Finite Group Databasebeta

← #27153632

Subgroups — #27153632

Order 12416.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261394C211yesmaximal, characteristic, complemented
2#20261092C211yesmaximal, characteristic, complemented
3#20261076C211yesmaximal, characteristic, complemented
4#20261053C211yesmaximal, characteristic, complemented
5#20261039C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260962C211yesmaximal, characteristic, complemented
8#20260950C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic, complemented
10#20260873C211yesmaximal, characteristic, complemented
11C4×#13095499C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14C2×#15690733C211yesmaximal, characteristic, complemented
15#553×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23#15690798C2211yescharacteristic
24#15690796—21no—
25#15690789C2211yescharacteristic
26#15690788—21no—
27#15690784C2211yescharacteristic
28#15690782—21no—
29C4×Dic194C2211yescharacteristic
30#15690770—21no—
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690733C2211yescharacteristic
36#15690733C2211yescharacteristic
37#15690733C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690733C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#126×C97—21no—
49#125×C97C2211yescharacteristic
50#123×C97—21no—
51#121×C97—21no—
52#121×C97C2211yescharacteristic
53#121×C97C2211yescharacteristic
54C4×D4×C97—21no—
55C4×D4×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57#117×C97C2211yescharacteristic
58C2×C4⋊C4×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—41no—
61C2×Dic194—22no—
62C2×Dic194—21no—
63C2×Dic194—41no—
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71#13095499—22no—
72#13095499—41no—
73#13095496—41no—
74C2×#11715895—41no—
75C2×#11715895—41no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C2×D4×C97—21no—
89C2×D4×C97—21no—
90C2×D4×C97D411yescharacteristic
91C2×D4×C97C2311yescharacteristic
92C22×C4×C97—21no—
93C22×C4×C97—21no—
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115D4×C97—22no—
116D4×C97—21no—
117D4×C97—21no—
118D4×C97—42no—
119D4×C97—81no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—81no—
128C2×C4×C97—81no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2×D411yescharacteristic
133C2×C4×C97C2411yescharacteristic
134C22⋊D97—42no—
135C22⋊D97—42no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—42no—
140D388—21no—
141D388—21no—
142C4×D97—42no—
143Dic194—21no—
144Dic194—21no—
145Dic194—21no—
146Dic194—42no—
147Dic194#4112yes—
148#11715895—21no—
149#11715895—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97#13512yes—
161C22×C97C22×D411yescharacteristic
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—41no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—21no—
168C4×C97—21no—
169C4×C97—41no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97C2×#4111yescharacteristic
173C4×C97H(5,𝔽2)11yescharacteristic
174Dic97—22no—
175Dic97—41no—
176Dic97—41no—
177C2×C97—22no—
178C2×C97—41no—
179C2×C97—81no—
180C2×C97#54611yescharacteristic
181C2×C97C2×#13511yescharacteristic
182C2×C97#57711yescharacteristic
183D97—81no—
184#3301—971nomaximal, Sylow
185C97#330111yescharacteristic, complemented, Sylow
186#553D9711yescharacteristic, complemented
187#548—971no—
188#488—971no—
189#485—971no—
190#481—971no—
191#479—971no—
192#466—971no—
193#465—971no—
194#463—971no—
195#449—971no—
196#448—971no—
197C4×SD16—971no—
198C8×D4—971no—
199#416—971no—
200C2×#103—971no—
201C2×#41—971no—
202C22×Q8—971no—
203C2×SD16—971no—
204C2×SD16—971no—
205C2×D8—971no—
206C22×C8—971no—
207C22×C8—971no—
208#126—21no—
209#125C2×D9711yescharacteristic
210#124—971no—
211#124—1941no—
212#123—21no—
213C22⋊Q8—971no—
214C22⋊Q8—1941no—
215#121—21no—
216#121—971no—
217#121—1941no—
218#121C2×D9711yescharacteristic
219#121C2×D9711yescharacteristic
220C4×Q8—971no—
221C4×D4—21no—
222C4×D4—1941no—
223C4×D4C2×D9711yescharacteristic
224C4×D4C2×D9711yescharacteristic
225#117C2×D9711yescharacteristic
226C2×C4⋊C4C2×D9711yescharacteristic
227#107—971no—
228#106—971no—
229#106—971no—
230C4⋊C8—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#103—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241C22⋊C8—971no—
242C22⋊C8—971no—
243C4×C8—971no—
244#41—3881no—
245#41—1942no—
246C2×Q8—971no—
247C2×Q8—971no—
248C2×Q8—3881no—
249C2×Q8—1942no—
250C2×Q8—1941no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—971no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4C22⋊D9711yescharacteristic
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264SD16—3881no—
265SD16—1942no—
266D8—3881no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—971no—
271C2×C8—3881no—
272C2×C8—3881no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—971no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4C22⋊D9711yescharacteristic
280C4⋊C4C22×D9711yescharacteristic
281C4⋊C4C22×D9711yescharacteristic
282C4⋊C4C22×D9711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4C22⋊D9711yescharacteristic
292C22⋊C4C22⋊D9711yescharacteristic
293C42—21no—
294C42—1941no—
295C42C22×D9711yescharacteristic
296C23—21no—
297C23—1941no—
298C23C2×C22⋊D9711yescharacteristic
299C23C2×C22⋊D9711yescharacteristic
300Q8—972no—
301Q8—3882no—
302Q8—1941no—
303Q8—1941no—
304Q8—1941no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—3882no—
311D4—3882no—
312D4—1941no—
313D4—1941no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—3882no—
324C2×C4—3882no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4C2×C22⋊D9711yescharacteristic
329C2×C4C2×C22⋊D9711yescharacteristic
330C2×C4C2×C22⋊D9711yescharacteristic
331C2×C4C2×C22⋊D9711yescharacteristic
332C2×C4C23×D9711yescharacteristic
333C8—1941no—
334C8—1941no—
335C8—1941no—
336C8—1941no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—3881no—
346C22—3881no—
347C22—3881no—
348C22#1569083012yes—
349C22C22×C22⋊D9711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3881no—
358C4—3881no—
359C4—1942no—
360C4#41×D9711yescharacteristic
361C4#1569085611yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—7761no—
366C2#2026137311yescharacteristic
367C2C2×#1569083011yescharacteristic
368C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#553×C97C2
socleC22×C97C22×D4
hypercenter#553D97
hypocenterC97#3301
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20261394C2semidirect
2. #20261092C2semidirect
3. #20261076C2semidirect
4. #20261053C2semidirect
5. #20261039C2semidirect
6. #20260969C2semidirect
7. #20260962C2semidirect
8. #20260950C2semidirect
9. #20260878C2semidirect
10. #20260873C2semidirect
11. C4×#13095499C2semidirect
12. D4×#11715895C2semidirect
13. #20260729C2semidirect
14. C2×#15690733C2semidirect
15. #553×C97C2semidirect
185. C97#3301semidirect
186. #553D97semidirect

Maximal subgroups

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

subgroupindex
#202613942
#202610922
#202610762
#202610532
#202610392
#202609692
#202609622
#202609502
#202608782
#202608732
C4×#130954992
D4×#117158952
#202607292
C2×#156907332
#553×C972
#330197