Finite Group Databasebeta

← #13181515

Subgroups — #13181515

Order 1728.

952 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 221 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×C3×#3830C211yesmaximal, characteristic
2C2×#88993C214yesmaximal, complemented
3#11724515C218yesmaximal, complemented
4#11724511C212yesmaximal, complemented
5#10596640—31nomaximal
6C2×C33×D4C411yescharacteristic
7C2×C3×#638C412yes—
8C22×C3×C3⋊Dic3C2212yes—
9C3×#3830C414yes—
10C3×#3830C2214yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C411yescharacteristic
13#88993C2214yes—
14#88993C2218yescomplemented
15C2×C3⋊#622C2218yescomplemented
16#88989C2214yescomplemented
17#88988C2214yescomplemented
18C2×#3830D311yescharacteristic
19C2×C3×#237—62no—
20C2×#3792—34no—
21#64287—38no—
22#64283—32no—
23C23×C33C2×C412yes—
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C33×D4C2×C414yes—
26C2×C33×C4C2×C411yescharacteristic
27C3×#638C2×C418yes—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C414yes—
30C2×C3×C3⋊Dic3C2314yes—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3C2×C412yes—
33C3×C4×C3⋊D3C2312yes—
34C3×#634C2×C414yes—
35C3⋊#622C2318yes—
36#5819—91nomaximal
37C2×C32×D4—22no—
38C2×C32×D4—42no—
39C2×C32×D4Dic311yescharacteristic
40C2×#638Dic312yes—
41C22×C3⋊Dic3C2×D312yes—
42#3830Dic314yes—
43#3830C2×D314yes—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×#634Dic311yescharacteristic
46C2×C3×#67—64no—
47C22×C3×Dic3—64no—
48C3×#237—68no—
49C3×#237—68no—
50C2×C3×C4×D3—62no—
51C2×C3×Dic6—62no—
52#3792—34no—
53#3792—38no—
54C2×#622—38no—
55#3788—34no—
56#3787—34no—
57C22×C33—42no—
58C22×C33C22×C411yescharacteristic
59C22×C33C22×C414yes—
60C2×C3×C3⋊D3—41no—
61C2×C3×C3⋊D3C22×C412yes—
62C33×C4C22×C412yes—
63C3×C3⋊Dic3C22×C414yes—
64C3×C3⋊Dic3C22×C411yescharacteristic
65C3×C3⋊Dic3C2411yescharacteristic
66C2×C3×#41—91no—
67C2×#237—62no—
68#948—98no—
69C2×#208—92no—
70C2×#208—94no—
71C23×C32—24no—
72C23×C32—44no—
73C23×C32C2×Dic312yes—
74C22×C3⋊D3C2×Dic311yescharacteristic
75C22×C3×D3—62no—
76C32×D4—28no—
77C32×D4—48no—
78C32×D4C2×Dic314yes—
79C2×C32×C4—22no—
80C2×C32×C4—42no—
81C2×C32×C4C2×Dic311yescharacteristic
82#638C2×Dic318yes—
83C2×C3⋊Dic3C2×Dic312yes—
84C2×C3⋊Dic3C2×Dic314yes—
85C2×C3⋊Dic3C22×D314yes—
86C2×C3⋊Dic3C22×D311yescharacteristic
87C4×C3⋊D3C2×Dic312yes—
88C4×C3⋊D3C22×D312yes—
89#634C2×Dic314yes—
90C3×#67—616no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—68no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—68no—
95C3×C4×D3—64no—
96C3×C4×D3—64no—
97C3×Dic6—68no—
98#622—38no—
99#568—271noSylow
100C2×C33—21no—
101C2×C33—24no—
102C2×C33C23×C411yescharacteristic
103C3×C3⋊D3—22no—
104C3×#41—94no—
105C3×#41—94no—
106C2×C3×Q8—91no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C2×C3×D4—92no—
110C2×C3×D4#15711yescharacteristic
111C22×C3×C4—92no—
112C22×C3×C4—91no—
113C2×#67—64no—
114C22×Dic3—64no—
115#237—68no—
116#237—68no—
117C2×C4×D3—62no—
118C2×Dic6—62no—
119#208—94no—
120#208—94no—
121#208—98no—
122C2×C3⋊C8—94no—
123C2×C3⋊C8—98no—
124C22×C32—22no—
125C22×C32—28no—
126C22×C32—42no—
127C22×C32—48no—
128C22×C32—42no—
129C22×C32—48no—
130C22×C32—88no—
131C22×C32C22×Dic311yescharacteristic
132C22×C32C22×Dic314yes—
133C2×C3⋊D3—41no—
134C2×C3⋊D3C22×Dic312yes—
135C2×C3×D3—64no—
136C2×C3×D3—124no—
137C32×C4—24no—
138C32×C4—44no—
139C32×C4C22×Dic312yes—
140C3⋊Dic3C22×Dic314yes—
141C3⋊Dic3C22×Dic311yescharacteristic
142C3⋊Dic3C23×D311yescharacteristic
143C3×Dic3—68no—
144C3×Dic3—62no—
145C3×Dic3—62no—
146C2×#41—91no—
147#131—278no—
148C2×M16—272no—
149C2×M16—274no—
150C33#56811yescharacteristic, complemented, Sylow
151C23×C3—24no—
152C23×C3—44no—
153C23×C3—91no—
154C23×C3C2×#15712yes—
155C22×D3—62no—
156C3×Q8—94no—
157C3×D4—28no—
158C3×D4—48no—
159C3×D4—98no—
160C3×D4C2×#15714yes—
161C2×C3×C4—22no—
162C2×C3×C4—42no—
163C2×C3×C4—94no—
164C2×C3×C4—91no—
165C2×C3×C4—92no—
166C2×C3×C4—94no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4C2×#15711yescharacteristic
170#67—616no—
171C2×Dic3—64no—
172C2×Dic3—68no—
173C2×Dic3—62no—
174C2×Dic3—68no—
175C4×D3—64no—
176C4×D3—64no—
177Dic6—68no—
178C3⋊C8—98no—
179C2×C32—21no—
180C2×C32—24no—
181C2×C32—22no—
182C2×C32—24no—
183C2×C32—44no—
184C2×C32—42no—
185C2×C32—48no—
186C2×C32—88no—
187C2×C32C23×Dic311yescharacteristic
188C3⋊D3—22no—
189C3×D3—124no—
190#41—94no—
191#41—94no—
192C2×Q8—91no—
193C2×D4—92no—
194C2×D4C3⋊#15711yescharacteristic
195C22×C4—92no—
196C22×C4—91no—
197M16—274no—
198M16—274no—
199M16—278no—
200C2×C8—274no—
201C2×C8—278no—
202C22×C3—22no—
203C22×C3—28no—
204C22×C3—42no—
205C22×C3—48no—
206C22×C3—42no—
207C22×C3—48no—
208C22×C3—88no—
209C22×C3—92no—
210C22×C3—361no—
211C22×C3C22×#15711yescharacteristic
212C22×C3C22×#15714yes—
213C2×D3—64no—
214C2×D3—124no—
215C3×C4—24no—
216C3×C4—44no—
217C3×C4—91no—
218C3×C4—91no—
219C3×C4—94no—
220C3×C4C22×#15712yes—
221Dic3—68no—
222Dic3—62no—
223Dic3—62no—
224C32—22no—
225C32—42no—
226C32#581911yescharacteristic, complemented
227C23—91no—
228C23C2×C3⋊#15712yes—
229Q8—94no—
230D4—98no—
231D4C2×C3⋊#15714yes—
232C2×C4—94no—
233C2×C4—91no—
234C2×C4—92no—
235C2×C4—94no—
236C2×C4—92no—
237C2×C4—92no—
238C2×C4C2×C3⋊#15711yescharacteristic
239C8—278no—
240C2×C3—24no—
241C2×C3—21no—
242C2×C3—24no—
243C2×C3—22no—
244C2×C3—48no—
245C2×C3—44no—
246C2×C3—42no—
247C2×C3—88no—
248C2×C3—182no—
249C2×C3C23×#15711yescharacteristic
250D3—124no—
251C22—42no—
252C22—92no—
253C22—361no—
254C22C22×C3⋊#15711yescharacteristic
255C22C22×C3⋊#15714yes—
256C4—91no—
257C4—91no—
258C4—94no—
259C4C22×C3⋊#15712yes—
260C3—22no—
261C3—42no—
262C3#1059664011yescharacteristic, complemented
263C2—21no—
264C2—24no—
265C2—182no—
266C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊#157
commutator, socleC2×C33C23×C4
FittingC2×C33×D4C4
prime radicalC2×C3×#3830C2
cosocleC3×C3⋊Dic3C24
hypercenterC2×D4C3⋊#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11724515C2semidirect
4. #11724511C2semidirect
14. #88993C22semidirect
15. C2×C3⋊#622C22semidirect
16. #88989C22semidirect
17. #88988C22semidirect
150. C33#568semidirect
226. C32#5819semidirect
262. C3#10596640semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#889932
#117245152
#117245112
#105966403
#58199