Finite Group Databasebeta

← #27153568

Subgroups — #27153568

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#20261423C211yesmaximal, characteristic, complemented
2#20261380C211yesmaximal, characteristic
3#20261092C211yesmaximal, characteristic, complemented
4#20261067C211yesmaximal, characteristic, complemented
5#20261062C211yesmaximal, characteristic, complemented
6#20260977C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260958C211yesmaximal, characteristic, complemented
9#20260898C211yesmaximal, characteristic, complemented
10#20260887C211yesmaximal, characteristic, complemented
11C4×#13095499C211yesmaximal, characteristic, complemented
12#20260800C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#478×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×Dic388C2211yescharacteristic, complemented
19C2×#13095500C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic, complemented
23#15690805—21no—
24#15690801—21no—
25#15690798C2211yescharacteristic
26#15690795—21no—
27#15690794—21no—
28#15690790C2211yescharacteristic
29#15690790C2211yescharacteristic
30#15690790C2211yescharacteristic, complemented
31#15690785C2211yescharacteristic
32#15690784—21no—
33#15690780—21no—
34C4×Dic194C2211yescharacteristic, complemented
35C4×D388C2211yescharacteristic
36#15690768—21no—
37#15690768C2211yescharacteristic, complemented
38C4⋊C4×D97C2211yescharacteristic
39C2×#13095488—21no—
40#15690741C2211yescharacteristic, complemented
41#15690735C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51C4×#11715895C2211yescharacteristic, complemented
52#125×C97C2211yescharacteristic, complemented
53C4×D4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60#13095510—22no—
61#13095510—41no—
62C2×Dic194C2311yescharacteristic
63C2×Dic194C2311yescharacteristic
64C2×D388—21no—
65C2×D388D411yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69D4×D97—22no—
70D4×D97—41no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75Dic388—41no—
76#13095500—41no—
77#13095499—22no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—41no—
81C8×D97—41no—
82C4⋊Dic97—21no—
83C4⋊Dic97—21no—
84C4⋊Dic97—21no—
85C4⋊Dic97D411yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488D411yescharacteristic
97#13095488D411yescharacteristic
98C4×Dic97—21no—
99C4×Dic97—21no—
100C2×D4×C97C2311yescharacteristic
101C22×C4×C97—21no—
102C2×C8×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C4⋊C4×C97—21no—
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C22⋊C4×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C22×D97—21no—
113C22×D97—42no—
114C22×D97—42no—
115C22×D97—81no—
116C22×D97C2×D411yescharacteristic
117C22×D97C2×D411yescharacteristic
118D4×C97—21no—
119D4×C97#4112yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—81no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C22⋊D97—42no—
128C22⋊D97—42no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—42no—
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135D388—22no—
136D388—21no—
137D388—21no—
138D388—81no—
139C4×D97—21no—
140C4×D97—21no—
141C4×D97—21no—
142C4×D97—21no—
143C4×D97—42no—
144Dic194—21no—
145Dic194—21no—
146Dic194—21no—
147Dic194—21no—
148C8×C97—21no—
149C8×C97—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—22no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97C22×D411yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—21no—
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—21no—
168C4×C97—41no—
169C4×C97—41no—
170C4×C97—41no—
171C4×C97C2×#4111yescharacteristic
172C4×C97H(5,𝔽2)11yescharacteristic
173Dic97—21no—
174Dic97—21no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97—42no—
179C2×C97#54611yescharacteristic
180C2×C97C2×#13511yescharacteristic
181C2×C97#57711yescharacteristic
182D97—22no—
183D97—41no—
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#478D9711yescharacteristic, complemented
193#467—971no—
194#465—971no—
195#464—971no—
196#452—971no—
197#450—971no—
198C4×SD16—971no—
199C8×D4—971no—
200#416—971no—
201C2×#102—971no—
202C2×#41—971no—
203C22×D4—971no—
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8—971no—
208C22×C8—971no—
209#126—1941no—
210#125—1941no—
211#125C2×D9711yescharacteristic, complemented
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#107C2×D9711yescharacteristic, complemented
229#106—971no—
230#106C2×D9711yescharacteristic, complemented
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#102C2×D9711yescharacteristic
241#102C2×D9711yescharacteristic
242C22⋊C8—971no—
243C22⋊C8—971no—
244C4×C8—971no—
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—1941no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4—1942no—
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—971no—
260C22×C4—971no—
261C22×C4—1941no—
262C22×C4—1941no—
263Q16—3881no—
264SD16—3881no—
265SD16—1942no—
266C2×C8—971no—
267C2×C8—971no—
268C2×C8—3881no—
269C2×C8—3881no—
270C2×C8C22×D9711yescharacteristic
271C2×C8C22×D9711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4C22×D9711yescharacteristic
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—1941no—
295C42—1941no—
296C42C22×D9711yescharacteristic
297C23—21no—
298C23—7761no—
299C23—971no—
300C23—971no—
301C23—3882no—
302C23—3882no—
303C23—1941no—
304Q8—1941no—
305Q8—1941no—
306Q8—1941no—
307Q8—1941no—
308D4—21no—
309D4—7761no—
310D4—3882no—
311D4—3882no—
312D4—1942no—
313D4—1941no—
314D4—1941no—
315D4#1309551212yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—3882no—
323C2×C4—3882no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4D4×D9711yescharacteristic
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1941no—
337C8—1941no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—7764no—
342C22—972no—
343C22—3881no—
344C22—3881no—
345C22—3881no—
346C22—1941no—
347C22—1941no—
348C22—1941no—
349C22—1941no—
350C22—1941no—
351C22C2×D4×D9711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3881no—
358C4—3881no—
359C4—3881no—
360C4—1941no—
361C4—1941no—
362C4C2×#1309551211yescharacteristic
363C4#1569085611yescharacteristic
364C2—42no—
365C2—7761no—
366C2—3881no—
367C2—1942no—
368C2#2026137611yescharacteristic
369C2#135×D9711yescharacteristic
370C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#478×C97C2
socleC22×C97C22×D4
prime radical#20261380C2
hypercenter#478D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20261423C2semidirect
3. #20261092C2semidirect
4. #20261067C2semidirect
5. #20261062C2semidirect
6. #20260977C2semidirect
7. #20260965C2semidirect
8. #20260958C2semidirect
9. #20260898C2semidirect
10. #20260887C2semidirect
11. C4×#13095499C2semidirect
12. #20260800C2semidirect
13. #20260728C2semidirect
14. #102×D97C2semidirect
15. #478×C97C2semidirect
18. C2×Dic388C22semidirect
22. C2×C8×D97C22semidirect
30. #15690790C22semidirect
34. C4×Dic194C22semidirect
37. #15690768C22semidirect
40. #15690741C22semidirect
41. #15690735C22semidirect
42. #15690733C22semidirect
43. #15690733C22semidirect
44. #15690733C22semidirect
50. #15690722C22semidirect
51. C4×#11715895C22semidirect
52. #125×C97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. C4⋊C8×C97C22semidirect
186. C97#3296semidirect
192. #478D97semidirect
211. #125C2×D97semidirect
228. #107C2×D97semidirect
230. #106C2×D97semidirect
231. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614232
#202613802
#202610922
#202610672
#202610622
#202609772
#202609652
#202609582
#202608982
#202608872
C4×#130954992
#202608002
#202607282
#102×D972
#478×C972
#329697