yes	—
43	C₄×C₈	Q₈	1	8	yes	complemented
44	#95	C₂×C₄	1	4	yes	—
45	#95	C₂×C₄	1	4	yes	—
46	#95	C₂×C₄	1	4	yes	—
47	C₂×Q₈	—	2	4	no	—
48	C₂×Q₈	—	4	4	no	—
49	C₂²×C₄	C₂²×C₄	1	4	yes	—
50	C₂²×C₄	C₂²×C₄	1	4	yes	—
51	C₂²×C₄	C₂²×C₄	1	2	yes	—
52	C₂²×C₄	C₂²×C₄	1	4	yes	—
53	C₂²×C₄	C₂⁴	1	1	yes	characteristic
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	—	2	8	no	—
59	C₂×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₈	C₂×Q₈	1	4	yes	—
62	C₂×C₈	C₂×Q₈	1	4	yes	—
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	2	8	no	—
65	C₄⋊C₄	—	4	8	no	—
66	C₄²	—	2	4	no	—
67	C₄²	—	2	4	no	—
68	C₄²	C₂×D₄	1	2	yes	—
69	C₄²	C₂×Q₈	1	4	yes	—
70	C₄²	#41	1	2	yes	—
71	C₄²	#41	1	4	yes	—
72	C₂³	C₂³×C₄	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	8	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	8	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	C₄×D₄	1	4	yes	—
82	C₂×C₄	C₄×Q₈	1	8	yes	—
83	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄	C₂²×Q₈	1	2	yes	—
85	C₂×C₄	C₂×#41	1	2	yes	—
86	C₂×C₄	C₂×#41	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	4	4	no	—
90	C₈	C₄⋊Q₈	1	4	yes	—
91	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
92	C₂²	C₂×C₄×Q₈	1	2	yes	—
93	C₂²	#568	1	4	yes	—
94	C₄	—	2	4	no	—
95	C₄	—	2	8	no	—
96	C₄	—	4	4	no	—
97	C₄	—	4	4	no	—
98	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₄	#532	1	1	yes	characteristic
100	C₄	#533	1	2	yes	—
101	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂	#2930	1	2	yes	—
103	C₂	#2961	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal

#2399

C₂

yes

maximal, characteristic

#1985

C₂

yes

maximal

#1940

C₂

yes

maximal

#1843

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal

#1726

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

—

#446

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#398

C₄

yes

—

#395

C₄

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

#376

C₂²

yes

characteristic

#336

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

complemented

#95

C₂×C₄

yes

—

#95

C₂×C₄

yes

—

#95

C₂×C₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

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

C₄×D₄

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂×#41

yes

—

C₂×C₄

C₂×#41

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₄⋊Q₈

yes

—

C₂²

C₂×C₄×D₄

yes

characteristic

C₂²

C₂×C₄×Q₈

yes

—

C₂²

#568

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×C₄⋊Q₈

yes

characteristic

C₄

#532

yes

characteristic

100

C₄

#533

yes

—

101

C₂

C₄×C₄⋊Q₈

yes

characteristic

102

C₂

#2930

yes

—

103

C₂

#2961

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₄

C₂⁴

commutator, socle, prime radical

C₂³

C₂³×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
43. C₄×C₈	Q₈	semidirect

H ≅ G/N

type

43. C₄×C₈

Q₈

semidirect

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

subgroup	index
C₂×#446	2
#2399	2
#1985	2
#1940	2
#1843	2
#1770	2
#1726	2

subgroup

index

Subgroups — #21716

Conjugacy classes of subgroups

Complements

Maximal subgroups