Finite Group Databasebeta

← #16781100

Subgroups — #16781100

Order 3328.

930 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 164 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
1#13130138C214yesmaximal, complemented
2#13130067C212yesmaximal
3C2×#11719517C211yesmaximal, characteristic
4C23×Q8×C13C411yescharacteristic
5#11719517—28no—
6C2×Q8×Dic13—22no—
7C2×Q8×Dic13C412yes—
8C2×Q8×Dic13C2212yes—
9C2×#88180—22no—
10C2×#88180C411yescharacteristic
11C2×#88180C2211yescharacteristic
12#11719119C2214yes—
13#11718305—22no—
14Q8⋊#1057—28no—
15#11718285—22no—
16C22×Q8×C13—24no—
17C22×Q8×C13—24no—
18C22×Q8×C13—21no—
19C22×Q8×C13—22no—
20C22×Q8×C13C814yescomplemented
21C22×Q8×C13C2×C412yes—
22C23×C4×C13—21no—
23C23×C4×C13C2×C411yescharacteristic
24C2×C22⋊Dic13—21no—
25C2×C22⋊Dic13D412yes—
26Q8×Dic13—216no—
27Q8×Dic13—24no—
28Q8×Dic13—24no—
29Q8×Dic13—24no—
30Q8×Dic13D418yes—
31#88180—216no—
32#88180—24no—
33#88180—24no—
34C2×C4⋊Dic13—22no—
35C2×C4⋊Dic13—22no—
36C2×C4⋊Dic13C2×C412yes—
37C2×C4⋊Dic13D412yes—
38C2×C4×Dic13—21no—
39C2×C4×Dic13—22no—
40C2×C4×Dic13—22no—
41C2×C4×Dic13C2×C411yescharacteristic
42C2×C4×Dic13C2311yescharacteristic
43C22⋊#1057—24no—
44C4⋊#1057—24no—
45C4×#1057—24no—
46#8323—131nomaximal, Sylow
47C24×C13C22⋊C411yescharacteristic
48C2×Q8×C13—24no—
49C2×Q8×C13—216no—
50C2×Q8×C13—24no—
51C2×Q8×C13—216no—
52C2×Q8×C13—24no—
53C2×Q8×C13—22no—
54C2×Q8×C13—22no—
55C2×Q8×C13—22no—
56C2×Q8×C13C22⋊C414yes—
57C2×Q8×C13C2×C814yes—
58C2×Q8×C13M1614yes—
59C22×C4×C13—24no—
60C22×C4×C13—24no—
61C22×C4×C13—24no—
62C22×C4×C13—22no—
63C22×C4×C13—22no—
64C22×C4×C13—22no—
65C22×C4×C13C22⋊C411yescharacteristic
66C22×C4×C13C2×C812yes—
67C22×C4×C13M1612yes—
68C22×C4×C13C22×C411yescharacteristic
69C22×Dic13—22no—
70C22×Dic13C22⋊C412yes—
71C22×Dic13C2×D412yes—
72C2×#1057—44no—
73C22⋊Dic13—24no—
74C22⋊Dic13—24no—
75C4⋊Dic13—28no—
76C4⋊Dic13—22no—
77C4⋊Dic13—28no—
78C4⋊Dic13—24no—
79C4⋊Dic13C22⋊C414yes—
80C4×Dic13—28no—
81C4×Dic13—24no—
82C4×Dic13—22no—
83C4×Dic13—28no—
84C4×Dic13C2×D414yes—
85C2×#519—131no—
86#1479—134no—
87#1458—132no—
88C23×C13—21no—
89C23×C13—21no—
90C23×C13—42no—
91C23×C13C22⋊C812yes—
92C23×C13C2×C22⋊C411yescharacteristic
93Q8×C13—216no—
94Q8×C13—24no—
95Q8×C13—24no—
96Q8×C13—24no—
97Q8×C13C22⋊C818yes—
98C2×C4×C13—216no—
99C2×C4×C13—24no—
100C2×C4×C13—21no—
101C2×C4×C13—24no—
102C2×C4×C13—24no—
103C2×C4×C13—21no—
104C2×C4×C13—24no—
105C2×C4×C13—24no—
106C2×C4×C13C22⋊C812yes—
107C2×C4×C13C2×C22⋊C412yes—
108C2×C4×C13C22×C811yescharacteristic
109C2×C4×C13C2×M1612yes—
110C2×C4×C13C2×M1611yescharacteristic
111C2×Dic13—22no—
112C2×Dic13—24no—
113C2×Dic13—42no—
114C2×Dic13—44no—
115C2×Dic13—42no—
116C2×Dic13#12014yes—
117#1057—84no—
118C23×Q8#26111yescharacteristic
119#519—268no—
120C2×C4×Q8—132no—
121C2×C4×Q8—132no—
122C2×C4×Q8—262no—
123C2×#117—131no—
124C2×#117—131no—
125C2×#117—262no—
126#431—134no—
127Q8⋊C8—268no—
128#324—262no—
129#323—262no—
130C22×C13—22no—
131C22×C13—22no—
132C22×C13—42no—
133C22×C13—42no—
134C22×C13—42no—
135C22×C13#37911yescharacteristic
136C22×C13C2×C22⋊C811yescharacteristic
137C22×C13#40711yescharacteristic
138C4×C13—28no—
139C4×C13—22no—
140C4×C13C2×C22⋊C812yes—
141C4×C13#40712yes—
142Dic13—24no—
143Dic13—42no—
144Dic13—44no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8#105714yescomplemented
150C22×Q8C2×#26112yes—
151C23×C4—21no—
152C23×C4C2×#26111yescharacteristic
153C4×Q8—138no—
154C4×Q8—264no—
155C4×Q8—264no—
156C4×Q8—2616no—
157C4×Q8—264no—
158#117—264no—
159#117—264no—
160#117—2616no—
161C2×C4⋊C4—132no—
162C2×C4⋊C4—132no—
163C2×C4⋊C4—262no—
164C2×C4⋊C4—262no—
165C2×C22⋊C4—132no—
166C2×C22⋊C4—261no—
167C2×C42—131no—
168C2×C42—131no—
169C2×C42—262no—
170C2×C42—262no—
171C2×C42—261no—
172C4⋊C8—264no—
173C22⋊C8—264no—
174C4×C8—264no—
175C2×C13—21no—
176C2×C13—21no—
177C2×C13—42no—
178C2×C13#177911yescharacteristic
179C2×C13#179612yes—
180C24C22⋊#26111yescharacteristic
181C2×Q8—24no—
182C2×Q8—216no—
183C2×Q8—24no—
184C2×Q8—216no—
185C2×Q8—24no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8C2×#105714yes—
190C2×Q8#612214yes—
191C2×Q8C22⋊#26114yes—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—132no—
199C22×C4—132no—
200C22×C4—262no—
201C22×C4C2×#105712yes—
202C22×C4#612212yes—
203C22×C4C22⋊#26111yescharacteristic
204C22×C4C22×#26111yescharacteristic
205C2×C8—524no—
206C4⋊C4—134no—
207C4⋊C4—264no—
208C4⋊C4—268no—
209C4⋊C4—268no—
210C4⋊C4—262no—
211C22⋊C4—264no—
212C22⋊C4—264no—
213C42—134no—
214C42—268no—
215C42—268no—
216C42—264no—
217C42—262no—
218C13#832311yescharacteristic, complemented, Sylow
219C23—21no—
220C23—21no—
221C23—42no—
222C23C22⋊#105712yes—
223C23C2×C22⋊#26111yescharacteristic
224Q8—216no—
225Q8—24no—
226Q8—24no—
227Q8—24no—
228Q8C22⋊#105718yes—
229C2×C4—216no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—134no—
238C2×C4—264no—
239C2×C4—262no—
240C2×C4—522no—
241C2×C4—524no—
242C2×C4—522no—
243C2×C4C22⋊#105712yes—
244C2×C4C22×#105711yescharacteristic
245C2×C4C2×#612212yes—
246C2×C4C2×#612211yescharacteristic
247C2×C4C2×C22⋊#26112yes—
248C8—1044no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×C22⋊#105711yescharacteristic
255C22#1171916911yescharacteristic
256C22#1171917111yescharacteristic
257C4—28no—
258C4—22no—
259C4—264no—
260C4—522no—
261C4—524no—
262C4C2×C22⋊#105712yes—
263C4#1171916912yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2#1313065811yescharacteristic
268C2#1313068812yes—
subgroupG/H
centerC22#11719171
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC23×Q8×C13C4
socleC22×C13#379
prime radicalC24×C13C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC23×Q8#261
hypocenterC13#8323
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #13130138C2semidirect
20. C22×Q8×C13C8semidirect
149. C22×Q8#1057semidirect
218. C13#8323semidirect

Maximal subgroups

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

subgroupindex
#131301382
#131300672
C2×#117195172
#832313