Finite Group Databasebeta

← #16797294

Subgroups — #16797294

Order 3328.

990 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×#11719128C211yesmaximal, characteristic
2C22×C4×#1057C211yesmaximal, characteristic
3#13130631C212yesmaximal, complemented
4#13130628C212yesmaximal, complemented
5#13130573C212yesmaximal
6C22⋊C4×#1057C212yesmaximal
7#13130555C212yesmaximal, complemented
8#13130493C212yesmaximal, complemented
9#13117227C211yesmaximal, characteristic
10C23×#1057C2211yescharacteristic
11C23×#1057C2211yescharacteristic
12C22×C4×Dic13C2211yescharacteristic
13C2×C22⋊#1057C2212yes—
14#11719128C418yescomplemented
15C2×#88112C2212yes—
16C2×#88110C2214yes—
17C2×#88110C2212yes—
18C2×C4×#1057C418yescomplemented
19C2×C4×#1057C2214yes—
20C2×C4×#1057C2211yescharacteristic
21C2×C4×#1057C2211yescharacteristic
22C2×C4×#1057C2211yescharacteristic
23C2×C4×#1057C2211yescharacteristic
24C13×#381C411yescharacteristic
25#11718872C411yescharacteristic
26#11718603C412yes—
27C22⋊C4×Dic13C2212yes—
28#11718312C412yes—
29#11718311C412yes—
30#11718310C2212yes—
31#11718309C2212yes—
32#11718290C2214yes—
33#11718290C2214yes—
34C23×C4×C13C2×C411yescharacteristic
35C23×Dic13C2×C411yescharacteristic
36C23×Dic13C2311yescharacteristic
37C22×#1057—24no—
38C22×#1057—24no—
39C22×#1057C2×C414yes—
40C22×#1057C2×C414yes—
41C22×#1057C2314yes—
42C22×#1057C2311yescharacteristic
43C22×#1057C2311yescharacteristic
44C22×#1057C2311yescharacteristic
45C22×#1057C2311yescharacteristic
46C2×C13×C22⋊C4C2×C412yes—
47C2×C22⋊Dic13C2×C412yes—
48C2×#6101C2×C414yes—
49C2×C4×Dic13C2×C411yescharacteristic
50C2×C4×Dic13C2×C411yescharacteristic
51C2×C4×Dic13C2×C414yes—
52C2×C4×Dic13C2×C414yes—
53C2×C4×Dic13C2314yes—
54C2×C4×Dic13C2311yescharacteristic
55C2×C4×Dic13C2311yescharacteristic
56C22⋊#1057C2×C418yes—
57#88112C2×C418yes—
58#88110—28no—
59#88110C2×C418yes—
60C4×#1057—28no—
61C4×#1057C2×C414yes—
62C4×#1057C2×C414yes—
63C4×#1057C2×C414yes—
64C4×#1057C2×C414yes—
65C13×#95C2×C412yes—
66C13×#95C2×C412yes—
67#88071C2×C412yes—
68#88071C2×C414yes—
69#88071C2×C412yes—
70#88071C2×C414yes—
71#20476—131nomaximal, Sylow
72C24×C13C22×C411yescharacteristic
73C22×C4×C13—24no—
74C22×C4×C13C4214yes—
75C22×C4×C13C22×C414yes—
76C22×C4×C13C22×C411yescharacteristic
77C22×C4×C13C22×C411yescharacteristic
78C22×Dic13—24no—
79C22×Dic13—22no—
80C22×Dic13—22no—
81C22×Dic13C4214yes—
82C22×Dic13C22×C414yes—
83C22×Dic13C22×C411yescharacteristic
84C22×Dic13C22×C411yescharacteristic
85C22×Dic13C22×C412yes—
86C22×Dic13C22×C412yes—
87C22×Dic13C22×C411yescharacteristic
88C22×Dic13C2411yescharacteristic
89C2×#1057—24no—
90C2×#1057—24no—
91C2×#1057—22no—
92C2×#1057—22no—
93C2×#1057—28no—
94C2×#1057—28no—
95C2×#1057C22×C412yes—
96C2×#1057C22×C412yes—
97C2×#1057C22×C418yes—
98C2×#1057C22×C412yes—
99C2×#1057C22×C412yes—
100C2×#1057#4114yes—
101C2×#1057#4114yes—
102C13×C22⋊C4—24no—
103C22⋊Dic13—24no—
104#6101—28no—
105C4×Dic13—24no—
106C4×Dic13—24no—
107C4×Dic13C4214yes—
108C4×Dic13C4214yes—
109C4×Dic13C22×C412yes—
110C4×Dic13C22×C412yes—
111C4×Dic13C22×C412yes—
112C4×Dic13C22×C412yes—
113C2×#432—131no—
114C22×C4×C8—131no—
115#2283—131no—
116#1770—132no—
117C8×C4⋊C4—132no—
118#1752—132no—
119C8×C22⋊C4—132no—
120#1746—132no—
121#1726—132no—
122C23×C13—22no—
123C23×C13—22no—
124C23×C13C2×C4212yes—
125C23×C13#13114yes—
126C23×C13C23×C411yescharacteristic
127C2×C4×C13—22no—
128C2×C4×C13—22no—
129C2×C4×C13—28no—
130C2×C4×C13—24no—
131C2×C4×C13—24no—
132C2×C4×C13—24no—
133C2×C4×C13C2×C4212yes—
134C2×C4×C13C2×C4212yes—
135C2×Dic13—22no—
136C2×Dic13—22no—
137C2×Dic13—22no—
138C2×Dic13—22no—
139C2×Dic13—24no—
140C2×Dic13—28no—
141C2×Dic13—24no—
142C2×Dic13—24no—
143C2×Dic13—24no—
144C2×Dic13C2×C4212yes—
145C2×Dic13C2×C4212yes—
146C2×Dic13C2×C4212yes—
147C2×Dic13#13114yes—
148C2×Dic13C23×C411yescharacteristic
149C2×Dic13C23×C411yescharacteristic
150C2×Dic13C2×#4112yes—
151C2×Dic13C2×#4112yes—
152#1057—22no—
153#1057—22no—
154#1057—28no—
155#1057#11714yes—
156#1057#11714yes—
157C23×C8—131no—
158C23×C8—131no—
159C22×C42—131no—
160#432—138no—
161C2×C4⋊C8—132no—
162C2×C22⋊C8—132no—
163C2×#97—132no—
164C2×#97—134no—
165C2×C4×C8—131no—
166C2×C4×C8—138no—
167C2×C4×C8—131no—
168C2×C4×C8—131no—
169C2×C4×C8—131no—
170C2×C4×C8—134no—
171#383—132no—
172#382—132no—
173#381—131no—
174#381#26111yescharacteristic
175#380—132no—
176C4×C4⋊C4—132no—
177C4×C22⋊C4—132no—
178#376—132no—
179#336—134no—
180#336—134no—
181C22×C13—22no—
182C22×C13—28no—
183C22×C13—22no—
184C22×C13#40514yes—
185C22×C13C22×C4211yescharacteristic
186C22×C13C4×#4112yes—
187C22×C13C2×#13112yes—
188C22×C13C2×#13112yes—
189C4×C13—22no—
190C4×C13—22no—
191C4×C13—44no—
192Dic13—22no—
193Dic13—22no—
194Dic13—44no—
195Dic13#40514yes—
196Dic13C2×#11711yescharacteristic
197Dic13C2×#11711yescharacteristic
198Dic13C4×#4112yes—
199C23×C4—131no—
200C23×C4—131no—
201C23×C4C2×#26111yescharacteristic
202C22×C8—131no—
203C22×C8—131no—
204C22×C8—131no—
205C22×C8—131no—
206C22×C8—134no—
207C22×C8—134no—
208C22×C8—134no—
209C22×C8—264no—
210C22×C8—264no—
211C2×C4⋊C4—134no—
212C2×C22⋊C4—132no—
213C2×C22⋊C4C2×#26112yes—
214C2×C42—134no—
215C2×C42—134no—
216C2×C42—131no—
217C2×C42—131no—
218C2×C42—131no—
219C2×C42—131no—
220C2×C42—134no—
221C4⋊C8—138no—
222C22⋊C8—138no—
223#97—138no—
224#97—268no—
225C4×C8—134no—
226C4×C8—134no—
227C4×C8—134no—
228C4×C8—134no—
229C4×C8—268no—
230#95—134no—
231#95—132no—
232#95—134no—
233#95—132no—
234#95C2×#26112yes—
235#95C2×#26112yes—
236C2×C13—24no—
237C2×C13C4×#11711yescharacteristic
238C2×C13C2×#40511yescharacteristic
239C2×C13C2×#40511yescharacteristic
240C2×C13#296614yes—
241C24C22×#26111yescharacteristic
242C22×C4—24no—
243C22×C4—131no—
244C22×C4—131no—
245C22×C4—132no—
246C22×C4—132no—
247C22×C4—134no—
248C22×C4—131no—
249C22×C4—131no—
250C22×C4—134no—
251C22×C4—262no—
252C22×C4—262no—
253C22×C4—264no—
254C22×C4C4×#26114yes—
255C22×C4C22×#26114yes—
256C22×C4C22×#26111yescharacteristic
257C22×C4C22×#26111yescharacteristic
258C2×C8—132no—
259C2×C8—132no—
260C2×C8—134no—
261C2×C8—134no—
262C2×C8—132no—
263C2×C8—132no—
264C2×C8—138no—
265C2×C8—268no—
266C2×C8—268no—
267C2×C8—264no—
268C2×C8—264no—
269C2×C8—262no—
270C2×C8—262no—
271C4⋊C4—268no—
272C22⋊C4—24no—
273C22⋊C4—264no—
274C42—132no—
275C42—132no—
276C42—132no—
277C42—132no—
278C42—134no—
279C42—134no—
280C42—264no—
281C42—264no—
282C13#2047611yescharacteristic, complemented, Sylow
283C23—22no—
284C23—22no—
285C23C2×C4×#26112yes—
286C23#8823814yes—
287C23C23×#26111yescharacteristic
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—131no—
295C2×C4—131no—
296C2×C4—132no—
297C2×C4—134no—
298C2×C4—132no—
299C2×C4—132no—
300C2×C4—132no—
301C2×C4—132no—
302C2×C4—262no—
303C2×C4—262no—
304C2×C4—264no—
305C2×C4—268no—
306C2×C4—262no—
307C2×C4—262no—
308C2×C4—264no—
309C2×C4—264no—
310C2×C4—264no—
311C2×C4C2×C4×#26112yes—
312C2×C4C2×C4×#26112yes—
313C8—134no—
314C8—134no—
315C8—262no—
316C8—262no—
317C8—268no—
318C22—22no—
319C22—28no—
320C22—22no—
321C22#1171906214yes—
322C22C22×C4×#26111yescharacteristic
323C22C2×#8823812yes—
324C22D13⋊#13112yes—
325C22#41×#26112yes—
326C4—22no—
327C4—22no—
328C4—44no—
329C4—131no—
330C4—131no—
331C4—132no—
332C4—134no—
333C4—262no—
334C4—262no—
335C4—524no—
336C2—24no—
337C2#117×#26111yescharacteristic
338C2C2×#1171906211yescharacteristic
339C2#1313139611yescharacteristic
340C2#1313165114yes—
subgroupG/H
center, FrattiniC23C23×#261
commutatorC22×C13C22×C42
FittingC13×#381C4
socleC23×C13C23×C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenter#381#261
hypocenterC13#20476
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #13130631C2semidirect
4. #13130628C2semidirect
7. #13130555C2semidirect
8. #13130493C2semidirect
14. #11719128C4semidirect
18. C2×C4×#1057C4semidirect
282. C13#20476semidirect

Maximal subgroups

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

subgroupindex
C2×#117191282
C22×C4×#10572
#131306312
#131306282
#131305732
C22⋊C4×#10572
#131305552
#131304932
#131172272
#2047613