Finite Group Databasebeta

← C4×#88993 #13173392

Subgroups — C4×#88993 #13173392

Order 1728.

608 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 121 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#11723102C212yesmaximal, complemented
4C4×C3⋊#622C212yesmaximal, complemented
5C4×#3792—31nomaximal
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9#88993C418yesdirect product factor
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#63777—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×C414yes—
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3⋊#622C2×C418yescomplemented
31C4×#208—91nomaximal
32C22×C32×C4—22no—
33C22×C32×C4—42no—
34C22×C32×C4Dic311yescharacteristic
35C22×C3⋊Dic3Dic311yescharacteristic
36C22×C3⋊Dic3C2×D311yescharacteristic
37C22×C3×Dic3—62no—
38C22×C3×Dic3—62no—
39#3792—38no—
40C2×#622—34no—
41C4×C3⋊Dic3Dic311yescharacteristic
42C4×C3⋊Dic3Dic311yescharacteristic
43C4×C3⋊Dic3C2×D311yescharacteristic
44C4×C3⋊Dic3C2×D311yescharacteristic
45C3×C4×Dic3—62no—
46C3×C4×Dic3—62no—
47C3×C4×Dic3—62no—
48C3×C4×Dic3—62no—
49C22×C33—21no—
50C22×C33—21no—
51C22×C33C4212yes—
52C22×C33C22×C411yescharacteristic
53C33×C4—21no—
54C33×C4M1612yescomplemented
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3C4212yes—
57C3×C3⋊Dic3M1612yescomplemented
58C3×C3⋊Dic3C22×C412yes—
59C2×C3×C42—91no—
60C2×C4×Dic3—62no—
61C2×#208—92no—
62#803—92no—
63C4×C3⋊C8—92no—
64C23×C32—22no—
65C23×C32—42no—
66C23×C32C2×Dic311yescharacteristic
67C2×C32×C4—22no—
68C2×C32×C4—28no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C2×C32×C4—48no—
72C2×C32×C4—42no—
73C2×C32×C4—42no—
74C2×C32×C4C2×Dic311yescharacteristic
75C2×C32×C4C2×Dic311yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3C4×D314yes—
78C2×C3⋊Dic3C2×Dic311yescharacteristic
79C2×C3⋊Dic3C2×Dic311yescharacteristic
80C2×C3⋊Dic3C2×Dic311yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—68no—
87C2×C3×Dic3—68no—
88#622—38no—
89C4×M16—271noSylow
90C2×C33—22no—
91C2×C33C2×C4211yescharacteristic
92C2×C33C2×M1611yescharacteristic
93C2×C33C2×M1611yescharacteristic
94C22×C3×C4—22no—
95C22×C3×C4—42no—
96C22×C3×C4—91no—
97C22×C3×C4—91no—
98C22×C3×C4#15711yescharacteristic
99C22×Dic3—62no—
100C22×Dic3—62no—
101C3×C42—91no—
102C3×C42—91no—
103C3×C42—91no—
104C3×C42—91no—
105C4×Dic3—62no—
106C4×Dic3—62no—
107C4×Dic3—62no—
108C4×Dic3—62no—
109#208—98no—
110C2×C3⋊C8—94no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32—22no—
116C22×C32—24no—
117C22×C32—44no—
118C22×C32—44no—
119C22×C32—44no—
120C22×C32—42no—
121C22×C32C4×Dic312yes—
122C22×C32C22×Dic311yescharacteristic
123C32×C4—21no—
124C32×C4—24no—
125C32×C4—24no—
126C32×C4—44no—
127C32×C4—44no—
128C32×C4#20812yescomplemented
129C3⋊Dic3—21no—
130C3⋊Dic3#20812yescomplemented
131C3⋊Dic3C4×Dic312yes—
132C3⋊Dic3C2×C4×D312yes—
133C3×Dic3—64no—
134C3×Dic3—64no—
135C3×Dic3—64no—
136C3×Dic3—64no—
137C2×M16—272no—
138C2×C42—91no—
139#97—272no—
140C4×C8—272no—
141C33C4×M1611yescharacteristic, complemented, Sylow
142C23×C3—22no—
143C23×C3—42no—
144C23×C3C2×#15711yescharacteristic
145C2×C3×C4—28no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—48no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—94no—
156C2×C3×C4—91no—
157C2×C3×C4—182no—
158C2×C3×C4C2×#15711yescharacteristic
159C2×C3×C4C2×#15711yescharacteristic
160C2×Dic3—68no—
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—68no—
166C3⋊C8—98no—
167C2×C32—22no—
168C2×C32—28no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—48no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32C2×#20811yescharacteristic
177C2×C32C2×#20811yescharacteristic
178C2×C32C2×C4×Dic311yescharacteristic
179C22×C4—91no—
180C22×C4—91no—
181C22×C4C3⋊#15711yescharacteristic
182M16—278no—
183C2×C8—274no—
184C42—91no—
185C42—91no—
186C42—91no—
187C42—91no—
188C22×C3—24no—
189C22×C3—24no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—22no—
193C22×C3—24no—
194C22×C3—44no—
195C22×C3—44no—
196C22×C3—42no—
197C22×C3—44no—
198C22×C3C4×#15712yes—
199C22×C3C22×#15711yescharacteristic
200C3×C4—24no—
201C3×C4—21no—
202C3×C4—24no—
203C3×C4—44no—
204C3×C4—44no—
205C3×C4—92no—
206C3×C4—92no—
207C3×C4—92no—
208C3×C4—181no—
209C3×C4#379212yescomplemented
210Dic3—64no—
211Dic3—64no—
212Dic3—64no—
213Dic3—64no—
214C32—22no—
215C32—42no—
216C32C4×#20811yescharacteristic, complemented
217C23C2×C3⋊#15711yescharacteristic
218C2×C4—22no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—94no—
223C2×C4—91no—
224C2×C4—182no—
225C2×C4C2×C3⋊#15711yescharacteristic
226C2×C4C2×C3⋊#15711yescharacteristic
227C8—278no—
228C2×C3—28no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—48no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—42no—
237C2×C3#6428311yescharacteristic
238C2×C3C2×C4×#15711yescharacteristic
239C2×C3C2×#379211yescharacteristic
240C22—21no—
241C22—21no—
242C22C4×C3⋊#15712yes—
243C22C22×C3⋊#15711yescharacteristic
244C4—21no—
245C4—92no—
246C4—92no—
247C4—92no—
248C4—181no—
249C4#8899312yesdirect product factor
250C3—22no—
251C3—42no—
252C3C4×#379211yescharacteristic, complemented
253C2—22no—
254C2#1172451111yescharacteristic
255C2C2×C4×C3⋊#15711yescharacteristic
256C2C2×#8899311yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723102C2semidirect
4. C4×C3⋊#622C2semidirect
9. #88993C4direct (N × H)
30. C3⋊#622C2×C4semidirect
54. C33×C4M16semidirect
57. C3×C3⋊Dic3M16semidirect
128. C32×C4#208semidirect
130. C3⋊Dic3#208semidirect
141. C33C4×M16semidirect
209. C3×C4#3792semidirect
216. C32C4×#208semidirect
249. C4#88993direct (N × H)
252. C3C4×#3792semidirect

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
#117231022
C4×C3⋊#6222
C4×#37923
C4×#2089