Finite Group Databasebeta

← #27153576

Subgroups — #27153576

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#20261426C211yesmaximal, characteristic, complemented
2#20261380C211yesmaximal, characteristic
3#20261090C211yesmaximal, characteristic, complemented
4#20261067C211yesmaximal, characteristic, complemented
5#20261061C211yesmaximal, characteristic, complemented
6#20261035C211yesmaximal, characteristic, complemented
7#20260975C211yesmaximal, characteristic, complemented
8#20260967C211yesmaximal, characteristic, complemented
9#20260957C211yesmaximal, characteristic, complemented
10#20260894C211yesmaximal, characteristic, complemented
11#20260884C211yesmaximal, characteristic, complemented
12C8×D388C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15C4×SD16×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095501C2211yescharacteristic, complemented
19C2×#13095498C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic, complemented
22#15690804—21no—
23#15690802C2211yescharacteristic, complemented
24#15690800—21no—
25#15690798C2211yescharacteristic
26#15690795—21no—
27#15690794—21no—
28#15690790C2211yescharacteristic
29#15690786C2211yescharacteristic
30#15690786C2211yescharacteristic, complemented
31#15690785C2211yescharacteristic
32#15690784—21no—
33#15690780—21no—
34C4×D388C2211yescharacteristic
35#15690768C2211yescharacteristic, complemented
36#15690767—21no—
37C4⋊C4×D97C2211yescharacteristic
38C2×#13095488—21no—
39#15690743C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41C4⋊#11715895C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690722C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095510—22no—
61#13095510—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—21no—
64C2×D388D411yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68D4×D97—22no—
69D4×D97—41no—
70C22×Dic97—21no—
71C2×C4×D97—21no—
72C2×C4×D97C2311yescharacteristic
73C2×C4×D97C2311yescharacteristic
74#13095501—41no—
75#13095498—41no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C8×D97—41no—
79C8×D97—41no—
80C4⋊Dic97—21no—
81C4⋊Dic97D411yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
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—
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97C2311yescharacteristic
101C22×C4×C97—21no—
102SD16×C97—22no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C22⋊C4×C97—21no—
109C42×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
118Q8×C97—21no—
119Q8×C97—21no—
120D4×C97—21no—
121D4×C97#4112yes—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—81no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—42no—
130C22⋊D97—42no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—42no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137D388—22no—
138D388—21no—
139D388—21no—
140D388—81no—
141C4×D97—21no—
142C4×D97—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—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#478—971no—
193#467—971no—
194#465—971no—
195#464—971no—
196#452—971no—
197#450—971no—
198C4×SD16D9711yescharacteristic, complemented
199C8×D4—971no—
200#416—971no—
201C2×#102—971no—
202C2×#41—971no—
203C22×D4—971no—
204C2×Q16—971no—
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⋊Q8—971no—
218#121—971no—
219#121—1941no—
220#120—1941no—
221C4×Q8C2×D9711yescharacteristic, complemented
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#106C2×D9711yescharacteristic, complemented
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⋊C8—971no—
244C4×C8C2×D9711yescharacteristic, complemented
245C24—1941no—
246#41—3881no—
247#41—1942no—
248C2×Q8—971no—
249C2×Q8C22×D9711yescharacteristic
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—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—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—1941no—
296C42C22×D9711yescharacteristic
297C23—21no—
298C23—7761no—
299C23—971no—
300C23—971no—
301C23—3882no—
302C23—3882no—
303C23—1941no—
304Q8—21no—
305Q8—21no—
306Q8—1941no—
307Q8—1941no—
308D4—21no—
309D4—7761no—
310D4—3882no—
311D4—3882no—
312D4—1942no—
313D4—1941no—
314D4—1941no—
315D4#1309551112yes—
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×#1309551111yescharacteristic
363C4#1569085611yescharacteristic
364C2—42no—
365C2—7761no—
366C2—3881no—
367C2—1942no—
368C2#2026138111yescharacteristic
369C2#135×D9711yescharacteristic
370C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
prime radical#20261380C2
hypercenterC4×SD16D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #20261426C2semidirect
3. #20261090C2semidirect
4. #20261067C2semidirect
5. #20261061C2semidirect
6. #20261035C2semidirect
7. #20260975C2semidirect
8. #20260967C2semidirect
9. #20260957C2semidirect
10. #20260894C2semidirect
11. #20260884C2semidirect
12. C8×D388C2semidirect
13. #20260728C2semidirect
14. #102×D97C2semidirect
15. C4×SD16×C97C2semidirect
18. C2×#13095501C22semidirect
21. C2×C8×D97C22semidirect
23. #15690802C22semidirect
30. #15690786C22semidirect
35. #15690768C22semidirect
39. #15690743C22semidirect
40. #15690741C22semidirect
41. C4⋊#11715895C22semidirect
42. #15690735C22semidirect
43. #15690735C22semidirect
44. #15690733C22semidirect
51. #15690722C22semidirect
53. C4×Q8×C97C22semidirect
55. #106×C97C22semidirect
56. #103×C97C22semidirect
58. C4×C8×C97C22semidirect
186. C97#3296semidirect
198. C4×SD16D97semidirect
221. C4×Q8C2×D97semidirect
230. #106C2×D97semidirect
235. #103C2×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
#202614262
#202613802
#202610902
#202610672
#202610612
#202610352
#202609752
#202609672
#202609572
#202608942
#202608842
C8×D3882
#202607282
#102×D972
C4×SD16×C972
#329697