Finite Group Databasebeta

← #27153592

Subgroups — #27153592

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261437C211yesmaximal, characteristic, complemented
2#20261376C211yesmaximal, characteristic, complemented
3#20261091C211yesmaximal, characteristic, complemented
4#20261068C211yesmaximal, characteristic, complemented
5#20261061C211yesmaximal, characteristic, complemented
6#20261034C211yesmaximal, characteristic, complemented
7#20260970C211yesmaximal, characteristic, complemented
8#20260952C211yesmaximal, characteristic, complemented
9#20260879C211yesmaximal, characteristic, complemented
10#20260868C211yesmaximal, characteristic, complemented
11#20260815C211yesmaximal, characteristic, complemented
12C8×C22⋊D97C211yesmaximal, characteristic, complemented
13#20260730C211yesmaximal, characteristic, complemented
14C2×#15690732C211yesmaximal, characteristic, complemented
15#465×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic, complemented
21#15690803—21no—
22#15690801C2211yescharacteristic, complemented
23#15690799C2211yescharacteristic
24#15690795—21no—
25#15690794—21no—
26#15690785—21no—
27#15690785C2211yescharacteristic
28#15690782—21no—
29#15690781C2211yescharacteristic, complemented
30#15690780—21no—
31D4×Dic97—21no—
32D4×Dic97C2211yescharacteristic, complemented
33#15690772C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35#15690768C2211yescharacteristic, complemented
36C2×C4⋊Dic97C2211yescharacteristic
37C22⋊C4×D97—21no—
38#15690743C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690739C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic
44#15690732C2211yescharacteristic
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic, complemented
53C22×C8×C97C2211yescharacteristic
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095512—22no—
61#13095512—41no—
62C2×D388—22no—
63C2×D388—21no—
64C2×D388—21no—
65C2×D388—21no—
66C2×D388—41no—
67C2×D388C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97D411yescharacteristic
71C22×Dic97—21no—
72C22×Dic97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97C2311yescharacteristic
76#13095496—22no—
77#13095496—41no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—41no—
81C4⋊Dic97—21no—
82C4⋊Dic97—21no—
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490D411yescharacteristic
87C22⋊Dic97—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97D411yescharacteristic
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488D411yescharacteristic
96C4×Dic97—21no—
97C4×Dic97C2311yescharacteristic
98C2×Q8×C97C2311yescharacteristic
99C2×D4×C97—21no—
100C2×D4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102SD16×C97—41no—
103C2×C8×C97—41no—
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C4⋊C4×C97—21no—
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C22⋊C4×C97—21no—
110C22⋊C4×C97—21no—
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—81no—
118C22×D97C2×D411yescharacteristic
119Q8×C97—21no—
120Q8×C97—21no—
121D4×C97—21no—
122D4×C97—21no—
123D4×C97—42no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—22no—
132C22⋊D97—81no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—81no—
136C2×Dic97—81no—
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139C2×Dic97C2×D411yescharacteristic
140D388—21no—
141D388—21no—
142D388—21no—
143D388—42no—
144D388—42no—
145D388—81no—
146D388#4112yes—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—42no—
150C4×D97—81no—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97#13512yes—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—22no—
164C2×D97—21no—
165C2×D97—21no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—84no—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97C2×#4111yescharacteristic
177C4×C97H(5,𝔽2)11yescharacteristic
178Dic97—21no—
179Dic97—21no—
180Dic97—21no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—81no—
185C2×C97#54611yescharacteristic
186C2×C97C2×#13511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—41no—
189D97—42no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
199#466—971no—
200#465D9711yescharacteristic, complemented
201#463—971no—
202#449—971no—
203#447—971no—
204C4×D8—971no—
205C8×D4—971no—
206#416—971no—
207C2×#102—971no—
208C2×#41—971no—
209C22×D4—971no—
210C2×SD16C2×D9711yescharacteristic, complemented
211C2×D8—971no—
212C2×D8—971no—
213C22×C8—971no—
214C22×C8C2×D9711yescharacteristic
215#126—1941no—
216#125—971no—
217#124—971no—
218#123—1941no—
219#123—1941no—
220C22⋊Q8C2×D9711yescharacteristic
221#121—971no—
222#121—971no—
223#121—1941no—
224#121—1941no—
225#121C2×D9711yescharacteristic
226#120—1941no—
227C4×D4—971no—
228C4×D4—971no—
229C4×D4—971no—
230C4×D4—1941no—
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
236#106C2×D9711yescharacteristic, complemented
237C4⋊C8—971no—
238#103—971no—
239#103—971no—
240#103C2×D9711yescharacteristic, complemented
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102C2×D9711yescharacteristic, complemented
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—21no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—3881no—
260C2×D4—1941no—
261C2×D4—1942no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1941no—
265C2×D4C22×D9711yescharacteristic
266C22×C4—971no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4C22×D9711yescharacteristic
272SD16—41no—
273D8—3881no—
274D8—1942no—
275C2×C8—41no—
276C2×C8—971no—
277C2×C8—971no—
278C2×C8—3881no—
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—971no—
293C22⋊C4—971no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C42—971no—
302C42—1941no—
303C23—21no—
304C23—7761no—
305C23—971no—
306C23—3882no—
307C23—3882no—
308C23—1941no—
309C23—1941no—
310C23D4×D9711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—7761no—
317D4—7761no—
318D4—972no—
319D4—3882no—
320D4—3882no—
321D4—1942no—
322D4—1941no—
323D4—1941no—
324D4—1941no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—7761no—
331C2×C4—7761no—
332C2×C4—7761no—
333C2×C4—971no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—3882no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4D4×D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1941no—
346C8—1941no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—7764no—
353C22—3881no—
354C22—3881no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—1941no—
359C22—1941no—
360C22—1942no—
361C22—1941no—
362C22#1569083112yes—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3881no—
369C4—3881no—
370C4—1941no—
371C4—1941no—
372C4—1941no—
373C4#41×D9711yescharacteristic
374C4#1569085511yescharacteristic
375C2—22no—
376C2—81no—
377C2—7761no—
378C2—3882no—
379C2—3881no—
380C2#2026137411yescharacteristic
381C2C2×#1569083111yescharacteristic
382C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#465×C97C2
socleC22×C97C22×D4
hypercenter#465D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20261437C2semidirect
2. #20261376C2semidirect
3. #20261091C2semidirect
4. #20261068C2semidirect
5. #20261061C2semidirect
6. #20261034C2semidirect
7. #20260970C2semidirect
8. #20260952C2semidirect
9. #20260879C2semidirect
10. #20260868C2semidirect
11. #20260815C2semidirect
12. C8×C22⋊D97C2semidirect
13. #20260730C2semidirect
14. C2×#15690732C2semidirect
15. #465×C97C2semidirect
20. C2×C8×D97C22semidirect
22. #15690801C22semidirect
29. #15690781C22semidirect
32. D4×Dic97C22semidirect
35. #15690768C22semidirect
38. #15690743C22semidirect
39. #15690743C22semidirect
40. #15690739C22semidirect
41. #15690734C22semidirect
42. #15690734C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
52. C2×SD16×C97C22semidirect
56. #106×C97C22semidirect
57. #103×C97C22semidirect
58. #102×C97C22semidirect
192. C97#3297semidirect
200. #465D97semidirect
210. C2×SD16C2×D97semidirect
236. #106C2×D97semidirect
240. #103C2×D97semidirect
247. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614372
#202613762
#202610912
#202610682
#202610612
#202610342
#202609702
#202609522
#202608792
#202608682
#202608152
C8×C22⋊D972
#202607302
C2×#156907322
#465×C972
#329797