Finite Group Databasebeta

← #44561866

Subgroups — #44561866

Order 32128.

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×#22548739C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632345C211yesmaximal, characteristic, complemented
4D4×D1004C211yesmaximal, characteristic, complemented
5#30632077C211yesmaximal, characteristic, complemented
6#30631936C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631888C211yesmaximal, characteristic, complemented
9#30631836C211yesmaximal, characteristic, complemented
10#30631833C211yesmaximal, characteristic, complemented
11#30631827C211yesmaximal, characteristic, complemented
12#30631819C211yesmaximal, characteristic, complemented
13#30631787C211yesmaximal, characteristic, complemented
14#30631757C211yesmaximal, characteristic, complemented
15#463×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic, complemented
18C22×D1004—21no—
19C2×D4×D251C2211yescharacteristic
20C2×D4×D251C2211yescharacteristic
21#22548739—24no—
22C2×#16860799C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic
25C2×#16860797C2211yescharacteristic, complemented
26C2×#16860797C2211yescharacteristic, complemented
27SD16×D251—24no—
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860795C2211yescharacteristic, complemented
30C2×#16860793C2211yescharacteristic, complemented
31C2×C8×D251C2211yescharacteristic, complemented
32#22548712C2211yescharacteristic
33#22548701—21no—
34#22548692—21no—
35#22548691C2211yescharacteristic, complemented
36#22548690C2211yescharacteristic
37#22548690C2211yescharacteristic
38#22548689—21no—
39#22548688—21no—
40#22548686C2211yescharacteristic, complemented
41#22548685—21no—
42#22548685—21no—
43C4×Dic502C2211yescharacteristic, complemented
44C4×D1004C2211yescharacteristic
45(C22⋊D251)⋊C4—21no—
46#22548648C2211yescharacteristic, complemented
47#22548638C2211yescharacteristic, complemented
48#22548637C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548626C2211yescharacteristic, complemented
54C2×SD16×C251C2211yescharacteristic
55C2×D8×C251C2211yescharacteristic
56#124×C251C2211yescharacteristic, complemented
57C4×D4×C251C2211yescharacteristic
58C4⋊C8×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860809—22no—
64#16860809—41no—
65C2×Dic502—21no—
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—41no—
69C2×D1004—21no—
70C2×D1004—21no—
71C2×D1004—21no—
72C2×D1004—21no—
73C2×D1004—44no—
74C2×D1004D411yescharacteristic
75C2×D1004C2311yescharacteristic
76C2×D1004C2311yescharacteristic
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79C2×C22⋊D251—21no—
80C2×C22⋊D251—21no—
81D4×D251—22no—
82D4×D251—22no—
83D4×D251—41no—
84D4×D251—41no—
85C22×Dic251—21no—
86C2×C4×D251—21no—
87C2×C4×D251C2311yescharacteristic
88C2×C4×D251C2311yescharacteristic
89#16860799—22no—
90#16860799—22no—
91#16860798D414yescomplemented
92#16860797—22no—
93#16860797—41no—
94#16860795—22no—
95#16860795—41no—
96#16860793—22no—
97C2×C251⋊C8C2311yescharacteristic
98C2×C251⋊C8C2311yescharacteristic
99C8×D251—22no—
100C4⋊Dic251D411yescharacteristic
101C4⋊Dic251C2311yescharacteristic
102Dic251⋊C4—21no—
103Dic251⋊C4—21no—
104C22⋊Dic251—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4D411yescharacteristic
109(C2×D251)⋊C4D411yescharacteristic
110C4×Dic251—21no—
111C2×Q8×C251C2311yescharacteristic
112C2×D4×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251—21no—
115SD16×C251—22no—
116D8×C251—22no—
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C4⋊C4×C251C2311yescharacteristic
120C22⋊C4×C251—21no—
121C22⋊C4×C251—21no—
122C22⋊C4×C251—21no—
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—81no—
131C22×D251—81no—
132C22×D251C2×D411yescharacteristic
133C22×D251C2×D411yescharacteristic
134Dic502—21no—
135Dic502—42no—
136Dic502C2×D412yescomplemented
137D1004—22no—
138D1004—21no—
139D1004—21no—
140D1004—21no—
141D1004—21no—
142D1004—42no—
143D1004—42no—
144D1004—42no—
145D1004—81no—
146C22⋊D251—42no—
147C22⋊D251—42no—
148C22⋊D251—42no—
149C22⋊D251—42no—
150C2×Dic251—21no—
151C2×Dic251—21no—
152C2×Dic251—42no—
153C2×Dic251C2×D411yescharacteristic
154C2×Dic251C2×D411yescharacteristic
155C4×D251—21no—
156C4×D251—21no—
157C4×D251—21no—
158C4×D251—21no—
159C4×D251—42no—
160C251⋊C8—21no—
161C251⋊C8C2×D412yescomplemented
162C23×C251—21no—
163C23×C251—21no—
164Q8×C251—21no—
165Q8×C251—21no—
166D4×C251—21no—
167D4×C251—21no—
168D4×C251—21no—
169D4×C251C2×D412yes—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—42no—
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251—21no—
178C22×C251—22no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251C22×D411yescharacteristic
185C2×D251—21no—
186C2×D251—21no—
187C2×D251—21no—
188C2×D251—21no—
189C2×D251—21no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—84no—
197C2×D251—84no—
198C2×D251AGL(1,ℤ/8)12yescomplemented
199C4×C251—21no—
200C4×C251—21no—
201C4×C251—41no—
202C4×C251—41no—
203C4×C251C22×D411yescharacteristic
204C4×C251H(5,𝔽2)11yescharacteristic
205Dic251—21no—
206Dic251—21no—
207Dic251—22no—
208Dic251—41no—
209C2×C251—42no—
210C2×C251—81no—
211C2×C251D4211yescharacteristic
212C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C251#57711yescharacteristic
214D251—22no—
215D251—41no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42—2511no—
224#496—2511no—
225#468—2511no—
226#466—2511no—
227#463D25111yescharacteristic, complemented
228#460—2511no—
229#450—2511no—
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435—2511no—
235C2×#41—2511no—
236C22×Q8—2511no—
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4—2511no—
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16—2511no—
247C2×SD16C2×D25111yescharacteristic
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8C2×D25111yescharacteristic
251C2×M16—2511no—
252C22×C8—2511no—
253#127—2511no—
254#124—5021no—
255#124C2×D25111yescharacteristic, complemented
256C22⋊Q8—5021no—
257C22⋊Q8—2511no—
258#121—5021no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121—2511no—
263#120—5021no—
264#120—5021no—
265C4×Q8—2511no—
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4C2×D25111yescharacteristic
269#107—2511no—
270C4⋊C8C2×D25111yescharacteristic, complemented
271#103—2511no—
272#103C2×D25111yescharacteristic, complemented
273#102—2511no—
274#102—2511no—
275#102—2511no—
276#102C2×D25111yescharacteristic, complemented
277C22⋊C8—2511no—
278C22⋊C8—2511no—
279#97—2511no—
280C24—5021no—
281C24—5021no—
282#41—10041no—
283#41—5022no—
284C2×Q8—10041no—
285C2×Q8—5021no—
286C2×Q8—5022no—
287C2×Q8—2511no—
288C2×Q8C22×D25111yescharacteristic
289C2×D4—10044no—
290C2×D4—10041no—
291C2×D4—10041no—
292C2×D4—5021no—
293C2×D4—5021no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—5022no—
297C2×D4—5022no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4C22×D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—21no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4—2511no—
312SD16—22no—
313SD16—10041no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—22no—
319D8—10041no—
320D8—5022no—
321M16—5022no—
322C2×C8—5022no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—5021no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4—2511no—
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4—2511no—
340C22⋊C4—2511no—
341C42—5021no—
342C42C22×D25111yescharacteristic
343C23—21no—
344C23—21no—
345C23—20081no—
346C23—20081no—
347C23—10042no—
348C23—10042no—
349C23—10042no—
350C23—10042no—
351C23—5021no—
352C23—5021no—
353C23—2511no—
354C23—2511no—
355Q8—21no—
356Q8—21no—
357Q8—10042no—
358Q8—5021no—
359Q8—2512no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—20081no—
364D4—10042no—
365D4—10042no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—10042no—
370D4—10042no—
371D4—5022no—
372D4—5021no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4D4×D25112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—10042no—
381C2×C4—10042no—
382C2×C4—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—2511no—
389C2×C4—2511no—
390C2×C4D4×D25111yescharacteristic
391C2×C4D4×D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—21no—
394C8—21no—
395C8—5021no—
396C8—2512no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—20084no—
404C22—20084no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—5021no—
412C22—5021no—
413C22—5021no—
414C22—5021no—
415C22—5021no—
416C22—2512no—
417C22C2×D4×D25111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—10041no—
423C4—5021no—
424C4—5022no—
425C4—5021no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876111yescharacteristic
428C2—42no—
429C2—81no—
430C2—20081no—
431C2—20081no—
432C2—10041no—
433C2—5022no—
434C2#3063232111yescharacteristic
435C2AGL(1,ℤ/8)×D25111yescharacteristic
436C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#463×C251C2
socleC22×C251C22×D4
hypercenter#463D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632345C2semidirect
4. D4×D1004C2semidirect
5. #30632077C2semidirect
6. #30631936C2semidirect
7. #30631922C2semidirect
8. #30631888C2semidirect
9. #30631836C2semidirect
10. #30631833C2semidirect
11. #30631827C2semidirect
12. #30631819C2semidirect
13. #30631787C2semidirect
14. #30631757C2semidirect
15. #463×C251C2semidirect
16. C2×#16860809C22semidirect
17. C2×Q8×D251C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860799C22semidirect
25. C2×#16860797C22semidirect
26. C2×#16860797C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860795C22semidirect
30. C2×#16860793C22semidirect
31. C2×C8×D251C22semidirect
35. #22548691C22semidirect
40. #22548686C22semidirect
43. C4×Dic502C22semidirect
46. #22548648C22semidirect
47. #22548638C22semidirect
48. #22548637C22semidirect
49. #22548637C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548626C22semidirect
56. #124×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #103×C251C22semidirect
60. #102×C251C22semidirect
91. #16860798D4semidirect
136. Dic502C2×D4semidirect
161. C251⋊C8C2×D4semidirect
198. C2×D251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
227. #463D251semidirect
255. #124C2×D251semidirect
270. C4⋊C8C2×D251semidirect
272. #103C2×D251semidirect
276. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×SD16×D2512
#306323452
D4×D10042
#306320772
#306319362
#306319222
#306318882
#306318362
#306318332
#306318272
#306318192
#306317872
#306317572
#463×C2512
#3265251