Finite Group Databasebeta

← #27153694

Subgroups — #27153694

Order 12416.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261388C211yesmaximal, characteristic, complemented
2#20261380C211yesmaximal, characteristic, complemented
3#20261097C211yesmaximal, characteristic, complemented
4#20261067C211yesmaximal, characteristic, complemented
5#20261055C211yesmaximal, characteristic, complemented
6#20260989C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260881C211yesmaximal, characteristic, complemented
10#20260870C211yesmaximal, characteristic, complemented
11#20260845C211yesmaximal, characteristic, complemented
12#20260805C211yesmaximal, characteristic, complemented
13#20260733C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#476×C97C211yesmaximal, characteristic, complemented
16C2×#13095512—21no—
17C2×#13095510C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19C2×#13095500C2211yescharacteristic, complemented
20C2×D776C2211yescharacteristic
21C2×#13095496C2211yescharacteristic
22C2×#13095494C2211yescharacteristic, complemented
23C2×C8×D97C2211yescharacteristic
24#15690798C2211yescharacteristic
25#15690795—21no—
26#15690795—21no—
27#15690794—21no—
28#15690790C2211yescharacteristic
29#15690786—21no—
30#15690785C2211yescharacteristic
31#15690785C2211yescharacteristic
32#15690785C2211yescharacteristic, complemented
33#15690784—21no—
34#15690780—21no—
35C4×D388C2211yescharacteristic
36C4×D388C2211yescharacteristic, complemented
37C4⋊C4×D97C2211yescharacteristic
38C4⋊C4×D97C2211yescharacteristic, complemented
39C2×#13095488—21no—
40#15690741C2211yescharacteristic, complemented
41#15690735C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic, complemented
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51#15690721C2211yescharacteristic, complemented
52C4⋊Q8×C97C2211yescharacteristic, complemented
53C4×D4×C97C2211yescharacteristic
54#106×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60#13095512—44no—
61#13095510—22no—
62#13095510—41no—
63C2×Dic194C2311yescharacteristic
64C2×D388—21no—
65C2×D388—21no—
66C2×D388—21no—
67C2×D388D411yescharacteristic
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72D4×D97—22no—
73D4×D97—41no—
74C22×Dic97—21no—
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77C2×C4×D97—21no—
78C2×C4×D97C2311yescharacteristic
79C2×C4×D97C2311yescharacteristic
80#13095500—41no—
81D776—41no—
82#13095496—22no—
83#13095494—41no—
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C8×D97—41no—
87C4⋊Dic97—21no—
88C4⋊Dic97D411yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91#13095490—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488D411yescharacteristic
100#13095488D411yescharacteristic
101C2×Q8×C97—21no—
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C4⋊C4×C97—21no—
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C42×C97C2311yescharacteristic
112C23×C97—21no—
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—81no—
118C22×D97C2×D411yescharacteristic
119C22×D97C2×D411yescharacteristic
120Q8×C97—42no—
121D4×C97—21no—
122D4×C97#4112yescomplemented
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—81no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—42no—
131C22⋊D97—42no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—42no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137D388—22no—
138D388—21no—
139D388—21no—
140D388—21no—
141D388—21no—
142D388—42no—
143D388—42no—
144D388—81no—
145C4×D97—21no—
146C4×D97—21no—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—42no—
150C4×D97—42no—
151C4×D97—42no—
152Dic194—21no—
153Dic194—21no—
154C8×C97—21no—
155C8×C97—21no—
156#11715895—21no—
157#11715895—21no—
158C22×C97—22no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—21no—
165C2×D97—21no—
166C2×D97—21no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—41no—
172C2×D97—41no—
173C2×D97—84no—
174C2×D97AGL(1,ℤ/8)12yes—
175C4×C97—21no—
176C4×C97—21no—
177C4×C97—41no—
178C4×C97—41no—
179C4×C97—41no—
180C4×C97C2×#4111yescharacteristic
181C4×C97H(5,𝔽2)11yescharacteristic
182Dic97—21no—
183Dic97—21no—
184Dic97—41no—
185Dic97—41no—
186C2×C97—42no—
187C2×C97#54611yescharacteristic
188C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C97#57711yescharacteristic
190D97—22no—
191D97—41no—
192D97—81no—
193D97—81no—
194#3306—971nomaximal, Sylow
195C97#330611yescharacteristic, complemented, Sylow
196#547—971no—
197#546—971no—
198#489—971no—
199#483—971no—
200#481—971no—
201#476D9711yescharacteristic, complemented
202#469—971no—
203#465—971no—
204#463—971no—
205#449—971no—
206#447—971no—
207#442—971no—
208#435—971no—
209#417—971no—
210C2×#102—971no—
211C2×#41—971no—
212C2×#41—1941no—
213C22×D4—971no—
214C2×SD16—971no—
215C2×D8—971no—
216C2×D8—971no—
217C2×M16—971no—
218C22×C8—971no—
219C4⋊Q8C2×D9711yescharacteristic, complemented
220#124—971no—
221#123—1941no—
222#123—1941no—
223#123—1941no—
224C22⋊Q8—971no—
225C22⋊Q8—1941no—
226#121—971no—
227#121—971no—
228#121—971no—
229#121—1941no—
230#120—1941no—
231C4×D4—971no—
232C4×D4—971no—
233C4×D4C2×D9711yescharacteristic
234C2×C4⋊C4—971no—
235C2×C4⋊C4—971no—
236C2×C22⋊C4—1941no—
237#106—971no—
238#106C2×D9711yescharacteristic, complemented
239#106C2×D9711yescharacteristic, complemented
240C4⋊C8C2×D9711yescharacteristic, complemented
241#103—971no—
242#103—971no—
243#103—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248#102—971no—
249#102C2×D9711yescharacteristic
250#102C2×D9711yescharacteristic
251C22⋊C8—971no—
252C22⋊C8—971no—
253#97—971no—
254C24—1941no—
255#41—3884no—
256#41—3881no—
257#41—1942no—
258C2×Q8—21no—
259C2×Q8—971no—
260C2×D4—971no—
261C2×D4—971no—
262C2×D4—971no—
263C2×D4—3881no—
264C2×D4—1941no—
265C2×D4—1941no—
266C2×D4—1941no—
267C2×D4—1941no—
268C2×D4—1941no—
269C2×D4—1942no—
270C2×D4C22×D9711yescharacteristic
271C22×C4—21no—
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—1941no—
275C22×C4—1941no—
276C22×C4—1941no—
277C22×C4—1941no—
278SD16—3881no—
279D8—3881no—
280D8—1942no—
281M16—3881no—
282C2×C8—971no—
283C2×C8—971no—
284C2×C8—3881no—
285C2×C8C22×D9711yescharacteristic
286C2×C8C22×D9711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—971no—
289C4⋊C4—971no—
290C4⋊C4—1941no—
291C4⋊C4—1941no—
292C4⋊C4—1941no—
293C4⋊C4C22×D9711yescharacteristic
294C4⋊C4C22×D9711yescharacteristic
295C4⋊C4C22×D9711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—971no—
298C22⋊C4—971no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C22⋊C4—1941no—
302C22⋊C4—1941no—
303C22⋊C4—1941no—
304C22⋊C4—1941no—
305C22⋊C4—1941no—
306C42C22×D9711yescharacteristic
307C23—21no—
308C23—7761no—
309C23—971no—
310C23—971no—
311C23—3882no—
312C23—3882no—
313C23—1941no—
314C23—1941no—
315Q8—42no—
316Q8—1941no—
317Q8—1941no—
318D4—21no—
319D4—7761no—
320D4—3882no—
321D4—3882no—
322D4—3882no—
323D4—3882no—
324D4—1942no—
325D4—1941no—
326D4—1941no—
327D4—1941no—
328D4—1941no—
329D4#1309551212yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—81no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—3882no—
337C2×C4—3882no—
338C2×C4—3882no—
339C2×C4—3882no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4D4×D9711yescharacteristic
347C2×C4D4×D9711yescharacteristic
348C2×C4C23×D9711yescharacteristic
349C8—21no—
350C8—21no—
351C8—1941no—
352C8—1941no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—7764no—
357C22—972no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22—3881no—
362C22—3881no—
363C22—3881no—
364C22—1941no—
365C22—1941no—
366C22—1941no—
367C22—1941no—
368C22—1941no—
369C22C2×D4×D9711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—3881no—
376C4—3881no—
377C4—1941no—
378C4—1941no—
379C4C2×#1309551211yescharacteristic
380C4#1569085611yescharacteristic
381C2—42no—
382C2—7761no—
383C2—7761no—
384C2—3881no—
385C2—1942no—
386C2#2026137611yescharacteristic
387C2AGL(1,ℤ/8)×D9711yescharacteristic
388C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#476×C97C2
socleC22×C97C22×D4
hypercenter#476D97
hypocenterC97#3306
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20261388C2semidirect
2. #20261380C2semidirect
3. #20261097C2semidirect
4. #20261067C2semidirect
5. #20261055C2semidirect
6. #20260989C2semidirect
7. #20260965C2semidirect
8. #20260951C2semidirect
9. #20260881C2semidirect
10. #20260870C2semidirect
11. #20260845C2semidirect
12. #20260805C2semidirect
13. #20260733C2semidirect
14. #102×D97C2semidirect
15. #476×C97C2semidirect
19. C2×#13095500C22semidirect
22. C2×#13095494C22semidirect
32. #15690785C22semidirect
36. C4×D388C22semidirect
38. C4⋊C4×D97C22semidirect
40. #15690741C22semidirect
41. #15690735C22semidirect
42. #15690733C22semidirect
43. #15690733C22semidirect
48. #15690730C22semidirect
50. #15690722C22semidirect
51. #15690721C22semidirect
52. C4⋊Q8×C97C22semidirect
54. #106×C97C22semidirect
55. #106×C97C22semidirect
56. C4⋊C8×C97C22semidirect
122. D4×C97#41semidirect
195. C97#3306semidirect
201. #476D97semidirect
219. C4⋊Q8C2×D97semidirect
238. #106C2×D97semidirect
239. #106C2×D97semidirect
240. C4⋊C8C2×D97semidirect
329. D4#13095512semidirect

Maximal subgroups

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

subgroupindex
#202613882
#202613802
#202610972
#202610672
#202610552
#202609892
#202609652
#202609512
#202608812
#202608702
#202608452
#202608052
#202607332
#102×D972
#476×C972
#330697