Finite Group Databasebeta

← #27153340

Subgroups — #27153340

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3#20261376C211yesmaximal, characteristic, complemented
4#20261373C211yesmaximal, characteristic, complemented
5#20261120C211yesmaximal, characteristic, complemented
6#20260984C211yesmaximal, characteristic, complemented
7#20260983C211yesmaximal, characteristic, complemented
8#20260953C211yesmaximal, characteristic, complemented
9#20260952C211yesmaximal, characteristic, complemented
10#20260896C211yesmaximal, characteristic, complemented
11#20260871C211yesmaximal, characteristic, complemented
12#20260870C211yesmaximal, characteristic, complemented
13#20260815C211yesmaximal, characteristic, complemented
14#20260810C211yesmaximal, characteristic, complemented
15#452×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C22×D388C2211yescharacteristic, complemented
18C2×D4×D97C2211yescharacteristic, complemented
19#15690837—24no—
20#15690835—24no—
21C2×#13095498C2211yescharacteristic, complemented
22C2×#13095498C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic, complemented
24C2×D776C2211yescharacteristic
25C2×D776C2211yescharacteristic
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095495C2211yescharacteristic, complemented
29C2×#13095494C2211yescharacteristic, complemented
30#15690799C2211yescharacteristic, complemented
31#15690799C2211yescharacteristic, complemented
32#15690795—21no—
33#15690793—21no—
34#15690786C2211yescharacteristic, complemented
35#15690785—21no—
36#15690785C2211yescharacteristic
37#15690784C2211yescharacteristic
38#15690782—21no—
39#15690780—21no—
40#15690778—21no—
41D4×Dic97C2211yescharacteristic, complemented
42#15690772C2211yescharacteristic, complemented
43C4×C22⋊D97C2211yescharacteristic
44C2×#13095488—21no—
45C22⋊C4×D97—21no—
46#15690743C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690734C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53C8×Dic97C2211yescharacteristic, complemented
54C2×#41×C97C2211yescharacteristic
55C2×Q16×C97C2211yescharacteristic, complemented
56C2×SD16×C97C2211yescharacteristic, complemented
57C22⋊Q8×C97C2211yescharacteristic, complemented
58#103×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⋊D97D411yescharacteristic
76D4×D97—22no—
77D4×D97—41no—
78C22×Dic97—21no—
79C2×C4×D97—21no—
80C2×C4×D97—21no—
81C2×C4×D97C2311yescharacteristic
82#13095498—22no—
83#13095498—22no—
84#13095498—22no—
85D776—22no—
86D776—41no—
87#13095496—22no—
88#13095496D414yescomplemented
89#13095495—22no—
90#13095494—22no—
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C4⋊Dic97C2311yescharacteristic
94#13095490—21no—
95#13095490D411yescharacteristic
96C22⋊Dic97—21no—
97C22⋊Dic97D411yescharacteristic
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488D411yescharacteristic
107C4×Dic97C2311yescharacteristic
108#41×C97—22no—
109#41×C97—41no—
110C2×Q8×C97C2311yescharacteristic
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97C2311yescharacteristic
114C22×C4×C97—21no—
115C22×C4×C97C2311yescharacteristic
116Q16×C97—41no—
117SD16×C97—22no—
118C2×C8×C97C2311yescharacteristic
119C2×C8×C97C2311yescharacteristic
120C4⋊C4×C97—21no—
121C4⋊C4×C97C2311yescharacteristic
122C22⋊C4×C97—21no—
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—81no—
133C22×D97—81no—
134C22×D97C2×D411yescharacteristic
135Q8×C97—21no—
136Q8×C97—21no—
137Q8×C97—21no—
138Q8×C97—21no—
139D4×C97—21no—
140D4×C97—42no—
141D4×C97C2×D412yes—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—42no—
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2411yescharacteristic
150C22⋊D97—41no—
151C22⋊D97—42no—
152C22⋊D97—81no—
153C2×Dic97—21no—
154C2×Dic97—81no—
155C2×Dic97C2×D411yescharacteristic
156C2×Dic97C2×D411yescharacteristic
157C2×Dic97C2×D411yescharacteristic
158D388—21no—
159D388—21no—
160D388—21no—
161D388—21no—
162D388—21no—
163D388—42no—
164D388—42no—
165D388—81no—
166D388C2×D412yes—
167C4×D97—21no—
168C4×D97—21no—
169C4×D97—42no—
170C4×D97—81no—
171C8×C97—21no—
172C8×C97—21no—
173#11715895—21no—
174#11715895C2×D412yes—
175C22×C97—22no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97C22×D411yescharacteristic
182C2×D97—21no—
183C2×D97—22no—
184C2×D97—21no—
185C2×D97—21no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—41no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—84no—
195C2×D97—84no—
196C4×C97—21no—
197C4×C97—41no—
198C4×C97—41no—
199C4×C97—41no—
200C4×C97C22×D411yescharacteristic
201C4×C97H(5,𝔽2)11yescharacteristic
202Dic97—21no—
203Dic97—21no—
204Dic97—21no—
205Dic97—41no—
206C2×C97—41no—
207C2×C97—42no—
208C2×C97D4211yescharacteristic
209C2×C97#57511yescharacteristic
210C2×C97#57611yescharacteristic
211D97—41no—
212D97—42no—
213D97—81no—
214D97—81no—
215#3273—971nomaximal, Sylow
216C97#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—971no—
218C2×AGL(1,ℤ/8)—971no—
219#546—971no—
220#546—971no—
221#495—971no—
222#468—971no—
223#468—971no—
224#463—971no—
225#463—971no—
226#452—971no—
227#452D9711yescharacteristic, complemented
228#447—971no—
229#447—971no—
230C4×D8—971no—
231#436—971no—
232C2×#41—971no—
233C2×#41C2×D9711yescharacteristic
234C22×D4—971no—
235C22×D4—971no—
236AGL(1,ℤ/8)—1944no—
237AGL(1,ℤ/8)—1944no—
238C2×Q16C2×D9711yescharacteristic, complemented
239C2×SD16—971no—
240C2×SD16—971no—
241C2×SD16—971no—
242C2×SD16C2×D9711yescharacteristic, complemented
243C2×D8—971no—
244C2×D8—971no—
245C2×D8—971no—
246C2×D8—971no—
247C2×M16—971no—
248C2×M16—971no—
249#124—971no—
250#124—971no—
251#123—1941no—
252#123—1941no—
253C22⋊Q8—971no—
254C22⋊Q8C2×D9711yescharacteristic, complemented
255#121—971no—
256#121—971no—
257#121—1941no—
258#121—1941no—
259#120—1941no—
260#120—1941no—
261C4×D4—971no—
262C4×D4—971no—
263C4×D4—971no—
264C2×C22⋊C4—1941no—
265C2×C22⋊C4—1941no—
266#107—971no—
267C4⋊C8—971no—
268#103—971no—
269#103C2×D9711yescharacteristic, complemented
270#102—971no—
271#102—971no—
272#102—971no—
273#102C2×D9711yescharacteristic, complemented
274C22⋊C8—971no—
275C22⋊C8C2×D9711yescharacteristic, complemented
276C4×C8—971no—
277C24—1941no—
278C24—1941no—
279#41—22no—
280#41—41no—
281#41—3881no—
282#41—1942no—
283C2×Q8C22×D9711yescharacteristic
284C2×Q8C22×D9711yescharacteristic
285C2×D4—21no—
286C2×D4—971no—
287C2×D4—971no—
288C2×D4—971no—
289C2×D4—971no—
290C2×D4—3881no—
291C2×D4—3881no—
292C2×D4—1941no—
293C2×D4—1942no—
294C2×D4—1941no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4—1942no—
298C2×D4—1941no—
299C2×D4C22×D9711yescharacteristic
300C22×C4—21no—
301C22×C4—971no—
302C22×C4—1941no—
303C22×C4—1941no—
304C22×C4—1941no—
305C22×C4C22×D9711yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—1942no—
309SD16—1942no—
310SD16—1942no—
311D8—974no—
312D8—3881no—
313D8—1942no—
314D8—1942no—
315M16—1942no—
316M16—1942no—
317C2×C8—971no—
318C2×C8—971no—
319C2×C8C22×D9711yescharacteristic
320C2×C8C22×D9711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—971no—
323C4⋊C4—971no—
324C4⋊C4—1941no—
325C4⋊C4C22×D9711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—971no—
328C22⋊C4—971no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C42—971no—
339C23—21no—
340C23—7761no—
341C23—7761no—
342C23—971no—
343C23—3882no—
344C23—3882no—
345C23—3882no—
346C23—3882no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23D4×D9711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—7761no—
358D4—7761no—
359D4—972no—
360D4—3882no—
361D4—3882no—
362D4—3882no—
363D4—3881no—
364D4—1941no—
365D4—1941no—
366D4—1941no—
367D4—1941no—
368D4—1941no—
369D4D4×D9712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—7761no—
377C2×C4—7761no—
378C2×C4—971no—
379C2×C4—971no—
380C2×C4—971no—
381C2×C4—3882no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4D4×D9711yescharacteristic
386C2×C4C23×D9711yescharacteristic
387C8—21no—
388C8—21no—
389C8—972no—
390C8—1941no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—7764no—
398C22—7764no—
399C22—3881no—
400C22—3881no—
401C22—3881no—
402C22—3881no—
403C22—3881no—
404C22—3881no—
405C22—3881no—
406C22—3881no—
407C22—1941no—
408C22—1941no—
409C22—1942no—
410C22—1941no—
411C22C2×D4×D9711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3881no—
417C4—1941no—
418C4—1941no—
419C4—1941no—
420C4C2×D4×D9711yescharacteristic
421C4#1569085511yescharacteristic
422C2—41no—
423C2—42no—
424C2—7761no—
425C2—7761no—
426C2—3881no—
427C2—3882no—
428C2#2026136711yescharacteristic
429C2#2026153411yescharacteristic
430C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#452×C97C2
socleC22×C97C22×D4
hypercenter#452D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690835C2semidirect
3. #20261376C2semidirect
4. #20261373C2semidirect
5. #20261120C2semidirect
6. #20260984C2semidirect
7. #20260983C2semidirect
8. #20260953C2semidirect
9. #20260952C2semidirect
10. #20260896C2semidirect
11. #20260871C2semidirect
12. #20260870C2semidirect
13. #20260815C2semidirect
14. #20260810C2semidirect
15. #452×C97C2semidirect
17. C22×D388C22semidirect
18. C2×D4×D97C22semidirect
21. C2×#13095498C22semidirect
22. C2×#13095498C22semidirect
23. C2×#13095498C22semidirect
28. C2×#13095495C22semidirect
29. C2×#13095494C22semidirect
30. #15690799C22semidirect
31. #15690799C22semidirect
34. #15690786C22semidirect
41. D4×Dic97C22semidirect
42. #15690772C22semidirect
46. #15690743C22semidirect
47. #15690739C22semidirect
48. #15690734C22semidirect
49. #15690732C22semidirect
50. #15690732C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. C8×Dic97C22semidirect
55. C2×Q16×C97C22semidirect
56. C2×SD16×C97C22semidirect
57. C22⋊Q8×C97C22semidirect
58. #103×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
88. #13095496D4semidirect
216. C97#3273semidirect
227. #452D97semidirect
238. C2×Q16C2×D97semidirect
242. C2×SD16C2×D97semidirect
254. C22⋊Q8C2×D97semidirect
269. #103C2×D97semidirect
273. #102C2×D97semidirect
275. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908352
#202613762
#202613732
#202611202
#202609842
#202609832
#202609532
#202609522
#202608962
#202608712
#202608702
#202608152
#202608102
#452×C972
#327397