Finite Group Databasebeta

← #27152087

Subgroups — #27152087

Order 12416.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261362C211yesmaximal, characteristic, complemented
2#20261303C211yesmaximal, characteristic
3#20261100C211yesmaximal, characteristic, complemented
4#20261098C211yesmaximal, characteristic, complemented
5#20261096C211yesmaximal, characteristic, complemented
6#20261091C211yesmaximal, characteristic, complemented
7#20261089C211yesmaximal, characteristic, complemented
8#20261083C211yesmaximal, characteristic, complemented
9#20261080C211yesmaximal, characteristic, complemented
10#20260738C211yesmaximal, characteristic, complemented
11#20260735C211yesmaximal, characteristic, complemented
12#20260730C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic, complemented
14#20260692C211yesmaximal, characteristic, complemented
15#489×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690809C2211yescharacteristic, complemented
21#15690808—21no—
22#15690807C2211yescharacteristic
23#15690806C2211yescharacteristic
24#15690802C2211yescharacteristic, complemented
25#15690801C2211yescharacteristic, complemented
26#15690800—21no—
27#15690799C2211yescharacteristic
28#15690788—21no—
29#15690786—21no—
30#15690783—21no—
31#15690783—21no—
32#15690780—21no—
33#15690780—21no—
34#15690768C2211yescharacteristic
35#15690767C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690734C2211yescharacteristic, complemented
38#15690734C2211yescharacteristic, complemented
39#15690732C2211yescharacteristic
40#15690732C2211yescharacteristic
41#15690732C2211yescharacteristic, complemented
42#15690732C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic
44#15690731C2211yescharacteristic
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic, complemented
48#15690730C2211yescharacteristic, complemented
49#15690720C2211yescharacteristic, complemented
50C4×#11715895C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C4⋊Q8×C97C2211yescharacteristic, complemented
53#124×C97C2211yescharacteristic
54#103×C97C2211yescharacteristic, complemented
55#103×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095512—22no—
61#13095512—22no—
62C2×Dic194—21no—
63C2×D388—21no—
64C2×D388—21no—
65C2×D388—21no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71D4×D97—22no—
72D4×D97—22no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97C2311yescharacteristic
76#13095494—22no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C8×D97—22no—
80C4⋊Dic97—21no—
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490D411yescharacteristic
89#13095490D411yescharacteristic
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488D411yescharacteristic
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97—21no—
98C2×Q8×C97C2311yescharacteristic
99C2×D4×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97—21no—
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C22×D97—21no—
110C22×D97—21no—
111C22×D97—42no—
112C22×D97—81no—
113C22×D97—81no—
114C22×D97C2×D411yescharacteristic
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—81no—
118D4×C97—21no—
119D4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C22⋊D97—42no—
127C22⋊D97—81no—
128C22⋊D97—81no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97C2×D411yescharacteristic
132C2×Dic97C2×D411yescharacteristic
133C2×Dic97C2×D411yescharacteristic
134D388—21no—
135D388—21no—
136D388—21no—
137D388—21no—
138D388—42no—
139D388—81no—
140D388—81no—
141C4×D97—42no—
142C4×D97—81no—
143C4×D97#4112yes—
144C4×D97#4112yes—
145Dic194—81no—
146C8×C97—21no—
147C8×C97—21no—
148#11715895—21no—
149#11715895—21no—
150C22×C97—41no—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97C22×D411yescharacteristic
154C2×D97—21no—
155C2×D97—21no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—84no—
163C2×D97#12412yes—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97C2×#4111yescharacteristic
171Dic97—21no—
172Dic97—21no—
173Dic97—41no—
174Dic97—41no—
175Dic97#12412yes—
176C2×C97—81no—
177C2×C97C2×#12411yescharacteristic
178C2×C97#57611yescharacteristic
179C2×C97#57711yescharacteristic
180D97—22no—
181D97—81no—
182D97—81no—
183#3136—971nomaximal, Sylow
184C97#313611yescharacteristic, complemented, Sylow
185#544—971no—
186#535—971no—
187#490—971no—
188#490—971no—
189#489—971no—
190#489D9711yescharacteristic, complemented
191#488—971no—
192#488—971no—
193#486—971no—
194#486—971no—
195#418—971no—
196#417—971no—
197#416—971no—
198C2×#102—971no—
199#405—971no—
200C2×#41—971no—
201C22×D4—971no—
202C2×M16—971no—
203C22×C8—971no—
204C4⋊Q8—971no—
205C4⋊Q8—1941no—
206C4⋊Q8C2×D9711yescharacteristic, complemented
207#127—971no—
208#127—971no—
209#125—971no—
210#125—971no—
211#125—1941no—
212#124—971no—
213#124C2×D9711yescharacteristic
214C22⋊Q8—1941no—
215C22⋊Q8—1941no—
216#121—1941no—
217#121—1941no—
218#120—1941no—
219#120—1941no—
220#117—971no—
221#117—971no—
222C2×C4⋊C4—971no—
223#103—971no—
224#103—971no—
225#103C2×D9711yescharacteristic, complemented
226#103C2×D9711yescharacteristic, complemented
227#102—971no—
228#102—971no—
229#102—971no—
230#102—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237#102C2×D9711yescharacteristic
238#102C2×D9711yescharacteristic
239#97—971no—
240#97C2×D9711yescharacteristic, complemented
241C4×C8—971no—
242C4×C8—971no—
243C24—1941no—
244#41—1942no—
245#41—1942no—
246C2×Q8—21no—
247C2×Q8—1941no—
248C2×Q8C22×D9711yescharacteristic
249C2×D4—971no—
250C2×D4—971no—
251C2×D4—1941no—
252C2×D4—1942no—
253C2×D4—1942no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4—1941no—
257C2×D4—1941no—
258C2×D4—1941no—
259C2×D4C22×D9711yescharacteristic
260C22×C4—971no—
261C22×C4—1941no—
262C22×C4—1941no—
263M16—1942no—
264C2×C8—971no—
265C2×C8—971no—
266C2×C8—1942no—
267C2×C8C22×D9711yescharacteristic
268C2×C8C22×D9711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—971no—
271C4⋊C4—971no—
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—1941no—
275C4⋊C4—1941no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4C22×D9711yescharacteristic
281C4⋊C4C22×D9711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—971no—
285C22⋊C4—971no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C42—971no—
290C42—1941no—
291C42C22×D9711yescharacteristic
292C23—21no—
293C23—7761no—
294C23—7761no—
295C23—971no—
296C23—3882no—
297C23—1941no—
298C23—1941no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—7761no—
303D4—21no—
304D4—21no—
305D4—7761no—
306D4—7761no—
307D4—7761no—
308D4—7761no—
309D4—3882no—
310D4—3882no—
311D4—1941no—
312D4—1941no—
313D4—1941no—
314D4—1941no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—7761no—
319C2×C4—971no—
320C2×C4—972no—
321C2×C4—972no—
322C2×C4—971no—
323C2×C4—971no—
324C2×C4—3882no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4D4×D9711yescharacteristic
328C2×C4D4×D9711yescharacteristic
329C2×C4C23×D9711yescharacteristic
330C8—21no—
331C8—21no—
332C8—1941no—
333C8—1941no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—7764no—
338C22—972no—
339C22—3881no—
340C22—3881no—
341C22—3881no—
342C22—3881no—
343C22—3881no—
344C22—3881no—
345C22—1941no—
346C22—1941no—
347C22C2×D4×D9711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—972no—
354C4—3881no—
355C4—3881no—
356C4—1941no—
357C4—1941no—
358C4#41×D9711yescharacteristic
359C4#41×D9711yescharacteristic
360C2—81no—
361C2—7761no—
362C2—7761no—
363C2—1942no—
364C2#124×D9711yescharacteristic
365C2#2026153911yescharacteristic
366C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#489×C97C2
socleC22×C97C22×D4
prime radical#20261303C2
hypercenter#489D97
hypocenterC97#3136
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20261362C2semidirect
3. #20261100C2semidirect
4. #20261098C2semidirect
5. #20261096C2semidirect
6. #20261091C2semidirect
7. #20261089C2semidirect
8. #20261083C2semidirect
9. #20261080C2semidirect
10. #20260738C2semidirect
11. #20260735C2semidirect
12. #20260730C2semidirect
13. #102×D97C2semidirect
14. #20260692C2semidirect
15. #489×C97C2semidirect
20. #15690809C22semidirect
24. #15690802C22semidirect
25. #15690801C22semidirect
37. #15690734C22semidirect
38. #15690734C22semidirect
41. #15690732C22semidirect
42. #15690732C22semidirect
47. #15690730C22semidirect
48. #15690730C22semidirect
49. #15690720C22semidirect
50. C4×#11715895C22semidirect
51. C8×Dic97C22semidirect
52. C4⋊Q8×C97C22semidirect
54. #103×C97C22semidirect
55. #103×C97C22semidirect
58. #97×C97C22semidirect
184. C97#3136semidirect
190. #489D97semidirect
206. C4⋊Q8C2×D97semidirect
225. #103C2×D97semidirect
226. #103C2×D97semidirect
240. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613622
#202613032
#202611002
#202610982
#202610962
#202610912
#202610892
#202610832
#202610802
#202607382
#202607352
#202607302
#102×D972
#202606922
#489×C972
#313697