Finite Group Databasebeta

← #16781710

Subgroups — #16781710

Order 3328.

778 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 104 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#13130735C212yesmaximal
2#13130086C212yesmaximal
3#13130084C212yesmaximal
4C22×#88143C211yesmaximal, characteristic
5C22×C13×C4⋊C4C411yescharacteristic
6C22×C4⋊Dic13C2211yescharacteristic
7C22×#6101C411yescharacteristic
8C22×#6101C411yescharacteristic
9C22×#6101C2211yescharacteristic
10C22×#6101C2211yescharacteristic
11C22×C4×Dic13C411yescharacteristic
12C2×#88143—24no—
13C2×#88143—24no—
14C2×#88143—24no—
15#11719169C2214yes—
16#11718291—24no—
17#11718286—24no—
18#11718285—24no—
19C23×C4×C13C2×C411yescharacteristic
20C23×C4×C13C2×C411yescharacteristic
21C23×C4×C13C2×C411yescharacteristic
22C23×Dic13C2×C411yescharacteristic
23C23×Dic13C2×C411yescharacteristic
24C23×Dic13C2×C411yescharacteristic
25C23×Dic13C2311yescharacteristic
26C2×#6122—24no—
27C2×C13×C4⋊C4—22no—
28C2×C13×C4⋊C4—22no—
29C2×C13×C4⋊C4—22no—
30C2×C4⋊Dic13—22no—
31C2×C4⋊Dic13—22no—
32C2×C4⋊Dic13D414yes—
33C2×#6101—22no—
34C2×#6101—22no—
35C2×#6101—22no—
36C2×#6101—22no—
37C2×#6101—22no—
38C2×#6101—22no—
39C2×#6101—22no—
40C2×#6101—22no—
41C2×#6101—22no—
42C2×#6101—22no—
43C2×#6101D414yes—
44C2×#6101D414yes—
45C2×C4×Dic13—22no—
46C2×C4×Dic13—22no—
47C2×C4×Dic13—22no—
48#88143—232no—
49C22⋊#1057—24no—
50C22⋊#1057—24no—
51#8554—131nomaximal, Sylow
52C24×C13C22×C411yescharacteristic
53C22×C4×C13—21no—
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—42no—
60C22×C4×C13—42no—
61C22×C4×C13—42no—
62C22×C4×C13C22⋊C412yes—
63C22×C4×C13C22⋊C412yes—
64C22×C4×C13C22⋊C412yes—
65C22×Dic13—21no—
66C22×Dic13—21no—
67C22×Dic13—24no—
68C22×Dic13—21no—
69C22×Dic13—21no—
70C22×Dic13—21no—
71C22×Dic13—21no—
72C22×Dic13—42no—
73C22×Dic13—42no—
74C22×Dic13—42no—
75C22×Dic13C22⋊C412yes—
76C22×Dic13C22⋊C412yes—
77C22×Dic13C22⋊C412yes—
78C22×Dic13C2×D412yes—
79C22×Dic13C2×D412yes—
80C22×Dic13C2×D412yes—
81#6122—48no—
82C2×#1057—44no—
83C13×C4⋊C4—28no—
84C4⋊Dic13—28no—
85#6101—28no—
86#6101—28no—
87#6101—28no—
88#6101—28no—
89C4×Dic13—28no—
90C22×#125—131no—
91#1821—132no—
92#1463—132no—
93#1463—132no—
94C23×C13—22no—
95C23×C13—22no—
96C23×C13—22no—
97C23×C13C2×C22⋊C411yescharacteristic
98C23×C13C2×C22⋊C411yescharacteristic
99C23×C13C2×C22⋊C411yescharacteristic
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13—22no—
103C2×C4×C13—24no—
104C2×C4×C13—42no—
105C2×C4×C13—42no—
106C2×C4×C13—42no—
107C2×C4×C13—42no—
108C2×C4×C13—42no—
109C2×C4×C13—42no—
110C2×C4×C13—42no—
111C2×C4×C13—42no—
112C2×Dic13—24no—
113C2×Dic13—24no—
114C2×Dic13—22no—
115C2×Dic13—24no—
116C2×Dic13—22no—
117C2×Dic13—22no—
118C2×Dic13—22no—
119C2×Dic13—42no—
120C2×Dic13—42no—
121C2×Dic13—42no—
122C2×Dic13—42no—
123C2×Dic13—42no—
124C2×Dic13—42no—
125C2×Dic13—42no—
126C2×Dic13—42no—
127C2×Dic13#10114yes—
128C2×Dic13#12014yes—
129#1057—84no—
130C2×#125—264no—
131C2×#125—264no—
132C2×#125—264no—
133C22×C4⋊C4—131no—
134C22×C4⋊C4—131no—
135C22×C4⋊C4—131no—
136C22×C4⋊C4—131no—
137C22×C4⋊C4—131no—
138C22×C4⋊C4#26111yescharacteristic
139C22×C42—131no—
140#407—134no—
141#343—264no—
142#324—264no—
143#324—264no—
144C22×C13—28no—
145C22×C13—21no—
146C22×C13—21no—
147C22×C13—21no—
148C22×C13—21no—
149C22×C13—21no—
150C22×C13—21no—
151C22×C13#37911yescharacteristic
152C22×C13#41012yes—
153C22×C13C2×#10112yes—
154C22×C13#41312yes—
155C4×C13—42no—
156C4×C13—42no—
157C4×C13—42no—
158Dic13—24no—
159Dic13—24no—
160Dic13—42no—
161Dic13—42no—
162C23×C4—131no—
163C23×C4—131no—
164C23×C4—131no—
165C23×C4—131no—
166C23×C4C2×#26111yescharacteristic
167C23×C4C2×#26111yescharacteristic
168C23×C4C2×#26111yescharacteristic
169C2×M16—264no—
170#125—2632no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—134no—
175C2×C4⋊C4—134no—
176C2×C4⋊C4—134no—
177C2×C4⋊C4—262no—
178C2×C4⋊C4—262no—
179C2×C4⋊C4—262no—
180C2×C4⋊C4—262no—
181C2×C4⋊C4—262no—
182C2×C4⋊C4—262no—
183C2×C4⋊C4—262no—
184C2×C4⋊C4—262no—
185C2×C4⋊C4—262no—
186C2×C4⋊C4—262no—
187C2×C4⋊C4—262no—
188C2×C4⋊C4—262no—
189C2×C42—262no—
190C2×C42—262no—
191C2×C42—262no—
192C22⋊C8—264no—
193C22⋊C8—264no—
194C2×C13—22no—
195C2×C13—22no—
196C2×C13—22no—
197C2×C13#178211yescharacteristic
198C2×C13#178411yescharacteristic
199C2×C13#178511yescharacteristic
200C24C22×#26111yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—132no—
211C22×C4—132no—
212C22×C4—132no—
213C22×C4—132no—
214C22×C4—132no—
215C22×C4—132no—
216C22×C4—264no—
217C22×C4—261no—
218C22×C4—261no—
219C22×C4—261no—
220C22×C4—261no—
221C22×C4—261no—
222C22×C4—261no—
223C22×C4—522no—
224C22×C4—522no—
225C22×C4—522no—
226C22×C4C22⋊#26112yes—
227C22×C4C22⋊#26112yes—
228C22×C4C22⋊#26112yes—
229M16—528no—
230C2×C8—524no—
231C4⋊C4—28no—
232C4⋊C4—268no—
233C4⋊C4—268no—
234C4⋊C4—268no—
235C4⋊C4—268no—
236C4⋊C4—268no—
237C42—268no—
238C13#855411yescharacteristic, complemented, Sylow
239C23—22no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊#26111yescharacteristic
243C23C2×C22⋊#26111yescharacteristic
244C23C2×C22⋊#26111yescharacteristic
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—134no—
258C2×C4—134no—
259C2×C4—264no—
260C2×C4—264no—
261C2×C4—262no—
262C2×C4—264no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—262no—
266C2×C4—522no—
267C2×C4—522no—
268C2×C4—522no—
269C2×C4—522no—
270C2×C4—522no—
271C2×C4—522no—
272C2×C4—522no—
273C2×C4—522no—
274C8—1044no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1171911612yes—
283C22#1171914812yes—
284C22D13⋊#10112yes—
285C22#1171917111yescharacteristic
286C4—42no—
287C4—42no—
288C4—42no—
289C4—264no—
290C4—264no—
291C4—522no—
292C4—522no—
293C2—22no—
294C2—22no—
295C2—22no—
296C2#1313066511yescharacteristic
297C2#1313067111yescharacteristic
298C2#1313067311yescharacteristic
subgroupG/H
centerC22#11719171
commutator, prime radicalC24×C13C22×C4
FrattiniC24C22×#261
FittingC22×C13×C4⋊C4C4
socleC22×C13#379
cosocleC23×Dic13C23
hypercenterC22×C4⋊C4#261
hypocenterC13#8554
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
238. C13#8554semidirect

Maximal subgroups

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

subgroupindex
#131307352
#131300862
#131300842
C22×#881432
#855413