Finite Group Databasebeta

← #27153220

Subgroups — #27153220

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C22×#13095496C211yesmaximal, characteristic, complemented
3#20261376C211yesmaximal, characteristic, complemented
4D4×C22⋊D97C211yesmaximal, characteristic, complemented
5#20261121C211yesmaximal, characteristic, complemented
6#20260987C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260952C211yesmaximal, characteristic, complemented
9#20260929C211yesmaximal, characteristic, complemented
10#20260879C211yesmaximal, characteristic, complemented
11#20260871C211yesmaximal, characteristic, complemented
12#20260870C211yesmaximal, characteristic, complemented
13#20260844C211yesmaximal, characteristic, complemented
14#20260803C211yesmaximal, characteristic, complemented
15#450×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C22×D388C2211yescharacteristic, complemented
18C22×C22⋊D97—21no—
19C2×D4×D97C2211yescharacteristic, complemented
20#15690837—24no—
21C2×#13095498C2211yescharacteristic, complemented
22C2×D776C2211yescharacteristic, complemented
23C2×D776C2211yescharacteristic, complemented
24C2×#13095496—24no—
25C2×#13095496C2211yescharacteristic
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095494C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic, complemented
31#15690806C2211yescharacteristic, complemented
32#15690799C2211yescharacteristic, complemented
33#15690795—21no—
34#15690785—21no—
35#15690785C2211yescharacteristic, complemented
36#15690784C2211yescharacteristic
37#15690782—21no—
38#15690781C2211yescharacteristic, complemented
39#15690780—21no—
40#15690779—21no—
41#15690778—21no—
42D4×Dic97C2211yescharacteristic, complemented
43#15690772C2211yescharacteristic, complemented
44C4×C22⋊D97C2211yescharacteristic
45C22⋊C4×D97—21no—
46#15690741C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690734C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic
55C2×SD16×C97C2211yescharacteristic, complemented
56C2×SD16×C97C2211yescharacteristic, complemented
57C22⋊Q8×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095512—22no—
64#13095512—41no—
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—21no—
68C2×D388—21no—
69C2×D388—41no—
70C2×D388C2311yescharacteristic
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77C2×C22⋊D97—21no—
78C2×C22⋊D97—44no—
79C2×C22⋊D97D411yescharacteristic
80D4×D97—22no—
81D4×D97—41no—
82C22×Dic97—21no—
83C2×C4×D97—21no—
84C2×C4×D97—21no—
85C2×C4×D97C2311yescharacteristic
86#13095498—22no—
87D776—22no—
88D776—41no—
89#13095496—22no—
90#13095496—22no—
91#13095496—22no—
92#13095496D414yescomplemented
93#13095494—22no—
94C2×#11715895—22no—
95C2×#11715895C2311yescharacteristic
96C2×#11715895C2311yescharacteristic
97C4⋊Dic97C2311yescharacteristic
98#13095490D411yescharacteristic
99C22⋊Dic97—21no—
100C22⋊Dic97—21no—
101C22⋊Dic97D411yescharacteristic
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488D411yescharacteristic
108C4×Dic97C2311yescharacteristic
109C24×C97—21no—
110C2×Q8×C97C2311yescharacteristic
111C2×D4×C97—22no—
112C2×D4×C97—21no—
113C2×D4×C97—41no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117SD16×C97—22no—
118SD16×C97—41no—
119C2×C8×C97C2311yescharacteristic
120C2×C8×C97C2311yescharacteristic
121C4⋊C4×C97—21no—
122C4⋊C4×C97C2311yescharacteristic
123C22⋊C4×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—42no—
127C23×C97—42no—
128C23×C97—81no—
129C23×C97C2×D411yescharacteristic
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—42no—
136C22×D97—42no—
137C22×D97—81no—
138C22×D97—81no—
139C22×D97C2×D411yescharacteristic
140Q8×C97—21no—
141Q8×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—42no—
146D4×C97C2×D412yes—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2411yescharacteristic
153C22⋊D97—22no—
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—81no—
159C2×Dic97—21no—
160C2×Dic97—42no—
161C2×Dic97C2×D411yescharacteristic
162C2×Dic97C2×D411yescharacteristic
163C2×Dic97C2×D411yescharacteristic
164D388—21no—
165D388—21no—
166D388—21no—
167D388—21no—
168D388—21no—
169D388—42no—
170D388—42no—
171D388—81no—
172D388C2×D412yescomplemented
173C4×D97—21no—
174C4×D97—21no—
175C4×D97—42no—
176C4×D97—81no—
177C8×C97—21no—
178C8×C97—21no—
179#11715895—21no—
180#11715895C2×D412yescomplemented
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—22no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—84no—
190C22×C97AGL(1,ℤ/8)12yescomplemented
191C22×C97C22×D411yescharacteristic
192C2×D97—21no—
193C2×D97—22no—
194C2×D97—21no—
195C2×D97—21no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—41no—
204C2×D97—84no—
205C2×D97—84no—
206C4×C97—21no—
207C4×C97—41no—
208C4×C97—41no—
209C4×C97C22×D411yescharacteristic
210C4×C97H(5,𝔽2)11yescharacteristic
211Dic97—21no—
212Dic97—21no—
213Dic97—21no—
214Dic97—41no—
215C2×C97—22no—
216C2×C97—42no—
217C2×C97—81no—
218C2×C97D4211yescharacteristic
219C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C97#57611yescharacteristic
221D97—41no—
222D97—42no—
223D97—81no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
228C22×D8—971no—
229#546—971no—
230D42—971no—
231#496—971no—
232#469—971no—
233#465—971no—
234#463—971no—
235#459—971no—
236#450D9711yescharacteristic, complemented
237#449—971no—
238#447—971no—
239#447—971no—
240#442—971no—
241#435—971no—
242C2×#41—971no—
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—1941no—
246C22×D4C2×D9711yescharacteristic
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16C2×D9711yescharacteristic, complemented
250C2×SD16C2×D9711yescharacteristic, complemented
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—971no—
256C2×D8—971no—
257C2×D8—1944no—
258C2×M16—971no—
259C22×C8—971no—
260#127—971no—
261#124—971no—
262#123—1941no—
263C22⋊Q8C2×D9711yescharacteristic, complemented
264#121—971no—
265#121—971no—
266#121—971no—
267#121—1941no—
268#121—1941no—
269#120—1941no—
270#120—1941no—
271#120—1941no—
272C4×D4—971no—
273C4×D4—971no—
274C4×D4—971no—
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8—971no—
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102C2×D9711yescharacteristic, complemented
283#102C2×D9711yescharacteristic, complemented
284C22⋊C8—971no—
285C22⋊C8C2×D9711yescharacteristic, complemented
286#97—971no—
287C24—21no—
288C24—1941no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8C22×D9711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—971no—
297C2×D4—971no—
298C2×D4—971no—
299C2×D4—971no—
300C2×D4—3881no—
301C2×D4—3884no—
302C2×D4—3881no—
303C2×D4—1941no—
304C2×D4—1942no—
305C2×D4—1941no—
306C2×D4—1941no—
307C2×D4—1941no—
308C2×D4—1941no—
309C2×D4—1941no—
310C2×D4—1941no—
311C2×D4—1942no—
312C2×D4—1941no—
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—971no—
316C22×C4—1941no—
317C22×C4—1941no—
318C22×C4—1941no—
319C22×C4C22×D9711yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—1942no—
323D8—974no—
324D8—3881no—
325D8—1942no—
326D8—1942no—
327D8—1942no—
328D8—1942no—
329M16—1942no—
330C2×C8—971no—
331C2×C8—971no—
332C2×C8—1942no—
333C2×C8C22×D9711yescharacteristic
334C2×C8C22×D9711yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—971no—
337C4⋊C4—971no—
338C4⋊C4C22×D9711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—971no—
341C22⋊C4—971no—
342C22⋊C4—1941no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4—1941no—
348C22⋊C4—1941no—
349C42—971no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—7761no—
356C23—7761no—
357C23—971no—
358C23—3882no—
359C23—3882no—
360C23—3882no—
361C23—3882no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23D4×D9711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—7761no—
373D4—7761no—
374D4—972no—
375D4—3882no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—1942no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387D4D4×D9712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—7761no—
393C2×C4—971no—
394C2×C4—971no—
395C2×C4—971no—
396C2×C4—3882no—
397C2×C4—3882no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4D4×D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—21no—
405C8—972no—
406C8—1941no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—7764no—
417C22—7764no—
418C22—3881no—
419C22—3881no—
420C22—3881no—
421C22—3881no—
422C22—3881no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—1941no—
427C22—1941no—
428C22—1942no—
429C22—1941no—
430C22#1569083712yescomplemented
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3881no—
436C4—1941no—
437C4—1941no—
438C4—1941no—
439C4C2×D4×D9711yescharacteristic
440C4#1569085511yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—7761no—
445C2—7761no—
446C2—3881no—
447C2—3882no—
448C2#2026136711yescharacteristic
449C2C2×#1569083711yescharacteristic
450C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
hypercenter#450D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C22×#13095496C2semidirect
3. #20261376C2semidirect
4. D4×C22⋊D97C2semidirect
5. #20261121C2semidirect
6. #20260987C2semidirect
7. #20260966C2semidirect
8. #20260952C2semidirect
9. #20260929C2semidirect
10. #20260879C2semidirect
11. #20260871C2semidirect
12. #20260870C2semidirect
13. #20260844C2semidirect
14. #20260803C2semidirect
15. #450×C97C2semidirect
16. C2×#13095512C22semidirect
17. C22×D388C22semidirect
19. C2×D4×D97C22semidirect
21. C2×#13095498C22semidirect
22. C2×D776C22semidirect
23. C2×D776C22semidirect
27. C2×#13095496C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095494C22semidirect
30. C22×#11715895C22semidirect
31. #15690806C22semidirect
32. #15690799C22semidirect
35. #15690785C22semidirect
38. #15690781C22semidirect
42. D4×Dic97C22semidirect
43. #15690772C22semidirect
46. #15690741C22semidirect
47. #15690739C22semidirect
48. #15690734C22semidirect
49. #15690732C22semidirect
50. #15690732C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×SD16×C97C22semidirect
57. C22⋊Q8×C97C22semidirect
58. #102×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
92. #13095496D4semidirect
172. D388C2×D4semidirect
180. #11715895C2×D4semidirect
190. C22×C97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
236. #450D97semidirect
249. C2×SD16C2×D97semidirect
250. C2×SD16C2×D97semidirect
263. C22⋊Q8C2×D97semidirect
282. #102C2×D97semidirect
283. #102C2×D97semidirect
285. C22⋊C8C2×D97semidirect
430. C22#15690837semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C22×#130954962
#202613762
D4×C22⋊D972
#202611212
#202609872
#202609662
#202609522
#202609292
#202608792
#202608712
#202608702
#202608442
#202608032
#450×C972
#326497