Finite Group Databasebeta

← C22×#88988 #13181508

Subgroups — C22×#88988 #13181508

Order 1728.

1256 conjugacy classes of subgroups; only the 341 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 341 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×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C3⋊#622C212yesmaximal, complemented
3C2×#88988C2112yesmaximal, direct product factor
4C22×#3787—31nomaximal
5C23×C3×C3⋊D3C411yescharacteristic
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×C4×C3⋊D3C416yes—
9C2×C3×C4×C3⋊D3C2216yes—
10C2×C3⋊#622C22112yescomplemented
11#88988C22116yesdirect product factor
12C22×C4×C3⋊D3D311yescharacteristic
13C22×C3×C4×D3—62no—
14C22×#622—32no—
15C2×#3787—312no—
16C23×C33C2×C411yescharacteristic
17C22×C3×C3⋊D3C818yescomplemented
18C22×C3×C3⋊D3C2×C416yes—
19C2×C33×C4C2×C416yes—
20C2×C3×C3⋊Dic3C2×C413yes—
21C2×C3×C3⋊Dic3C2313yes—
22C3×C4×C3⋊D3C2×C4112yes—
23C3×C4×C3⋊D3C2314yes—
24C3⋊#622C2318yescomplemented
25C23×C3⋊C8—91nomaximal
26C23×C3⋊D3Dic311yescharacteristic
27C23×C3×D3—62no—
28C22×C32×C4—22no—
29C22×C32×C4—42no—
30C22×C32×C4Dic311yescharacteristic
31C22×C3⋊Dic3C2×D311yescharacteristic
32C2×C4×C3⋊D3Dic316yes—
33C2×C4×C3⋊D3C2×D316yes—
34C22×C3×Dic3—62no—
35C2×C3×C4×D3—612no—
36C2×C3×C4×D3—612no—
37C2×#622—312no—
38#3787—316no—
39C22×C33C2×C814yescomplemented
40C22×C33C22×C413yes—
41C2×C3×C3⋊D3C2×C8124yescomplemented
42C2×C3×C3⋊D3C22×C414yes—
43C33×C4C22×C414yes—
44C3×C3⋊Dic3C22×C413yes—
45C3×C3⋊Dic3C2411yescharacteristic
46C23×C3×C4—91no—
47C22×C4×D3—62no—
48C22×C3⋊C8—912no—
49C22×C3⋊C8—92no—
50C23×C32—22no—
51C23×C32—42no—
52C23×C32C2×Dic311yescharacteristic
53C22×C3⋊D3C3⋊C818yescomplemented
54C22×C3⋊D3C2×Dic316yes—
55C22×C3×D3—616no—
56C22×C3×D3—612no—
57C2×C32×C4—212no—
58C2×C32×C4—412no—
59C2×C32×C4C2×Dic316yes—
60C2×C3⋊Dic3C2×Dic313yes—
61C2×C3⋊Dic3C22×D313yes—
62C4×C3⋊D3C2×Dic3112yes—
63C4×C3⋊D3C22×D314yes—
64C2×C3×Dic3—66no—
65C2×C3×Dic3—66no—
66C3×C4×D3—624no—
67C3×C4×D3—68no—
68#622—38no—
69C23×C8—271noSylow
70C2×C33C22×C816yescomplemented
71C2×C33C23×C411yescharacteristic
72C3×C3⋊D3C22×C818yescomplemented
73C24×C3—91no—
74C23×D3—62no—
75C22×C3×C4—22no—
76C22×C3×C4—42no—
77C22×C3×C4—91no—
78C22×C3×C4—96no—
79C22×C3×C4—96no—
80C22×C3×C4#15711yescharacteristic
81C22×Dic3—62no—
82C2×C4×D3—612no—
83C2×C4×D3—612no—
84C2×C3⋊C8—916no—
85C2×C3⋊C8—912no—
86C22×C32—26no—
87C22×C32—28no—
88C22×C32—48no—
89C22×C32—46no—
90C22×C32C2×C3⋊C814yescomplemented
91C22×C32C22×Dic313yes—
92C2×C3⋊D3C2×C3⋊C8124yescomplemented
93C2×C3⋊D3C22×Dic314yes—
94C2×C3×D3—648no—
95C2×C3×D3—68no—
96C32×C4—28no—
97C32×C4—48no—
98C32×C4C22×Dic314yes—
99C3⋊Dic3C22×Dic313yes—
100C3⋊Dic3C23×D311yescharacteristic
101C3×Dic3—66no—
102C3×Dic3—62no—
103C23×C4—91no—
104C22×C8—2712no—
105C22×C8—272no—
106C33C23×C811yescharacteristic, complemented, Sylow
107C23×C3—22no—
108C23×C3—42no—
109C23×C3—96no—
110C23×C3—98no—
111C23×C3C2×#15711yescharacteristic
112C22×D3—616no—
113C22×D3—612no—
114C2×C3×C4—212no—
115C2×C3×C4—412no—
116C2×C3×C4—93no—
117C2×C3×C4—93no—
118C2×C3×C4—94no—
119C2×C3×C4—912no—
120C2×C3×C4C2×#15716yes—
121C2×Dic3—66no—
122C2×Dic3—66no—
123C4×D3—624no—
124C4×D3—68no—
125C3⋊C8—98no—
126C2×C32—212no—
127C2×C32—22no—
128C2×C32—412no—
129C2×C32—42no—
130C2×C32C22×C3⋊C816yescomplemented
131C2×C32C23×Dic311yescharacteristic
132C3⋊D3C22×C3⋊C818yescomplemented
133C3×D3—616no—
134C24—91no—
135C22×C4—91no—
136C22×C4—96no—
137C22×C4—96no—
138C22×C4C3⋊#15711yescharacteristic
139C2×C8—2716no—
140C2×C8—2712no—
141C22×C3—26no—
142C22×C3—28no—
143C22×C3—46no—
144C22×C3—48no—
145C22×C3—924no—
146C22×C3—94no—
147C22×C3#378714yescomplemented
148C22×C3C22×#15713yes—
149C2×D3—68no—
150C2×D3—648no—
151C3×C4—28no—
152C3×C4—48no—
153C3×C4—91no—
154C3×C4—93no—
155C3×C4C22×#15714yes—
156Dic3—66no—
157Dic3—62no—
158C32—22no—
159C32—42no—
160C32C23×C3⋊C811yescharacteristic, complemented
161C23—96no—
162C23—98no—
163C23C2×C3⋊#15711yescharacteristic
164C2×C4—93no—
165C2×C4—93no—
166C2×C4—94no—
167C2×C4—912no—
168C2×C4C2×C3⋊#15716yes—
169C8—278no—
170C2×C3—22no—
171C2×C3—212no—
172C2×C3—42no—
173C2×C3—412no—
174C2×C3—98no—
175C2×C3C2×#378716yescomplemented
176C2×C3C23×#15711yescharacteristic
177D3—616no—
178C22—94no—
179C22—924no—
180C22#8898814yesdirect product factor
181C22C22×C3⋊#15713yes—
182C4—91no—
183C4—93no—
184C4C22×C3⋊#15714yes—
185C3—22no—
186C3—42no—
187C3C22×#378711yescharacteristic, complemented
188C2—98no—
189C2C2×#8898816yesdirect product factor
190C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊#157
commutator, hypocenterC33C23×C8
FrattiniC2C23×C3⋊#157
FittingC22×C33×C4C4
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC3×C3⋊Dic3C24
LCSG ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. C2×#88988C2direct (N × H)
10. C2×C3⋊#622C22semidirect
11. #88988C22direct (N × H)
17. C22×C3×C3⋊D3C8semidirect
24. C3⋊#622C23semidirect
39. C22×C33C2×C8semidirect
41. C2×C3×C3⋊D3C2×C8semidirect
53. C22×C3⋊D3C3⋊C8semidirect
70. C2×C33C22×C8semidirect
72. C3×C3⋊D3C22×C8semidirect
90. C22×C32C2×C3⋊C8semidirect
92. C2×C3⋊D3C2×C3⋊C8semidirect
106. C33C23×C8semidirect
130. C2×C32C22×C3⋊C8semidirect
132. C3⋊D3C22×C3⋊C8semidirect
147. C22×C3#3787semidirect
160. C32C23×C3⋊C8semidirect
175. C2×C3C2×#3787semidirect
180. C22#88988direct (N × H)
187. C3C22×#3787semidirect
189. C2C2×#88988direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×C3⋊#6222
C2×#889882
C22×#37873
C23×C3⋊C89