Finite Group Databasebeta

← #13173442

Subgroups — #13173442

Order 1728.

1120 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88993C212yesmaximal, complemented
3#11723112C214yesmaximal, complemented
4#10593845—31nomaximal
5C24×C33C411yescharacteristic
6C22×C3×C3⋊Dic3—24no—
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C412yes—
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11#88993—24no—
12C2×C3⋊#622C2214yescomplemented
13C23×C3⋊Dic3D311yescharacteristic
14C23×C3×Dic3—62no—
15C2×#3792—32no—
16C22⋊#622—34no—
17C23×C33—22no—
18C23×C33—24no—
19C23×C33C2×C411yescharacteristic
20C23×C33C2×C412yes—
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3—24no—
23C2×C3×C3⋊Dic3—24no—
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3D414yes—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3⋊#622—24no—
29#5207—91nomaximal
30C24×C32—22no—
31C24×C32—42no—
32C24×C32Dic311yescharacteristic
33C22×C3⋊Dic3—24no—
34C22×C3⋊Dic3Dic311yescharacteristic
35C22×C3⋊Dic3Dic312yes—
36C22×C3⋊Dic3C2×D312yes—
37C22×C3⋊Dic3C2×D311yescharacteristic
38C22×C3×Dic3—64no—
39C22×C3×Dic3—62no—
40C22×C3×Dic3—62no—
41C22×C3×Dic3—64no—
42C22×C3×Dic3—616no—
43#3792—64no—
44C2×#622—34no—
45C22×C33—28no—
46C22×C33—22no—
47C22×C33—22no—
48C22×C33—22no—
49C22×C33C22⋊C412yes—
50C22×C33M1614yescomplemented
51C22×C33C22×C411yescharacteristic
52C3×C3⋊Dic3—22no—
53C3×C3⋊Dic3C22⋊C412yes—
54C3×C3⋊Dic3C2×D412yes—
55C23×C3×C4—91no—
56C23×Dic3—62no—
57C2×#208—92no—
58#830—94no—
59C23×C32—24no—
60C23×C32—22no—
61C23×C32—216no—
62C23×C32—28no—
63C23×C32—24no—
64C23×C32—22no—
65C23×C32—48no—
66C23×C32—416no—
67C23×C32—42no—
68C23×C32—44no—
69C23×C32C2×Dic312yes—
70C23×C32C2×Dic311yescharacteristic
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—24no—
73C2×C3⋊Dic3—24no—
74C2×C3⋊Dic3C2×Dic312yes—
75C2×C3⋊Dic3C2×Dic311yescharacteristic
76C2×C3⋊Dic3#6714yes—
77C2×C3⋊Dic3C22×D311yescharacteristic
78C2×C3×Dic3—62no—
79C2×C3×Dic3—64no—
80C2×C3×Dic3—68no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—616no—
83C2×C3×Dic3—616no—
84C2×C3×Dic3—68no—
85#622—64no—
86#407—271noSylow
87C2×C33—22no—
88C2×C33—24no—
89C2×C33C2×C22⋊C411yescharacteristic
90C2×C33C2×M1612yes—
91C24×C3—22no—
92C24×C3—42no—
93C24×C3#15711yescharacteristic
94C22×C3×C4—91no—
95C22×C3×C4—92no—
96C22×C3×C4—92no—
97C22×C3×C4—91no—
98C22×C3×C4—184no—
99C22×Dic3—64no—
100C22×Dic3—62no—
101C22×Dic3—62no—
102C22×Dic3—64no—
103C22×Dic3—616no—
104#208—184no—
105C2×C3⋊C8—94no—
106C22×C32—28no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—22no—
110C22×C32—28no—
111C22×C32—232no—
112C22×C32—28no—
113C22×C32—28no—
114C22×C32—22no—
115C22×C32—28no—
116C22×C32—24no—
117C22×C32—48no—
118C22×C32—432no—
119C22×C32—48no—
120C22×C32—48no—
121C22×C32—44no—
122C22×C32—48no—
123C22×C32—42no—
124C22×C32#20814yescomplemented
125C22×C32#21712yes—
126C22×C32C22×Dic311yescharacteristic
127C3⋊Dic3—22no—
128C3⋊Dic3#21712yes—
129C3⋊Dic3C2×#6712yes—
130C3×Dic3—64no—
131C3×Dic3—64no—
132C3×Dic3—68no—
133C23×C4—91no—
134C2×M16—272no—
135C22⋊C8—274no—
136C33#40711yescharacteristic, complemented, Sylow
137C23×C3—28no—
138C23×C3—216no—
139C23×C3—24no—
140C23×C3—22no—
141C23×C3—24no—
142C23×C3—22no—
143C23×C3—416no—
144C23×C3—48no—
145C23×C3—44no—
146C23×C3—42no—
147C23×C3C2×#15712yes—
148C23×C3C2×#15711yescharacteristic
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—94no—
152C2×C3×C4—92no—
153C2×C3×C4—184no—
154C2×C3×C4—184no—
155C2×C3×C4—182no—
156C2×Dic3—62no—
157C2×Dic3—64no—
158C2×Dic3—68no—
159C2×Dic3—62no—
160C2×Dic3—616no—
161C2×Dic3—616no—
162C2×Dic3—68no—
163C3⋊C8—184no—
164C2×C32—24no—
165C2×C32—22no—
166C2×C32—216no—
167C2×C32—28no—
168C2×C32—24no—
169C2×C32—22no—
170C2×C32—48no—
171C2×C32—416no—
172C2×C32—42no—
173C2×C32—44no—
174C2×C32C2×#20812yes—
175C2×C32C2×#21711yescharacteristic
176C24C3⋊#15711yescharacteristic
177C22×C4—91no—
178C22×C4—92no—
179C22×C4—92no—
180C22×C4—91no—
181C22×C4—184no—
182M16—544no—
183C2×C8—274no—
184C22×C3—28no—
185C22×C3—28no—
186C22×C3—232no—
187C22×C3—28no—
188C22×C3—28no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—28no—
194C22×C3—24no—
195C22×C3—432no—
196C22×C3—48no—
197C22×C3—48no—
198C22×C3—48no—
199C22×C3—42no—
200C22×C3—48no—
201C22×C3—44no—
202C22×C3#379214yescomplemented
203C22×C3C22⋊#15712yes—
204C22×C3C22×#15711yescharacteristic
205C3×C4—92no—
206C3×C4—92no—
207C3×C4—182no—
208Dic3—64no—
209Dic3—64no—
210Dic3—68no—
211C32—22no—
212C32—42no—
213C32#520711yescharacteristic, complemented
214C23—24no—
215C23—22no—
216C23C2×C3⋊#15712yes—
217C23C2×C3⋊#15711yescharacteristic
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—94no—
221C2×C4—92no—
222C2×C4—184no—
223C2×C4—184no—
224C2×C4—182no—
225C8—544no—
226C2×C3—216no—
227C2×C3—28no—
228C2×C3—24no—
229C2×C3—22no—
230C2×C3—24no—
231C2×C3—22no—
232C2×C3—416no—
233C2×C3—48no—
234C2×C3—44no—
235C2×C3—42no—
236C2×C3C2×#379212yes—
237C2×C3C2×C22⋊#15711yescharacteristic
238C22—28no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22#8899314yescomplemented
243C22#8899412yes—
244C22C22×C3⋊#15711yescharacteristic
245C4—92no—
246C4—92no—
247C4—182no—
248C3—22no—
249C3—42no—
250C3#1059384511yescharacteristic, complemented
251C2—24no—
252C2—22no—
253C2C2×#8899312yes—
254C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
Fitting, prime radicalC24×C33C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC24C3⋊#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11723112C2semidirect
12. C2×C3⋊#622C22semidirect
50. C22×C33M16semidirect
124. C22×C32#208semidirect
136. C33#407semidirect
202. C22×C3#3792semidirect
213. C32#5207semidirect
242. C22#88993semidirect
250. C3#10593845semidirect

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
#117231122
#105938453
#52079