Finite Group Databasebeta

← #27152300

Subgroups — #27152300

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261269C211yesmaximal, characteristic, complemented
2#20261134C211yesmaximal, characteristic, complemented
3#20261117C211yesmaximal, characteristic, complemented
4#20261092C211yesmaximal, characteristic, complemented
5#20261089C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic
8#20260966C211yesmaximal, characteristic, complemented
9#20260962C211yesmaximal, characteristic, complemented
10C4×#13095501C211yesmaximal, characteristic, complemented
11C4×#13095498C211yesmaximal, characteristic, complemented
12C4×#13095499C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14C2×C4×#11715895C211yesmaximal, characteristic, complemented
15#529×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22#15690804—21no—
23#15690802C2211yescharacteristic
24#15690798C2211yescharacteristic
25#15690793—21no—
26#15690789C2211yescharacteristic
27#15690784C2211yescharacteristic
28C4×Dic194C2211yescharacteristic
29C4×D388C2211yescharacteristic
30C4×C22⋊D97—21no—
31#15690767—21no—
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34#15690741C2211yescharacteristic
35#15690741C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C4×#11715895C2211yescharacteristic
45C4×#11715895C2211yescharacteristic
46C4×#11715895C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#126×C97—21no—
49#125×C97C2211yescharacteristic
50#124×C97C2211yescharacteristic
51#123×C97—21no—
52C22⋊Q8×C97C2211yescharacteristic
53#121×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55C4×D4×C97—21no—
56C4×D4×C97C2211yescharacteristic
57#117×C97—21no—
58C2×C42×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C4×D97—21no—
63#13095501—22no—
64#13095499—22no—
65#13095498—22no—
66#13095496—22no—
67C2×#11715895—22no—
68C2×#11715895—22no—
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84C4×Dic97—21no—
85C2×Q8×C97C2311yescharacteristic
86C2×D4×C97—21no—
87C2×D4×C97C2311yescharacteristic
88C22×C4×C97—21no—
89C22×C4×C97D411yescharacteristic
90C22×C4×C97D411yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C42×C97—21no—
104C42×C97D411yescharacteristic
105C42×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113D4×C97—21no—
114D4×C97—21no—
115D4×C97—42no—
116C2×C4×C97—22no—
117C2×C4×C97—22no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—42no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—42no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134D388—21no—
135D388—21no—
136C4×D97—42no—
137Dic194—21no—
138Dic194—21no—
139#11715895—21no—
140#11715895—21no—
141#11715895#4112yes—
142#11715895#4112yes—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—41no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97#13512yes—
149C22×C97C22×D411yescharacteristic
150C2×D97—41no—
151C2×D97—41no—
152C2×D97—41no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—41no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97#13512yes—
161C4×C97C2×#4111yescharacteristic
162C4×C97C2×#4111yescharacteristic
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166C2×C97—22no—
167C2×C97—81no—
168C2×C97#53211yescharacteristic
169C2×C97C2×#13511yescharacteristic
170C2×C97C2×#13511yescharacteristic
171D97—81no—
172#3168—971nomaximal, Sylow
173C97#316811yescharacteristic, complemented, Sylow
174#529—971no—
175#529D9711yescharacteristic, complemented
176#499—971no—
177#495—971no—
178#488—971no—
179#488—971no—
180#467—971no—
181#466—971no—
182#465—971no—
183#465—971no—
184C4×Q16—971no—
185C4×SD16—971no—
186C4×SD16—971no—
187C4×D8—971no—
188C2×C4×C8—971no—
189C2×Q16—971no—
190C2×SD16—971no—
191C2×SD16—971no—
192C2×D8—971no—
193C22×C8—971no—
194C22×C8—971no—
195#126—21no—
196#126—1941no—
197#125—971no—
198#125C2×D9711yescharacteristic
199#124—971no—
200#124C2×D9711yescharacteristic
201#123—21no—
202#123—1941no—
203C22⋊Q8—971no—
204C22⋊Q8C2×D9711yescharacteristic
205#121—971no—
206#121C2×D9711yescharacteristic
207C4×Q8—971no—
208C4×Q8C2×D9711yescharacteristic
209C4×D4—21no—
210C4×D4—971no—
211C4×D4—1941no—
212C4×D4C2×D9711yescharacteristic
213#117—21no—
214#117—1941no—
215C2×C42C2×D9711yescharacteristic
216#107—971no—
217#107—971no—
218#106—971no—
219#106—971no—
220#103—971no—
221#103—971no—
222#103—971no—
223#103—971no—
224#102—971no—
225#102—971no—
226#102—971no—
227#102—971no—
228C4×C8—971no—
229C4×C8—971no—
230C4×C8—971no—
231C4×C8—971no—
232C2×Q8—971no—
233C2×Q8C22×D9711yescharacteristic
234C2×D4—21no—
235C2×D4—971no—
236C2×D4—1941no—
237C2×D4C22×D9711yescharacteristic
238C22×C4—21no—
239C22×C4—1941no—
240C22×C4C22⋊D9711yescharacteristic
241C22×C4C22⋊D9711yescharacteristic
242C22×C4C22×D9711yescharacteristic
243Q16—1942no—
244SD16—1942no—
245SD16—1942no—
246D8—1942no—
247C2×C8—971no—
248C2×C8—971no—
249C2×C8—971no—
250C2×C8—971no—
251C2×C8—1942no—
252C2×C8—1942no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4—1941no—
262C4⋊C4—1941no—
263C4⋊C4C22×D9711yescharacteristic
264C4⋊C4C22×D9711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C42—21no—
276C42—1941no—
277C42C22⋊D9711yescharacteristic
278C42C22⋊D9711yescharacteristic
279C42C22×D9711yescharacteristic
280C42C22×D9711yescharacteristic
281C23—21no—
282C23—1941no—
283C23C2×C22⋊D9711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—1941no—
287Q8—1941no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—3882no—
292D4—1941no—
293D4—1941no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3882no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C2×C22⋊D9711yescharacteristic
307C2×C4C2×C22⋊D9711yescharacteristic
308C2×C4C2×C22⋊D9711yescharacteristic
309C2×C4C2×C22⋊D9711yescharacteristic
310C2×C4C2×C22⋊D9711yescharacteristic
311C2×C4C23×D9711yescharacteristic
312C8—972no—
313C8—972no—
314C8—1941no—
315C8—1941no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—3881no—
322C22—3881no—
323C22—3881no—
324C22#1569083012yes—
325C22C22×C22⋊D9711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3881no—
334C4—3881no—
335C4—3881no—
336C4#1569083012yes—
337C4#41×D9711yescharacteristic
338C4#41×D9711yescharacteristic
339C2—22no—
340C2—81no—
341C2—7761no—
342C2#2026128111yescharacteristic
343C2C2×#1569083011yescharacteristic
344C2C2×#1569083011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#529×C97C2
socleC22×C97C22×D4
prime radical#20260969C2
hypercenter#529D97
hypocenterC97#3168
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20261269C2semidirect
2. #20261134C2semidirect
3. #20261117C2semidirect
4. #20261092C2semidirect
5. #20261089C2semidirect
6. #20260974C2semidirect
8. #20260966C2semidirect
9. #20260962C2semidirect
10. C4×#13095501C2semidirect
11. C4×#13095498C2semidirect
12. C4×#13095499C2semidirect
13. C4×#13095496C2semidirect
14. C2×C4×#11715895C2semidirect
15. #529×C97C2semidirect
173. C97#3168semidirect
175. #529D97semidirect

Maximal subgroups

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

subgroupindex
#202612692
#202611342
#202611172
#202610922
#202610892
#202609742
#202609692
#202609662
#202609622
C4×#130955012
C4×#130954982
C4×#130954992
C4×#130954962
C2×C4×#117158952
#529×C972
#316897