Group #84480 — subgroups

Subgroups — C₃×#2616 #84480

Order 384.

816 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 232 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×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
3	C₃×#395	C₂	1	2	yes	maximal, complemented
4	C₃×#394	C₂	1	6	yes	maximal, complemented
5	C₃×#380	C₂	1	3	yes	maximal, complemented
6	#2616	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#41	—	2	12	no	—
8	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×D₄	C₂²	1	3	yes	—
11	C₂³×C₃×C₄	C₂²	1	3	yes	—
12	C₃×C₂²⋊Q₈	—	2	12	no	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₂×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
17	C₃×#95	C₂²	1	6	yes	—
18	C₂²×#41	C₂×C₃	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈	C₂×C₃	1	3	yes	complemented
20	#395	C₂×C₃	1	2	yes	complemented
21	#394	C₂×C₃	1	6	yes	complemented
22	#380	C₂×C₃	1	3	yes	complemented
23	C₂⁴×C₃	C₂³	1	3	yes	—
24	C₃×#41	—	2	32	no	—
25	C₂×C₃×Q₈	—	2	6	no	—
26	C₂×C₃×Q₈	—	2	6	no	—
27	C₂×C₃×D₄	—	2	12	no	—
28	C₂×C₃×D₄	—	2	6	no	—
29	C₂²×C₃×C₄	—	2	12	no	—
30	C₂²×C₃×C₄	D₄	1	12	yes	—
31	C₂²×C₃×C₄	C₂³	1	6	yes	—
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	3	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₃×C₄⋊C₄	—	2	6	no	—
36	C₃×C₄⋊C₄	—	2	12	no	—
37	C₃×C₂²⋊C₄	—	2	12	no	—
38	C₃×C₂²⋊C₄	—	4	6	no	—
39	C₂×#41	—	2	12	no	—
40	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
42	C₂²×D₄	C₂²×C₃	1	3	yes	—
43	C₂³×C₄	C₂²×C₃	1	3	yes	—
44	C₂²⋊Q₈	—	2	12	no	—
45	C₂×C₄⋊C₄	C₂²×C₃	1	3	yes	—
46	C₂×C₄⋊C₄	C₂²×C₃	1	6	yes	—
47	C₂×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
48	C₂×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
49	#95	C₂²×C₃	1	6	yes	—
50	C₂³×C₃	—	2	6	no	—
51	C₂³×C₃	—	2	3	no	—
52	C₂³×C₃	—	2	6	no	—
53	C₂³×C₃	—	2	3	no	—
54	C₂³×C₃	C₂×D₄	1	6	yes	—
55	C₂³×C₃	C₂⁴	1	1	yes	characteristic
56	C₃×Q₈	—	2	8	no	—
57	C₃×Q₈	—	4	4	no	—
58	C₃×D₄	—	2	24	no	—
59	C₂×C₃×C₄	—	2	6	no	—
60	C₂×C₃×C₄	—	2	24	no	—
61	C₂×C₃×C₄	—	2	6	no	—
62	C₂×C₃×C₄	—	2	6	no	—
63	C₂×C₃×C₄	—	2	6	no	—
64	C₂×C₃×C₄	—	2	6	no	—
65	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
66	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
67	C₂⁴	C₂³×C₃	1	3	yes	—
68	#41	—	2	32	no	—
69	C₂×Q₈	—	2	6	no	—
70	C₂×Q₈	—	2	6	no	—
71	C₂×D₄	—	2	12	no	—
72	C₂×D₄	—	2	6	no	—
73	C₂²×C₄	—	2	12	no	—
74	C₂²×C₄	C₃×D₄	1	12	yes	—
75	C₂²×C₄	C₂³×C₃	1	6	yes	—
76	C₂²×C₄	C₂³×C₃	1	2	yes	—
77	C₂²×C₄	C₂³×C₃	1	3	yes	—
78	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₄⋊C₄	—	2	6	no	—
80	C₄⋊C₄	—	2	12	no	—
81	C₂²⋊C₄	—	2	12	no	—
82	C₂²⋊C₄	—	4	6	no	—
83	C₂²×C₃	—	2	6	no	—
84	C₂²×C₃	—	2	6	no	—
85	C₂²×C₃	—	2	6	no	—
86	C₂²×C₃	—	4	6	no	—
87	C₂²×C₃	—	4	6	no	—
88	C₂²×C₃	C₂²×D₄	1	3	yes	—
89	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂²×C₃	2₋¹⁺⁴	1	3	yes	—
91	C₃×C₄	—	2	6	no	—
92	C₃×C₄	—	4	2	no	—
93	C₃×C₄	—	4	6	no	—
94	C₃×C₄	#120	1	4	yes	—
95	C₂³	—	2	6	no	—
96	C₂³	—	2	3	no	—
97	C₂³	—	2	6	no	—
98	C₂³	—	2	3	no	—
99	C₂³	C₂×C₃×D₄	1	6	yes	—
100	C₂³	C₂⁴×C₃	1	1	yes	characteristic
101	Q₈	—	2	8	no	—
102	Q₈	—	4	4	no	—
103	D₄	—	2	24	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	6	no	—
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
111	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
112	C₂×C₃	—	4	6	no	—
113	C₂×C₃	C₂×#120	1	1	yes	characteristic
114	C₂×C₃	#536	1	3	yes	—
115	C₂×C₃	#537	1	3	yes	—
116	C₂²	—	2	6	no	—
117	C₂²	—	2	6	no	—
118	C₂²	—	2	6	no	—
119	C₂²	—	4	6	no	—
120	C₂²	—	4	6	no	—
121	C₂²	C₂²×C₃×D₄	1	3	yes	—
122	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²	C₃×2₋¹⁺⁴	1	3	yes	—
124	C₄	—	2	6	no	—
125	C₄	—	4	2	no	—
126	C₄	—	4	6	no	—
127	C₄	C₃×#120	1	4	yes	—
128	C₃	#2616	1	1	yes	characteristic, direct product factor, Sylow
129	C₂	—	4	6	no	—
130	C₂	C₂×C₃×#120	1	1	yes	characteristic
131	C₂	C₃×#536	1	3	yes	—
132	C₂	C₃×#537	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×#41

C₂

yes

maximal, characteristic

C₂×C₃×C₂²⋊Q₈

C₂

yes

maximal, complemented

C₃×#395

C₂

yes

maximal, complemented

C₃×#394

C₂

yes

maximal, complemented

C₃×#380

C₂

yes

maximal, complemented

#2616

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×#41

—

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂²×C₃×D₄

C₂²

yes

—

C₂³×C₃×C₄

C₂²

yes

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂²×#41

C₂×C₃

yes

characteristic

C₂×C₂²⋊Q₈

C₂×C₃

yes

complemented

#395

C₂×C₃

yes

complemented

#394

C₂×C₃

yes

complemented

#380

C₂×C₃

yes

complemented

C₂⁴×C₃

C₂³

yes

—

C₃×#41

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₂×#41

—

C₂²×Q₈

C₂²×C₃

yes

characteristic

C₂²×Q₈

C₂²×C₃

yes

characteristic

C₂²×D₄

C₂²×C₃

yes

—

C₂³×C₄

C₂²×C₃

yes

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×D₄

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂⁴

C₂³×C₃

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×D₄

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#120

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×C₃×D₄

yes

—

100

C₂³

C₂⁴×C₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₃×D₄

yes

—

111

C₂×C₄

C₂×C₃×D₄

yes

—

112

C₂×C₃

—

113

C₂×C₃

C₂×#120

yes

characteristic

114

C₂×C₃

#536

yes

—

115

C₂×C₃

#537

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×C₃×D₄

yes

—

122

C₂²

C₃×H(5,𝔽₂)

yes

characteristic

123

C₂²

C₃×2₋¹⁺⁴

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₃×#120

yes

—

128

C₃

#2616

yes

characteristic, direct product factor, Sylow

129

C₂

—

130

C₂

C₂×C₃×#120

yes

characteristic

131

C₂

C₃×#536

yes

—

132

C₂

C₃×#537

yes

—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃

prime radical

C₂²×C₃×#41

C₂

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
3. C₃×#395	C₂	semidirect
4. C₃×#394	C₂	semidirect
5. C₃×#380	C₂	semidirect
6. #2616	C₃	direct (N × H)
19. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
20. #395	C₂×C₃	semidirect
21. #394	C₂×C₃	semidirect
22. #380	C₂×C₃	semidirect
128. C₃	#2616	direct (N × H)

H ≅ G/N

type

2. C₂×C₃×C₂²⋊Q₈

C₂

semidirect

3. C₃×#395

C₂

semidirect

4. C₃×#394

C₂

semidirect

5. C₃×#380

C₂

semidirect

6. #2616

C₃

direct (N × H)

19. C₂×C₂²⋊Q₈

C₂×C₃

semidirect

20. #395

C₂×C₃

semidirect

21. #394

C₂×C₃

semidirect

22. #380

C₂×C₃

semidirect

128. C₃

#2616

direct (N × H)

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

subgroup	index
C₂²×C₃×#41	2
C₂×C₃×C₂²⋊Q₈	2
C₃×#395	2
C₃×#394	2
C₃×#380	2
#2616	3

subgroup

index

C₂²×C₃×#41

C₂×C₃×C₂²⋊Q₈

Subgroups — C₃×#2616 #84480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2616 #84480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2616 #84480