Finite Group Databasebeta

← #44560193

Subgroups — #44560193

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632317C211yesmaximal, characteristic, complemented
2#30632159C211yesmaximal, characteristic
3#30631990C211yesmaximal, characteristic, complemented
4#30631985C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631977C211yesmaximal, characteristic, complemented
7#30631976C211yesmaximal, characteristic, complemented
8#30631968C211yesmaximal, characteristic, complemented
9#30631965C211yesmaximal, characteristic, complemented
10#30631726C211yesmaximal, characteristic, complemented
11#30631724C211yesmaximal, characteristic, complemented
12#30631712C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic
14#102×D251C211yesmaximal, characteristic
15#478×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860793C2211yescharacteristic
18C2×C8×D251C2211yescharacteristic
19#22548715C2211yescharacteristic, complemented
20#22548714—21no—
21#22548714C2211yescharacteristic, complemented
22#22548706C2211yescharacteristic, complemented
23#22548705—21no—
24#22548695—21no—
25#22548693—21no—
26C4×D1004C2211yescharacteristic
27D4×Dic251C2211yescharacteristic
28#22548677C2211yescharacteristic
29C4×C22⋊D251—21no—
30(C22⋊D251)⋊C4—21no—
31#22548673C2211yescharacteristic
32#22548673C2211yescharacteristic
33#22548672C2211yescharacteristic
34C4⋊C4×D251C2211yescharacteristic
35C2×(C2×D251)⋊C4—21no—
36C22⋊C4×D251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548649C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43C4⋊(C251⋊C8)C2211yescharacteristic, complemented
44Dic251⋊C8C2211yescharacteristic, complemented
45Dic251⋊C8C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic
50#22548635C2211yescharacteristic
51#22548635C2211yescharacteristic
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—
60C2×Dic502—21no—
61C2×Dic502—21no—
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64D4×D251—22no—
65D4×D251Q814yescomplemented
66C22×Dic251—21no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251C2311yescharacteristic
71#16860793—22no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C8×D251—22no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84Dic251⋊C4D411yescharacteristic
85C22⋊Dic251—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4D411yescharacteristic
91(C2×D251)⋊C4D411yescharacteristic
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251C2311yescharacteristic
95C2×D4×C251C2311yescharacteristic
96C22×C4×C251—21no—
97C2×C8×C251C2311yescharacteristic
98C2×C8×C251C2311yescharacteristic
99C4⋊C4×C251—21no—
100C4⋊C4×C251—21no—
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C22⋊C4×C251—21no—
105C42×C251C2311yescharacteristic
106C22×D251—21no—
107C22×D251—24no—
108C22×D251—81no—
109C22×D251—81no—
110C22×D251C2×D411yescharacteristic
111Dic502—81no—
112Dic502—81no—
113D1004—21no—
114D1004C2×Q812yes—
115C22⋊D251—24no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—81no—
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C4×D251—81no—
124C4×D251—81no—
125C4×D251—81no—
126C4×D251C2×Q812yes—
127C4×D251#4112yes—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251—21no—
131D4×C251—21no—
132D4×C251C2×Q812yes—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—81no—
137C2×C4×C251C2×D411yescharacteristic
138C2×C4×C251C2×D411yescharacteristic
139C2×C4×C251C2411yescharacteristic
140C8×C251—21no—
141C8×C251—21no—
142C22×C251—22no—
143C22×C251—41no—
144C22×C251—41no—
145C22×C251C22×D411yescharacteristic
146C2×D251—21no—
147C2×D251—21no—
148C2×D251—22no—
149C2×D251—41no—
150C2×D251—41no—
151C2×D251—84no—
152C2×D251C22⋊Q812yes—
153C4×C251—21no—
154C4×C251—21no—
155C4×C251—41no—
156C4×C251—41no—
157C4×C251—41no—
158C4×C251C22×Q811yescharacteristic
159C4×C251C2×#4111yescharacteristic
160Dic251—21no—
161Dic251—21no—
162Dic251—41no—
163Dic251—41no—
164Dic251—41no—
165Dic251C22⋊Q812yes—
166C2×C251—42no—
167C2×C251C2×C22⋊Q811yescharacteristic
168C2×C251#57611yescharacteristic
169C2×C251#57711yescharacteristic
170D251—22no—
171D251—42no—
172C251#308111yescharacteristic, complemented, Sylow
173#3081—2511nomaximal, Sylow
174#544—2511no—
175#518—2511no—
176#478—2511no—
177#478—2511no—
178#478—2511no—
179#478D25111yescharacteristic, complemented
180#476—2511no—
181#476—2511no—
182#474—2511no—
183#474—2511no—
184#428—2511no—
185#427—2511no—
186#424—2511no—
187#418—2511no—
188C2×#102—2511no—
189C22×D4—2511no—
190C2×M16—2511no—
191C22×C8—2511no—
192C4⋊Q8—5021no—
193C4⋊Q8—2511no—
194C4⋊Q8—2511no—
195#125—5021no—
196#125—2511no—
197#125C2×D25111yescharacteristic, complemented
198C22⋊Q8—5021no—
199C22⋊Q8—5021no—
200C4×D4—5021no—
201C4×D4—5021no—
202C4×D4—2511no—
203C4×D4—2511no—
204C4×D4—2511no—
205C4×D4C2×D25111yescharacteristic
206#117—2511no—
207#117—2511no—
208#117—2511no—
209C2×C4⋊C4—2511no—
210C2×C22⋊C4—5021no—
211C2×C22⋊C4—5021no—
212#107—2511no—
213#107—2511no—
214#107—2511no—
215#107C2×D25111yescharacteristic, complemented
216#106—2511no—
217#106—2511no—
218#106—2511no—
219#106C2×D25111yescharacteristic, complemented
220C4⋊C8—2511no—
221C4⋊C8—2511no—
222C4⋊C8—2511no—
223C4⋊C8C2×D25111yescharacteristic, complemented
224#102—2511no—
225#102—2511no—
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102C2×D25111yescharacteristic
231#102C2×D25111yescharacteristic
232C24—5021no—
233C2×Q8—5021no—
234C2×Q8—5021no—
235C2×D4—5022no—
236C2×D4—5021no—
237C2×D4—2514no—
238C2×D4—2511no—
239C2×D4C22×D25111yescharacteristic
240C22×C4—21no—
241C22×C4—5021no—
242C22×C4—5021no—
243C22×C4—5021no—
244C22×C4—5021no—
245C22×C4—2511no—
246M16—5022no—
247C2×C8—5022no—
248C2×C8—2511no—
249C2×C8—2511no—
250C2×C8C22×D25111yescharacteristic
251C2×C8C22×D25111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—5021no—
255C4⋊C4—5021no—
256C4⋊C4—5021no—
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—2511no—
260C4⋊C4—2511no—
261C4⋊C4—2511no—
262C4⋊C4—2511no—
263C4⋊C4—2511no—
264C4⋊C4C22×D25111yescharacteristic
265C4⋊C4C22×D25111yescharacteristic
266C4⋊C4C22×D25111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5021no—
269C22⋊C4—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—2511no—
274C22⋊C4—2511no—
275C42—5021no—
276C42—5021no—
277C42—2511no—
278C42C22×D25111yescharacteristic
279C23—21no—
280C23—20081no—
281C23—20081no—
282C23—5024no—
283C23—5021no—
284C23—2511no—
285Q8—20081no—
286Q8—20081no—
287D4—21no—
288D4—5024no—
289D4—5021no—
290D4—2512no—
291D4Q8×D25112yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—20081no—
297C2×C4—20081no—
298C2×C4—20081no—
299C2×C4—20081no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—2511no—
304C2×C4—2511no—
305C2×C4—2511no—
306C2×C4—2512no—
307C2×C4—2512no—
308C2×C4D4×D25111yescharacteristic
309C2×C4D4×D25111yescharacteristic
310C2×C4C23×D25111yescharacteristic
311C8—21no—
312C8—21no—
313C8—5021no—
314C8—5021no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—20084no—
319C22—10041no—
320C22—10041no—
321C22—5022no—
322C22—5021no—
323C22—5021no—
324C22—2512no—
325C22C2×D4×D25111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—10041no—
332C4—10041no—
333C4—10041no—
334C4—5021no—
335C4—5021no—
336C4—2512no—
337C4C2×Q8×D25111yescharacteristic
338C4#41×D25111yescharacteristic
339C2—42no—
340C2—10042no—
341C2—5022no—
342C2C22⋊Q8×D25111yescharacteristic
343C2#3063248911yescharacteristic
344C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#478×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#478D251
hypocenterC251#3081
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. #30632317C2semidirect
3. #30631990C2semidirect
4. #30631985C2semidirect
5. #30631984C2semidirect
6. #30631977C2semidirect
7. #30631976C2semidirect
8. #30631968C2semidirect
9. #30631965C2semidirect
10. #30631726C2semidirect
11. #30631724C2semidirect
12. #30631712C2semidirect
15. #478×C251C2semidirect
19. #22548715C22semidirect
21. #22548714C22semidirect
22. #22548706C22semidirect
37. #22548649C22semidirect
38. #22548649C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. #22548646C22semidirect
42. #22548646C22semidirect
43. C4⋊(C251⋊C8)C22semidirect
44. Dic251⋊C8C22semidirect
45. Dic251⋊C8C22semidirect
52. #125×C251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. C4⋊C8×C251C22semidirect
65. D4×D251Q8semidirect
172. C251#3081semidirect
179. #478D251semidirect
197. #125C2×D251semidirect
215. #107C2×D251semidirect
219. #106C2×D251semidirect
223. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323172
#306321592
#306319902
#306319852
#306319842
#306319772
#306319762
#306319682
#306319652
#306317262
#306317242
#306317122
#306316902
#102×D2512
#478×C2512
#3081251