Finite Group Databasebeta

← C22×#88993 #13181527

Subgroups — C22×#88993 #13181527

Order 1728.

1336 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 261 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
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88993C2112yesmaximal, direct product factor
3C22×C3⋊#622C212yesmaximal, complemented
4C22×#3792—31nomaximal
5C24×C33C411yescharacteristic
6C22×C3×C3⋊Dic3C416yes—
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2216yes—
10#88993C22116yesdirect product factor
11C2×C3⋊#622C22112yescomplemented
12C23×C3⋊Dic3D311yescharacteristic
13C23×C3×Dic3—62no—
14C2×#3792—312no—
15C22×#622—32no—
16C23×C33—24no—
17C23×C33C2×C416yes—
18C23×C33C2×C411yescharacteristic
19C2×C3×C3⋊Dic3C2×C4112yes—
20C2×C3×C3⋊Dic3C2×C413yes—
21C2×C3×C3⋊Dic3C2×C416yes—
22C2×C3×C3⋊Dic3C2313yes—
23C2×C3×C3⋊Dic3C2314yes—
24C3⋊#622C2318yescomplemented
25C22×#208—91nomaximal
26C24×C32—22no—
27C24×C32—42no—
28C24×C32Dic311yescharacteristic
29C22×C3⋊Dic3Dic316yes—
30C22×C3⋊Dic3Dic311yescharacteristic
31C22×C3⋊Dic3C2×D311yescharacteristic
32C22×C3⋊Dic3C2×D316yes—
33C22×C3×Dic3—612no—
34C22×C3×Dic3—62no—
35C22×C3×Dic3—62no—
36C22×C3×Dic3—612no—
37#3792—316no—
38C2×#622—312no—
39C22×C33—212no—
40C22×C33M1614yescomplemented
41C22×C33C22×C414yes—
42C22×C33C22×C413yes—
43C3×C3⋊Dic3C22×C413yes—
44C3×C3⋊Dic3C22×C414yes—
45C3×C3⋊Dic3C2411yescharacteristic
46C23×C3×C4—91no—
47C23×Dic3—62no—
48C2×#208—912no—
49C22×C3⋊C8—92no—
50C23×C32—24no—
51C23×C32—216no—
52C23×C32—212no—
53C23×C32—22no—
54C23×C32—416no—
55C23×C32—412no—
56C23×C32—42no—
57C23×C32C2×Dic316yes—
58C23×C32C2×Dic311yescharacteristic
59C2×C3⋊Dic3C2×Dic3112yes—
60C2×C3⋊Dic3C2×Dic313yes—
61C2×C3⋊Dic3C2×Dic316yes—
62C2×C3⋊Dic3C22×D313yes—
63C2×C3⋊Dic3C22×D314yes—
64C2×C3×Dic3—624no—
65C2×C3×Dic3—66no—
66C2×C3×Dic3—612no—
67C2×C3×Dic3—66no—
68C2×C3×Dic3—68no—
69#622—38no—
70C22×M16—271noSylow
71C2×C33—24no—
72C2×C33C2×M1616yescomplemented
73C2×C33C23×C411yescharacteristic
74C24×C3—22no—
75C24×C3—42no—
76C24×C3#15711yescharacteristic
77C22×C3×C4—91no—
78C22×C3×C4—96no—
79C22×C3×C4—96no—
80C22×C3×C4—91no—
81C22×Dic3—612no—
82C22×Dic3—62no—
83C22×Dic3—612no—
84C22×Dic3—62no—
85#208—916no—
86C2×C3⋊C8—912no—
87C22×C32—212no—
88C22×C32—248no—
89C22×C32—28no—
90C22×C32—26no—
91C22×C32—28no—
92C22×C32—448no—
93C22×C32—48no—
94C22×C32—48no—
95C22×C32—46no—
96C22×C32#20814yescomplemented
97C22×C32C22×Dic314yes—
98C22×C32C22×Dic313yes—
99C3⋊Dic3C22×Dic313yes—
100C3⋊Dic3C22×Dic314yes—
101C3⋊Dic3C23×D311yescharacteristic
102C3×Dic3—66no—
103C3×Dic3—68no—
104C3×Dic3—62no—
105C23×C4—91no—
106C2×M16—2712no—
107C22×C8—272no—
108C33C22×M1611yescharacteristic, complemented, Sylow
109C23×C3—216no—
110C23×C3—24no—
111C23×C3—212no—
112C23×C3—22no—
113C23×C3—416no—
114C23×C3—412no—
115C23×C3—42no—
116C23×C3C2×#15716yes—
117C23×C3C2×#15711yescharacteristic
118C2×C3×C4—93no—
119C2×C3×C4—94no—
120C2×C3×C4—912no—
121C2×C3×C4—93no—
122C2×C3×C4—96no—
123C2×Dic3—624no—
124C2×Dic3—612no—
125C2×Dic3—66no—
126C2×Dic3—66no—
127C2×Dic3—68no—
128C3⋊C8—98no—
129C2×C32—24no—
130C2×C32—216no—
131C2×C32—212no—
132C2×C32—22no—
133C2×C32—416no—
134C2×C32—412no—
135C2×C32—42no—
136C2×C32C2×#20816yescomplemented
137C2×C32C23×Dic311yescharacteristic
138C24C3⋊#15711yescharacteristic
139C22×C4—96no—
140C22×C4—96no—
141C22×C4—91no—
142C22×C4—91no—
143M16—2716no—
144C2×C8—2712no—
145C22×C3—248no—
146C22×C3—212no—
147C22×C3—28no—
148C22×C3—26no—
149C22×C3—28no—
150C22×C3—448no—
151C22×C3—48no—
152C22×C3—46no—
153C22×C3—48no—
154C22×C3#379214yescomplemented
155C22×C3C22×#15714yes—
156C22×C3C22×#15713yes—
157C3×C4—91no—
158C3×C4—93no—
159C3×C4—94no—
160Dic3—66no—
161Dic3—68no—
162Dic3—62no—
163C32—22no—
164C32—42no—
165C32C22×#20811yescharacteristic, complemented
166C23—24no—
167C23C2×C3⋊#15716yes—
168C23C2×C3⋊#15711yescharacteristic
169C2×C4—93no—
170C2×C4—94no—
171C2×C4—912no—
172C2×C4—96no—
173C2×C4—93no—
174C8—278no—
175C2×C3—216no—
176C2×C3—24no—
177C2×C3—22no—
178C2×C3—212no—
179C2×C3—416no—
180C2×C3—42no—
181C2×C3—412no—
182C2×C3C2×#379216yescomplemented
183C2×C3C23×#15711yescharacteristic
184C22—212no—
185C22#8899314yesdirect product factor
186C22C22×C3⋊#15714yes—
187C22C22×C3⋊#15713yes—
188C4—91no—
189C4—93no—
190C4—94no—
191C3—22no—
192C3—42no—
193C3C22×#379211yescharacteristic, complemented
194C2—24no—
195C2C2×#8899316yesdirect product factor
196C2C23×C3⋊#15711yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C23×C4
FrattiniC2C23×C3⋊#157
Fitting, prime radicalC24×C33C4
socleC23×C33C2×C4
cosocleC3×C3⋊Dic3C24
hypercenterC24C3⋊#157
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2direct (N × H)
3. C22×C3⋊#622C2semidirect
10. #88993C22direct (N × H)
11. C2×C3⋊#622C22semidirect
24. C3⋊#622C23semidirect
40. C22×C33M16semidirect
72. C2×C33C2×M16semidirect
96. C22×C32#208semidirect
108. C33C22×M16semidirect
136. C2×C32C2×#208semidirect
154. C22×C3#3792semidirect
165. C32C22×#208semidirect
182. C2×C3C2×#3792semidirect
185. C22#88993direct (N × H)
193. C3C22×#3792semidirect
195. C2C2×#88993direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×#889932
C22×C3⋊#6222
C22×#37923
C22×#2089