Finite Group Databasebeta

← #13179617

Subgroups — #13179617

Order 1728.

4844 conjugacy classes of subgroups; only the 891 normal subgroups are shown.

normal subgroups only

This group has 4844 conjugacy classes of subgroups — too many to draw. Only its 891 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C212yesmaximal, complemented
2C2×C33×C22⋊C4C211yesmaximal, characteristic, complemented
3#11723971C214yesmaximal, complemented
4#11723927C214yesmaximal, complemented
5#11723923C214yesmaximal, complemented
6#10595964—313nomaximal
7C24×C33C2211yescharacteristic
8C23×C3⋊(C3⋊D3)C2212yescomplemented
9C22×C33×C4C2212yescomplemented
10C2×#6312—28no—
11C2×#6312C2214yes—
12C2×#6312C2214yescomplemented
13C22×C3⋊(C3⋊Dic3)C2212yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2214yescomplemented
15C33×C22⋊C4C2214yescomplemented
16#88859C2218yescomplemented
17C4⋊(C3⋊(C3⋊Dic3))C2214yescomplemented
18C22×#638—326no—
19C2×C32×C22⋊C4D3113yescomplemented
20#64139—352no—
21#64095—352no—
22#64091—352no—
23C23×C33—42no—
24C23×C33D414yescomplemented
25C23×C33C2312yes—
26C23×C33C2311yescharacteristic
27C22×C3⋊(C3⋊D3)—24no—
28C22×C3⋊(C3⋊D3)—44no—
29C22×C3⋊(C3⋊D3)C2314yes—
30C2×C33×C4—42no—
31C2×C33×C4C2314yes—
32#6312—28no—
33#6312—28no—
34C2×C3⋊(C3⋊Dic3)—24no—
35C2×C3⋊(C3⋊Dic3)C2314yes—
36C4⋊(C3⋊(C3⋊D3))—44no—
37#5493—913no—
38C24×C32C2×D3113yes—
39C23×C3⋊D3—326no—
40C22×C32×C4C2×D3126yescomplemented
41C2×#638—352no—
42C2×#638—352no—
43C2×#638—6104no—
44C22×C3⋊Dic3—326no—
45C2×C4⋊(C3⋊D3)—352no—
46C32×C22⋊C4C2×D3152yescomplemented
47#3752—3104no—
48C4⋊(C3⋊Dic3)—352no—
49C22×C33—22no—
50C22×C33—24no—
51C22×C33—44no—
52C22×C33C2×D414yes—
53C22×C33C2×D412yes—
54C22×C33C2411yescharacteristic
55C2×C3⋊(C3⋊D3)—24no—
56C2×C3⋊(C3⋊D3)—28no—
57C2×C3⋊(C3⋊D3)—48no—
58C33×C4—24no—
59C3⋊(C3⋊Dic3)—24no—
60C22×#67—926no—
61C2×C3×C22⋊C4C3⋊D3113yescomplemented
62#930—952no—
63#886—952no—
64#882—952no—
65C23×C32—426no—
66C23×C32D12152yescomplemented
67C23×C32C22×D3126yes—
68C23×C32C22×D3113yes—
69C22×C3⋊D3—352no—
70C22×C3⋊D3—652no—
71C22×C3⋊D3—1252no—
72C2×C32×C4—426no—
73C2×C32×C4C22×D3152yes—
74#638—6104no—
75#638—6104no—
76C2×C3⋊Dic3—352no—
77C2×C3⋊Dic3—652no—
78C4⋊(C3⋊D3)—1252no—
79#534—271noSylow
80C2×C33—24no—
81C2×C33—22no—
82C2×C33C22×D411yescharacteristic
83C2×C33H(5,𝔽2)12yes—
84C3⋊(C3⋊D3)—44no—
85C24×C3C2×C3⋊D3113yes—
86C23×D3—926no—
87C22×C3×C4C2×C3⋊D3126yescomplemented
88C2×#67—952no—
89C2×#67—952no—
90C2×#67—18104no—
91C22×Dic3—926no—
92C2×D12—952no—
93C3×C22⋊C4C2×C3⋊D3152yescomplemented
94(C2×D3)⋊C4—9104no—
95C4⋊Dic3—952no—
96C22×C32—226no—
97C22×C32—252no—
98C22×C32—452no—
99C22×C32C2×D12152yes—
100C22×C32C2×D12126yes—
101C22×C32C23×D3113yes—
102C2×C3⋊D3—652no—
103C2×C3⋊D3—6104no—
104C2×C3⋊D3—12104no—
105C32×C4—252no—
106C3⋊Dic3—652no—
107C22×D4—272no—
108#123—274no—
109#121—274no—
110#120—274no—
111C2×C22⋊C4C3⋊(C3⋊D3)11yescharacteristic, complemented
112C33#53411yescharacteristic, complemented, Sylow
113C23×C3—426no—
114C23×C3C4⋊(C3⋊D3)152yescomplemented
115C23×C3C22×C3⋊D3126yes—
116C23×C3C22×C3⋊D3113yes—
117C22×D3—952no—
118C22×D3—1852no—
119C22×D3—3652no—
120C2×C3×C4—426no—
121C2×C3×C4C22×C3⋊D3152yes—
122#67—18104no—
123#67—18104no—
124C2×Dic3—952no—
125C2×Dic3—1852no—
126D12—3652no—
127C2×C32—252no—
128C2×C32—226no—
129C2×C32C22×D12113yes—
130C2×C32#1004126yes—
131C3⋊D3—1252no—
132C24—272no—
133C24C2×C3⋊(C3⋊D3)11yescharacteristic
134C2×D4—274no—
135C2×D4—274no—
136C2×D4—274no—
137C2×D4—548no—
138C22×C4—272no—
139C22×C4C2×C3⋊(C3⋊D3)12yescomplemented
140C4⋊C4—274no—
141C22⋊C4—278no—
142C22⋊C4C2×C3⋊(C3⋊D3)14yescomplemented
143C22×C3—226no—
144C22×C3—252no—
145C22×C3—452no—
146C22×C3C2×C4⋊(C3⋊D3)152yes—
147C22×C3C2×C4⋊(C3⋊D3)126yes—
148C22×C3C23×C3⋊D3113yes—
149C2×D3—1852no—
150C2×D3—18104no—
151C2×D3—36104no—
152C3×C4—252no—
153Dic3—1852no—
154C32#5493113yescomplemented
155C23—42no—
156C23—274no—
157C23—544no—
158C23—1084no—
159C23C4⋊(C3⋊(C3⋊D3))14yescomplemented
160C23C22×C3⋊(C3⋊D3)12yes—
161C23C22×C3⋊(C3⋊D3)11yescharacteristic
162D4—548no—
163D4—548no—
164D4—1084no—
165C2×C4—42no—
166C2×C4—274no—
167C2×C4—544no—
168C2×C4C22×C3⋊(C3⋊D3)14yes—
169C2×C3—252no—
170C2×C3—226no—
171C2×C3C22×C4⋊(C3⋊D3)113yes—
172C2×C3#64361126yes—
173D3—3652no—
174C22—22no—
175C22—24no—
176C22—44no—
177C22—544no—
178C22—548no—
179C22—1088no—
180C22C2×C4⋊(C3⋊(C3⋊D3))14yes—
181C22C2×C4⋊(C3⋊(C3⋊D3))12yes—
182C22C23×C3⋊(C3⋊D3)11yescharacteristic
183C4—24no—
184C4—544no—
185C3#10595964113yescomplemented
186C2—24no—
187C2—22no—
188C2—1084no—
189C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
190C2#1172467912yes—
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C22⋊C4C2
hypercenterC2×C22⋊C4C3⋊(C3⋊D3)
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#6312C2semidirect
2. C2×C33×C22⋊C4C2semidirect
3. #11723971C2semidirect
4. #11723927C2semidirect
5. #11723923C2semidirect
8. C23×C3⋊(C3⋊D3)C22semidirect
9. C22×C33×C4C22semidirect
12. C2×#6312C22semidirect
13. C22×C3⋊(C3⋊Dic3)C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C33×C22⋊C4C22semidirect
16. #88859C22semidirect
17. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
19. C2×C32×C22⋊C4D3semidirect
24. C23×C33D4semidirect
40. C22×C32×C4C2×D3semidirect
46. C32×C22⋊C4C2×D3semidirect
61. C2×C3×C22⋊C4C3⋊D3semidirect
66. C23×C32D12semidirect
87. C22×C3×C4C2×C3⋊D3semidirect
93. C3×C22⋊C4C2×C3⋊D3semidirect
111. C2×C22⋊C4C3⋊(C3⋊D3)semidirect
112. C33#534semidirect
114. C23×C3C4⋊(C3⋊D3)semidirect
139. C22×C4C2×C3⋊(C3⋊D3)semidirect
142. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
154. C32#5493semidirect
159. C23C4⋊(C3⋊(C3⋊D3))semidirect
185. C3#10595964semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×C33×C22⋊C42
#117239712
#117239272
#117239232
#105959643