Group #68701 — subgroups

Subgroups — #68701

Order 384.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#5125	C₂	1	1	yes	maximal, characteristic
2	#5118	C₂	1	1	yes	maximal, characteristic
3	C₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	#4533	C₂	1	2	yes	maximal
5	#4532	C₂	1	2	yes	maximal
6	#2102	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₃×#117	C₂²	1	2	yes	—
9	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×#208	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	1	yes	characteristic
12	#819	—	2	2	no	—
13	Dic₃⋊C₈	—	2	2	no	—
14	#430	—	3	1	no	—
15	#423	—	3	1	no	—
16	#344	—	3	2	no	—
17	#339	—	3	2	no	—
18	H(3,ℤ/4)	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	—	2	1	no	—
21	C₃×M₁₆	—	2	2	no	—
22	C₂×C₃×C₈	D₄	1	2	yes	—
23	C₃×C₄⋊C₄	—	2	2	no	—
24	C₃×C₂²⋊C₄	—	2	2	no	—
25	C₃×C₄²	—	2	2	no	—
26	C₄×Dic₃	—	2	1	no	—
27	C₄×Dic₃	Q₈	1	1	yes	characteristic
28	C₄×Dic₃	Q₈	1	1	yes	characteristic
29	#208	—	2	2	no	—
30	#208	—	2	2	no	—
31	#208	—	2	2	no	—
32	C₂×C₃⋊C₈	—	2	1	no	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×M₁₆	—	3	2	no	—
35	C₂×M₁₆	—	3	1	no	—
36	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
37	#117	C₂×D₃	1	2	yes	—
38	C₂×C₄²	—	3	1	no	—
39	#108	—	6	2	no	—
40	C₄⋊C₈	—	6	2	no	—
41	C₂³×C₃	#41	1	1	yes	characteristic
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃×C₄	#41	1	2	yes	—
46	C₂×C₃×C₄	#41	1	2	yes	—
47	C₂×Dic₃	—	2	1	no	—
48	C₂×Dic₃	—	2	1	no	—
49	C₂×Dic₃	—	4	2	no	—
50	C₃×C₈	—	2	2	no	—
51	C₃⋊C₈	—	4	1	no	—
52	C₃⋊C₈	—	4	2	no	—
53	C₂²×C₄	—	6	1	no	—
54	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	6	2	no	—
57	M₁₆	—	6	2	no	—
58	M₁₆	—	6	2	no	—
59	C₂×C₈	—	6	1	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	D₁₂	1	2	yes	—
62	C₄⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	2	2	no	—
64	C₄²	—	2	2	no	—
65	C₄²	—	3	1	no	—
66	C₄²	—	3	1	no	—
67	C₄²	—	6	1	no	—
68	C₂²×C₃	—	4	1	no	—
69	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₃	#126	1	2	yes	—
71	C₃×C₄	—	8	2	no	—
72	C₃×C₄	#123	1	1	yes	characteristic
73	C₃×C₄	#124	1	1	yes	characteristic
74	C₃×C₄	#125	1	2	yes	—
75	Dic₃	—	2	2	no	—
76	Dic₃	—	4	1	no	—
77	C₂³	#237	1	1	yes	characteristic
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	6	1	no	—
80	C₂×C₄	—	6	1	no	—
81	C₂×C₄	—	12	2	no	—
82	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
83	C₂×C₄	#235	1	2	yes	—
84	C₂×C₄	#237	1	2	yes	—
85	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	12	2	no	—
88	C₈	—	12	1	no	—
89	C₂×C₃	—	2	1	no	—
90	C₂×C₃	—	2	2	no	—
91	C₂×C₃	#400	1	1	yes	characteristic
92	C₂²	—	4	1	no	—
93	C₂²	#879	1	2	yes	—
94	C₂²	#897	1	1	yes	characteristic
95	C₄	—	6	2	no	—
96	C₄	—	8	2	no	—
97	C₄	—	12	1	no	—
98	C₄	#875	1	1	yes	characteristic
99	C₄	#886	1	1	yes	characteristic
100	C₄	#889	1	2	yes	—
101	C₃	#2102	1	1	yes	characteristic, complemented, Sylow
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#4659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5125

C₂

yes

maximal, characteristic

#5118

C₂

yes

maximal, characteristic

C₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic

#4533

C₂

yes

maximal

#4532

C₂

yes

maximal

#2102

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₃×#117

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂×#208

C₂²

yes

characteristic

#819

—

Dic₃⋊C₈

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

Q₈

yes

characteristic

C₄×Dic₃

Q₈

yes

—

—

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

#117

C₂×D₃

yes

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₃

#126

yes

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

Dic₃

—

C₂³

#237

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

#235

yes

—

C₂×C₄

#237

yes

—

C₂×C₄

D₃×Q₈

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#400

yes

characteristic

C₂²

—

C₂²

#879

yes

—

C₂²

#897

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#889

yes

—

101

C₃

#2102

yes

characteristic, complemented, Sylow

102

C₂

—

103

C₂

—

104

C₂

#4659

yes

characteristic

	subgroup	G/H
center	C₄	#875
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×H(3,ℤ/4)	C₂
socle	C₂×C₃	#400
prime radical	C₂³×C₃	#41
hypercenter	H(3,ℤ/4)	D₃
hypocenter	C₃	#2102
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#875

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
101. C₃	#2102	semidirect

H ≅ G/N

type

101. C₃

#2102

semidirect

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

subgroup	index
#5125	2
#5118	2
C₃×H(3,ℤ/4)	2
#4533	2
#4532	2
#2102	3

subgroup

index

Subgroups — #68701

Conjugacy classes of subgroups

Complements

Maximal subgroups