Finite Group Databasebeta

← #16797333

Subgroups — #16797333

Order 3328.

950 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719119C211yesmaximal, characteristic
2C2×C4×#6122C211yesmaximal, characteristic
3#13130632C212yesmaximal, complemented
4C4⋊C4×#1057C212yesmaximal, complemented
5#13130578C214yesmaximal
6#13130555C212yesmaximal, complemented
7#13130493C212yesmaximal, complemented
8#13117231C211yesmaximal, characteristic
9C22×#6122C2212yes—
10C22×C4×Dic13C2211yescharacteristic
11C2×C22⋊#1057C2212yes—
12C4×#6122C418yescomplemented
13C2×#88110C2214yes—
14C2×#88110C2211yescharacteristic
15C2×#88110C2211yescharacteristic
16#11719119C418yescomplemented
17C2×C4⋊#1057C2211yescharacteristic
18C2×C4⋊#1057C2211yescharacteristic
19C2×C4×#1057C2214yes—
20C2×C4×#1057C2212yes—
21C2×C4×#1057C2212yes—
22C13×#380C411yescharacteristic
23#11718873C411yescharacteristic
24#11718638C412yes—
25C4⋊C4×Dic13C2212yes—
26#11718604C412yes—
27C22⋊C4×Dic13C2212yes—
28#11718312C412yes—
29#11718309C2212yes—
30#11718290C2214yes—
31#11718290C2214yes—
32C23×C4×C13C2×C411yescharacteristic
33C23×Dic13C2×C411yescharacteristic
34C23×Dic13C2311yescharacteristic
35C2×#6122—24no—
36C2×#6122—24no—
37C2×#6122C2×C418yes—
38C22×#1057C2312yes—
39C22×#1057C2312yes—
40C22×#1057C2312yes—
41C22×#1057C2312yes—
42C2×C13×C4⋊C4C2×C412yes—
43C2×C13×C22⋊C4C2×C412yes—
44C2×C22⋊Dic13C2×C412yes—
45C2×C4⋊Dic13C2×C412yes—
46C2×C4×Dic13C2×C411yescharacteristic
47C2×C4×Dic13C2×C411yescharacteristic
48C2×C4×Dic13C2×C414yes—
49C2×C4×Dic13C2×C414yes—
50C2×C4×Dic13C2314yes—
51C2×C4×Dic13C2311yescharacteristic
52C2×C4×Dic13C2311yescharacteristic
53C22⋊#1057C2×C418yes—
54#88110—28no—
55#88110C2×C414yes—
56#88110C2×C414yes—
57C4⋊#1057C2×C414yes—
58C4⋊#1057C2×C414yes—
59C4×#1057—28no—
60C4×#1057C2×C418yes—
61C4×#1057C2×C418yes—
62C13×#95C2×C412yes—
63#88071C2×C412yes—
64#88071C2×C414yes—
65#88071C2×C414yes—
66#88071C2×C414yes—
67#20482—131nomaximal, Sylow
68C24×C13C22×C411yescharacteristic
69C22×C4×C13—24no—
70C22×C4×C13C4214yes—
71C22×C4×C13C22×C414yes—
72C22×C4×C13C22×C411yescharacteristic
73C22×C4×C13C22×C411yescharacteristic
74C22×Dic13—22no—
75C22×Dic13—22no—
76C22×Dic13—41no—
77C22×Dic13—41no—
78C22×Dic13C4214yes—
79C22×Dic13C22×C414yes—
80C22×Dic13C22×C411yescharacteristic
81C22×Dic13C22×C411yescharacteristic
82C22×Dic13C22×C411yescharacteristic
83C22×Dic13C22×C412yes—
84C22×Dic13C22×C412yes—
85C22×Dic13C2411yescharacteristic
86#6122—216no—
87C2×#1057—24no—
88C2×#1057—24no—
89C2×#1057—22no—
90C2×#1057—22no—
91C2×#1057C22×C414yes—
92C2×#1057C22×C414yes—
93C2×#1057C22×C414yes—
94C2×#1057C22×C414yes—
95C2×#1057#4114yes—
96C2×#1057#4114yes—
97C13×C4⋊C4—24no—
98C13×C22⋊C4—42no—
99C22⋊Dic13—42no—
100C4⋊Dic13—24no—
101C4×Dic13—24no—
102C4×Dic13—24no—
103C4×Dic13C4214yes—
104C4×Dic13C4214yes—
105C4×Dic13C22×C412yes—
106C4×Dic13C22×C412yes—
107C4×Dic13C22×C412yes—
108C4×Dic13C22×C412yes—
109C2×#431—131no—
110C2×C4×M16—131no—
111#2283—131no—
112#1770—132no—
113C8×C4⋊C4—132no—
114#1753—134no—
115#1746—132no—
116#1726—132no—
117C23×C13—22no—
118C23×C13—22no—
119C23×C13—21no—
120C23×C13—21no—
121C23×C13C2×C4212yes—
122C23×C13C23×C411yescharacteristic
123C2×C4×C13—22no—
124C2×C4×C13—28no—
125C2×C4×C13—24no—
126C2×C4×C13—24no—
127C2×C4×C13—24no—
128C2×C4×C13C2×C4212yes—
129C2×C4×C13C2×C4212yes—
130C2×C4×C13#13114yes—
131C2×Dic13—22no—
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—21no—
135C2×Dic13—21no—
136C2×Dic13—24no—
137C2×Dic13—22no—
138C2×Dic13—22no—
139C2×Dic13—24no—
140C2×Dic13—24no—
141C2×Dic13—24no—
142C2×Dic13—44no—
143C2×Dic13C2×C4212yes—
144C2×Dic13C2×C4212yes—
145C2×Dic13C2×C4212yes—
146C2×Dic13C23×C411yescharacteristic
147C2×Dic13C23×C411yescharacteristic
148C2×Dic13C2×#4112yes—
149C2×Dic13C2×#4112yes—
150#1057—24no—
151#1057—24no—
152#1057—24no—
153#1057#11718yes—
154C22×M16—132no—
155C22×C42—131no—
156#431—138no—
157C2×C4⋊C8—131no—
158C2×C4⋊C8—131no—
159C2×C22⋊C8—132no—
160C4×M16—138no—
161C2×#97—131no—
162C2×#97—131no—
163C2×#97—134no—
164C2×C4×C8—132no—
165C2×C4×C8—132no—
166C2×C4×C8—134no—
167#383—132no—
168#382—132no—
169#381—132no—
170#380—131no—
171#380#26111yescharacteristic
172C4×C4⋊C4—132no—
173C4×C22⋊C4—132no—
174#376—132no—
175#336—134no—
176#336—134no—
177C22×C13—22no—
178C22×C13—22no—
179C22×C13—22no—
180C22×C13—42no—
181C22×C13—42no—
182C22×C13C22×C4211yescharacteristic
183C22×C13C4×#4112yes—
184C22×C13C2×#13112yes—
185C22×C13#56811yescharacteristic
186C22×C13#56811yescharacteristic
187C4×C13—22no—
188C4×C13—44no—
189C4×C13#40514yes—
190Dic13—22no—
191Dic13—22no—
192Dic13—22no—
193Dic13—44no—
194Dic13C2×#11712yes—
195Dic13C4×#4111yescharacteristic
196Dic13C4×#4111yescharacteristic
197C23×C4—131no—
198C23×C4—131no—
199C23×C4C2×#26111yescharacteristic
200C2×M16—138no—
201C2×M16—264no—
202C2×M16—264no—
203C22×C8—132no—
204C22×C8—132no—
205C22×C8—132no—
206C22×C8—132no—
207C2×C4⋊C4—132no—
208C2×C4⋊C4C2×#26112yes—
209C2×C22⋊C4—132no—
210C2×C22⋊C4C2×#26112yes—
211C2×C42—134no—
212C2×C42—134no—
213C2×C42—131no—
214C2×C42—131no—
215C2×C42—131no—
216C2×C42—131no—
217C2×C42—134no—
218C4⋊C8—134no—
219C4⋊C8—134no—
220C22⋊C8—138no—
221#97—134no—
222#97—134no—
223#97—268no—
224C4×C8—138no—
225C4×C8—138no—
226C4×C8—268no—
227#95—132no—
228#95—134no—
229#95—134no—
230#95—134no—
231#95C2×#26112yes—
232C2×C13—42no—
233C2×C13C4×#11711yescharacteristic
234C2×C13C2×#40511yescharacteristic
235C2×C13#287311yescharacteristic
236C2×C13#296912yes—
237C2×C13#296912yes—
238C24C22×#26111yescharacteristic
239C22×C4—24no—
240C22×C4—131no—
241C22×C4—132no—
242C22×C4—132no—
243C22×C4—131no—
244C22×C4—131no—
245C22×C4—131no—
246C22×C4—134no—
247C22×C4—134no—
248C22×C4—262no—
249C22×C4—262no—
250C22×C4—521no—
251C22×C4—521no—
252C22×C4C4×#26114yes—
253C22×C4C22×#26114yes—
254C22×C4C22×#26111yescharacteristic
255C22×C4C22×#26111yescharacteristic
256M16—2616no—
257C2×C8—134no—
258C2×C8—134no—
259C2×C8—134no—
260C2×C8—134no—
261C2×C8—134no—
262C2×C8—134no—
263C2×C8—264no—
264C2×C8—264no—
265C2×C8—262no—
266C2×C8—262no—
267C4⋊C4—24no—
268C4⋊C4—264no—
269C22⋊C4—42no—
270C22⋊C4—522no—
271C42—132no—
272C42—132no—
273C42—132no—
274C42—132no—
275C42—134no—
276C42—134no—
277C42—264no—
278C42—264no—
279C13#2048211yescharacteristic, complemented, Sylow
280C23—22no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23C2×C4×#26112yes—
285C23C23×#26111yescharacteristic
286C2×C4—22no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—131no—
292C2×C4—131no—
293C2×C4—132no—
294C2×C4—132no—
295C2×C4—132no—
296C2×C4—132no—
297C2×C4—132no—
298C2×C4—264no—
299C2×C4—262no—
300C2×C4—262no—
301C2×C4—261no—
302C2×C4—261no—
303C2×C4—261no—
304C2×C4—261no—
305C2×C4—262no—
306C2×C4—264no—
307C2×C4—264no—
308C2×C4—264no—
309C2×C4—524no—
310C2×C4C2×C4×#26112yes—
311C2×C4C2×C4×#26112yes—
312C2×C4#8823814yes—
313C8—138no—
314C8—264no—
315C8—264no—
316C8—264no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22C22×C4×#26111yescharacteristic
323C22C2×#8823812yes—
324C22#1171962111yescharacteristic
325C22#1171962511yescharacteristic
326C22#41×#26112yes—
327C4—22no—
328C4—44no—
329C4—132no—
330C4—131no—
331C4—131no—
332C4—262no—
333C4—262no—
334C4—262no—
335C4—524no—
336C4#1171906214yes—
337C2—42no—
338C2#117×#26111yescharacteristic
339C2C2×#1171906211yescharacteristic
340C2#1313140211yescharacteristic
341C2#1313166712yes—
342C2#1313166912yes—
subgroupG/H
center, FrattiniC23C23×#261
commutatorC22×C13C22×C42
FittingC13×#380C4
socleC23×C13C23×C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenter#380#261
hypocenterC13#20482
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #13130632C2semidirect
4. C4⋊C4×#1057C2semidirect
6. #13130555C2semidirect
7. #13130493C2semidirect
12. C4×#6122C4semidirect
16. #11719119C4semidirect
279. C13#20482semidirect

Maximal subgroups

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

subgroupindex
C2×#117191192
C2×C4×#61222
#131306322
C4⋊C4×#10572
#131305782
#131305552
#131304932
#131172312
#2048213