Finite Group Databasebeta

← #27153572

Subgroups — #27153572

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#20261424C211yesmaximal, characteristic, complemented
2#20261373C211yesmaximal, characteristic
3#20261091C211yesmaximal, characteristic, complemented
4#20261064C211yesmaximal, characteristic, complemented
5#20261059C211yesmaximal, characteristic, complemented
6#20261037C211yesmaximal, characteristic, complemented
7#20260975C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260960C211yesmaximal, characteristic, complemented
10#20260897C211yesmaximal, characteristic, complemented
11#20260887C211yesmaximal, characteristic, complemented
12#20260826C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#416×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095501C2211yescharacteristic, complemented
18C2×#13095500C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690803—21no—
22#15690802—21no—
23#15690801C2211yescharacteristic, complemented
24#15690799C2211yescharacteristic
25#15690793—21no—
26#15690792—21no—
27#15690790C2211yescharacteristic
28#15690789C2211yescharacteristic, complemented
29#15690786C2211yescharacteristic
30#15690784C2211yescharacteristic
31#15690782—21no—
32#15690778—21no—
33#15690775C2211yescharacteristic, complemented
34D4×Dic97C2211yescharacteristic
35C4×C22⋊D97C2211yescharacteristic
36#15690767—21no—
37C4⋊C4×D97C2211yescharacteristic
38C2×#13095488—21no—
39#15690743C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic, complemented
52C2×#41×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic, complemented
54#117×C97C2211yescharacteristic, complemented
55#103×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C2×C22⋊D97D411yescharacteristic
65D4×D97—22no—
66D4×D97—41no—
67C22×Dic97—21no—
68C2×C4×D97—21no—
69C2×C4×D97C2311yescharacteristic
70#13095501—41no—
71#13095500—41no—
72#13095499—22no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C8×D97—41no—
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490D411yescharacteristic
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88C22⋊Dic97D411yescharacteristic
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97#41×C97—22no—
98#41×C97—41no—
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—21no—
101C2×D4×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97C2311yescharacteristic
104C2×C8×C97—41no—
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C22⋊C4×C97—21no—
110C42×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—
120D4×C97—21no—
121D4×C97—42no—
122D4×C97#4112yes—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—42no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—22no—
132C22⋊D97—42no—
133C22⋊D97—81no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—81no—
138C2×Dic97C2×D411yescharacteristic
139C2×Dic97C2×D411yescharacteristic
140C2×Dic97C2×D411yescharacteristic
141D388—21no—
142D388—21no—
143C4×D97—21no—
144C4×D97—21no—
145C4×D97—42no—
146Dic194—21no—
147Dic194—21no—
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#452—971no—
197#450—971no—
198C4×SD16—971no—
199C8×D4—971no—
200#416D9711yescharacteristic, complemented
201C2×#102—971no—
202C2×#41C2×D9711yescharacteristic
203C22×D4—971no—
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8—971no—
208C22×C8C2×D9711yescharacteristic, complemented
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⋊Q8—971no—
218#121—971no—
219#121—1941no—
220#120—1941no—
221C4×Q8—971no—
222C4×D4—971no—
223C4×D4—971no—
224#117—1941no—
225#117C2×D9711yescharacteristic, complemented
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#103C2×D9711yescharacteristic, complemented
235#103C2×D9711yescharacteristic, complemented
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102C2×D9711yescharacteristic
241#102C2×D9711yescharacteristic
242C22⋊C8—971no—
243C22⋊C8—971no—
244C4×C8—971no—
245C24—1941no—
246#41—22no—
247#41—41no—
248C2×Q8—971no—
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—3881no—
264SD16—3881no—
265SD16—1942no—
266C2×C8—41no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—3881no—
270C2×C8C22×D9711yescharacteristic
271C2×C8C22×D9711yescharacteristic
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—1941no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4C22×D9711yescharacteristic
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—21no—
295C42—971no—
296C42—1941no—
297C23—21no—
298C23—7761no—
299C23—971no—
300C23—3882no—
301C23—3882no—
302C23—1941no—
303C23D4×D9711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1941no—
307Q8—1941no—
308D4—21no—
309D4—42no—
310D4—7761no—
311D4—3882no—
312D4—1942no—
313D4—1941no—
314D4—1941no—
315D4#1309551112yes—
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×#1309551111yescharacteristic
363C4#1569085511yescharacteristic
364C2—41no—
365C2—42no—
366C2—7761no—
367C2—1942no—
368C2#2026137511yescharacteristic
369C2#135×D9711yescharacteristic
370C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
prime radical#20261373C2
hypercenter#416D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20261424C2semidirect
3. #20261091C2semidirect
4. #20261064C2semidirect
5. #20261059C2semidirect
6. #20261037C2semidirect
7. #20260975C2semidirect
8. #20260964C2semidirect
9. #20260960C2semidirect
10. #20260897C2semidirect
11. #20260887C2semidirect
12. #20260826C2semidirect
13. C8×C22⋊D97C2semidirect
14. #102×D97C2semidirect
15. #416×C97C2semidirect
17. C2×#13095501C22semidirect
23. #15690801C22semidirect
28. #15690789C22semidirect
33. #15690775C22semidirect
39. #15690743C22semidirect
40. #15690742C22semidirect
41. #15690741C22semidirect
42. #15690739C22semidirect
43. #15690735C22semidirect
44. #15690734C22semidirect
49. C22⋊#11715895C22semidirect
51. C8×Dic97C22semidirect
53. C22×C8×C97C22semidirect
54. #117×C97C22semidirect
55. #103×C97C22semidirect
56. #103×C97C22semidirect
186. C97#3296semidirect
200. #416D97semidirect
208. C22×C8C2×D97semidirect
225. #117C2×D97semidirect
234. #103C2×D97semidirect
235. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614242
#202613732
#202610912
#202610642
#202610592
#202610372
#202609752
#202609642
#202609602
#202608972
#202608872
#202608262
C8×C22⋊D972
#102×D972
#416×C972
#329697