Group #84135 — subgroups

Subgroups — C₃×C₄²×D₄ #84135

Order 384.

924 conjugacy classes of subgroups; only the 576 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 576 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×C₃×C₄×D₄	C₂	1	3	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₃×C₄×C₂²⋊C₄	C₂	1	6	yes	maximal, complemented
5	C₃×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄²×D₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₄	C₂²	1	6	yes	—
9	C₃×C₄×D₄	C₄	1	24	yes	direct product factor
10	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
12	C₂×C₃×C₄²	—	2	8	no	—
13	C₂×C₃×C₄²	C₂²	1	3	yes	—
14	C₂×C₃×C₄²	C₂²	1	6	yes	—
15	C₂×C₃×C₄²	C₂²	1	3	yes	—
16	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#95	C₂²	1	6	yes	—
18	C₂×C₄×D₄	C₂×C₃	1	3	yes	—
19	C₂²×C₄²	C₂×C₃	1	2	yes	complemented
20	C₄×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
21	C₄×C₂²⋊C₄	C₂×C₃	1	6	yes	complemented
22	C₄³	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂⁴×C₃	C₂³	1	2	yes	—
24	C₂×C₃×D₄	C₂×C₄	1	12	yes	—
25	C₂²×C₃×C₄	—	2	12	no	—
26	C₂²×C₃×C₄	—	2	12	no	—
27	C₂²×C₃×C₄	C₂×C₄	1	24	yes	complemented
28	C₂²×C₃×C₄	C₂³	1	3	yes	—
29	C₂²×C₃×C₄	C₂³	1	6	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	3	yes	—
32	C₃×C₄⋊C₄	C₂×C₄	1	12	yes	complemented
33	C₃×C₂²⋊C₄	C₂×C₄	1	24	yes	complemented
34	C₃×C₄²	—	2	12	no	—
35	C₃×C₄²	—	2	6	no	—
36	C₃×C₄²	C₂×C₄	1	12	yes	complemented
37	C₃×C₄²	D₄	1	4	yes	direct product factor
38	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
39	C₂³×C₄	C₂²×C₃	1	6	yes	—
40	C₄×D₄	C₃×C₄	1	24	yes	direct product factor
41	C₂×C₄⋊C₄	C₂²×C₃	1	3	yes	—
42	C₂×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
43	C₂×C₄²	—	2	8	no	—
44	C₂×C₄²	C₂²×C₃	1	3	yes	—
45	C₂×C₄²	C₂²×C₃	1	6	yes	—
46	C₂×C₄²	C₂²×C₃	1	3	yes	—
47	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
48	#95	C₂²×C₃	1	6	yes	—
49	C₂³×C₃	—	2	2	no	—
50	C₂³×C₃	—	2	6	no	—
51	C₂³×C₃	C₂²×C₄	1	12	yes	—
52	C₂³×C₃	C₂⁴	1	1	yes	characteristic
53	C₃×D₄	C₄²	1	16	yes	direct product factor
54	C₂×C₃×C₄	—	2	6	no	—
55	C₂×C₃×C₄	—	2	6	no	—
56	C₂×C₃×C₄	—	2	6	no	—
57	C₂×C₃×C₄	—	2	48	no	—
58	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
59	C₂×C₃×C₄	C₂²×C₄	1	12	yes	—
60	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
62	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
63	C₂×C₃×C₄	#41	1	6	yes	—
64	C₂⁴	C₂³×C₃	1	2	yes	—
65	C₂×D₄	C₂×C₃×C₄	1	12	yes	—
66	C₂²×C₄	—	2	12	no	—
67	C₂²×C₄	—	2	12	no	—
68	C₂²×C₄	C₂×C₃×C₄	1	24	yes	complemented
69	C₂²×C₄	C₂³×C₃	1	3	yes	—
70	C₂²×C₄	C₂³×C₃	1	6	yes	—
71	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂²×C₄	C₂³×C₃	1	3	yes	—
73	C₄⋊C₄	C₂×C₃×C₄	1	12	yes	complemented
74	C₂²⋊C₄	C₂×C₃×C₄	1	24	yes	complemented
75	C₄²	—	2	12	no	—
76	C₄²	—	2	6	no	—
77	C₄²	C₂×C₃×C₄	1	12	yes	complemented
78	C₄²	C₃×D₄	1	4	yes	direct product factor
79	C₂²×C₃	—	2	12	no	—
80	C₂²×C₃	—	2	12	no	—
81	C₂²×C₃	C₂×C₄²	1	8	yes	complemented
82	C₂²×C₃	C₂³×C₄	1	3	yes	—
83	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₃	C₂×#41	1	3	yes	—
85	C₃×C₄	—	2	12	no	—
86	C₃×C₄	—	2	6	no	—
87	C₃×C₄	C₂×C₄²	1	4	yes	complemented
88	C₃×C₄	C₄×D₄	1	12	yes	direct product factor
89	C₂³	—	2	2	no	—
90	C₂³	—	2	6	no	—
91	C₂³	C₂²×C₃×C₄	1	12	yes	—
92	C₂³	C₂⁴×C₃	1	1	yes	characteristic
93	D₄	C₃×C₄²	1	16	yes	direct product factor
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	48	no	—
98	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
99	C₂×C₄	C₂²×C₃×C₄	1	12	yes	—
100	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
101	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
102	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
103	C₂×C₄	C₃×#41	1	6	yes	—
104	C₂×C₃	—	2	8	no	—
105	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
106	C₂×C₃	C₂×C₄×D₄	1	3	yes	—
107	C₂×C₃	C₄×#41	1	3	yes	—
108	C₂²	—	2	12	no	—
109	C₂²	—	2	12	no	—
110	C₂²	C₂×C₃×C₄²	1	8	yes	complemented
111	C₂²	C₂³×C₃×C₄	1	3	yes	—
112	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
113	C₂²	C₂×C₃×#41	1	3	yes	—
114	C₄	—	2	12	no	—
115	C₄	—	2	6	no	—
116	C₄	C₂×C₃×C₄²	1	4	yes	complemented
117	C₄	C₃×C₄×D₄	1	12	yes	direct product factor
118	C₃	C₄²×D₄	1	1	yes	characteristic, direct product factor, Sylow
119	C₂	—	2	8	no	—
120	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
121	C₂	C₂×C₃×C₄×D₄	1	3	yes	—
122	C₂	C₃×C₄×#41	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄×D₄

C₂

yes

maximal

C₂²×C₃×C₄²

C₂

yes

maximal, complemented

C₃×C₄×C₄⋊C₄

C₂

yes

maximal, complemented

C₃×C₄×C₂²⋊C₄

C₂

yes

maximal, complemented

C₃×C₄³

C₂

yes

maximal, characteristic, complemented

C₄²×D₄

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×D₄

C₂²

yes

characteristic

C₂³×C₃×C₄

C₂²

yes

—

C₃×C₄×D₄

C₄

yes

direct product factor

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×#95

C₂²

yes

—

C₂×C₄×D₄

C₂×C₃

yes

—

C₂²×C₄²

C₂×C₃

yes

complemented

C₄×C₄⋊C₄

C₂×C₃

yes

complemented

C₄×C₂²⋊C₄

C₂×C₃

yes

complemented

C₄³

C₂×C₃

yes

characteristic, complemented

C₂⁴×C₃

C₂³

yes

—

C₂×C₃×D₄

C₂×C₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

complemented

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

complemented

C₃×C₂²⋊C₄

C₂×C₄

yes

complemented

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

complemented

C₃×C₄²

D₄

yes

direct product factor

C₂²×D₄

C₂²×C₃

yes

characteristic

C₂³×C₄

C₂²×C₃

yes

—

C₄×D₄

C₃×C₄

yes

direct product factor

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

#95

C₂²×C₃

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂²×C₄

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×D₄

C₄²

yes

direct product factor

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂⁴

C₂³×C₃

yes

—

C₂×D₄

C₂×C₃×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₃×C₄

yes

complemented

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₂²×C₄

C₂³×C₃

yes

—

C₄⋊C₄

C₂×C₃×C₄

yes

complemented

C₂²⋊C₄

C₂×C₃×C₄

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂×C₃×C₄

yes

complemented

C₄²

C₃×D₄

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄²

yes

complemented

C₂²×C₃

C₂³×C₄

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₄²

yes

complemented

C₃×C₄

C₄×D₄

yes

direct product factor

C₂³

—

C₂³

—

C₂³

C₂²×C₃×C₄

yes

—

C₂³

C₂⁴×C₃

yes

characteristic

D₄

C₃×C₄²

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₃×C₄

yes

—

C₂×C₄

C₂²×C₃×C₄

yes

—

100

C₂×C₄

C₂²×C₃×C₄

yes

—

101

C₂×C₄

C₂²×C₃×C₄

yes

—

102

C₂×C₄

C₂×C₃×D₄

yes

—

103

C₂×C₄

C₃×#41

yes

—

104

C₂×C₃

—

105

C₂×C₃

C₂²×C₄²

yes

characteristic

106

C₂×C₃

C₂×C₄×D₄

yes

—

107

C₂×C₃

C₄×#41

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₃×C₄²

yes

complemented

111

C₂²

C₂³×C₃×C₄

yes

—

112

C₂²

C₂²×C₃×D₄

yes

characteristic

113

C₂²

C₂×C₃×#41

yes

—

114

C₄

—

115

C₄

—

116

C₄

C₂×C₃×C₄²

yes

complemented

117

C₄

C₃×C₄×D₄

yes

direct product factor

118

C₃

C₄²×D₄

yes

characteristic, direct product factor, Sylow

119

C₂

—

120

C₂

C₂²×C₃×C₄²

yes

characteristic

121

C₂

C₂×C₃×C₄×D₄

yes

—

122

C₂

C₃×C₄×#41

yes

—

	subgroup	G/H
center	C₂×C₃×C₄²	C₂²
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ G

subgroup

G/H

center

C₂×C₃×C₄²

C₂²

commutator

C₂

C₂²×C₃×C₄²

Frattini, cosocle

C₂³

C₂⁴×C₃

socle

C₂³×C₃

C₂⁴

prime radical

C₂²×C₃×D₄

C₂²

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
2. C₂²×C₃×C₄²	C₂	semidirect
3. C₃×C₄×C₄⋊C₄	C₂	semidirect
4. C₃×C₄×C₂²⋊C₄	C₂	semidirect
5. C₃×C₄³	C₂	semidirect
6. C₄²×D₄	C₃	direct (N × H)
9. C₃×C₄×D₄	C₄	direct (N × H)
19. C₂²×C₄²	C₂×C₃	semidirect
20. C₄×C₄⋊C₄	C₂×C₃	semidirect
21. C₄×C₂²⋊C₄	C₂×C₃	semidirect
22. C₄³	C₂×C₃	semidirect
27. C₂²×C₃×C₄	C₂×C₄	semidirect
32. C₃×C₄⋊C₄	C₂×C₄	semidirect
33. C₃×C₂²⋊C₄	C₂×C₄	semidirect
36. C₃×C₄²	C₂×C₄	semidirect
37. C₃×C₄²	D₄	direct (N × H)
40. C₄×D₄	C₃×C₄	direct (N × H)
53. C₃×D₄	C₄²	direct (N × H)
68. C₂²×C₄	C₂×C₃×C₄	semidirect
73. C₄⋊C₄	C₂×C₃×C₄	semidirect
74. C₂²⋊C₄	C₂×C₃×C₄	semidirect
77. C₄²	C₂×C₃×C₄	semidirect
78. C₄²	C₃×D₄	direct (N × H)
81. C₂²×C₃	C₂×C₄²	semidirect
87. C₃×C₄	C₂×C₄²	semidirect
88. C₃×C₄	C₄×D₄	direct (N × H)
93. D₄	C₃×C₄²	direct (N × H)
110. C₂²	C₂×C₃×C₄²	semidirect
116. C₄	C₂×C₃×C₄²	semidirect
117. C₄	C₃×C₄×D₄	direct (N × H)
118. C₃	C₄²×D₄	direct (N × H)

H ≅ G/N

type

2. C₂²×C₃×C₄²

C₂

semidirect

3. C₃×C₄×C₄⋊C₄

C₂

semidirect

4. C₃×C₄×C₂²⋊C₄

C₂

semidirect

5. C₃×C₄³

C₂

semidirect

6. C₄²×D₄

C₃

direct (N × H)

9. C₃×C₄×D₄

C₄

direct (N × H)

19. C₂²×C₄²

C₂×C₃

semidirect

20. C₄×C₄⋊C₄

C₂×C₃

semidirect

21. C₄×C₂²⋊C₄

C₂×C₃

semidirect

22. C₄³

C₂×C₃

semidirect

27. C₂²×C₃×C₄

C₂×C₄

semidirect

32. C₃×C₄⋊C₄

C₂×C₄

semidirect

33. C₃×C₂²⋊C₄

C₂×C₄

semidirect

36. C₃×C₄²

C₂×C₄

semidirect

37. C₃×C₄²

D₄

direct (N × H)

40. C₄×D₄

C₃×C₄

direct (N × H)

53. C₃×D₄

C₄²

direct (N × H)

68. C₂²×C₄

C₂×C₃×C₄

semidirect

73. C₄⋊C₄

C₂×C₃×C₄

semidirect

74. C₂²⋊C₄

C₂×C₃×C₄

semidirect

77. C₄²

C₂×C₃×C₄

semidirect

78. C₄²

C₃×D₄

direct (N × H)

81. C₂²×C₃

C₂×C₄²

semidirect

87. C₃×C₄

C₂×C₄²

semidirect

88. C₃×C₄

C₄×D₄

direct (N × H)

93. D₄

C₃×C₄²

direct (N × H)

110. C₂²

C₂×C₃×C₄²

semidirect

116. C₄

C₂×C₃×C₄²

semidirect

117. C₄

C₃×C₄×D₄

direct (N × H)

118. C₃

C₄²×D₄

direct (N × H)

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

subgroup	index
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₄²	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×C₄³	2
C₄²×D₄	3

subgroup

index

C₂×C₃×C₄×D₄

C₂²×C₃×C₄²

C₃×C₄×C₄⋊C₄

C₃×C₄×C₂²⋊C₄

C₃×C₄³

C₄²×D₄

Subgroups — C₃×C₄²×D₄ #84135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C42×D4 #84135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×D₄ #84135