Group #33909 — subgroups

Subgroups — #33909

Order 256.

234 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3342	C₂	1	2	yes	maximal
2	#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	#2237	C₂	1	4	yes	maximal, complemented
4	#2225	C₂	1	2	yes	maximal, complemented
5	#2216	C₂	1	2	yes	maximal, complemented
6	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#2158	C₂	1	2	yes	maximal, complemented
9	#556	—	2	2	no	—
10	C₂×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	2	yes	—
12	#500	C₂²	1	2	yes	—
13	#485	—	2	4	no	—
14	#475	—	2	2	no	—
15	#467	C₂²	1	4	yes	—
16	C₄×Q₁₆	C₂²	1	2	yes	—
17	C₈×D₄	C₂²	1	1	yes	characteristic
18	#427	C₂²	1	4	yes	—
19	#367	C₂²	1	4	yes	—
20	#366	C₂²	1	1	yes	characteristic
21	#366	C₂²	1	2	yes	—
22	#366	C₂²	1	2	yes	—
23	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	#358	C₂²	1	2	yes	—
25	#358	C₂²	1	4	yes	—
26	C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆	C₂³	1	2	yes	—
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₄⋊Q₈	—	2	2	no	—
31	#126	—	4	4	no	—
32	#124	—	4	2	no	—
33	C₂²⋊Q₈	—	2	4	no	—
34	C₄×Q₈	—	2	2	no	—
35	C₄×Q₈	—	4	2	no	—
36	C₄×D₄	D₄	1	1	yes	characteristic
37	#117	—	2	4	no	—
38	Q₃₂	—	2	2	no	—
39	C₂×C₁₆	—	2	4	no	—
40	C₂×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₆	C₂³	1	1	yes	characteristic
43	#107	—	2	2	no	—
44	#107	C₂³	1	4	yes	—
45	#107	C₂³	1	2	yes	—
46	#106	—	2	4	no	—
47	C₄⋊C₈	D₄	1	1	yes	characteristic
48	#103	—	2	2	no	—
49	#103	—	2	4	no	—
50	C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂×D₄	D₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	2	yes	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₈	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	4	4	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	D₈	1	1	yes	characteristic
69	C₂²⋊C₄	—	4	4	no	—
70	C₂²⋊C₄	—	4	4	no	—
71	C₂²⋊C₄	D₈	1	2	yes	—
72	C₄²	—	4	4	no	—
73	C₄²	—	4	2	no	—
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₁₆	—	2	2	no	—
76	C₁₆	—	2	1	no	—
77	C₁₆	#41	1	2	yes	—
78	C₂³	C₂×D₈	1	2	yes	—
79	Q₈	—	4	2	no	—
80	Q₈	—	4	2	no	—
81	Q₈	—	8	4	no	—
82	D₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	4	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	C₂×D₈	1	2	yes	—
89	C₂×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
92	C₈	—	2	2	no	—
93	C₈	—	2	1	no	—
94	C₈	C₂×#41	1	1	yes	characteristic
95	C₈	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂²	—	2	2	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	C₂²×D₈	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	1	no	—
102	C₄	—	8	2	no	—
103	C₄	—	8	4	no	—
104	C₄	—	8	2	no	—
105	C₄	#508	1	2	yes	—
106	C₄	#547	1	1	yes	characteristic
107	C₄	#578	1	1	yes	characteristic
108	C₂	—	4	2	no	—
109	C₂	#3334	1	1	yes	characteristic
110	C₂	C₂×#508	1	1	yes	characteristic
111	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3342

C₂

yes

maximal

#2252

C₂

yes

maximal, characteristic, complemented

#2237

C₂

yes

maximal, complemented

#2225

C₂

yes

maximal, complemented

#2216

C₂

yes

maximal, complemented

C₄×Q₃₂

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆

C₂

yes

maximal, characteristic

#2158

C₂

yes

maximal, complemented

#556

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

#500

C₂²

yes

—

#485

—

#475

—

#467

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₈×D₄

C₂²

yes

characteristic

#427

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

#358

C₂²

yes

—

#358

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

D₄

yes

characteristic

#117

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

C₄⋊C₈

D₄

yes

characteristic

#103

—

#103

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#41

yes

—

C₂³

C₂×D₈

yes

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

—

C₈

C₂×#41

yes

characteristic

C₈

2₋¹⁺⁴

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₈

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#508

yes

—

106

C₄

#547

yes

characteristic

107

C₄

#578

yes

characteristic

108

C₂

—

109

C₂

#3334

yes

characteristic

110

C₂

C₂×#508

yes

characteristic

111

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂×D₄

D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2252	C₂	semidirect
3. #2237	C₂	semidirect
4. #2225	C₂	semidirect
5. #2216	C₂	semidirect
6. C₄×Q₃₂	C₂	semidirect
8. #2158	C₂	semidirect

H ≅ G/N

type

2. #2252

C₂

semidirect

3. #2237

C₂

semidirect

4. #2225

C₂

semidirect

5. #2216

C₂

semidirect

6. C₄×Q₃₂

C₂

semidirect

8. #2158

C₂

semidirect

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

subgroup	index
#3342	2
#2252	2
#2237	2
#2225	2
#2216	2
C₄×Q₃₂	2
D₄×C₁₆	2
#2158	2

subgroup

index

C₄×Q₃₂

D₄×C₁₆

#2158

Subgroups — #33909

Conjugacy classes of subgroups

Complements

Maximal subgroups