Group #33449 — subgroups

Subgroups — C₂×#2176 #33449

Order 256.

335 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆	C₂	1	2	yes	maximal
3	#2176	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367	C₂	1	1	yes	maximal, characteristic
5	C₂×#358	C₂	1	2	yes	maximal
6	C₂×#346	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₄×Q₈	—	2	2	no	—
9	C₂×Q₃₂	C₄	1	8	yes	complemented
10	C₂×Q₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×Q₁₆	—	2	4	no	—
14	C₄×Q₁₆	C₂²	1	8	yes	—
15	C₂×#107	C₂²	1	2	yes	—
16	C₂×#103	—	2	2	no	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#367	C₂²	1	4	yes	—
19	#358	C₂²	1	8	yes	—
20	#346	C₂²	1	4	yes	—
21	C₂²×Q₈	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	4	no	—
24	C₂×Q₁₆	C₂×C₄	1	8	yes	—
25	C₂×Q₁₆	C₂³	1	4	yes	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	C₄×Q₈	—	2	8	no	—
29	C₄×Q₈	—	4	4	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	4	2	no	—
32	C₂×C₄²	—	4	2	no	—
33	C₂×C₄²	D₄	1	1	yes	characteristic
34	Q₃₂	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆	—	2	2	no	—
36	C₂×C₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₁₆	C₂³	1	2	yes	—
38	C₂×C₁₆	C₂³	1	2	yes	—
39	#107	—	2	2	no	—
40	#107	C₂³	1	4	yes	—
41	#103	—	2	8	no	—
42	C₄×C₈	D₄	1	2	yes	—
43	C₄×C₈	C₂³	1	2	yes	—
44	C₂×Q₈	—	2	8	no	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	4	no	—
48	C₂²×C₄	—	4	2	no	—
49	C₂²×C₄	—	4	2	no	—
50	C₂²×C₄	D₈	1	2	yes	—
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	Q₁₆	—	2	8	no	—
53	Q₁₆	—	2	4	no	—
54	Q₁₆	C₂²×C₄	1	8	yes	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	C₂²×C₄	1	2	yes	—
57	C₂×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	#41	1	2	yes	—
60	C₂×C₈	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄	—	2	4	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	8	no	—
64	C₄²	—	4	8	no	—
65	C₄²	D₈	1	2	yes	—
66	C₄²	C₂×D₄	1	2	yes	—
67	C₁₆	—	2	2	no	—
68	C₁₆	C₂²×C₄	1	4	yes	—
69	C₂³	C₂×D₈	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	Q₈	—	4	4	no	—
72	Q₈	—	4	8	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	—	4	8	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	8	4	no	—
78	C₂×C₄	C₄×D₄	1	2	yes	—
79	C₂×C₄	C₂×D₈	1	4	yes	—
80	C₂×C₄	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄	#135	1	2	yes	—
82	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	C₄×D₄	1	2	yes	—
85	C₈	C₂³×C₄	1	1	yes	characteristic
86	C₈	C₂×#41	1	1	yes	characteristic
87	C₂²	C₄×D₈	1	2	yes	—
88	C₂²	#510	1	4	yes	—
89	C₂²	C₂²×D₈	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	4	8	no	—
92	C₄	—	8	4	no	—
93	C₄	C₄×D₈	1	2	yes	—
94	C₄	C₂×C₄×D₄	1	1	yes	characteristic
95	C₄	C₂×#135	1	1	yes	characteristic
96	C₂	#2176	1	4	yes	direct product factor
97	C₂	C₂×C₄×D₈	1	1	yes	characteristic
98	C₂	C₂×#510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic

C₂×C₄×Q₁₆

C₂

yes

maximal

#2176

C₂

yes

maximal, direct product factor

C₂×#367

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal

C₂×#346

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄×Q₈

—

C₂×Q₃₂

C₄

yes

complemented

C₂×Q₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

#367

C₂²

yes

—

#358

C₂²

yes

—

#346

C₂²

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

Q₃₂

C₂×C₄

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

#107

—

#107

C₂³

yes

—

#103

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₈

yes

—

C₄²

C₂×D₄

yes

—

C₁₆

—

C₁₆

C₂²×C₄

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

#135

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

C₄×D₄

yes

—

C₈

C₂³×C₄

yes

characteristic

C₈

C₂×#41

yes

characteristic

C₂²

C₄×D₈

yes

—

C₂²

#510

yes

—

C₂²

C₂²×D₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₄×D₈

yes

—

C₄

C₂×C₄×D₄

yes

C₄

yes

C₂

yes

direct product factor

C₂

C₂×C₄×D₈

yes

characteristic

C₂

C₂×#510

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator

C₈

C₂³×C₄

Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #2176	C₂	direct (N × H)
9. C₂×Q₃₂	C₄	semidirect
34. Q₃₂	C₂×C₄	semidirect
96. C₂	#2176	direct (N × H)

H ≅ G/N

type

3. #2176

C₂

direct (N × H)

9. C₂×Q₃₂

C₄

semidirect

34. Q₃₂

C₂×C₄

semidirect

96. C₂

#2176

direct (N × H)

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

subgroup	index
C₂²×Q₃₂	2
C₂×C₄×Q₁₆	2
#2176	2
C₂×#367	2
C₂×#358	2
C₂×#346	2

subgroup

index

C₂²×Q₃₂

C₂×C₄×Q₁₆

Subgroups — C₂×#2176 #33449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2176 #33449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2176 #33449