Finite Group Databasebeta

← #13173395

Subgroups — #13173395

Order 1728.

960 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 101 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
2C2×#88993C212yesmaximal, complemented
3#11723103C214yesmaximal, complemented
4#10593802—31nomaximal
5C23×C3×C3⋊D3C411yescharacteristic
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×C4×C3⋊D3—24no—
9C2×C3×C4×C3⋊D3C412yes—
10C2×C3×C4×C3⋊D3C2212yes—
11#88993—24no—
12C2×C3⋊#622C2214yescomplemented
13C22×C4×C3⋊D3D311yescharacteristic
14C22×C3×C4×D3—62no—
15C2×#3792—32no—
16#63778—34no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3—22no—
19C22×C3×C3⋊D3—24no—
20C22×C3×C3⋊D3C2×C412yes—
21C2×C33×C4—22no—
22C2×C33×C4C2×C412yes—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3D414yes—
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3—24no—
27C3×C4×C3⋊D3—24no—
28C3⋊#622—24no—
29#5207—91nomaximal
30C23×C3⋊D3Dic311yescharacteristic
31C23×C3×D3—62no—
32C22×C32×C4—22no—
33C22×C32×C4—42no—
34C22×C32×C4Dic311yescharacteristic
35C22×C3⋊Dic3C2×D311yescharacteristic
36C2×C4×C3⋊D3—24no—
37C2×C4×C3⋊D3Dic312yes—
38C2×C4×C3⋊D3C2×D312yes—
39C22×C3×Dic3—62no—
40C2×C3×C4×D3—64no—
41C2×C3×C4×D3—64no—
42C2×C3×C4×D3—616no—
43#3792—64no—
44C2×#622—34no—
45C22×C33—22no—
46C22×C33C22⋊C412yes—
47C22×C33C22×C411yescharacteristic
48C2×C3×C3⋊D3—28no—
49C2×C3×C3⋊D3—22no—
50C2×C3×C3⋊D3—22no—
51C2×C3×C3⋊D3M1614yescomplemented
52C33×C4—22no—
53C3×C3⋊Dic3C22⋊C412yes—
54C3×C3⋊Dic3C2×D412yes—
55C23×C3×C4—91no—
56C22×C4×D3—62no—
57C2×#208—92no—
58#830—94no—
59C23×C32—22no—
60C23×C32—42no—
61C23×C32C2×Dic311yescharacteristic
62C22×C3⋊D3—22no—
63C22×C3⋊D3—24no—
64C22×C3⋊D3C2×Dic312yes—
65C22×C3×D3—64no—
66C22×C3×D3—616no—
67C22×C3×D3—68no—
68C2×C32×C4—22no—
69C2×C32×C4—28no—
70C2×C32×C4—24no—
71C2×C32×C4—48no—
72C2×C32×C4—44no—
73C2×C32×C4C2×Dic312yes—
74C2×C3⋊Dic3C2×Dic311yescharacteristic
75C2×C3⋊Dic3#6714yes—
76C2×C3⋊Dic3C22×D311yescharacteristic
77C4×C3⋊D3—24no—
78C4×C3⋊D3—24no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—68no—
81C2×C3×Dic3—62no—
82C3×C4×D3—616no—
83C3×C4×D3—616no—
84#622—64no—
85#407—271noSylow
86C2×C33—22no—
87C2×C33C2×C22⋊C411yescharacteristic
88C2×C33C2×M1612yes—
89C3×C3⋊D3—24no—
90C24×C3—91no—
91C23×D3—62no—
92C22×C3×C4—22no—
93C22×C3×C4—42no—
94C22×C3×C4—91no—
95C22×C3×C4—92no—
96C22×C3×C4—92no—
97C22×C3×C4—184no—
98C22×C3×C4#15711yescharacteristic
99C22×Dic3—62no—
100C2×C4×D3—64no—
101C2×C4×D3—64no—
102C2×C4×D3—616no—
103#208—184no—
104C2×C3⋊C8—94no—
105C22×C32—22no—
106C22×C32—28no—
107C22×C32—22no—
108C22×C32—24no—
109C22×C32—48no—
110C22×C32—44no—
111C22×C32—42no—
112C22×C32#21712yes—
113C22×C32C22×Dic311yescharacteristic
114C2×C3⋊D3—28no—
115C2×C3⋊D3—22no—
116C2×C3⋊D3—22no—
117C2×C3⋊D3#20814yescomplemented
118C2×C3×D3—68no—
119C2×C3×D3—68no—
120C2×C3×D3—632no—
121C2×C3×D3—68no—
122C32×C4—22no—
123C32×C4—28no—
124C32×C4—48no—
125C3⋊Dic3#21712yes—
126C3⋊Dic3C2×#6712yes—
127C3×Dic3—64no—
128C3×Dic3—64no—
129C23×C4—91no—
130C2×M16—272no—
131C22⋊C8—274no—
132C33#40711yescharacteristic, complemented, Sylow
133C23×C3—22no—
134C23×C3—42no—
135C23×C3—92no—
136C23×C3—184no—
137C23×C3—182no—
138C23×C3C2×#15711yescharacteristic
139C22×D3—64no—
140C22×D3—68no—
141C22×D3—616no—
142C2×C3×C4—28no—
143C2×C3×C4—22no—
144C2×C3×C4—24no—
145C2×C3×C4—48no—
146C2×C3×C4—44no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—94no—
150C2×C3×C4—184no—
151C2×C3×C4—184no—
152C2×C3×C4C2×#15712yes—
153C2×Dic3—62no—
154C2×Dic3—68no—
155C2×Dic3—62no—
156C4×D3—616no—
157C4×D3—616no—
158C3⋊C8—184no—
159C2×C32—22no—
160C2×C32—28no—
161C2×C32—24no—
162C2×C32—22no—
163C2×C32—48no—
164C2×C32—42no—
165C2×C32—44no—
166C2×C32C2×#20812yes—
167C2×C32C2×#21711yescharacteristic
168C3⋊D3—24no—
169C3×D3—616no—
170C24—91no—
171C22×C4—91no—
172C22×C4—92no—
173C22×C4—92no—
174C22×C4—184no—
175C22×C4C3⋊#15711yescharacteristic
176M16—544no—
177C2×C8—274no—
178C22×C3—28no—
179C22×C3—22no—
180C22×C3—22no—
181C22×C3—24no—
182C22×C3—48no—
183C22×C3—42no—
184C22×C3—44no—
185C22×C3—94no—
186C22×C3—188no—
187C22×C3—182no—
188C22×C3—182no—
189C22×C3C22⋊#15712yes—
190C22×C3C22×#15711yescharacteristic
191C2×D3—68no—
192C2×D3—68no—
193C2×D3—632no—
194C2×D3—68no—
195C3×C4—28no—
196C3×C4—22no—
197C3×C4—48no—
198C3×C4—92no—
199C3×C4—92no—
200Dic3—64no—
201Dic3—64no—
202C32—22no—
203C32—42no—
204C32#520711yescharacteristic, complemented
205C23—92no—
206C23—184no—
207C23—182no—
208C23C2×C3⋊#15711yescharacteristic
209C2×C4—22no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—94no—
213C2×C4—184no—
214C2×C4—184no—
215C2×C4C2×C3⋊#15712yes—
216C8—544no—
217C2×C3—28no—
218C2×C3—22no—
219C2×C3—24no—
220C2×C3—22no—
221C2×C3—48no—
222C2×C3—44no—
223C2×C3—42no—
224C2×C3—184no—
225C2×C3#6428312yes—
226C2×C3C2×C22⋊#15711yescharacteristic
227D3—616no—
228C22—22no—
229C22—94no—
230C22—188no—
231C22—182no—
232C22—182no—
233C22#8899412yes—
234C22C22×C3⋊#15711yescharacteristic
235C4—22no—
236C4—92no—
237C4—92no—
238C3—22no—
239C3—42no—
240C3#1059380211yescharacteristic, complemented
241C2—22no—
242C2—184no—
243C2#1172451112yes—
244C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11723103C2semidirect
12. C2×C3⋊#622C22semidirect
51. C2×C3×C3⋊D3M16semidirect
117. C2×C3⋊D3#208semidirect
132. C33#407semidirect
204. C32#5207semidirect
240. C3#10593802semidirect

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
C2×#889932
#117231032
#105938023
#52079