Finite Group Databasebeta

← #27150761

Subgroups — #27150761

Order 12416.

456 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×D97C211yesmaximal, characteristic
2#20260809C211yesmaximal, characteristic, complemented
3#20260808C211yesmaximal, characteristic, complemented
4#20260807C211yesmaximal, characteristic, complemented
5#20260806C211yesmaximal, characteristic, complemented
6#20260805C211yesmaximal, characteristic, complemented
7#20260804C211yesmaximal, characteristic, complemented
8#20260803C211yesmaximal, characteristic, complemented
9#20260791C211yesmaximal, characteristic, complemented
10#20260790C211yesmaximal, characteristic, complemented
11#20260698C211yesmaximal, characteristic, complemented
12#20260697C211yesmaximal, characteristic, complemented
13C2×#15690722C211yesmaximal, characteristic, complemented
14C22⋊C8×D97C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C411yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C2×#13095495C2211yescharacteristic, complemented
20C2×#13095494C2211yescharacteristic, complemented
21C2×#13095494C2211yescharacteristic, complemented
22C22×#11715895C2211yescharacteristic, complemented
23C2×C8×D97C2211yescharacteristic, complemented
24C2×C8×D97C2211yescharacteristic, complemented
25C4×D388C2211yescharacteristic
26D4×Dic97C411yescharacteristic
27#15690772C411yescharacteristic
28C4×C22⋊D97C2211yescharacteristic
29C4×C22⋊D97C2211yescharacteristic
30#15690770C411yescharacteristic
31#15690770C411yescharacteristic
32C4⋊C4×D97C411yescharacteristic
33C22⋊C4×D97C411yescharacteristic
34C22⋊C4×D97C411yescharacteristic
35C42×D97C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic, complemented
37#15690739C2211yescharacteristic, complemented
38#15690739C2211yescharacteristic, complemented
39C22⋊#11715895C2211yescharacteristic, complemented
40C22⋊#11715895C2211yescharacteristic, complemented
41#15690722C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic, complemented
46#15690722C2211yescharacteristic, complemented
47#15690722C2211yescharacteristic, complemented
48#15690722C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50#15690720C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52C2×M16×C97C2211yescharacteristic, complemented
53C22×C8×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic, complemented
56C22⋊C8×C97C2211yescharacteristic, complemented
57C22⋊C8×C97C2211yescharacteristic, complemented
58#97×C97C2211yescharacteristic, complemented
59C23×D97C2×C411yescharacteristic
60C23×D97C2×C411yescharacteristic
61C2×D388C2×C411yescharacteristic
62C2×C22⋊D97C2×C411yescharacteristic
63C2×C22⋊D97C2×C411yescharacteristic
64D4×D97—24no—
65C22×Dic97C2×C411yescharacteristic
66C22×Dic97C2×C411yescharacteristic
67C2×C4×D97—41no—
68C2×C4×D97—41no—
69C2×C4×D97—41no—
70C2×C4×D97—41no—
71C2×C4×D97C2×C411yescharacteristic
72C2×C4×D97C2×C411yescharacteristic
73C2×C4×D97C2×C411yescharacteristic
74C2×C4×D97C2×C411yescharacteristic
75C2×C4×D97C2×C411yescharacteristic
76C2×C4×D97C2×C411yescharacteristic
77C2×C4×D97C2311yescharacteristic
78C2×C4×D97C2311yescharacteristic
79C2×C4×D97C2311yescharacteristic
80#13095495—41no—
81#13095494—41no—
82#13095494—41no—
83C2×#11715895—41no—
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C8×D97—41no—
89C8×D97—41no—
90C4⋊Dic97C2×C411yescharacteristic
91#13095490C2×C411yescharacteristic
92#13095490C2×C411yescharacteristic
93C22⋊Dic97C2×C411yescharacteristic
94C22⋊Dic97C2×C411yescharacteristic
95#13095488C2×C411yescharacteristic
96#13095488C2×C411yescharacteristic
97#13095488C2×C411yescharacteristic
98#13095488C2×C411yescharacteristic
99C4×Dic97C2×C411yescharacteristic
100C4×Dic97C2×C411yescharacteristic
101C4×Dic97C2311yescharacteristic
102C2×D4×C97C2×C411yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105M16×C97—41no—
106C2×C8×C97—41no—
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C2×C8×C97C2311yescharacteristic
110C2×C8×C97C2311yescharacteristic
111C4⋊C4×C97C2×C411yescharacteristic
112C22⋊C4×C97C2×C411yescharacteristic
113C22⋊C4×C97C2×C411yescharacteristic
114C42×C97C2311yescharacteristic
115C23×C97C22×C411yescharacteristic
116C23×C97C22×C411yescharacteristic
117C22×D97—22no—
118C22×D97—41no—
119C22×D97—41no—
120C22×D97—41no—
121C22×D97—41no—
122C22×D97M1614yescomplemented
123C22×D97C22×C411yescharacteristic
124C22×D97C22×C411yescharacteristic
125C22×D97C22×C411yescharacteristic
126D4×C97—22no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97C22×C411yescharacteristic
132C2×C4×C97C22×C411yescharacteristic
133C2×C4×C97C22×C411yescharacteristic
134C2×C4×C97C22×C411yescharacteristic
135C2×C4×C97C2411yescharacteristic
136C22⋊D97—22no—
137C22⋊D97—22no—
138C2×Dic97—41no—
139C2×Dic97—41no—
140C2×Dic97C22×C411yescharacteristic
141C2×Dic97C22×C411yescharacteristic
142C2×Dic97C22×C411yescharacteristic
143C2×Dic97C22×C411yescharacteristic
144C2×Dic97C22×C411yescharacteristic
145D388—22no—
146C4×D97—22no—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—21no—
150C4×D97—21no—
151C4×D97—21no—
152C4×D97—21no—
153C4×D97—41no—
154C4×D97—41no—
155C4×D97—41no—
156C4×D97—41no—
157C4×D97—41no—
158C8×C97—21no—
159C8×C97—21no—
160C8×C97—21no—
161C8×C97—21no—
162#11715895—21no—
163#11715895—21no—
164#11715895—21no—
165#11715895—21no—
166C22×C97—21no—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97C2×M1612yes—
172C22×C97C23×C411yescharacteristic
173C2×D97—21no—
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—44no—
181C2×D97—44no—
182C2×D97C2×M1612yes—
183C2×D97C2×M1612yes—
184C4×C97—21no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—21no—
188C4×C97H(5,𝔽2)11yescharacteristic
189C4×C97H(5,𝔽2)11yescharacteristic
190Dic97—21no—
191Dic97—21no—
192Dic97—21no—
193Dic97—21no—
194Dic97—21no—
195C2×C97—22no—
196C2×C97—41no—
197C2×C97#51811yescharacteristic
198C2×C97C22×M1611yescharacteristic
199C2×C97#56811yescharacteristic
200D97—22no—
201D97—22no—
202D97—41no—
203#2973—971nomaximal, Sylow
204C97#297311yescharacteristic, complemented, Sylow
205C2×C4×D4—971no—
206#435—971no—
207#435—971no—
208#435—971no—
209#435—971no—
210#435—971no—
211#435—971no—
212#435—971no—
213#435D9711yescharacteristic, complemented
214#432—971no—
215#432—971no—
216#407—971no—
217#407—971no—
218C2×C22⋊C8—971no—
219C2×C22⋊C8—971no—
220C22×D4—971no—
221C23×C4—971no—
222C23×C4—971no—
223C2×M16—971no—
224C2×M16—971no—
225C2×M16—971no—
226C2×M16C2×D9711yescharacteristic, complemented
227C22×C8—971no—
228C22×C8—971no—
229C22×C8—971no—
230C22×C8C2×D9711yescharacteristic, complemented
231C4×D4—971no—
232C4×D4—971no—
233C4×D4—971no—
234C4×D4—971no—
235C4×D4—971no—
236C4×D4—971no—
237C4×D4—971no—
238C4×D4C2×D9711yescharacteristic
239C2×C4⋊C4—971no—
240C2×C22⋊C4—971no—
241C2×C22⋊C4—971no—
242C2×C42—971no—
243C4⋊C8—971no—
244C4⋊C8—971no—
245C4⋊C8—971no—
246C4⋊C8C2×D9711yescharacteristic, complemented
247C22⋊C8—971no—
248C22⋊C8—971no—
249C22⋊C8—971no—
250C22⋊C8—971no—
251C22⋊C8—971no—
252C22⋊C8—971no—
253C22⋊C8—971no—
254C22⋊C8—971no—
255C22⋊C8—971no—
256C22⋊C8—971no—
257C22⋊C8C2×D9711yescharacteristic, complemented
258C22⋊C8C2×D9711yescharacteristic, complemented
259#97—971no—
260#97—971no—
261#97—971no—
262#97C2×D9711yescharacteristic, complemented
263C24—971no—
264C24—971no—
265C2×D4—971no—
266C2×D4—971no—
267C2×D4—971no—
268C2×D4—1944no—
269C2×D4C4×D9711yescharacteristic
270C22×C4—971no—
271C22×C4—971no—
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—3881no—
282C22×C4—3881no—
283C22×C4—3881no—
284C22×C4—3881no—
285C22×C4C22×D9711yescharacteristic
286C22×C4C22×D9711yescharacteristic
287M16—41no—
288M16—3881no—
289M16—3881no—
290M16—3881no—
291C2×C8—41no—
292C2×C8—971no—
293C2×C8—971no—
294C2×C8—971no—
295C2×C8—971no—
296C2×C8—3881no—
297C2×C8—3881no—
298C2×C8—3881no—
299C2×C8C22×D9711yescharacteristic
300C2×C8C22×D9711yescharacteristic
301C2×C8C22×D9711yescharacteristic
302C2×C8C22×D9711yescharacteristic
303C4⋊C4—971no—
304C4⋊C4—971no—
305C4⋊C4—971no—
306C4⋊C4C4×D9711yescharacteristic
307C22⋊C4—971no—
308C22⋊C4—971no—
309C22⋊C4—971no—
310C22⋊C4—971no—
311C22⋊C4—971no—
312C22⋊C4—971no—
313C22⋊C4C4×D9711yescharacteristic
314C22⋊C4C4×D9711yescharacteristic
315C42—971no—
316C42—971no—
317C42—971no—
318C42C22×D9711yescharacteristic
319C23—974no—
320C23—971no—
321C23—971no—
322C23—971no—
323C23—3881no—
324C23—3881no—
325C23—3881no—
326C23—3881no—
327C23—1942no—
328C23C2×C4×D9711yescharacteristic
329C23C2×C4×D9711yescharacteristic
330D4—22no—
331D4—1942no—
332D4—1942no—
333D4—1942no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—971no—
339C2×C4—971no—
340C2×C4—971no—
341C2×C4—971no—
342C2×C4—971no—
343C2×C4—3881no—
344C2×C4—3881no—
345C2×C4—3881no—
346C2×C4—3881no—
347C2×C4—3881no—
348C2×C4—3881no—
349C2×C4—3881no—
350C2×C4—1942no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4C2×C4×D9711yescharacteristic
358C2×C4C2×C4×D9711yescharacteristic
359C2×C4C2×C4×D9711yescharacteristic
360C2×C4C2×C4×D9711yescharacteristic
361C2×C4C23×D9711yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8—21no—
366C8—1941no—
367C8—1941no—
368C8—1941no—
369C8—1941no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—972no—
376C22—972no—
377C22—3884no—
378C22—3884no—
379C22—1941no—
380C22—1941no—
381C22—1941no—
382C22—1941no—
383C22—1941no—
384C22—1941no—
385C22—1941no—
386C22M16×D9712yes—
387C22C22×C4×D9711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—1941no—
393C4—1941no—
394C4—1941no—
395C4—1941no—
396C4—1941no—
397C4#1569085511yescharacteristic
398C4#1569085611yescharacteristic
399C2—22no—
400C2—41no—
401C2—3881no—
402C2—1942no—
403C2—1942no—
404C2#2026120711yescharacteristic
405C2C2×M16×D9711yescharacteristic
406C2#2026150011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#435×C97C2
prime radicalC2×D4×D97C4
cosocleC2×C4×C97C24
hypercenter#435D97
hypocenterC97#2973
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #20260809C2semidirect
3. #20260808C2semidirect
4. #20260807C2semidirect
5. #20260806C2semidirect
6. #20260805C2semidirect
7. #20260804C2semidirect
8. #20260803C2semidirect
9. #20260791C2semidirect
10. #20260790C2semidirect
11. #20260698C2semidirect
12. #20260697C2semidirect
13. C2×#15690722C2semidirect
14. C22⋊C8×D97C2semidirect
15. #435×C97C2semidirect
19. C2×#13095495C22semidirect
20. C2×#13095494C22semidirect
21. C2×#13095494C22semidirect
22. C22×#11715895C22semidirect
23. C2×C8×D97C22semidirect
24. C2×C8×D97C22semidirect
36. C4⋊#11715895C22semidirect
37. #15690739C22semidirect
38. #15690739C22semidirect
39. C22⋊#11715895C22semidirect
40. C22⋊#11715895C22semidirect
45. #15690722C22semidirect
46. #15690722C22semidirect
47. #15690722C22semidirect
48. #15690722C22semidirect
49. #15690721C22semidirect
50. #15690720C22semidirect
51. #15690720C22semidirect
52. C2×M16×C97C22semidirect
53. C22×C8×C97C22semidirect
55. C4⋊C8×C97C22semidirect
56. C22⋊C8×C97C22semidirect
57. C22⋊C8×C97C22semidirect
58. #97×C97C22semidirect
122. C22×D97M16semidirect
204. C97#2973semidirect
213. #435D97semidirect
226. C2×M16C2×D97semidirect
230. C22×C8C2×D97semidirect
246. C4⋊C8C2×D97semidirect
257. C22⋊C8C2×D97semidirect
258. C22⋊C8C2×D97semidirect
262. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C4×D4×D972
#202608092
#202608082
#202608072
#202608062
#202608052
#202608042
#202608032
#202607912
#202607902
#202606982
#202606972
C2×#156907222
C22⋊C8×D972
#435×C972
#297397