Finite Group Databasebeta

← #11724403

Subgroups — #11724403

Order 864.

1122 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C2×D33C212yesmaximal, complemented
2C4×#1113C211yesmaximal, characteristic, complemented
3C3⋊D3×D12C213yesmaximal, complemented
4C3×#3802C213yesmaximal, complemented
5#88952C216yesmaximal, complemented
6D32×D4—33nomaximal
7C2×#1113C2211yescharacteristic
8C2×D3×C3⋊D3C2216yescomplemented
9C2×C3×D32C2216yescomplemented
10D33—28no—
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
12C3×C4×C3⋊D3C2213yescomplemented
13C32×D12C2213yescomplemented
14#6294C2213yescomplemented
15#6291C2216yescomplemented
16C3×#625C2216yescomplemented
17C22×D32—36no—
18D4×C3⋊D3—33no—
19C3×D3×D4—36no—
20#3811—36no—
21D3×#67—312no—
22#3802D313yescomplemented
23D3×D12—36no—
24D3×D12—63no—
25C4×D32—33no—
26C2×C3⋊(C3⋊D3)C2312yescomplemented
27C2×C3×C3⋊D3C2313yes—
28C2×C32×D3C2316yescomplemented
29#1113D414yescomplemented
30D3×C3⋊D3—212no—
31C3×D32—212no—
32C33×C4C2311yescharacteristic, complemented
33C3×C3⋊Dic3C2313yescomplemented
34C2×D3×D4—93no—
35C22×C3⋊D3—36no—
36C22×C3×D3—312no—
37C2×D32—33no—
38C2×D32—312no—
39C2×D32—66no—
40C2×D32—624no—
41C2×D32C2×D316yescomplemented
42C32×D4—33no—
43#638—36no—
44C4⋊(C3⋊D3)—33no—
45C4⋊(C3⋊D3)—63no—
46C4⋊(C3⋊D3)—121no—
47C4×C3⋊D3C2×D313yescomplemented
48C3×#67—312no—
49C3×D12—23no—
50C3×D12C2×D316yescomplemented
51C3×C4×D3—36no—
52C3×C4×D3—63no—
53#626—312no—
54#626—66no—
55#625C2×D316yescomplemented
56#624—33no—
57D3×Dic3—36no—
58C2×C33C2411yescharacteristic
59C3⋊(C3⋊D3)—22no—
60C3×C3⋊D3C2×D416yescomplemented
61C32×D3—26no—
62C23×D3—96no—
63C2×C3×D4—93no—
64C2×#67—96no—
65D3×D4—36no—
66D3×D4—96no—
67D3×D4—912no—
68D3×D4—183no—
69C2×D12—93no—
70C2×C4×D3—93no—
71C22×C32—36no—
72C2×C3⋊D3—36no—
73C2×C3⋊D3—66no—
74C2×C3⋊D3—612no—
75C2×C3⋊D3—122no—
76C2×C3⋊D3C22×D313yes—
77C2×C3×D3—26no—
78C2×C3×D3—36no—
79C2×C3×D3—63no—
80C2×C3×D3—624no—
81C2×C3×D3C22×D3112yescomplemented
82D32—212no—
83D32—312no—
84D32—624no—
85D32—1212no—
86C32×C4—23no—
87C32×C4—41no—
88C32×C4C22×D313yescomplemented
89C3⋊Dic3C22×D313yescomplemented
90C3×Dic3—36no—
91C3×Dic3—63no—
92C22×D4—271noSylow
93C33C22×D411yescharacteristic, complemented, Sylow
94C23×C3—96no—
95C22×D3—312no—
96C22×D3—93no—
97C22×D3—912no—
98C22×D3—96no—
99C22×D3—186no—
100C22×D3—1824no—
101C3×D4—36no—
102C3×D4—63no—
103C3×D4—96no—
104C2×C3×C4—93no—
105#67—312no—
106#67—912no—
107#67—186no—
108C2×Dic3—93no—
109D12—96no—
110D12—93no—
111D12—183no—
112D12—361no—
113D12D3213yescomplemented
114C4×D3—36no—
115C4×D3—63no—
116C4×D3—96no—
117C2×C32—23no—
118C2×C32—41no—
119C2×C32—66no—
120C2×C32C23×D313yes—
121C3⋊D3—66no—
122C3⋊D3—126no—
123C3⋊D3—242no—
124C3⋊D3D3×D416yescomplemented
125C3×D3—212no—
126C3×D3—312no—
127C3×D3—46no—
128C3×D3—66no—
129C24—272no—
130C2×D4—93no—
131C2×D4—273no—
132C2×D4—276no—
133C22×C4—271no—
134C22×C3—312no—
135C22×C3—66no—
136C22×C3—93no—
137C22×C3—1812no—
138C2×D3—36no—
139C2×D3—63no—
140C2×D3—624no—
141C2×D3—912no—
142C2×D3—96no—
143C2×D3—186no—
144C2×D3—1812no—
145C2×D3—1824no—
146C2×D3—362no—
147C2×D3—3612no—
148C2×D3C2×D3216yescomplemented
149C3×C4—23no—
150C3×C4—41no—
151C3×C4—93no—
152C3×C4C2×D3213yescomplemented
153Dic3—36no—
154Dic3—63no—
155C32—23no—
156C32—41no—
157C32C2×D3×D413yescomplemented
158C23—96no—
159C23—276no—
160C23—271no—
161C23—548no—
162D4—33no—
163D4—96no—
164D4—276no—
165D4—271no—
166C2×C4—93no—
167C2×C4—273no—
168C2×C3—23no—
169C2×C3—41no—
170C2×C3—612no—
171C2×C3—96no—
172C2×C3—126no—
173C2×C3C22×D3213yes—
174D3—26no—
175D3—312no—
176D3—66no—
177D3—186no—
178D3—366no—
179D3—722no—
180C22—36no—
181C22—93no—
182C22—1812no—
183C22—274no—
184C22—272no—
185C22—5412no—
186C4—93no—
187C4D3311yescharacteristic, complemented
188C3—23no—
189C3—41no—
190C3D32×D413yescomplemented
191C2—66no—
192C2—96no—
193C2—542no—
194C2C2×D3311yescharacteristic
subgroupG/H
center, FrattiniC2C2×D33
commutator, socle, cosocleC2×C33C24
FittingC33×C4C23
hypercenterC4D33
hypocenterC33C22×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D33C2semidirect
2. C4×#1113C2semidirect
3. C3⋊D3×D12C2semidirect
4. C3×#3802C2semidirect
5. #88952C2semidirect
8. C2×D3×C3⋊D3C22semidirect
9. C2×C3×D32C22semidirect
11. C4⋊(C3⋊(C3⋊D3))C22semidirect
12. C3×C4×C3⋊D3C22semidirect
13. C32×D12C22semidirect
14. #6294C22semidirect
15. #6291C22semidirect
16. C3×#625C22semidirect
22. #3802D3semidirect
26. C2×C3⋊(C3⋊D3)C23semidirect
28. C2×C32×D3C23semidirect
29. #1113D4semidirect
32. C33×C4C23semidirect
33. C3×C3⋊Dic3C23semidirect
41. C2×D32C2×D3semidirect
47. C4×C3⋊D3C2×D3semidirect
50. C3×D12C2×D3semidirect
55. #625C2×D3semidirect
60. C3×C3⋊D3C2×D4semidirect
81. C2×C3×D3C22×D3semidirect
88. C32×C4C22×D3semidirect
89. C3⋊Dic3C22×D3semidirect
93. C33C22×D4semidirect
113. D12D32semidirect
124. C3⋊D3D3×D4semidirect
148. C2×D3C2×D32semidirect
152. C3×C4C2×D32semidirect
157. C32C2×D3×D4semidirect
187. C4D33semidirect
190. C3D32×D4semidirect

Maximal subgroups

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

subgroupindex
C2×D332
C4×#11132
C3⋊D3×D122
C3×#38022
#889522
D32×D43