Finite Group Databasebeta

← #27150780

Subgroups — #27150780

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261244C211yesmaximal, characteristic
2#20260811C211yesmaximal, characteristic, complemented
3#20260809C211yesmaximal, characteristic, complemented
4#20260807C211yesmaximal, characteristic, complemented
5#20260805C211yesmaximal, characteristic, complemented
6#20260804C211yesmaximal, characteristic, complemented
7C8×D388C211yesmaximal, characteristic, complemented
8#20260798C211yesmaximal, characteristic, complemented
9#20260795C211yesmaximal, characteristic, complemented
10#20260794C211yesmaximal, characteristic, complemented
11#20260703C211yesmaximal, characteristic, complemented
12#20260700C211yesmaximal, characteristic, complemented
13#20260698C211yesmaximal, characteristic, complemented
14C22⋊C8×D97C211yesmaximal, characteristic, complemented
15#436×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19C2×#13095494C2211yescharacteristic, complemented
20C22×#11715895C2211yescharacteristic, complemented
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic, complemented
24#15690795C411yescharacteristic
25#15690792C411yescharacteristic
26#15690788C411yescharacteristic
27#15690787C411yescharacteristic
28#15690783C411yescharacteristic
29#15690782C411yescharacteristic
30#15690780C411yescharacteristic
31#15690778C411yescharacteristic
32C4×D388C2211yescharacteristic
33C4×C22⋊D97C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35#15690767C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic, complemented
37#15690739C2211yescharacteristic, complemented
38#15690739C2211yescharacteristic, complemented
39C22⋊#11715895C2211yescharacteristic
40C22⋊#11715895C2211yescharacteristic, complemented
41#15690722C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic
46#15690722C2211yescharacteristic
47#15690722C2211yescharacteristic, complemented
48#15690722C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×M16×C97C2211yescharacteristic
53C2×M16×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic, complemented
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97C2×C411yescharacteristic
60#13095511—41no—
61#13095511—41no—
62C2×Dic194C2×C411yescharacteristic
63C2×D388C2×C411yescharacteristic
64C2×D388C2×C411yescharacteristic
65C2×C22⋊D97C2×C411yescharacteristic
66C2×C22⋊D97C2×C411yescharacteristic
67C2×C22⋊D97C2×C411yescharacteristic
68C2×C22⋊D97C2×C411yescharacteristic
69C22×Dic97C2×C411yescharacteristic
70C2×C4×D97—41no—
71C2×C4×D97—41no—
72C2×C4×D97C2×C411yescharacteristic
73C2×C4×D97C2×C411yescharacteristic
74C2×C4×D97C2311yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2311yescharacteristic
77#13095495—41no—
78#13095494—41no—
79C2×#11715895—41no—
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C2×#11715895C2311yescharacteristic
84C8×D97—41no—
85C8×D97—41no—
86C8×D97—41no—
87C4⋊Dic97C2×C411yescharacteristic
88#13095490C2×C411yescharacteristic
89#13095490C2×C411yescharacteristic
90#13095490C2×C411yescharacteristic
91#13095490C2×C411yescharacteristic
92C22⋊Dic97C2×C411yescharacteristic
93C22⋊Dic97C2×C411yescharacteristic
94#13095488C2×C411yescharacteristic
95#13095488C2×C411yescharacteristic
96#13095488C2×C411yescharacteristic
97#13095488C2×C411yescharacteristic
98#13095488C2×C411yescharacteristic
99#13095488C2×C411yescharacteristic
100C4×Dic97C2311yescharacteristic
101C2×D4×C97C2×C411yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104M16×C97—41no—
105M16×C97—41no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C2×C8×C97C2311yescharacteristic
110C4⋊C4×C97C2×C411yescharacteristic
111C22⋊C4×C97C2×C411yescharacteristic
112C22⋊C4×C97C2×C411yescharacteristic
113C42×C97C2311yescharacteristic
114C23×C97C22×C411yescharacteristic
115C23×C97C22×C411yescharacteristic
116C22×D97—22no—
117C22×D97—41no—
118C22×D97—41no—
119C22×D97C22×C411yescharacteristic
120C22×D97C22×C411yescharacteristic
121C22×D97C22×C411yescharacteristic
122D4×C97—41no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97C22×C411yescharacteristic
128C2×C4×C97C22×C411yescharacteristic
129C2×C4×C97C22×C411yescharacteristic
130C2×C4×C97C22×C411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—22no—
133C22⋊D97—41no—
134C22⋊D97—41no—
135C22⋊D97—41no—
136C2×Dic97—41no—
137C2×Dic97C22×C411yescharacteristic
138C2×Dic97C22×C411yescharacteristic
139C2×Dic97C22×C411yescharacteristic
140C2×Dic97C22×C411yescharacteristic
141C2×Dic97C22×C411yescharacteristic
142D388—22no—
143D388—41no—
144C4×D97—21no—
145C4×D97—21no—
146C4×D97—21no—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—21no—
150C4×D97—41no—
151C4×D97—41no—
152Dic194—41no—
153C8×C97—21no—
154C8×C97—21no—
155C8×C97—21no—
156C8×C97—21no—
157#11715895—21no—
158#11715895—21no—
159#11715895—21no—
160#11715895—21no—
161C22×C97—21no—
162C22×C97—21no—
163C22×C97—21no—
164C22×C97—21no—
165C22×C97—21no—
166C22×C97—21no—
167C22×C97C23×C411yescharacteristic
168C2×D97—21no—
169C2×D97—21no—
170C2×D97—21no—
171C2×D97—21no—
172C2×D97—21no—
173C2×D97—21no—
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—44no—
177C2×D97#13112yes—
178C4×C97—21no—
179C4×C97—21no—
180C4×C97—21no—
181C4×C97—21no—
182C4×C97H(5,𝔽2)11yescharacteristic
183C4×C97H(5,𝔽2)11yescharacteristic
184Dic97—21no—
185Dic97—21no—
186Dic97—21no—
187Dic97—21no—
188Dic97—21no—
189C2×C97—41no—
190C2×C97—41no—
191C2×C97#51811yescharacteristic
192C2×C97C2×#13111yescharacteristic
193C2×C97#56811yescharacteristic
194D97—22no—
195D97—41no—
196D97—41no—
197#2976—971nomaximal, Sylow
198C97#297611yescharacteristic, complemented, Sylow
199#525—971no—
200#436—971no—
201#436D9711yescharacteristic, complemented
202#435—971no—
203#435—971no—
204#435—971no—
205#435—971no—
206C8×D4—971no—
207C8×D4—971no—
208#433—971no—
209#433—971no—
210#408—971no—
211#408—971no—
212#407—971no—
213C2×C22⋊C8—971no—
214C2×#41—971no—
215C23×C4—971no—
216C2×M16—971no—
217C2×M16—971no—
218C2×M16C2×D9711yescharacteristic
219C2×M16C2×D9711yescharacteristic, complemented
220C22×C8—971no—
221C22×C8—971no—
222C22×C8—971no—
223C22×C8—971no—
224#123—971no—
225#123—971no—
226C22⋊Q8—971no—
227C22⋊Q8—971no—
228#121—971no—
229#121—971no—
230#120—971no—
231#120—971no—
232C4×D4—971no—
233C4×D4—971no—
234C4×D4—971no—
235C4×D4C2×D9711yescharacteristic
236#117—971no—
237C4⋊C8—971no—
238C4⋊C8—971no—
239C4⋊C8—971no—
240C4⋊C8C2×D9711yescharacteristic, complemented
241C22⋊C8—971no—
242C22⋊C8—971no—
243C22⋊C8—971no—
244C22⋊C8—971no—
245C22⋊C8—971no—
246C22⋊C8—971no—
247C22⋊C8—971no—
248C22⋊C8—971no—
249C22⋊C8—971no—
250C22⋊C8—971no—
251C22⋊C8C2×D9711yescharacteristic
252C22⋊C8C2×D9711yescharacteristic, complemented
253#97—971no—
254#97—971no—
255C4×C8—971no—
256C4×C8C2×D9711yescharacteristic, complemented
257C24—971no—
258#41—3881no—
259#41—3881no—
260C2×Q8—971no—
261C2×D4—971no—
262C2×D4—971no—
263C2×D4—971no—
264C2×D4—971no—
265C2×D4—971no—
266C2×D4—971no—
267C2×D4C4×D9711yescharacteristic
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—971no—
271C22×C4—971no—
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—3881no—
275C22×C4—3881no—
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278M16—41no—
279M16—41no—
280M16—3881no—
281M16—3881no—
282C2×C8—971no—
283C2×C8—971no—
284C2×C8—971no—
285C2×C8—971no—
286C2×C8—3881no—
287C2×C8—3881no—
288C2×C8—3881no—
289C2×C8—3881no—
290C2×C8C22×D9711yescharacteristic
291C2×C8C22×D9711yescharacteristic
292C2×C8C22×D9711yescharacteristic
293C2×C8C22×D9711yescharacteristic
294C4⋊C4—971no—
295C4⋊C4—971no—
296C4⋊C4—971no—
297C4⋊C4—971no—
298C4⋊C4—971no—
299C4⋊C4C4×D9711yescharacteristic
300C22⋊C4—971no—
301C22⋊C4—971no—
302C22⋊C4—971no—
303C22⋊C4—971no—
304C22⋊C4—971no—
305C22⋊C4—971no—
306C22⋊C4—971no—
307C22⋊C4—971no—
308C22⋊C4C4×D9711yescharacteristic
309C22⋊C4C4×D9711yescharacteristic
310C42—971no—
311C42C22×D9711yescharacteristic
312C23—971no—
313C23—971no—
314C23—971no—
315C23—3881no—
316C23—3881no—
317C23—1942no—
318C23C2×C4×D9711yescharacteristic
319C23C2×C4×D9711yescharacteristic
320Q8—3881no—
321D4—41no—
322D4—3881no—
323D4—3881no—
324D4—3881no—
325D4—3881no—
326D4—1942no—
327D4—1942no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—971no—
333C2×C4—971no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—971no—
337C2×C4—3881no—
338C2×C4—3881no—
339C2×C4—3881no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4C2×C4×D9711yescharacteristic
347C2×C4C2×C4×D9711yescharacteristic
348C2×C4C2×C4×D9711yescharacteristic
349C2×C4C2×C4×D9711yescharacteristic
350C2×C4C23×D9711yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—1941no—
356C8—1941no—
357C8—1941no—
358C8—1941no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—972no—
366C22—3884no—
367C22—1941no—
368C22—1941no—
369C22—1941no—
370C22—1941no—
371C22—1941no—
372C22—1941no—
373C22—1941no—
374C22—1941no—
375C22C22×C4×D9711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—1941no—
381C4—1941no—
382C4—1941no—
383C4—1941no—
384C4—1941no—
385C4#1569085511yescharacteristic
386C4#1569085611yescharacteristic
387C2—41no—
388C2—41no—
389C2—3881no—
390C2—3881no—
391C2—1942no—
392C2#2026120711yescharacteristic
393C2#131×D9711yescharacteristic
394C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#436×C97C2
prime radical#20261244C2
cosocleC2×C4×C97C24
hypercenter#436D97
hypocenterC97#2976
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #436

Complements

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

NH ≅ G/Ntype
2. #20260811C2semidirect
3. #20260809C2semidirect
4. #20260807C2semidirect
5. #20260805C2semidirect
6. #20260804C2semidirect
7. C8×D388C2semidirect
8. #20260798C2semidirect
9. #20260795C2semidirect
10. #20260794C2semidirect
11. #20260703C2semidirect
12. #20260700C2semidirect
13. #20260698C2semidirect
14. C22⋊C8×D97C2semidirect
15. #436×C97C2semidirect
19. C2×#13095494C22semidirect
20. C22×#11715895C22semidirect
23. C2×C8×D97C22semidirect
36. C4⋊#11715895C22semidirect
37. #15690739C22semidirect
38. #15690739C22semidirect
40. C22⋊#11715895C22semidirect
47. #15690722C22semidirect
48. #15690722C22semidirect
49. #15690721C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
53. C2×M16×C97C22semidirect
55. C4⋊C8×C97C22semidirect
57. C22⋊C8×C97C22semidirect
58. C4×C8×C97C22semidirect
198. C97#2976semidirect
201. #436D97semidirect
219. C2×M16C2×D97semidirect
240. C4⋊C8C2×D97semidirect
252. C22⋊C8C2×D97semidirect
256. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612442
#202608112
#202608092
#202608072
#202608052
#202608042
C8×D3882
#202607982
#202607952
#202607942
#202607032
#202607002
#202606982
C22⋊C8×D972
#436×C972
#297697