Finite Group Databasebeta

← #27153573

Subgroups — #27153573

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#20261425C211yesmaximal, characteristic, complemented
2#20261381C211yesmaximal, characteristic
3#20261090C211yesmaximal, characteristic, complemented
4#20261066C211yesmaximal, characteristic, complemented
5#20261060C211yesmaximal, characteristic, complemented
6#20261036C211yesmaximal, characteristic, complemented
7#20260976C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260958C211yesmaximal, characteristic, complemented
10#20260899C211yesmaximal, characteristic, complemented
11#20260888C211yesmaximal, characteristic, complemented
12C4×#13095500C211yesmaximal, characteristic, complemented
13#20260732C211yesmaximal, characteristic, complemented
14C2×#15690732C211yesmaximal, characteristic, complemented
15C8×D4×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×Dic388C2211yescharacteristic, complemented
19C2×#13095500C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21#15690803—21no—
22#15690802C2211yescharacteristic, complemented
23#15690800—21no—
24#15690798C2211yescharacteristic
25#15690795—21no—
26#15690791—21no—
27#15690789C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690789C2211yescharacteristic, complemented
30#15690784C2211yescharacteristic
31#15690783—21no—
32#15690778—21no—
33C4×Dic194C2211yescharacteristic, complemented
34C4×D388C2211yescharacteristic
35#15690769C2211yescharacteristic, complemented
36#15690766—21no—
37C2×C4⋊Dic97C2211yescharacteristic
38C22⋊C4×D97—21no—
39#15690744C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, complemented
41#15690742C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690735C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52C22×C8×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic, complemented
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×Dic194C2311yescharacteristic
64C2×D388—22no—
65C2×D388—21no—
66C2×D388—41no—
67C2×D388C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74Dic388—41no—
75#13095500—22no—
76#13095500—41no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97—21no—
90C22⋊Dic97—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C2×D4×C97D411yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C2×C8×C97—41no—
102C2×C8×C97—41no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—81no—
118D4×C97—22no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—42no—
129C22⋊D97—81no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—42no—
137D388—21no—
138D388—21no—
139D388—21no—
140D388—42no—
141D388#4112yes—
142C4×D97—42no—
143C4×D97—81no—
144Dic194—21no—
145Dic194—21no—
146Dic194—21no—
147Dic194—21no—
148C8×C97—21no—
149C8×C97—21no—
150C8×C97—21no—
151C8×C97—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97#13512yes—
158C22×C97C22×D411yescharacteristic
159C2×D97—22no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—41no—
165C2×D97—84no—
166C4×C97—21no—
167C4×C97—21no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97C2×#4111yescharacteristic
171C4×C97H(5,𝔽2)11yescharacteristic
172Dic97—41no—
173Dic97—41no—
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97—22no—
179C2×C97—41no—
180C2×C97#54611yescharacteristic
181C2×C97C2×#13511yescharacteristic
182C2×C97#57711yescharacteristic
183D97—42no—
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×D4D9711yescharacteristic, complemented
200#416—971no—
201C2×#102—971no—
202C2×#41—971no—
203C22×D4—971no—
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8C2×D9711yescharacteristic
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×D4C2×D9711yescharacteristic
224#117—971no—
225#117—1941no—
226C2×C4⋊C4—971no—
227C2×C22⋊C4—1941no—
228#107—971no—
229#106—971no—
230#106—971no—
231C4⋊C8C2×D9711yescharacteristic, complemented
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241#102—971no—
242C22⋊C8C2×D9711yescharacteristic
243C22⋊C8C2×D9711yescharacteristic, complemented
244C4×C8C2×D9711yescharacteristic, complemented
245C24—1941no—
246#41—3881no—
247#41—1942no—
248C2×Q8—971no—
249C2×Q8—971no—
250C2×D4—971no—
251C2×D4—971no—
252C2×D4—3881no—
253C2×D4—1942no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4—1941no—
257C2×D4D38811yescharacteristic
258C22×C4—1941no—
259C22×C4—1941no—
260C22×C4—1941no—
261C22×C4C22×D9711yescharacteristic
262C22×C4C22×D9711yescharacteristic
263Q16—3881no—
264SD16—3881no—
265SD16—1942no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D9711yescharacteristic
269C2×C8C22×D9711yescharacteristic
270C2×C8C22×D9711yescharacteristic
271C2×C8C22×D9711yescharacteristic
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4—1941no—
283C4⋊C4D38811yescharacteristic
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4D38811yescharacteristic
293C22⋊C4D38811yescharacteristic
294C42—1941no—
295C42—1941no—
296C42C22×D9711yescharacteristic
297C23—7761no—
298C23—3882no—
299C23—3882no—
300C23—1941no—
301C23—1941no—
302C23C2×D38811yescharacteristic
303C23C2×D38811yescharacteristic
304Q8—1941no—
305Q8—1941no—
306Q8—1941no—
307Q8—1941no—
308D4—22no—
309D4—7761no—
310D4—972no—
311D4—3882no—
312D4—3882no—
313D4—1941no—
314D4—1941no—
315D4—1941no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—7761no—
321C2×C4—3882no—
322C2×C4—3882no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4C2×D38811yescharacteristic
330C2×C4C2×D38811yescharacteristic
331C2×C4C2×D38811yescharacteristic
332C2×C4C2×D38811yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—7764no—
344C22—3881no—
345C22—3881no—
346C22—3881no—
347C22—3881no—
348C22—3881no—
349C22—1942no—
350C22#1569083212yes—
351C22C22×D38811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—3881no—
357C4—3881no—
358C4—3881no—
359C4—3881no—
360C4—3881no—
361C4—3881no—
362C4#41×D9711yescharacteristic
363C4#1569085511yescharacteristic
364C2—22no—
365C2—41no—
366C2—7761no—
367C2—3882no—
368C2#2026138011yescharacteristic
369C2C2×#1569083211yescharacteristic
370C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC8×D4×C97C2
socleC22×C97C22×D4
prime radical#20261381C2
hypercenterC8×D4D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20261425C2semidirect
3. #20261090C2semidirect
4. #20261066C2semidirect
5. #20261060C2semidirect
6. #20261036C2semidirect
7. #20260976C2semidirect
8. #20260964C2semidirect
9. #20260958C2semidirect
10. #20260899C2semidirect
11. #20260888C2semidirect
12. C4×#13095500C2semidirect
13. #20260732C2semidirect
14. C2×#15690732C2semidirect
15. C8×D4×C97C2semidirect
18. C2×Dic388C22semidirect
22. #15690802C22semidirect
29. #15690789C22semidirect
33. C4×Dic194C22semidirect
35. #15690769C22semidirect
39. #15690744C22semidirect
40. #15690742C22semidirect
41. #15690742C22semidirect
42. #15690735C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
45. #15690735C22semidirect
53. C22×C8×C97C22semidirect
55. C4⋊C8×C97C22semidirect
57. C22⋊C8×C97C22semidirect
58. C4×C8×C97C22semidirect
186. C97#3296semidirect
199. C8×D4D97semidirect
208. C22×C8C2×D97semidirect
231. C4⋊C8C2×D97semidirect
243. C22⋊C8C2×D97semidirect
244. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614252
#202613812
#202610902
#202610662
#202610602
#202610362
#202609762
#202609642
#202609582
#202608992
#202608882
C4×#130955002
#202607322
C2×#156907322
C8×D4×C972
#329697