Finite Group Databasebeta

← #27153238

Subgroups — #27153238

Order 12416.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690838C211yesmaximal, characteristic, complemented
2C2×SD16×D97C211yesmaximal, characteristic, complemented
3#20261395C211yesmaximal, characteristic, complemented
4#20261369C211yesmaximal, characteristic, complemented
5#20261121C211yesmaximal, characteristic, complemented
6#20260972C211yesmaximal, characteristic, complemented
7#20260953C211yesmaximal, characteristic, complemented
8#20260934C211yesmaximal, characteristic, complemented
9#20260885C211yesmaximal, characteristic, complemented
10#20260878C211yesmaximal, characteristic, complemented
11#20260875C211yesmaximal, characteristic, complemented
12#20260870C211yesmaximal, characteristic, complemented
13#20260834C211yesmaximal, characteristic, complemented
14#20260804C211yesmaximal, characteristic, complemented
15#468×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic, complemented
17C2×Q8×D97C2211yescharacteristic, complemented
18C22×D388C2211yescharacteristic
19C2×D4×D97—21no—
20C2×D4×D97C2211yescharacteristic
21#15690838—24no—
22C2×#13095500C2211yescharacteristic
23C2×#13095500C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×#13095498C2211yescharacteristic
26C2×#13095498C2211yescharacteristic, complemented
27SD16×D97—24no—
28C2×D776C2211yescharacteristic
29C2×D776C2211yescharacteristic, complemented
30C2×#13095496C2211yescharacteristic, complemented
31C2×C8×D97C2211yescharacteristic, complemented
32#15690806C2211yescharacteristic
33#15690799C2211yescharacteristic, complemented
34#15690796—21no—
35#15690788—21no—
36#15690785C2211yescharacteristic
37#15690785C2211yescharacteristic, complemented
38#15690783—21no—
39#15690783—21no—
40#15690780—21no—
41#15690778—21no—
42#15690775C2211yescharacteristic, complemented
43#15690772—21no—
44#15690772C2211yescharacteristic
45C4×C22⋊D97C2211yescharacteristic
46#15690739C2211yescharacteristic, complemented
47#15690733C2211yescharacteristic, complemented
48#15690730C2211yescharacteristic
49#15690730C2211yescharacteristic, complemented
50#15690730C2211yescharacteristic, complemented
51C22⋊#11715895C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×M16×C97C2211yescharacteristic, complemented
56C22⋊Q8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic
58#107×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095511—22no—
64#13095511—41no—
65C2×Dic194—21no—
66C2×Dic194C2311yescharacteristic
67Q8×D97—22no—
68Q8×D97—41no—
69C2×D388—22no—
70C2×D388—21no—
71C2×D388—21no—
72C2×D388—21no—
73C2×D388—41no—
74C2×D388C2311yescharacteristic
75C2×D388C2311yescharacteristic
76C2×D388C2311yescharacteristic
77C2×C22⋊D97—21no—
78C2×C22⋊D97—21no—
79C2×C22⋊D97—21no—
80C2×C22⋊D97—21no—
81C2×C22⋊D97D411yescharacteristic
82D4×D97—22no—
83D4×D97—41no—
84D4×D97—44no—
85C2×C4×D97—21no—
86C2×C4×D97—21no—
87C2×C4×D97—21no—
88C2×C4×D97C2311yescharacteristic
89#13095500—22no—
90#13095500D414yescomplemented
91#13095499—22no—
92#13095498—22no—
93#13095498—41no—
94D776—22no—
95D776—22no—
96#13095496—41no—
97C2×#11715895C2311yescharacteristic
98C2×#11715895C2311yescharacteristic
99C8×D97—22no—
100#13095490—21no—
101#13095490D411yescharacteristic
102C22⋊Dic97D411yescharacteristic
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488D411yescharacteristic
109C4×Dic97—21no—
110C4×Dic97C2311yescharacteristic
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97C2311yescharacteristic
114C22×C4×C97C2311yescharacteristic
115SD16×C97—22no—
116M16×C97—22no—
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C4⋊C4×C97—21no—
120C4⋊C4×C97C2311yescharacteristic
121C4⋊C4×C97C2311yescharacteristic
122C22⋊C4×C97—21no—
123C22⋊C4×C97—21no—
124C23×C97—21no—
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—81no—
134C22×D97—81no—
135C22×D97C2×D411yescharacteristic
136Q8×C97—21no—
137Q8×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—42no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97C2×D411yescharacteristic
147C2×C4×C97C2411yescharacteristic
148C22⋊D97—22no—
149C22⋊D97—42no—
150C22⋊D97—42no—
151C22⋊D97—42no—
152C22⋊D97—81no—
153C2×Dic97—21no—
154C2×Dic97C2×D411yescharacteristic
155C2×Dic97C2×D411yescharacteristic
156C2×Dic97C2×D411yescharacteristic
157D388—21no—
158D388—21no—
159D388—21no—
160D388—21no—
161D388—21no—
162D388—42no—
163D388—42no—
164D388—42no—
165D388C2×D412yes—
166C4×D97—21no—
167C4×D97—21no—
168C4×D97—42no—
169C4×D97—42no—
170C4×D97—42no—
171Dic194—21no—
172Dic194—42no—
173Dic194C2×D412yescomplemented
174C8×C97—21no—
175C8×C97C2×D412yescomplemented
176#11715895—21no—
177#11715895—21no—
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97C22×D411yescharacteristic
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—22no—
188C2×D97—41no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—84no—
197C2×D97—84no—
198C2×D97AGL(1,ℤ/8)12yescomplemented
199C4×C97—21no—
200C4×C97—41no—
201C4×C97—41no—
202C4×C97—41no—
203C4×C97C22×D411yescharacteristic
204C4×C97H(5,𝔽2)11yescharacteristic
205Dic97—22no—
206Dic97—21no—
207Dic97—21no—
208Dic97—21no—
209C2×C97—41no—
210C2×C97—81no—
211C2×C97D4211yescharacteristic
212C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C97#57711yescharacteristic
214D97—22no—
215D97—42no—
216D97—81no—
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16—971no—
222#548—971no—
223D42—971no—
224#496—971no—
225#468D9711yescharacteristic, complemented
226#466—971no—
227#463—971no—
228#460—971no—
229#450—971no—
230#449—971no—
231#448—971no—
232#447—971no—
233#440—971no—
234#435—971no—
235C2×#41—971no—
236C22×Q8—971no—
237C22×D4—971no—
238C22×D4—971no—
239C22×D4—1941no—
240AGL(1,ℤ/8)—1944no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16—971no—
245C2×SD16—971no—
246C2×SD16—1944no—
247C2×SD16C2×D9711yescharacteristic, complemented
248C2×D8—971no—
249C2×D8—971no—
250C2×D8—971no—
251C2×M16C2×D9711yescharacteristic, complemented
252C22×C8—971no—
253#127—971no—
254#124—971no—
255#124—1941no—
256C22⋊Q8—1941no—
257C22⋊Q8C2×D9711yescharacteristic, complemented
258#121—971no—
259#121—971no—
260#121—1941no—
261#121—1941no—
262#121C2×D9711yescharacteristic
263#120—1941no—
264#120—1941no—
265C4×Q8—971no—
266C4×D4—971no—
267C4×D4—971no—
268C4×D4—1941no—
269#107C2×D9711yescharacteristic, complemented
270C4⋊C8—971no—
271#103—971no—
272#103C2×D9711yescharacteristic, complemented
273#102—971no—
274#102—971no—
275#102—971no—
276#102C2×D9711yescharacteristic, complemented
277C22⋊C8—971no—
278C22⋊C8—971no—
279#97—971no—
280C24—1941no—
281C24—1941no—
282#41—3881no—
283#41—1942no—
284C2×Q8—971no—
285C2×Q8—3881no—
286C2×Q8—1941no—
287C2×Q8—1942no—
288C2×Q8C22×D9711yescharacteristic
289C2×D4—21no—
290C2×D4—971no—
291C2×D4—971no—
292C2×D4—971no—
293C2×D4—971no—
294C2×D4—3881no—
295C2×D4—3884no—
296C2×D4—3881no—
297C2×D4—1942no—
298C2×D4—1941no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4—1942no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4C22×D9711yescharacteristic
307C22×C4—971no—
308C22×C4—1941no—
309C22×C4—1941no—
310C22×C4—1941no—
311C22×C4C22×D9711yescharacteristic
312SD16—22no—
313SD16—974no—
314SD16—3881no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—3881no—
319D8—1942no—
320D8—1942no—
321M16—22no—
322C2×C8—971no—
323C2×C8—971no—
324C2×C8—1942no—
325C2×C8C22×D9711yescharacteristic
326C2×C8C22×D9711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—971no—
329C4⋊C4—1941no—
330C4⋊C4C22×D9711yescharacteristic
331C4⋊C4C22×D9711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—971no—
335C22⋊C4—971no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4—1941no—
340C22⋊C4—1941no—
341C42—971no—
342C42—1941no—
343C23—21no—
344C23—7761no—
345C23—7761no—
346C23—971no—
347C23—3882no—
348C23—3882no—
349C23—3882no—
350C23—3882no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23D4×D9711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—972no—
358Q8—3882no—
359Q8—1941no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—7761no—
364D4—972no—
365D4—3882no—
366D4—3882no—
367D4—3882no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—1942no—
372D4—1941no—
373D4—1941no—
374D4—1941no—
375D4—1941no—
376D4—1941no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—971no—
383C2×C4—971no—
384C2×C4—971no—
385C2×C4—3882no—
386C2×C4—3882no—
387C2×C4—3882no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4D4×D9711yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—21no—
394C8—1941no—
395C8—1941no—
396C8D4×D9712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—7764no—
404C22—7764no—
405C22—972no—
406C22—3881no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—1942no—
415C22—1941no—
416C22—1941no—
417C22C2×D4×D9711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—1941no—
423C4—1941no—
424C4—1941no—
425C4—1942no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085611yescharacteristic
428C2—41no—
429C2—81no—
430C2—7761no—
431C2—7761no—
432C2—3882no—
433C2—1942no—
434C2#2026136911yescharacteristic
435C2AGL(1,ℤ/8)×D9711yescharacteristic
436C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#468×C97C2
socleC22×C97C22×D4
hypercenter#468D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. C2×SD16×D97C2semidirect
3. #20261395C2semidirect
4. #20261369C2semidirect
5. #20261121C2semidirect
6. #20260972C2semidirect
7. #20260953C2semidirect
8. #20260934C2semidirect
9. #20260885C2semidirect
10. #20260878C2semidirect
11. #20260875C2semidirect
12. #20260870C2semidirect
13. #20260834C2semidirect
14. #20260804C2semidirect
15. #468×C97C2semidirect
16. C2×#13095511C22semidirect
17. C2×Q8×D97C22semidirect
23. C2×#13095500C22semidirect
24. C2×#13095499C22semidirect
26. C2×#13095498C22semidirect
29. C2×D776C22semidirect
30. C2×#13095496C22semidirect
31. C2×C8×D97C22semidirect
33. #15690799C22semidirect
37. #15690785C22semidirect
42. #15690775C22semidirect
46. #15690739C22semidirect
47. #15690733C22semidirect
49. #15690730C22semidirect
50. #15690730C22semidirect
51. C22⋊#11715895C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
54. C2×SD16×C97C22semidirect
55. C2×M16×C97C22semidirect
56. C22⋊Q8×C97C22semidirect
58. #107×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
90. #13095500D4semidirect
173. Dic194C2×D4semidirect
175. C8×C97C2×D4semidirect
198. C2×D97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
225. #468D97semidirect
247. C2×SD16C2×D97semidirect
251. C2×M16C2×D97semidirect
257. C22⋊Q8C2×D97semidirect
269. #107C2×D97semidirect
272. #103C2×D97semidirect
276. #102C2×D97semidirect
396. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×SD16×D972
#202613952
#202613692
#202611212
#202609722
#202609532
#202609342
#202608852
#202608782
#202608752
#202608702
#202608342
#202608042
#468×C972
#326597