Finite Group Databasebeta

← #44562383

Subgroups — #44562383

Order 32128.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632375C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic
3#30632053C211yesmaximal, characteristic, complemented
4#30632013C211yesmaximal, characteristic, complemented
5#30632002C211yesmaximal, characteristic, complemented
6#30631934C211yesmaximal, characteristic, complemented
7#30631917C211yesmaximal, characteristic, complemented
8#30631888C211yesmaximal, characteristic, complemented
9#30631839C211yesmaximal, characteristic, complemented
10#30631839C211yesmaximal, characteristic, complemented
11#30631787C211yesmaximal, characteristic, complemented
12#30631757C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#478×C251C211yesmaximal, characteristic, complemented
16C22×D1004—21no—
17C2×D4×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19C2×#16860799C2211yescharacteristic, complemented
20C2×#16860799C2211yescharacteristic, complemented
21C2×#16860798C2211yescharacteristic
22C2×#16860793C2211yescharacteristic, complemented
23C2×C8×D251C2211yescharacteristic, complemented
24#22548712C2211yescharacteristic
25#22548710—21no—
26#22548706—21no—
27#22548700—21no—
28#22548695C2211yescharacteristic, complemented
29#22548695C2211yescharacteristic, complemented
30#22548690C2211yescharacteristic
31#22548690C2211yescharacteristic
32#22548689—21no—
33#22548685—21no—
34#22548685—21no—
35C4×Dic502C2211yescharacteristic, complemented
36C4×D1004C2211yescharacteristic
37#22548673—21no—
38#22548673C2211yescharacteristic, complemented
39C4⋊C4×D251C2211yescharacteristic, complemented
40#22548648C2211yescharacteristic, complemented
41#22548638C2211yescharacteristic, complemented
42#22548637C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic, complemented
44#22548636C2211yescharacteristic
45#22548635C2211yescharacteristic, complemented
46#22548635C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49(C2×D251)⋊C8C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51#22548626C2211yescharacteristic, complemented
52#125×C251C2211yescharacteristic, complemented
53C4×D4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56C4⋊C8×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60C23×D251—21no—
61C2×Dic502C2311yescharacteristic
62C2×D1004—21no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—44no—
67C2×D1004D411yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72D4×D251—22no—
73D4×D251—22no—
74D4×D251—41no—
75D4×D251—41no—
76C2×C4×D251—21no—
77C2×C4×D251C2311yescharacteristic
78C2×C4×D251C2311yescharacteristic
79#16860799—41no—
80#16860799—41no—
81#16860798—22no—
82#16860793—41no—
83C2×C251⋊C8C2311yescharacteristic
84C2×C251⋊C8C2311yescharacteristic
85C8×D251—41no—
86C4⋊Dic251—21no—
87C4⋊Dic251—21no—
88C4⋊Dic251D411yescharacteristic
89C4⋊Dic251C2311yescharacteristic
90Dic251⋊C4—21no—
91Dic251⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4D411yescharacteristic
98(C2×D251)⋊C4D411yescharacteristic
99C4×Dic251—21no—
100C4×Dic251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—42no—
118C22×D251—81no—
119C22×D251—81no—
120C22×D251C2×D411yescharacteristic
121C22×D251C2×D411yescharacteristic
122Dic502—21no—
123Dic502—21no—
124D1004—22no—
125D1004—21no—
126D1004—21no—
127D1004—21no—
128D1004—21no—
129D1004—42no—
130D1004—42no—
131D1004—42no—
132D1004—81no—
133C22⋊D251—42no—
134C22⋊D251—42no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251C2×D411yescharacteristic
138C2×Dic251C2×D411yescharacteristic
139C4×D251—21no—
140C4×D251—21no—
141C4×D251—21no—
142C4×D251—21no—
143C4×D251—42no—
144C251⋊C8—21no—
145C251⋊C8—21no—
146C23×C251—21no—
147D4×C251—21no—
148D4×C251#4112yes—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—42no—
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2411yescharacteristic
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—22no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251C22×D411yescharacteristic
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—21no—
165C2×D251—21no—
166C2×D251—21no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—84no—
174C2×D251—84no—
175C2×D251AGL(1,ℤ/8)12yes—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251C2×#4111yescharacteristic
182C4×C251H(5,𝔽2)11yescharacteristic
183Dic251—21no—
184Dic251—21no—
185Dic251—41no—
186Dic251—41no—
187C2×C251—42no—
188C2×C251#54611yescharacteristic
189C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C251#57711yescharacteristic
191D251—22no—
192D251—41no—
193D251—81no—
194D251—81no—
195C251#331011yescharacteristic, complemented, Sylow
196#3310—2511nomaximal, Sylow
197#552—2511no—
198D42—2511no—
199#490—2511no—
200#482—2511no—
201#480—2511no—
202#478D25111yescharacteristic, complemented
203#468—2511no—
204#465—2511no—
205#460—2511no—
206#450—2511no—
207#450—2511no—
208#440—2511no—
209#435—2511no—
210#418—2511no—
211C2×#102—2511no—
212C22×D4—5021no—
213C22×D4—2511no—
214C22×D4—2511no—
215C2×SD16—2511no—
216C2×SD16—2511no—
217C2×SD16—2511no—
218C2×M16—2511no—
219C22×C8—2511no—
220#127—2511no—
221#126—5021no—
222#125—5021no—
223#125C2×D25111yescharacteristic, complemented
224#123—5021no—
225C22⋊Q8—2511no—
226C22⋊Q8—2511no—
227#121—5021no—
228#121—2511no—
229#121—2511no—
230#120—5021no—
231#120—5021no—
232C4×Q8—2511no—
233C4×D4—2511no—
234C4×D4C2×D25111yescharacteristic
235#117—5021no—
236#117—2511no—
237C2×C4⋊C4—2511no—
238#107—2511no—
239#107C2×D25111yescharacteristic, complemented
240#106C2×D25111yescharacteristic, complemented
241C4⋊C8C2×D25111yescharacteristic, complemented
242#103—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102C2×D25111yescharacteristic
251#102C2×D25111yescharacteristic
252C22⋊C8—2511no—
253C22⋊C8—2511no—
254#97—2511no—
255C24—5021no—
256C24—5021no—
257C2×Q8—2511no—
258C2×D4—10044no—
259C2×D4—10041no—
260C2×D4—10041no—
261C2×D4—5021no—
262C2×D4—5021no—
263C2×D4—5022no—
264C2×D4—5022no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—5021no—
268C2×D4—5021no—
269C2×D4—2511no—
270C2×D4—2511no—
271C2×D4—2511no—
272C2×D4C22×D25111yescharacteristic
273C22×C4—21no—
274C22×C4—5021no—
275C22×C4—2511no—
276C22×C4—2511no—
277SD16—10041no—
278SD16—10041no—
279SD16—5022no—
280M16—10041no—
281C2×C8—10041no—
282C2×C8—2511no—
283C2×C8—2511no—
284C2×C8C22×D25111yescharacteristic
285C2×C8C22×D25111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—5021no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4C22×D25111yescharacteristic
295C4⋊C4C22×D25111yescharacteristic
296C4⋊C4C22×D25111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—2511no—
304C22⋊C4—2511no—
305C42—5021no—
306C42—5021no—
307C42C22×D25111yescharacteristic
308C23—21no—
309C23—20081no—
310C23—20081no—
311C23—10042no—
312C23—10042no—
313C23—10042no—
314C23—10042no—
315C23—5021no—
316C23—5021no—
317C23—2511no—
318C23—2511no—
319Q8—5021no—
320Q8—5021no—
321D4—21no—
322D4—20081no—
323D4—10042no—
324D4—10042no—
325D4—10042no—
326D4—10042no—
327D4—10042no—
328D4—5022no—
329D4—5021no—
330D4—5021no—
331D4—5021no—
332D4—5021no—
333D4#1686081112yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—10042no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—2511no—
346C2×C4—2511no—
347C2×C4D4×D25111yescharacteristic
348C2×C4D4×D25111yescharacteristic
349C2×C4C23×D25111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5021no—
353C8—5021no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—20084no—
358C22—20084no—
359C22—10041no—
360C22—10041no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—5021no—
366C22—5021no—
367C22—5021no—
368C22—5021no—
369C22—5021no—
370C22—2512no—
371C22C2×D4×D25111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—10041no—
378C4—10041no—
379C4—5021no—
380C4—5021no—
381C4C2×#1686081111yescharacteristic
382C4#2254876111yescharacteristic
383C2—42no—
384C2—20081no—
385C2—20081no—
386C2—10041no—
387C2—5022no—
388C2#3063232811yescharacteristic
389C2AGL(1,ℤ/8)×D25111yescharacteristic
390C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#478×C251C2
socleC22×C251C22×D4
prime radicalD4×D1004C2
hypercenter#478D251
hypocenterC251#3310
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #30632375C2semidirect
3. #30632053C2semidirect
4. #30632013C2semidirect
5. #30632002C2semidirect
6. #30631934C2semidirect
7. #30631917C2semidirect
8. #30631888C2semidirect
9. #30631839C2semidirect
10. #30631839C2semidirect
11. #30631787C2semidirect
12. #30631757C2semidirect
13. #30631690C2semidirect
14. #102×D251C2semidirect
15. #478×C251C2semidirect
19. C2×#16860799C22semidirect
20. C2×#16860799C22semidirect
22. C2×#16860793C22semidirect
23. C2×C8×D251C22semidirect
28. #22548695C22semidirect
29. #22548695C22semidirect
35. C4×Dic502C22semidirect
38. #22548673C22semidirect
39. C4⋊C4×D251C22semidirect
40. #22548648C22semidirect
41. #22548638C22semidirect
42. #22548637C22semidirect
43. #22548637C22semidirect
45. #22548635C22semidirect
46. #22548635C22semidirect
47. #22548635C22semidirect
48. #22548635C22semidirect
49. (C2×D251)⋊C8C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. #22548626C22semidirect
52. #125×C251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. C4⋊C8×C251C22semidirect
195. C251#3310semidirect
202. #478D251semidirect
223. #125C2×D251semidirect
239. #107C2×D251semidirect
240. #106C2×D251semidirect
241. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323752
D4×D10042
#306320532
#306320132
#306320022
#306319342
#306319172
#306318882
#306318392
#306317872
#306317572
#306316902
#102×D2512
#478×C2512
#3310251