Finite Group Databasebeta

← #27153564

Subgroups — #27153564

Order 12416.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261419C211yesmaximal, characteristic, complemented
2#20261373C211yesmaximal, characteristic
3#20261091C211yesmaximal, characteristic, complemented
4#20261064C211yesmaximal, characteristic, complemented
5#20261060C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20260973C211yesmaximal, characteristic, complemented
8#20260966C211yesmaximal, characteristic, complemented
9#20260959C211yesmaximal, characteristic, complemented
10#20260884C211yesmaximal, characteristic, complemented
11SD16×Dic97C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13#20260731C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#452×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C22×#11715895C2211yescharacteristic, complemented
19C2×C8×D97C2211yescharacteristic
20#15690803—21no—
21#15690801C2211yescharacteristic, complemented
22#15690800—21no—
23#15690799C2211yescharacteristic
24#15690793—21no—
25#15690792—21no—
26#15690786C2211yescharacteristic
27#15690786C2211yescharacteristic
28#15690784C2211yescharacteristic
29#15690782—21no—
30#15690778—21no—
31Q8×Dic97C2211yescharacteristic, complemented
32D4×Dic97C2211yescharacteristic
33C4×C22⋊D97C2211yescharacteristic
34#15690769C2211yescharacteristic, complemented
35#15690768—21no—
36C4⋊C4×D97C2211yescharacteristic
37C2×#13095488—21no—
38#15690744C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic, complemented
52C2×#41×C97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C22⋊D97—21no—
63C2×C22⋊D97D411yescharacteristic
64D4×D97—22no—
65D4×D97—41no—
66C22×Dic97—21no—
67C2×C4×D97—21no—
68C2×C4×D97C2311yescharacteristic
69#13095498—41no—
70C2×#11715895—41no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C8×D97—41no—
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97D411yescharacteristic
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488D411yescharacteristic
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97C2311yescharacteristic
96#41×C97—22no—
97#41×C97—41no—
98C2×Q8×C97C2311yescharacteristic
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—21no—
101C2×D4×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97C2311yescharacteristic
104Q16×C97—41no—
105SD16×C97—22no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—42no—
115C22×D97—42no—
116C22×D97—81no—
117C22×D97C2×D411yescharacteristic
118Q8×C97—21no—
119Q8×C97—21no—
120Q8×C97—21no—
121Q8×C97—21no—
122D4×C97—21no—
123D4×C97—42no—
124D4×C97#4112yes—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—42no—
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2411yescharacteristic
133C22⋊D97—22no—
134C22⋊D97—42no—
135C22⋊D97—81no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—81no—
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142C2×Dic97C2×D411yescharacteristic
143D388—21no—
144D388—21no—
145C4×D97—21no—
146C4×D97—21no—
147C4×D97—42no—
148C8×C97—21no—
149C8×C97—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—22no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97C22×D411yescharacteristic
159C2×D97—21no—
160C2×D97—21no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—84no—
165C2×D97#13512yes—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170C4×C97C2×#4111yescharacteristic
171C4×C97H(5,𝔽2)11yescharacteristic
172Dic97—21no—
173Dic97—21no—
174Dic97—21no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97—42no—
179C2×C97—41no—
180C2×C97#54611yescharacteristic
181C2×C97C2×#13511yescharacteristic
182C2×C97#57711yescharacteristic
183D97—22no—
184D97—81no—
185#3296—971nomaximal, Sylow
186C97#329611yescharacteristic, complemented, Sylow
187#552—971no—
188#546—971no—
189#488—971no—
190#483—971no—
191#482—971no—
192#478—971no—
193#467—971no—
194#465—971no—
195#464—971no—
196#452D9711yescharacteristic, complemented
197#450—971no—
198C4×SD16—971no—
199C8×D4—971no—
200#416—971no—
201C2×#102—971no—
202C2×#41C2×D9711yescharacteristic
203C22×D4—971no—
204C2×Q16C2×D9711yescharacteristic, complemented
205C2×SD16—971no—
206C2×SD16C2×D9711yescharacteristic
207C22×C8—971no—
208C22×C8—971no—
209#126—1941no—
210#125—971no—
211#125—1941no—
212#124—971no—
213#123—1941no—
214#123—1941no—
215C22⋊Q8—971no—
216C22⋊Q8—971no—
217C22⋊Q8C2×D9711yescharacteristic, complemented
218#121—971no—
219#121—1941no—
220#120—1941no—
221C4×Q8—971no—
222C4×D4—971no—
223C4×D4—971no—
224#117—971no—
225#117—1941no—
226C2×C4⋊C4—971no—
227C2×C22⋊C4—1941no—
228#107—971no—
229#106—971no—
230#106—971no—
231C4⋊C8—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103C2×D9711yescharacteristic, complemented
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241#102C2×D9711yescharacteristic
242C22⋊C8—971no—
243C22⋊C8C2×D9711yescharacteristic, complemented
244C4×C8—971no—
245C24—1941no—
246#41—22no—
247#41—41no—
248C2×Q8C22×D9711yescharacteristic
249C2×Q8C22×D9711yescharacteristic
250C2×D4—21no—
251C2×D4—971no—
252C2×D4—971no—
253C2×D4—3881no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4—1942no—
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—971no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4C22×D9711yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—3881no—
266C2×C8—971no—
267C2×C8—971no—
268C2×C8—3881no—
269C2×C8—3881no—
270C2×C8C22×D9711yescharacteristic
271C2×C8C22×D9711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—971no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4—1941no—
283C4⋊C4C22×D9711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—971no—
286C22⋊C4—971no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C42—971no—
295C42—1941no—
296C42—1941no—
297C23—21no—
298C23—7761no—
299C23—971no—
300C23—3882no—
301C23—3882no—
302C23—1941no—
303C23D4×D9711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—42no—
310D4—7761no—
311D4—3882no—
312D4—1942no—
313D4—1941no—
314D4—1941no—
315D4#1309551012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—7761no—
323C2×C4—971no—
324C2×C4—971no—
325C2×C4—971no—
326C2×C4—3882no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1941no—
337C8—1941no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—7764no—
345C22—972no—
346C22—3881no—
347C22—3881no—
348C22—3881no—
349C22—1941no—
350C22—1941no—
351C22C2×D4×D9711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3881no—
357C4—3881no—
358C4—3881no—
359C4—1941no—
360C4—1941no—
361C4—1941no—
362C4C2×#1309551011yescharacteristic
363C4#1569085511yescharacteristic
364C2—41no—
365C2—42no—
366C2—7761no—
367C2—1942no—
368C2#2026137211yescharacteristic
369C2#135×D9711yescharacteristic
370C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#452×C97C2
socleC22×C97C22×D4
prime radical#20261373C2
hypercenter#452D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #20261419C2semidirect
3. #20261091C2semidirect
4. #20261064C2semidirect
5. #20261060C2semidirect
6. #20261033C2semidirect
7. #20260973C2semidirect
8. #20260966C2semidirect
9. #20260959C2semidirect
10. #20260884C2semidirect
11. SD16×Dic97C2semidirect
12. #20260798C2semidirect
13. #20260731C2semidirect
14. #102×D97C2semidirect
15. #452×C97C2semidirect
18. C22×#11715895C22semidirect
21. #15690801C22semidirect
31. Q8×Dic97C22semidirect
34. #15690769C22semidirect
38. #15690744C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690734C22semidirect
43. #15690734C22semidirect
44. #15690734C22semidirect
51. C8×Dic97C22semidirect
53. C2×Q16×C97C22semidirect
55. C22⋊Q8×C97C22semidirect
56. #103×C97C22semidirect
58. C22⋊C8×C97C22semidirect
186. C97#3296semidirect
196. #452D97semidirect
204. C2×Q16C2×D97semidirect
217. C22⋊Q8C2×D97semidirect
235. #103C2×D97semidirect
243. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614192
#202613732
#202610912
#202610642
#202610602
#202610332
#202609732
#202609662
#202609592
#202608842
SD16×Dic972
#202607982
#202607312
#102×D972
#452×C972
#329697