Finite Group Databasebeta

← #44562171

Subgroups — #44562171

Order 32128.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632337C211yesmaximal, characteristic, complemented
2D4×C22⋊D251C211yesmaximal, characteristic, complemented
3#30632032C211yesmaximal, characteristic, complemented
4#30632001C211yesmaximal, characteristic, complemented
5#30631999C211yesmaximal, characteristic, complemented
6#30631965C211yesmaximal, characteristic, complemented
7#30631921C211yesmaximal, characteristic, complemented
8#30631920C211yesmaximal, characteristic, complemented
9#30631880C211yesmaximal, characteristic, complemented
10#30631819C211yesmaximal, characteristic, complemented
11D8×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13C2×#22548636C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16C2×#16860809—21no—
17C22×C22⋊D251—21no—
18C2×D4×D251C2211yescharacteristic
19C2×#16860795C2211yescharacteristic, complemented
20C22×C251⋊C8C2211yescharacteristic, complemented
21C2×C8×D251C2211yescharacteristic, complemented
22#22548714C2211yescharacteristic, complemented
23#22548711C2211yescharacteristic
24#22548697—21no—
25#22548696—21no—
26#22548692—21no—
27#22548689C2211yescharacteristic
28#22548687—21no—
29#22548686C2211yescharacteristic
30#22548686C2211yescharacteristic, complemented
31#22548684—21no—
32#22548683—21no—
33D4×Dic251C2211yescharacteristic
34D4×Dic251C2211yescharacteristic, complemented
35C4×C22⋊D251C2211yescharacteristic
36C2×C4⋊Dic251C2211yescharacteristic, complemented
37C4⋊C4×D251C2211yescharacteristic, complemented
38#22548649C2211yescharacteristic, complemented
39#22548649C2211yescharacteristic, complemented
40#22548648C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548637C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic, complemented
44#22548636C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic, complemented
47#22548636C2211yescharacteristic, complemented
48#22548636C2211yescharacteristic, complemented
49#22548635C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×D8×C251C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic, complemented
55#121×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860809—44no—
61C2×Dic502—21no—
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—44no—
70C2×C22⋊D251D411yescharacteristic
71D4×D251—22no—
72D4×D251—41no—
73C22×Dic251—21no—
74C22×Dic251—21no—
75C22×Dic251—21no—
76C2×C4×D251—21no—
77C2×C4×D251C2311yescharacteristic
78#16860795—41no—
79C2×C251⋊C8—41no—
80C2×C251⋊C8C2311yescharacteristic
81C2×C251⋊C8C2311yescharacteristic
82C8×D251—41no—
83C4⋊Dic251—21no—
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86C4⋊Dic251C2311yescharacteristic
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251D411yescharacteristic
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97C4×Dic251C2311yescharacteristic
98C24×C251—21no—
99C2×D4×C251—22no—
100C2×D4×C251—21no—
101C2×D4×C251—21no—
102C2×D4×C251—41no—
103C2×D4×C251C2311yescharacteristic
104C2×D4×C251C2311yescharacteristic
105C2×D4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107D8×C251—22no—
108D8×C251—41no—
109C2×C8×C251C2311yescharacteristic
110C2×C8×C251C2311yescharacteristic
111C4⋊C4×C251C2311yescharacteristic
112C22⋊C4×C251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—81no—
117C22×D251C2×D411yescharacteristic
118Dic502—42no—
119D1004—21no—
120D1004—21no—
121C22⋊D251—42no—
122C22⋊D251—42no—
123C22⋊D251—42no—
124C22⋊D251—42no—
125C22⋊D251—42no—
126C22⋊D251—81no—
127C22⋊D251D814yescomplemented
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—42no—
132C2×Dic251—42no—
133C2×Dic251—42no—
134C2×Dic251C2×D411yescharacteristic
135C2×Dic251C2×D411yescharacteristic
136C2×Dic251C2×D411yescharacteristic
137C4×D251—21no—
138C4×D251—21no—
139C4×D251—42no—
140C251⋊C8—21no—
141C251⋊C8—21no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—42no—
146C23×C251—42no—
147C23×C251—81no—
148C23×C251C2×D411yescharacteristic
149D4×C251—21no—
150D4×C251—21no—
151D4×C251—21no—
152D4×C251—21no—
153D4×C251—21no—
154D4×C251—42no—
155D4×C251—42no—
156D4×C251#4112yescomplemented
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251C2×D411yescharacteristic
161C2×C4×C251C2411yescharacteristic
162C8×C251—21no—
163C8×C251—21no—
164C22×C251—21no—
165C22×C251—21no—
166C22×C251—22no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251—41no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—84no—
176C22×C251C2×D812yes—
177C22×C251C22×D411yescharacteristic
178C2×D251—21no—
179C2×D251—21no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—84no—
184C2×D251C2×D812yes—
185C4×C251—21no—
186C4×C251—41no—
187C4×C251C2×#4111yescharacteristic
188C4×C251H(5,𝔽2)11yescharacteristic
189Dic251—21no—
190Dic251—21no—
191Dic251—41no—
192Dic251—41no—
193Dic251—41no—
194Dic251C2×D812yes—
195C2×C251—22no—
196C2×C251—42no—
197C2×C251—81no—
198C2×C251—81no—
199C2×C251#54611yescharacteristic
200C2×C251C22×D811yescharacteristic
201C2×C251#57711yescharacteristic
202D251—22no—
203D251—81no—
204C251#329411yescharacteristic, complemented, Sylow
205#3294—2511nomaximal, Sylow
206#547—2511no—
207D42—2511no—
208#486—2511no—
209#480—2511no—
210#480—2511no—
211#474—2511no—
212#466—2511no—
213#466—2511no—
214#459—2511no—
215#447—2511no—
216#447D25111yescharacteristic, complemented
217C4×D8—2511no—
218C8×D4—2511no—
219C2×#102—2511no—
220C2×#102—2511no—
221C2×#41—5021no—
222C22×D4—5021no—
223C22×D4—2511no—
224C22×D4C2×D25111yescharacteristic
225C2×D8—2511no—
226C2×D8C2×D25111yescharacteristic
227C2×D8C2×D25111yescharacteristic, complemented
228C22×C8—2511no—
229C22×C8—2511no—
230C4⋊Q8—2511no—
231#127—2511no—
232#123—5021no—
233#123—5021no—
234C22⋊Q8—5021no—
235#121—5021no—
236#121—2511no—
237#121—2511no—
238#121—2511no—
239#121C2×D25111yescharacteristic, complemented
240#120—5021no—
241#120—5021no—
242C4×D4—2511no—
243C4×D4—2511no—
244C4×D4—2511no—
245C2×C4⋊C4—2511no—
246C2×C4⋊C4—2511no—
247#107—2511no—
248#107—2511no—
249#107—2511no—
250C4⋊C8—2511no—
251#102—2511no—
252#102—2511no—
253#102—2511no—
254#102—2511no—
255#102—2511no—
256#102—2511no—
257#102—2511no—
258#102—2511no—
259#102C2×D25111yescharacteristic
260#102C2×D25111yescharacteristic, complemented
261C22⋊C8—2511no—
262C22⋊C8C2×D25111yescharacteristic, complemented
263C4×C8—2511no—
264C24—21no—
265C24—5021no—
266#41—10044no—
267C2×Q8—5021no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—10044no—
273C2×D4—10041no—
274C2×D4—5021no—
275C2×D4—5021no—
276C2×D4—5021no—
277C2×D4—5022no—
278C2×D4—5021no—
279C2×D4—5021no—
280C2×D4—5021no—
281C2×D4—2511no—
282C2×D4—2511no—
283C2×D4C22×D25111yescharacteristic
284C2×D4C22×D25111yescharacteristic
285C2×D4C22×D25111yescharacteristic
286C22×C4—5021no—
287C22×C4—5021no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—2511no—
291C22×C4C22×D25111yescharacteristic
292D8—22no—
293D8—41no—
294D8—10041no—
295C2×C8—10041no—
296C2×C8—10041no—
297C2×C8—2511no—
298C2×C8—2511no—
299C2×C8C22×D25111yescharacteristic
300C2×C8C22×D25111yescharacteristic
301C4⋊C4—5021no—
302C4⋊C4—5021no—
303C4⋊C4—5021no—
304C4⋊C4—2511no—
305C4⋊C4—2511no—
306C4⋊C4—2511no—
307C4⋊C4—2511no—
308C4⋊C4C22×D25111yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—5021no—
311C22⋊C4—5021no—
312C22⋊C4—5021no—
313C22⋊C4—5021no—
314C22⋊C4—5021no—
315C22⋊C4—2511no—
316C22⋊C4—2511no—
317C42—2511no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—20081no—
325C23—10042no—
326C23—10042no—
327C23—5021no—
328C23—2511no—
329C23D4×D25111yescharacteristic
330Q8—10042no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—20081no—
339D4—10042no—
340D4—10042no—
341D4—10042no—
342D4—10042no—
343D4—10042no—
344D4—5021no—
345D4—5021no—
346D4—2514no—
347D4#1686080912yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—10042no—
352C2×C4—10042no—
353C2×C4—10042no—
354C2×C4—10042no—
355C2×C4—5021no—
356C2×C4—5021no—
357C2×C4—5021no—
358C2×C4—5021no—
359C2×C4—5021no—
360C2×C4—2511no—
361C2×C4—2511no—
362C2×C4—2511no—
363C2×C4D4×D25111yescharacteristic
364C2×C4C23×D25111yescharacteristic
365C8—21no—
366C8—21no—
367C8—5021no—
368C8—5021no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—20084no—
382C22—10041no—
383C22—10041no—
384C22—10041no—
385C22—5021no—
386C22—5021no—
387C22—2512no—
388C22D8×D25112yes—
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—41no—
392C4—10041no—
393C4—10041no—
394C4—10041no—
395C4—5021no—
396C4—5021no—
397C4—2512no—
398C4C2×#1686080911yescharacteristic
399C4#2254876011yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—20081no—
405C2—5022no—
406C2#3063232411yescharacteristic
407C2C2×D8×D25111yescharacteristic
408C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3294
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #30632337C2semidirect
2. D4×C22⋊D251C2semidirect
3. #30632032C2semidirect
4. #30632001C2semidirect
5. #30631999C2semidirect
6. #30631965C2semidirect
7. #30631921C2semidirect
8. #30631920C2semidirect
9. #30631880C2semidirect
10. #30631819C2semidirect
11. D8×Dic251C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. C2×#22548636C2semidirect
14. #102×D251C2semidirect
15. #447×C251C2semidirect
19. C2×#16860795C22semidirect
20. C22×C251⋊C8C22semidirect
21. C2×C8×D251C22semidirect
22. #22548714C22semidirect
30. #22548686C22semidirect
34. D4×Dic251C22semidirect
36. C2×C4⋊Dic251C22semidirect
37. C4⋊C4×D251C22semidirect
38. #22548649C22semidirect
39. #22548649C22semidirect
40. #22548648C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548637C22semidirect
43. #22548637C22semidirect
46. #22548636C22semidirect
47. #22548636C22semidirect
48. #22548636C22semidirect
49. #22548635C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
54. C2×D8×C251C22semidirect
55. #121×C251C22semidirect
57. #102×C251C22semidirect
58. C22⋊C8×C251C22semidirect
127. C22⋊D251D8semidirect
156. D4×C251#41semidirect
204. C251#3294semidirect
216. #447D251semidirect
227. C2×D8C2×D251semidirect
239. #121C2×D251semidirect
260. #102C2×D251semidirect
262. C22⋊C8C2×D251semidirect
347. D4#16860809semidirect

Maximal subgroups

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

subgroupindex
#306323372
D4×C22⋊D2512
#306320322
#306320012
#306319992
#306319652
#306319212
#306319202
#306318802
#306318192
D8×Dic2512
(C22⋊D251)⋊C82
C2×#225486362
#102×D2512
#447×C2512
#3294251