Finite Group Databasebeta

← #13173399

Subgroups — #13173399

Order 1728.

576 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 105 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×#88993C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3#11723104C212yesmaximal, complemented
4C4⋊(C3⋊#622)C212yesmaximal, complemented
5#10593805—31nomaximal
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9#88993—24no—
10C2×C3⋊#622C2214yes—
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×#3792—32no—
16C2×C4×C3⋊Dic3D311yescharacteristic
17C2×C3×C4×Dic3—62no—
18#63779—32no—
19C4⋊#622—32no—
20C23×C33C2×C411yescharacteristic
21C2×C33×C4—22no—
22C2×C33×C4C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3D412yes—
29C2×C3×C3⋊Dic3Q812yes—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3⋊#622—24no—
32#4924—91nomaximal
33C22×C32×C4—22no—
34C22×C32×C4—42no—
35C22×C32×C4Dic311yescharacteristic
36C22×C3⋊Dic3Dic311yescharacteristic
37C22×C3⋊Dic3C2×D311yescharacteristic
38C22×C3×Dic3—62no—
39C22×C3×Dic3—62no—
40#3792—64no—
41C2×#622—34no—
42C4×C3⋊Dic3Dic311yescharacteristic
43C4×C3⋊Dic3Dic311yescharacteristic
44C4×C3⋊Dic3C2×D311yescharacteristic
45C4×C3⋊Dic3C2×D311yescharacteristic
46C3×C4×Dic3—62no—
47C3×C4×Dic3—62no—
48C3×C4×Dic3—62no—
49C3×C4×Dic3—62no—
50C22×C33—21no—
51C22×C33—21no—
52C22×C33C4⋊C412yes—
53C22×C33C22×C411yescharacteristic
54C33×C4—21no—
55C33×C4M1612yescomplemented
56C3×C3⋊Dic3—21no—
57C3×C3⋊Dic3C4⋊C412yes—
58C3×C3⋊Dic3M1612yescomplemented
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3C2×Q811yescharacteristic
61C2×C3×C42—91no—
62C2×C4×Dic3—62no—
63C2×#208—92no—
64C4⋊(C3⋊C8)—92no—
65C4⋊(C3⋊C8)—92no—
66C23×C32—22no—
67C23×C32—42no—
68C23×C32C2×Dic311yescharacteristic
69C2×C32×C4—22no—
70C2×C32×C4—28no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—48no—
74C2×C32×C4—42no—
75C2×C32×C4—42no—
76C2×C32×C4C2×Dic311yescharacteristic
77C2×C32×C4C2×Dic311yescharacteristic
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3Dic612yes—
80C2×C3⋊Dic3D1212yes—
81C2×C3⋊Dic3C2×Dic311yescharacteristic
82C2×C3⋊Dic3C2×Dic311yescharacteristic
83C2×C3⋊Dic3C2×Dic311yescharacteristic
84C2×C3⋊Dic3C22×D311yescharacteristic
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—68no—
92#622—64no—
93#423—271noSylow
94C2×C33—22no—
95C2×C33C2×C4⋊C411yescharacteristic
96C2×C33C2×M1611yescharacteristic
97C2×C33C2×M1611yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—42no—
100C22×C3×C4—91no—
101C22×C3×C4—91no—
102C22×C3×C4#15711yescharacteristic
103C22×Dic3—62no—
104C22×Dic3—62no—
105C3×C42—91no—
106C3×C42—91no—
107C3×C42—91no—
108C3×C42—91no—
109C4×Dic3—62no—
110C4×Dic3—62no—
111C4×Dic3—62no—
112C4×Dic3—62no—
113#208—184no—
114C2×C3⋊C8—94no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—24no—
118C22×C32—24no—
119C22×C32—22no—
120C22×C32—24no—
121C22×C32—44no—
122C22×C32—44no—
123C22×C32—44no—
124C22×C32—42no—
125C22×C32C4⋊Dic312yes—
126C22×C32C22×Dic311yescharacteristic
127C32×C4—21no—
128C32×C4—24no—
129C32×C4—24no—
130C32×C4—44no—
131C32×C4—44no—
132C32×C4#20812yescomplemented
133C3⋊Dic3—21no—
134C3⋊Dic3#20812yescomplemented
135C3⋊Dic3C4⋊Dic312yes—
136C3⋊Dic3C2×Dic611yescharacteristic
137C3⋊Dic3C2×D1211yescharacteristic
138C3×Dic3—64no—
139C3×Dic3—64no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—64no—
143C2×M16—272no—
144C2×C42—91no—
145C4⋊C8—272no—
146C4⋊C8—272no—
147C33#42311yescharacteristic, complemented, Sylow
148C23×C3—22no—
149C23×C3—42no—
150C23×C3C2×#15711yescharacteristic
151C2×C3×C4—28no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—48no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—92no—
162C2×C3×C4—92no—
163C2×C3×C4—91no—
164C2×C3×C4—182no—
165C2×C3×C4C2×#15711yescharacteristic
166C2×C3×C4C2×#15711yescharacteristic
167C2×Dic3—64no—
168C2×Dic3—64no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—68no—
174C3⋊C8—184no—
175C2×C32—22no—
176C2×C32—28no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—48no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32—42no—
184C2×C32C2×#20811yescharacteristic
185C2×C32C2×#20811yescharacteristic
186C2×C32C2×C4⋊Dic311yescharacteristic
187C22×C4—91no—
188C22×C4—91no—
189C22×C4C3⋊#15711yescharacteristic
190M16—544no—
191C2×C8—274no—
192C42—91no—
193C42—91no—
194C42—91no—
195C42—91no—
196C22×C3—24no—
197C22×C3—24no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—22no—
201C22×C3—24no—
202C22×C3—44no—
203C22×C3—44no—
204C22×C3—42no—
205C22×C3—44no—
206C22×C3C4⋊#15712yes—
207C22×C3C22×#15711yescharacteristic
208C3×C4—24no—
209C3×C4—21no—
210C3×C4—24no—
211C3×C4—44no—
212C3×C4—44no—
213C3×C4—92no—
214C3×C4—91no—
215C3×C4—91no—
216C3×C4—92no—
217C3×C4—181no—
218C3×C4#379212yescomplemented
219Dic3—64no—
220Dic3—64no—
221Dic3—62no—
222Dic3—62no—
223Dic3—64no—
224C32—22no—
225C32—42no—
226C32#492411yescharacteristic, complemented
227C23C2×C3⋊#15711yescharacteristic
228C2×C4—22no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—91no—
235C2×C4—182no—
236C2×C4C2×C3⋊#15711yescharacteristic
237C2×C4C2×C3⋊#15711yescharacteristic
238C8—544no—
239C2×C3—28no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—22no—
244C2×C3—48no—
245C2×C3—42no—
246C2×C3—42no—
247C2×C3—42no—
248C2×C3#6428311yescharacteristic
249C2×C3C2×C4⋊#15711yescharacteristic
250C2×C3C2×#379211yescharacteristic
251C22—21no—
252C22—21no—
253C22C4⋊(C3⋊#157)12yes—
254C22C22×C3⋊#15711yescharacteristic
255C4—21no—
256C4—92no—
257C4—91no—
258C4—91no—
259C4—92no—
260C4—181no—
261C4#8899312yescomplemented
262C3—22no—
263C3—42no—
264C3#1059380511yescharacteristic, complemented
265C2—22no—
266C2#1172451111yescharacteristic
267C2C2×C4⋊(C3⋊#157)11yescharacteristic
268C2C2×#8899311yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#423
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723104C2semidirect
4. C4⋊(C3⋊#622)C2semidirect
55. C33×C4M16semidirect
58. C3×C3⋊Dic3M16semidirect
132. C32×C4#208semidirect
134. C3⋊Dic3#208semidirect
147. C33#423semidirect
218. C3×C4#3792semidirect
226. C32#4924semidirect
261. C4#88993semidirect
264. C3#10593805semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C2×C3×C4×C3⋊Dic32
#117231042
C4⋊(C3⋊#622)2
#105938053
#49249