Finite Group Databasebeta

← #16786750

Subgroups — #16786750

Order 3328.

686 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 112 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#13130067C212yesmaximal, complemented
2#13129764C214yesmaximal, complemented
3#13117418C211yesmaximal, characteristic
4C23×D4×C13C411yescharacteristic
5C2×#88180C411yescharacteristic
6C2×#88180C2211yescharacteristic
7#11719119—22no—
8#11718908—22no—
9#11718882—22no—
10#11718882C412yes—
11#11718882C2212yes—
12#11718305—22no—
13#11718285C2214yescomplemented
14C25×C13—21no—
15C22×D4×C13—21no—
16C22×D4×C13—22no—
17C22×D4×C13—42no—
18C22×D4×C13—42no—
19C22×D4×C13C814yescomplemented
20C22×D4×C13C2×C412yes—
21C23×C4×C13C2×C411yescharacteristic
22C2×C22⋊Dic13—21no—
23C2×C22⋊Dic13—24no—
24C2×C22⋊Dic13—24no—
25C2×C22⋊Dic13D412yes—
26#88180—42no—
27#88180—42no—
28C2×C4⋊Dic13—21no—
29C2×C4⋊Dic13C2×C411yescharacteristic
30C2×C4⋊Dic13C2311yescharacteristic
31C2×C4×Dic13C2×C412yes—
32C2×C4×Dic13D412yes—
33C22⋊#1057—24no—
34C4⋊#1057—42no—
35C4×#1057—42no—
36#11433—131nomaximal, Sylow
37C24×C13—22no—
38C24×C13—24no—
39C24×C13—42no—
40C24×C13—42no—
41C24×C13—82no—
42C24×C13—82no—
43C24×C13C22⋊C411yescharacteristic
44C2×D4×C13—24no—
45C2×D4×C13—22no—
46C2×D4×C13—42no—
47C2×D4×C13—42no—
48C2×D4×C13—48no—
49C2×D4×C13—48no—
50C2×D4×C13—42no—
51C2×D4×C13—42no—
52C2×D4×C13C2×C814yes—
53C22×C4×C13—22no—
54C22×C4×C13—22no—
55C22×C4×C13C22⋊C411yescharacteristic
56C22×C4×C13C2×C812yes—
57C22×C4×C13M1612yes—
58C22×C4×C13C22×C411yescharacteristic
59C22×Dic13—22no—
60C22×Dic13C22⋊C412yes—
61C22×Dic13C2×D412yes—
62C2×#1057—44no—
63C22⋊Dic13—42no—
64C22⋊Dic13—42no—
65C22⋊Dic13—84no—
66C22⋊Dic13—84no—
67C4⋊Dic13—41no—
68C4⋊Dic13—41no—
69C4⋊Dic13—42no—
70C4×Dic13—22no—
71C4×Dic13—22no—
72C4×Dic13—42no—
73#2309—131no—
74#1458—132no—
75#1318—134no—
76C23×C13—21no—
77C23×C13—21no—
78C23×C13—24no—
79C23×C13—41no—
80C23×C13—41no—
81C23×C13—44no—
82C23×C13—44no—
83C23×C13—48no—
84C23×C13—42no—
85C23×C13—42no—
86C23×C13—88no—
87C23×C13—88no—
88C23×C13—84no—
89C23×C13—84no—
90C23×C13C22⋊C812yes—
91C23×C13C2×C22⋊C411yescharacteristic
92D4×C13—44no—
93D4×C13—44no—
94D4×C13—48no—
95C2×C4×C13—21no—
96C2×C4×C13—21no—
97C2×C4×C13—24no—
98C2×C4×C13—24no—
99C2×C4×C13C22⋊C812yes—
100C2×C4×C13#9912yes—
101C2×C4×C13#10012yes—
102C2×C4×C13C22×C811yescharacteristic
103C2×C4×C13C2×M1611yescharacteristic
104C2×Dic13—22no—
105C2×Dic13—22no—
106C2×Dic13—42no—
107C2×Dic13—42no—
108C2×Dic13—42no—
109C2×Dic13—42no—
110C2×Dic13—42no—
111#1057—84no—
112C23×D4#26111yescharacteristic
113C2×#117—131no—
114C2×#117—131no—
115#431—262no—
116#390—132no—
117#390—132no—
118#390—262no—
119#379—262no—
120#324—134no—
121#323—262no—
122C22×C13—21no—
123C22×C13—21no—
124C22×C13—22no—
125C22×C13—42no—
126C22×C13—42no—
127C22×C13—48no—
128C22×C13—41no—
129C22×C13—41no—
130C22×C13—44no—
131C22×C13—84no—
132C22×C13—816no—
133C22×C13—82no—
134C22×C13—82no—
135C22×C13C2×C22⋊C811yescharacteristic
136C22×C13C2×#9911yescharacteristic
137C22×C13C2×#10011yescharacteristic
138C4×C13—21no—
139C4×C13—21no—
140C4×C13#32314yes—
141Dic13—42no—
142Dic13—42no—
143Dic13—82no—
144C25—21no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4#105714yescomplemented
150C22×D4C2×#26112yes—
151C23×C4C2×#26111yescharacteristic
152#117—522no—
153#117—522no—
154C2×C4⋊C4—131no—
155C2×C4⋊C4—131no—
156C2×C4⋊C4—261no—
157C2×C22⋊C4—132no—
158C2×C22⋊C4—261no—
159C2×C22⋊C4—264no—
160C2×C22⋊C4—264no—
161C2×C42—132no—
162C2×C42—132no—
163C4⋊C8—522no—
164C22⋊C8—264no—
165C4×C8—522no—
166C2×C13—21no—
167C2×C13—21no—
168C2×C13—42no—
169C2×C13—84no—
170C2×C13C2×#32311yescharacteristic
171C2×C13#212811yescharacteristic
172C2×C13#213411yescharacteristic
173C24—22no—
174C24—24no—
175C24—42no—
176C24—42no—
177C24—82no—
178C24—82no—
179C24C22⋊#26111yescharacteristic
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—48no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4C2×#105714yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—132no—
192C22×C4—132no—
193C22×C4—262no—
194C22×C4C2×#105712yes—
195C22×C4#612212yes—
196C22×C4C22⋊#26111yescharacteristic
197C22×C4C22×#26111yescharacteristic
198C2×C8—524no—
199C4⋊C4—521no—
200C4⋊C4—521no—
201C4⋊C4—522no—
202C22⋊C4—522no—
203C22⋊C4—522no—
204C22⋊C4—1044no—
205C22⋊C4—1044no—
206C42—262no—
207C42—262no—
208C42—522no—
209C13#1143311yescharacteristic, complemented, Sylow
210C23—21no—
211C23—21no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—88no—
221C23—88no—
222C23—84no—
223C23—84no—
224C23C22⋊#105712yes—
225C23C2×C22⋊#26111yescharacteristic
226D4—44no—
227D4—44no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—262no—
234C2×C4—262no—
235C2×C4—522no—
236C2×C4—522no—
237C2×C4—522no—
238C2×C4—522no—
239C2×C4—522no—
240C2×C4#8811912yes—
241C2×C4C22⋊#105712yes—
242C2×C4#8812112yes—
243C2×C4C22×#105711yescharacteristic
244C2×C4C2×#612211yescharacteristic
245C8—1044no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—48no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—84no—
256C22—816no—
257C22—82no—
258C22—82no—
259C22C2×#8811911yescharacteristic
260C22C2×C22⋊#105711yescharacteristic
261C22C2×#8812111yescharacteristic
262C4—21no—
263C4—21no—
264C4—522no—
265C4—522no—
266C4—1042no—
267C4#1171830514yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—84no—
272C2C2×#1171830511yescharacteristic
273C2#1313111511yescharacteristic
274C2#1313113511yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
Fitting, prime radicalC23×D4×C13C4
socleC22×C13C2×#99
cosocleC2×C4⋊Dic13C23
hypercenterC23×D4#261
hypocenterC13#11433
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #13130067C2semidirect
2. #13129764C2semidirect
13. #11718285C22semidirect
19. C22×D4×C13C8semidirect
149. C22×D4#1057semidirect
209. C13#11433semidirect

Maximal subgroups

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

subgroupindex
#131300672
#131297642
#131174182
#1143313