Finite Group Databasebeta

← #27153232

Subgroups — #27153232

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×#15690835C211yesmaximal, characteristic, complemented
2C22×#13095498C211yesmaximal, characteristic, complemented
3D4×D388C211yesmaximal, characteristic, complemented
4#20261125C211yesmaximal, characteristic, complemented
5#20260983C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260935C211yesmaximal, characteristic, complemented
9#20260885C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260874C211yesmaximal, characteristic, complemented
12#20260868C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×D4×D97—21no—
19#15690835—24no—
20C2×#13095498—24no—
21C2×#13095498C2211yescharacteristic
22C2×#13095498C2211yescharacteristic
23C2×#13095498C2211yescharacteristic, complemented
24C2×#13095498C2211yescharacteristic, complemented
25C2×#13095498C2211yescharacteristic, complemented
26C2×#13095498C2211yescharacteristic, complemented
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095496C2211yescharacteristic, complemented
30C2×#13095495C2211yescharacteristic, complemented
31C22×#11715895C2211yescharacteristic, complemented
32#15690807C2211yescharacteristic
33#15690785—21no—
34#15690784C2211yescharacteristic
35#15690784C2211yescharacteristic
36#15690780—21no—
37#15690780—21no—
38C4×D388C2211yescharacteristic
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic, complemented
44#15690730C2211yescharacteristic
45#15690730C2211yescharacteristic, complemented
46#15690730C2211yescharacteristic, complemented
47C22⋊#11715895C2211yescharacteristic, complemented
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C2×#41×C97C2211yescharacteristic, complemented
51C22×Q8×C97C2211yescharacteristic, complemented
52#124×C97—21no—
53#124×C97C2211yescharacteristic, complemented
54C22⋊Q8×C97—21no—
55C22⋊Q8×C97C2211yescharacteristic, complemented
56#121×C97—21no—
57#121×C97C2211yescharacteristic, complemented
58C4×Q8×C97C2211yescharacteristic, complemented
59C4×D4×C97—21no—
60C4×D4×C97C2211yescharacteristic
61C23×D97—21no—
62C23×D97—21no—
63C2×D388—22no—
64C2×D388—22no—
65C2×D388—21no—
66C2×D388—21no—
67C2×D388—21no—
68C2×D388—21no—
69C2×D388—41no—
70C2×D388—41no—
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×D388C2311yescharacteristic
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76D4×D97—44no—
77C2×C4×D97—21no—
78#13095498—22no—
79#13095498—22no—
80#13095498—22no—
81#13095498—22no—
82#13095498—41no—
83#13095498D414yescomplemented
84#13095496—22no—
85#13095496—22no—
86#13095496—41no—
87#13095495—22no—
88C2×#11715895—22no—
89C2×#11715895C2311yescharacteristic
90C2×#11715895C2311yescharacteristic
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C4⋊Dic97C2311yescharacteristic
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#41×C97—22no—
98#41×C97—41no—
99C2×Q8×C97—21no—
100C2×Q8×C97—22no—
101C2×Q8×C97—41no—
102C2×Q8×C97C2311yescharacteristic
103C2×Q8×C97C2311yescharacteristic
104C2×D4×C97—21no—
105C2×D4×C97—21no—
106C2×D4×C97D411yescharacteristic
107C2×D4×C97C2311yescharacteristic
108C22×C4×C97—21no—
109C22×C4×C97—21no—
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C4⋊C4×C97—21no—
113C4⋊C4×C97—21no—
114C4⋊C4×C97D411yescharacteristic
115C4⋊C4×C97C2311yescharacteristic
116C22⋊C4×C97—21no—
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97—21no—
123C42×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97C2×D411yescharacteristic
126C23×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135C22×D97—81no—
136Q8×C97—21no—
137Q8×C97—21no—
138Q8×C97—21no—
139Q8×C97—42no—
140Q8×C97C2×D412yescomplemented
141D4×C97—22no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—42no—
145D4×C97—42no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—21no—
153C2×C4×C97—42no—
154C2×C4×C97—42no—
155C2×C4×C97C2×D411yescharacteristic
156C2×C4×C97C2×D411yescharacteristic
157C2×C4×C97C2×D411yescharacteristic
158C2×C4×C97C2×D411yescharacteristic
159C2×C4×C97C2411yescharacteristic
160C22⋊D97—42no—
161C22⋊D97—42no—
162C2×Dic97—21no—
163D388—21no—
164D388—21no—
165D388—21no—
166D388—21no—
167D388—21no—
168D388—42no—
169D388—42no—
170D388—42no—
171D388—81no—
172D388C2×D412yes—
173C4×D97—42no—
174#11715895—21no—
175#11715895—21no—
176#11715895—21no—
177#11715895C2×D412yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97AGL(1,ℤ/8)12yescomplemented
187C22×C97C22×D411yescharacteristic
188C2×D97—22no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—84no—
198C2×D97—84no—
199C4×C97—21no—
200C4×C97—21no—
201C4×C97—22no—
202C4×C97—21no—
203C4×C97—21no—
204C4×C97—41no—
205C4×C97—41no—
206C4×C97C22×D411yescharacteristic
207C4×C97H(5,𝔽2)11yescharacteristic
208Dic97—41no—
209C2×C97—22no—
210C2×C97—41no—
211C2×C97—81no—
212C2×C97D4211yescharacteristic
213C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C97#57711yescharacteristic
215D97—42no—
216D97—81no—
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16—971no—
222#548D9711yescharacteristic, complemented
223D42—971no—
224#496—971no—
225#468—971no—
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×#41C2×D9711yescharacteristic, complemented
236C22×Q8C2×D9711yescharacteristic, complemented
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—971no—
247C2×SD16—1944no—
248C2×D8—971no—
249C2×D8—971no—
250C2×D8—971no—
251C2×M16—971no—
252C22×C8—971no—
253#127—971no—
254#124—21no—
255#124C2×D9711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D9711yescharacteristic, complemented
258#121—21no—
259#121—971no—
260#121—971no—
261#121—1941no—
262#121C2×D9711yescharacteristic, complemented
263#120—1941no—
264#120—1941no—
265C4×Q8C2×D9711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—971no—
268C4×D4C2×D9711yescharacteristic
269#107—971no—
270C4⋊C8—971no—
271#103—971no—
272#103—971no—
273#102—971no—
274#102—971no—
275#102—971no—
276#102—971no—
277C22⋊C8—971no—
278C22⋊C8—971no—
279#97—971no—
280C24—1941no—
281C24—1941no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D9711yescharacteristic
288C2×Q8C22×D9711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—971no—
292C2×D4—971no—
293C2×D4—971no—
294C2×D4—3881no—
295C2×D4—3881no—
296C2×D4—3884no—
297C2×D4—1942no—
298C2×D4—1942no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4C22⋊D9711yescharacteristic
306C2×D4C22×D9711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—1941no—
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312SD16—974no—
313SD16—3881no—
314SD16—1942no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—3881no—
319D8—1942no—
320D8—1942no—
321M16—1942no—
322C2×C8—971no—
323C2×C8—971no—
324C2×C8—971no—
325C2×C8—971no—
326C2×C8—1942no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—971no—
330C4⋊C4C22⋊D9711yescharacteristic
331C4⋊C4C22×D9711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4C22⋊D9711yescharacteristic
340C22⋊C4C22⋊D9711yescharacteristic
341C42—21no—
342C42C22×D9711yescharacteristic
343C23—21no—
344C23—7761no—
345C23—7761no—
346C23—3882no—
347C23—3882no—
348C23—3882no—
349C23—3882no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23C2×C22⋊D9711yescharacteristic
354C23C2×C22⋊D9711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D9712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—7761no—
366D4—972no—
367D4—3882no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—3882no—
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—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—3882no—
387C2×C4—1941no—
388C2×C4C2×C22⋊D9711yescharacteristic
389C2×C4C2×C22⋊D9711yescharacteristic
390C2×C4C2×C22⋊D9711yescharacteristic
391C2×C4C2×C22⋊D9711yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—972no—
394C8—1941no—
395C8—1941no—
396C8—1941no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—7764no—
406C22—7764no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—3881no—
415C22—1942no—
416C22#1569083512yescomplemented
417C22C22×C22⋊D9711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—3881no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—7761no—
432C2—7761no—
433C2—3882no—
434C2D4×C22⋊D9711yescharacteristic
435C2C2×#1569083511yescharacteristic
436C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#548×C97C2
socleC22×C97C22×D4
hypercenter#548D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C22×#13095498C2semidirect
3. D4×D388C2semidirect
4. #20261125C2semidirect
5. #20260983C2semidirect
6. #20260969C2semidirect
7. #20260947C2semidirect
8. #20260935C2semidirect
9. #20260885C2semidirect
10. #20260880C2semidirect
11. #20260874C2semidirect
12. #20260868C2semidirect
13. #20260833C2semidirect
14. #20260807C2semidirect
15. #548×C97C2semidirect
23. C2×#13095498C22semidirect
24. C2×#13095498C22semidirect
25. C2×#13095498C22semidirect
26. C2×#13095498C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095496C22semidirect
30. C2×#13095495C22semidirect
31. C22×#11715895C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690734C22semidirect
43. #15690731C22semidirect
45. #15690730C22semidirect
46. #15690730C22semidirect
47. C22⋊#11715895C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
50. C2×#41×C97C22semidirect
51. C22×Q8×C97C22semidirect
53. #124×C97C22semidirect
55. C22⋊Q8×C97C22semidirect
57. #121×C97C22semidirect
58. C4×Q8×C97C22semidirect
83. #13095498D4semidirect
140. Q8×C97C2×D4semidirect
177. #11715895C2×D4semidirect
186. C22×C97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
222. #548D97semidirect
235. C2×#41C2×D97semidirect
236. C22×Q8C2×D97semidirect
255. #124C2×D97semidirect
257. C22⋊Q8C2×D97semidirect
262. #121C2×D97semidirect
265. C4×Q8C2×D97semidirect
359. Q8D4×D97semidirect
416. C22#15690835semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C22×#130954982
D4×D3882
#202611252
#202609832
#202609692
#202609472
#202609352
#202608852
#202608802
#202608742
#202608682
#202608332
#202608072
#548×C972
#326597