Finite Group Databasebeta

← #16800792

Subgroups — #16800792

Order 3328.

990 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131313C212yesmaximal
2#13131027C214yesmaximal
3#13131026C214yesmaximal
4#13130734C212yesmaximal
5#13130733C212yesmaximal
6C2×#11719394C211yesmaximal, characteristic
7C2×C13×#125C411yescharacteristic, complemented
8#11719394—28no—
9C2×#88151C2212yes—
10C2×#88150C2212yes—
11C2×#88146C2211yescharacteristic
12C2×C4⋊C4×D13C2211yescharacteristic
13C2×#88144C411yescharacteristic, complemented
14C2×#88143C412yescomplemented
15C2×#88142C412yescomplemented
16C2×#88142C411yescharacteristic, complemented
17C2×#88126C2211yescharacteristic
18C2×#88123C411yescharacteristic, complemented
19#11719136C2214yes—
20#11719135C2214yes—
21C4×C22⋊#261C2214yes—
22#11719082—28no—
23#11718289C2214yes—
24#11718289C2218yes—
25#11718289C2214yes—
26C22×C4×D13C2311yescharacteristic
27C22×C4×D13C2312yes—
28C22×Dic26C2×C412yes—
29C22×Dic26C2×C412yes—
30C2×C22⋊#261C2318yes—
31C2×C4⋊#261—24no—
32C2×C4×#261—24no—
33C13×#125—24no—
34C2×C13×C4⋊C4C2×C412yes—
35C2×C13×C4⋊C4C2×C412yes—
36C2×C13×C4⋊C4C2×C411yescharacteristic
37C2×C13×C4⋊C4C2×C411yescharacteristic
38C2×C42×C13C2×C411yescharacteristic
39C2×#6103C2311yescharacteristic
40C2×#6103C2311yescharacteristic
41C2×#6103C2312yes—
42C2×C4⋊Dic13C2×C411yescharacteristic
43C2×C4⋊Dic13C2×C412yes—
44C2×C4⋊Dic13C2×C412yes—
45C2×C4⋊Dic13C2×C411yescharacteristic
46C2×#6101C2×C411yescharacteristic
47C2×#6101C2×C411yescharacteristic
48C2×#6101C2×C412yes—
49C2×#6101C2×C412yes—
50C2×#6101C2×C412yes—
51C2×C4×Dic13C2×C411yescharacteristic
52C2×C4×Dic13C2×C412yes—
53#88151—24no—
54#88151—24no—
55#88150—24no—
56#88150—24no—
57#88146—24no—
58C4⋊C4×D13D418yes—
59#88144—24no—
60#88143—28no—
61#88142—24no—
62#88142—28no—
63#88126—24no—
64#88123—24no—
65(C2×#261)⋊C4—24no—
66(C2×#261)⋊C4—28no—
67(C2×#261)⋊C4—24no—
68#21828—131nomaximal, Sylow
69C23×D13C2411yescharacteristic
70C22×#261—28no—
71C22×C4×C13C22×C411yescharacteristic
72C22×C4×C13C22×C412yes—
73C22×C4×C13C22×C411yescharacteristic
74C22×C4×C13C22×C411yescharacteristic
75C22×C4×C13C22×C412yes—
76C22×Dic13C22×C411yescharacteristic
77C22×Dic13C22×C411yescharacteristic
78C22×Dic13C22×C411yescharacteristic
79C22×Dic13C22×C412yes—
80C22×Dic13C22×C412yes—
81C2×C4×D13—24no—
82C2×C4×D13—82no—
83C2×C4×D13C2×D414yes—
84C2×C4×D13C2×D414yes—
85C2×C4×D13C2×D414yes—
86C2×Dic26—24no—
87C2×Dic26—24no—
88C2×Dic26—82no—
89C2×Dic26—82no—
90C22⋊#261—216no—
91C4⋊#261—84no—
92C4×#261—84no—
93C13×C4⋊C4—24no—
94C13×C4⋊C4—24no—
95C13×C4⋊C4—22no—
96C13×C4⋊C4C22⋊C414yes—
97C42×C13—22no—
98#6103—22no—
99#6103—22no—
100#6103—22no—
101#6103—22no—
102C4⋊Dic13—22no—
103C4⋊Dic13—24no—
104C4⋊Dic13—24no—
105C4⋊Dic13C22⋊C414yes—
106#6101—24no—
107#6101—22no—
108#6101—22no—
109#6101—24no—
110#6101C22⋊C418yes—
111C4×Dic13—22no—
112C4×Dic13—24no—
113C2×#544—131no—
114#2313—132no—
115#1996—134no—
116#1996—134no—
117#1820—132no—
118#1820—132no—
119C23×C13C23×C411yescharacteristic
120C22×D13—41no—
121C22×D13—41no—
122C22×D13C22×D412yes—
123C22×D132−1+414yes—
124C2×#261—48no—
125C2×#261—48no—
126C2×#261—48no—
127C2×C4×C13—22no—
128C2×C4×C13—22no—
129C2×C4×C13—21no—
130C2×C4×C13—21no—
131C2×C4×C13—42no—
132C2×C4×C13—41no—
133C2×C4×C13—42no—
134C2×C4×C13—41no—
135C2×C4×C13C2×C22⋊C412yes—
136C2×C4×C13C2×C22⋊C414yes—
137C2×Dic13—22no—
138C2×Dic13—21no—
139C2×Dic13—21no—
140C2×Dic13—22no—
141C2×Dic13—42no—
142C2×Dic13—42no—
143C2×Dic13—41no—
144C2×Dic13—41no—
145C2×Dic13C2×C22⋊C412yes—
146C2×Dic13C2×C22⋊C414yes—
147C4×D13—28no—
148C4×D13—44no—
149C4×D13—44no—
150Dic26—84no—
151Dic26—84no—
152#544—268no—
153C2×C4⋊Q8—132no—
154C2×C4⋊Q8—131no—
155C2×C4⋊Q8—131no—
156C2×#125—132no—
157C2×#125—131no—
158C2×#125#26111yescharacteristic, complemented
159C2×C22⋊Q8—132no—
160C2×C22⋊Q8—132no—
161C2×#117—131no—
162C2×#117—131no—
163C22×C4⋊C4—131no—
164#387—268no—
165#381—134no—
166#380—134no—
167C4×C22⋊C4—134no—
168#342—134no—
169#342—134no—
170#342—138no—
171C22×C13—21no—
172C22×C13—21no—
173C22×C13C22×C22⋊C411yescharacteristic
174C22×C13#51912yes—
175C2×D13—22no—
176C2×D13—22no—
177C2×D13—44no—
178C2×D13#53614yes—
179#261—88no—
180C4×C13—22no—
181C4×C13—24no—
182C4×C13—41no—
183C4×C13—42no—
184C4×C13—41no—
185Dic13—22no—
186Dic13—24no—
187Dic13—42no—
188Dic13—41no—
189Dic13—41no—
190C22×Q8—132no—
191C22×Q8—132no—
192C23×C4—131no—
193C23×C4—132no—
194C4⋊Q8—264no—
195C4⋊Q8—268no—
196C4⋊Q8—264no—
197#125—24no—
198#125—268no—
199#125—264no—
200C22⋊Q8—264no—
201C22⋊Q8—264no—
202C22⋊Q8—264no—
203C22⋊Q8—264no—
204#117—264no—
205#117—264no—
206C2×C4⋊C4—132no—
207C2×C4⋊C4—131no—
208C2×C4⋊C4—138no—
209C2×C4⋊C4—131no—
210C2×C4⋊C4—131no—
211C2×C4⋊C4—131no—
212C2×C4⋊C4—132no—
213C2×C4⋊C4—132no—
214C2×C4⋊C4—132no—
215C2×C4⋊C4—132no—
216C2×C4⋊C4—264no—
217C2×C4⋊C4C2×#26111yescharacteristic
218C2×C4⋊C4C2×#26112yes—
219C2×C4⋊C4C2×#26112yes—
220C2×C4⋊C4C2×#26111yescharacteristic
221C2×C22⋊C4—131no—
222C2×C22⋊C4—132no—
223C2×C22⋊C4—131no—
224C2×C22⋊C4—138no—
225C2×C42—132no—
226C2×C42—131no—
227C2×C42—264no—
228C2×C42C2×#26111yescharacteristic
229#95—264no—
230#95—264no—
231#95—268no—
232C2×C13—22no—
233C2×C13#231011yescharacteristic
234C2×C13#288311yescharacteristic
235C2×C13#288411yescharacteristic
236D13—24no—
237C24—131no—
238C2×Q8—264no—
239C2×Q8—264no—
240C2×Q8—1042no—
241C2×Q8—1042no—
242C22×C4—131no—
243C22×C4—132no—
244C22×C4—134no—
245C22×C4—134no—
246C22×C4—134no—
247C22×C4—131no—
248C22×C4—131no—
249C22×C4—132no—
250C22×C4—264no—
251C22×C4—268no—
252C22×C4—1042no—
253C22×C4C22×#26111yescharacteristic
254C22×C4C22×#26112yes—
255C22×C4C22×#26111yescharacteristic
256C22×C4C22×#26111yescharacteristic
257C22×C4C22×#26112yes—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—138no—
262C4⋊C4—134no—
263C4⋊C4—262no—
264C4⋊C4—262no—
265C4⋊C4—264no—
266C4⋊C4—264no—
267C4⋊C4—264no—
268C4⋊C4—262no—
269C4⋊C4—264no—
270C4⋊C4—1044no—
271C4⋊C4C22⋊#26114yes—
272C22⋊C4—262no—
273C22⋊C4—262no—
274C22⋊C4—262no—
275C22⋊C4—262no—
276C22⋊C4—2616no—
277C42—22no—
278C42—264no—
279C42—262no—
280C42—1044no—
281C13#2182811yescharacteristic, complemented, Sylow
282C23—132no—
283C23—134no—
284C23—521no—
285C23—521no—
286C23C23×#26111yescharacteristic
287Q8—1044no—
288Q8—1044no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—132no—
298C2×C4—134no—
299C2×C4—268no—
300C2×C4—262no—
301C2×C4—261no—
302C2×C4—262no—
303C2×C4—261no—
304C2×C4—524no—
305C2×C4—524no—
306C2×C4—522no—
307C2×C4—521no—
308C2×C4—521no—
309C2×C4—522no—
310C2×C4—528no—
311C2×C4—528no—
312C2×C4—528no—
313C2×C4C2×C22⋊#26112yes—
314C2×C4C2×C22⋊#26114yes—
315C22—21no—
316C22—21no—
317C22—134no—
318C22—262no—
319C22—262no—
320C22—524no—
321C22#1171962712yes—
322C22C22×C22⋊#26111yescharacteristic
323C4—22no—
324C4—24no—
325C4—41no—
326C4—41no—
327C4—42no—
328C4—262no—
329C4—264no—
330C4—521no—
331C4—522no—
332C4—521no—
333C4—1048no—
334C2—22no—
335C2—264no—
336C2#1313131011yescharacteristic
337C2#1313143611yescharacteristic
338C2#1313143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#261
commutatorC23×C13C23×C4
FrattiniC23C23×#261
FittingC2×C13×#125C4
socleC22×C13C22×C22⋊C4
prime radical, cosocleC23×D13C24
hypercenterC2×#125#261
hypocenterC13#21828
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
7. C2×C13×#125C4semidirect
13. C2×#88144C4semidirect
14. C2×#88143C4semidirect
15. C2×#88142C4semidirect
16. C2×#88142C4semidirect
18. C2×#88123C4semidirect
158. C2×#125#261semidirect
281. C13#21828semidirect

Maximal subgroups

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

subgroupindex
#131313132
#131310272
#131310262
#131307342
#131307332
C2×#117193942
#2182813