Finite Group Databasebeta

← #16797330

Subgroups — #16797330

Order 3328.

950 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719051C211yesmaximal, characteristic
2C2×C4×#6118C211yesmaximal, characteristic
3#13130632C212yesmaximal, complemented
4C4⋊C4×#1057C212yesmaximal, complemented
5#13130577C214yesmaximal
6#13130555C212yesmaximal, complemented
7#13130493C212yesmaximal, complemented
8#13117229C211yesmaximal, characteristic
9C22×#6118C2212yes—
10C2×C42×D13C2211yescharacteristic
11C2×#88111C2212yes—
12C2×#88110C2214yes—
13C2×#88110C2211yescharacteristic
14C2×#88110C2211yescharacteristic
15C2×C4⋊#1057C2211yescharacteristic
16C2×C4⋊#1057C2211yescharacteristic
17C2×C4×#1057C2214yes—
18C2×C4×#1057C2212yes—
19C2×C4×#1057C2212yes—
20#11719051C418yescomplemented
21C4×#6118C418yescomplemented
22C13×#382C411yescharacteristic
23#11718648C412yes—
24C4⋊C4×Dic13C2212yes—
25#11718597C411yescharacteristic
26#11718327C2212yes—
27#11718325C412yes—
28#11718312C412yes—
29#11718309C2212yes—
30#11718290C2214yes—
31#11718290C2214yes—
32C22×C4×D13C2×C411yescharacteristic
33C22×C4×D13C2×C411yescharacteristic
34C22×C4×D13C2311yescharacteristic
35C22×#1057C2312yes—
36C22×#1057C2312yes—
37C22×#1057C2312yes—
38C22×#1057C2312yes—
39C2×#6118—24no—
40C2×#6118—24no—
41C2×#6118C2×C418yes—
42C2×C13×C4⋊C4C2×C412yes—
43C2×C42×C13C2×C411yescharacteristic
44C2×#6103C2×C414yes—
45C2×C4⋊Dic13C2×C412yes—
46C2×C4×Dic13C2×C411yescharacteristic
47C2×C4×Dic13C2314yes—
48C2×C4×Dic13C2311yescharacteristic
49C2×C4×Dic13C2311yescharacteristic
50C42×D13C2×C414yes—
51C42×D13C2×C414yes—
52#88111C2×C418yes—
53#88110—28no—
54#88110C2×C414yes—
55#88110C2×C414yes—
56C4⋊#1057C2×C414yes—
57C4⋊#1057C2×C414yes—
58C4×#1057—28no—
59C4×#1057C2×C418yes—
60C4×#1057C2×C418yes—
61C13×#95C2×C412yes—
62C13×#95C2×C412yes—
63#88071C2×C414yes—
64#88071C2×C412yes—
65#88071C2×C414yes—
66#88071C2×C412yes—
67#20482—131nomaximal, Sylow
68C23×D13C22×C411yescharacteristic
69C22×C4×C13C22×C414yes—
70C22×C4×C13C22×C411yescharacteristic
71C22×C4×C13C22×C411yescharacteristic
72C22×C4×C13C22×C411yescharacteristic
73C22×Dic13C22×C414yes—
74C22×Dic13C22×C411yescharacteristic
75C22×Dic13C22×C411yescharacteristic
76C22×Dic13C2411yescharacteristic
77C2×C4×D13—22no—
78C2×C4×D13—22no—
79C2×C4×D13—22no—
80C2×C4×D13—22no—
81C2×C4×D13—42no—
82C2×C4×D13C4214yes—
83C2×C4×D13C4214yes—
84C2×C4×D13C22×C412yes—
85C2×C4×D13C22×C412yes—
86C2×#1057—24no—
87C2×#1057—24no—
88C2×#1057—22no—
89C2×#1057—22no—
90C2×#1057C22×C414yes—
91C2×#1057C22×C414yes—
92C2×#1057C22×C414yes—
93C2×#1057C22×C414yes—
94C2×#1057#4114yes—
95C2×#1057#4114yes—
96#6118—216no—
97C13×C4⋊C4—24no—
98C42×C13C4214yes—
99#6103—44no—
100C4⋊Dic13—24no—
101C4×Dic13—24no—
102C4×Dic13—24no—
103C4×Dic13C4214yes—
104C4×Dic13C22×C412yes—
105C4×Dic13C22×C412yes—
106C4×Dic13C22×C412yes—
107C4×Dic13C22×C412yes—
108C2×#431—131no—
109C2×C4×M16—131no—
110#2283—131no—
111#1770—132no—
112C8×C4⋊C4—132no—
113#1753—134no—
114#1746—132no—
115#1726—132no—
116C23×C13C23×C411yescharacteristic
117C22×D13—22no—
118C22×D13—21no—
119C22×D13—21no—
120C22×D13—22no—
121C22×D13C2×C4212yes—
122C2×C4×C13—22no—
123C2×C4×C13—22no—
124C2×C4×C13—24no—
125C2×C4×C13—24no—
126C2×C4×C13—24no—
127C2×C4×C13C2×C4212yes—
128C2×C4×C13C2×C4212yes—
129C2×C4×C13C2×C4212yes—
130C2×C4×C13#13112yes—
131C2×C4×C13#13112yes—
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—22no—
135C2×Dic13—24no—
136C2×Dic13—24no—
137C2×Dic13—24no—
138C2×Dic13C2×C4212yes—
139C2×Dic13C2×C4212yes—
140C2×Dic13C23×C411yescharacteristic
141C2×Dic13C23×C411yescharacteristic
142C2×Dic13C2×#4112yes—
143C2×Dic13C2×#4112yes—
144C4×D13—28no—
145C4×D13—24no—
146C4×D13—22no—
147C4×D13—22no—
148C4×D13—44no—
149#1057—24no—
150#1057—24no—
151#1057—24no—
152#1057#11718yes—
153C22×M16—132no—
154C22×C42—131no—
155#431—138no—
156C2×C4⋊C8—131no—
157C2×C4⋊C8—131no—
158C2×C22⋊C8—132no—
159C4×M16—138no—
160C2×#97—131no—
161C2×#97—131no—
162C2×#97—134no—
163C2×C4×C8—132no—
164C2×C4×C8—132no—
165C2×C4×C8—134no—
166#383—132no—
167#382—131no—
168#382#26111yescharacteristic
169#381—132no—
170#380—132no—
171C4×C4⋊C4—132no—
172C4×C22⋊C4—132no—
173#376—132no—
174#336—134no—
175#336—134no—
176C22×C13C22×C4211yescharacteristic
177C22×C13C4×#4112yes—
178C22×C13C2×#13111yescharacteristic
179C22×C13C2×#13111yescharacteristic
180C22×C13#56812yes—
181C2×D13—22no—
182C2×D13—22no—
183C2×D13—22no—
184C2×D13—42no—
185C2×D13—42no—
186C4×C13—22no—
187C4×C13—22no—
188C4×C13—44no—
189C4×C13#40514yes—
190Dic13—22no—
191Dic13—22no—
192Dic13—44no—
193Dic13C2×#11712yes—
194Dic13C4×#4111yescharacteristic
195Dic13C4×#4111yescharacteristic
196C23×C4—131no—
197C23×C4—131no—
198C23×C4—131no—
199C2×M16—138no—
200C2×M16—264no—
201C2×M16—264no—
202C22×C8—132no—
203C22×C8—132no—
204C22×C8—132no—
205C22×C8—132no—
206C2×C4⋊C4—132no—
207C2×C4⋊C4C2×#26112yes—
208C2×C22⋊C4—134no—
209C2×C42—131no—
210C2×C42—131no—
211C2×C42—131no—
212C2×C42—134no—
213C2×C42—134no—
214C2×C42—134no—
215C2×C42C2×#26111yescharacteristic
216C4⋊C8—134no—
217C4⋊C8—134no—
218C22⋊C8—138no—
219#97—134no—
220#97—134no—
221#97—268no—
222C4×C8—138no—
223C4×C8—138no—
224C4×C8—268no—
225#95—134no—
226#95—132no—
227#95—132no—
228#95—134no—
229#95C2×#26112yes—
230#95C2×#26112yes—
231C2×C13C4×#11711yescharacteristic
232C2×C13C2×#40511yescharacteristic
233C2×C13#287311yescharacteristic
234C2×C13#296912yes—
235C2×C13#296912yes—
236D13—42no—
237C24—131no—
238C22×C4—131no—
239C22×C4—131no—
240C22×C4—131no—
241C22×C4—132no—
242C22×C4—132no—
243C22×C4—134no—
244C22×C4—134no—
245C22×C4—134no—
246C22×C4—262no—
247C22×C4—262no—
248C22×C4—262no—
249C22×C4—262no—
250C22×C4—522no—
251C22×C4C22×#26114yes—
252C22×C4C22×#26111yescharacteristic
253C22×C4C22×#26111yescharacteristic
254C22×C4C22×#26111yescharacteristic
255M16—2616no—
256C2×C8—134no—
257C2×C8—134no—
258C2×C8—134no—
259C2×C8—134no—
260C2×C8—134no—
261C2×C8—134no—
262C2×C8—264no—
263C2×C8—264no—
264C2×C8—262no—
265C2×C8—262no—
266C4⋊C4—24no—
267C4⋊C4—264no—
268C22⋊C4—524no—
269C42—132no—
270C42—132no—
271C42—132no—
272C42—132no—
273C42—134no—
274C42—264no—
275C42—264no—
276C42C4×#26114yes—
277C13#2048211yescharacteristic, complemented, Sylow
278C23—132no—
279C23—261no—
280C23—261no—
281C23—262no—
282C23—262no—
283C23C23×#26111yescharacteristic
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—131no—
290C2×C4—131no—
291C2×C4—132no—
292C2×C4—132no—
293C2×C4—132no—
294C2×C4—132no—
295C2×C4—264no—
296C2×C4—262no—
297C2×C4—262no—
298C2×C4—261no—
299C2×C4—261no—
300C2×C4—262no—
301C2×C4—268no—
302C2×C4—264no—
303C2×C4—264no—
304C2×C4—264no—
305C2×C4—524no—
306C2×C4C2×C4×#26112yes—
307C2×C4C2×C4×#26112yes—
308C2×C4C2×C4×#26112yes—
309C2×C4#8823812yes—
310C2×C4#8824012yes—
311C8—138no—
312C8—264no—
313C8—264no—
314C8—264no—
315C22—262no—
316C22—262no—
317C22—262no—
318C22—522no—
319C22—522no—
320C22C22×C4×#26111yescharacteristic
321C22C2×#8823811yescharacteristic
322C22C2×#8824011yescharacteristic
323C22#1171962912yes—
324C22#41×#26112yes—
325C4—22no—
326C4—22no—
327C4—44no—
328C4—132no—
329C4—131no—
330C4—131no—
331C4—262no—
332C4—262no—
333C4—524no—
334C4#1171907214yes—
335C2—522no—
336C2#117×#26111yescharacteristic
337C2C2×#1171907211yescharacteristic
338C2#1313140111yescharacteristic
339C2#1313166812yes—
340C2#1313167212yes—
subgroupG/H
center, FrattiniC23C23×#261
commutatorC22×C13C22×C42
FittingC13×#382C4
socleC23×C13C23×C4
prime radicalC23×D13C22×C4
cosocleC22×Dic13C24
hypercenter#382#261
hypocenterC13#20482
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #13130632C2semidirect
4. C4⋊C4×#1057C2semidirect
6. #13130555C2semidirect
7. #13130493C2semidirect
20. #11719051C4semidirect
21. C4×#6118C4semidirect
277. C13#20482semidirect

Maximal subgroups

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

subgroupindex
C2×#117190512
C2×C4×#61182
#131306322
C4⋊C4×#10572
#131305772
#131305552
#131304932
#131172292
#2048213