Finite Group Databasebeta

← #44562208

Subgroups — #44562208

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632384C211yesmaximal, characteristic, complemented
2#30632325C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632021C211yesmaximal, characteristic, complemented
5#30632012C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631921C211yesmaximal, characteristic, complemented
8#30631913C211yesmaximal, characteristic, complemented
9#30631904C211yesmaximal, characteristic, complemented
10#30631819C211yesmaximal, characteristic, complemented
11D8×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#449×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic, complemented
19C2×C8×D251C2211yescharacteristic
20#22548708—21no—
21#22548706C2211yescharacteristic, complemented
22#22548704C2211yescharacteristic
23#22548698—21no—
24#22548696—21no—
25#22548691C2211yescharacteristic
26#22548689C2211yescharacteristic
27#22548687—21no—
28#22548686—21no—
29#22548686C2211yescharacteristic
30#22548683—21no—
31D4×Dic251C2211yescharacteristic
32D4×Dic251C2211yescharacteristic, complemented
33C4×C22⋊D251C2211yescharacteristic
34(C22⋊D251)⋊C4—21no—
35#22548674C2211yescharacteristic, complemented
36C4⋊C4×D251C2211yescharacteristic
37C2×(C2×D251)⋊C4—21no—
38#22548649C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic, complemented
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic, complemented
52C2×#41×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic, complemented
54C2×D8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C2×D1004C2311yescharacteristic
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251D411yescharacteristic
65D4×D251—22no—
66D4×D251—41no—
67C22×Dic251—21no—
68C22×Dic251—21no—
69C2×C4×D251—21no—
70C2×C4×D251C2311yescharacteristic
71#16860795—41no—
72C2×C251⋊C8—41no—
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C8×D251—41no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88C22⋊Dic251D411yescharacteristic
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96#41×C251—22no—
97#41×C251—41no—
98C2×Q8×C251C2311yescharacteristic
99C2×D4×C251—21no—
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251C2311yescharacteristic
105SD16×C251—41no—
106D8×C251—22no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116D1004—21no—
117D1004—21no—
118C22⋊D251—22no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C22⋊D251—81no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—81no—
126C2×Dic251—81no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—81no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251C2×D411yescharacteristic
138Q8×C251—21no—
139Q8×C251—21no—
140D4×C251—21no—
141D4×C251—21no—
142D4×C251—21no—
143D4×C251—42no—
144D4×C251—81no—
145D4×C251#4112yes—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—42no—
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—22no—
157C22×C251—21no—
158C22×C251—21no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251C22×D411yescharacteristic
165C2×D251—21no—
166C2×D251—21no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—84no—
171C2×D251#13512yes—
172C4×C251—21no—
173C4×C251—41no—
174C4×C251—41no—
175C4×C251C2×#4111yescharacteristic
176C4×C251H(5,𝔽2)11yescharacteristic
177Dic251—21no—
178Dic251—21no—
179Dic251—21no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—42no—
184C2×C251—41no—
185C2×C251—81no—
186C2×C251#54611yescharacteristic
187C2×C251C2×#13511yescharacteristic
188C2×C251#57611yescharacteristic
189D251—22no—
190D251—81no—
191C251#329711yescharacteristic, complemented, Sylow
192#3297—2511nomaximal, Sylow
193#553—2511no—
194#546—2511no—
195#488—2511no—
196#484—2511no—
197#482—2511no—
198#478—2511no—
199#466—2511no—
200#465—2511no—
201#463—2511no—
202#449D25111yescharacteristic, complemented
203#447—2511no—
204C4×D8—2511no—
205C8×D4—2511no—
206#416—2511no—
207C2×#102—2511no—
208C2×#41C2×D25111yescharacteristic
209C22×D4—2511no—
210C2×SD16C2×D25111yescharacteristic, complemented
211C2×D8—2511no—
212C2×D8C2×D25111yescharacteristic
213C22×C8—2511no—
214C22×C8—2511no—
215#126—5021no—
216#125—2511no—
217#124—2511no—
218#123—5021no—
219#123—5021no—
220C22⋊Q8—2511no—
221#121—5021no—
222#121—5021no—
223#121—2511no—
224#121—2511no—
225#121C2×D25111yescharacteristic, complemented
226#120—5021no—
227C4×D4—5021no—
228C4×D4—2511no—
229C4×D4—2511no—
230C4×D4—2511no—
231#117—2511no—
232C2×C4⋊C4—2511no—
233C2×C22⋊C4—5021no—
234#107—2511no—
235#107—2511no—
236#106—2511no—
237C4⋊C8—2511no—
238#103—2511no—
239#103—2511no—
240#103C2×D25111yescharacteristic, complemented
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102C2×D25111yescharacteristic
248C22⋊C8—2511no—
249C22⋊C8C2×D25111yescharacteristic, complemented
250C4×C8—2511no—
251C24—5021no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—10041no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—5022no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4—2511no—
264C2×D4C22×D25111yescharacteristic
265C2×D4C22×D25111yescharacteristic
266C22×C4—21no—
267C22×C4—5021no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4—2511no—
271C22×C4C22×D25111yescharacteristic
272SD16—41no—
273D8—22no—
274D8—10041no—
275C2×C8—10041no—
276C2×C8—10041no—
277C2×C8—2511no—
278C2×C8—2511no—
279C2×C8C22×D25111yescharacteristic
280C2×C8C22×D25111yescharacteristic
281C4⋊C4—5021no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—5021no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4—2511no—
288C4⋊C4—2511no—
289C4⋊C4C22×D25111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—5021no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4—2511no—
300C22⋊C4—2511no—
301C42—5021no—
302C42—2511no—
303C23—21no—
304C23—21no—
305C23—20081no—
306C23—10042no—
307C23—10042no—
308C23—5021no—
309C23—2511no—
310C23D4×D25111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—20081no—
319D4—10042no—
320D4—10042no—
321D4—5022no—
322D4—5021no—
323D4—5021no—
324D4#1686080912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—20081no—
331C2×C4—20081no—
332C2×C4—20081no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—2511no—
339C2×C4—2511no—
340C2×C4—2511no—
341C2×C4D4×D25111yescharacteristic
342C2×C4C23×D25111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5021no—
346C8—5021no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—20084no—
357C22—10041no—
358C22—10041no—
359C22—10041no—
360C22—5021no—
361C22—5021no—
362C22—2512no—
363C22C2×D4×D25111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—10041no—
368C4—10041no—
369C4—10041no—
370C4—5021no—
371C4—5021no—
372C4—5021no—
373C4C2×#1686080911yescharacteristic
374C4#2254876011yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—20081no—
379C2—5022no—
380C2#3063232411yescharacteristic
381C2#135×D25111yescharacteristic
382C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#449×C251C2
socleC22×C251C22×D4
hypercenter#449D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #30632384C2semidirect
2. #30632325C2semidirect
3. #30632043C2semidirect
4. #30632021C2semidirect
5. #30632012C2semidirect
6. #30631989C2semidirect
7. #30631921C2semidirect
8. #30631913C2semidirect
9. #30631904C2semidirect
10. #30631819C2semidirect
11. D8×Dic251C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631683C2semidirect
14. #102×D251C2semidirect
15. #449×C251C2semidirect
18. C22×C251⋊C8C22semidirect
21. #22548706C22semidirect
32. D4×Dic251C22semidirect
35. #22548674C22semidirect
38. #22548649C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548639C22semidirect
43. #22548639C22semidirect
48. #22548636C22semidirect
51. C8×Dic251C22semidirect
53. C2×SD16×C251C22semidirect
55. #121×C251C22semidirect
56. #103×C251C22semidirect
58. C22⋊C8×C251C22semidirect
191. C251#3297semidirect
202. #449D251semidirect
210. C2×SD16C2×D251semidirect
225. #121C2×D251semidirect
240. #103C2×D251semidirect
249. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323842
#306323252
#306320432
#306320212
#306320122
#306319892
#306319212
#306319132
#306319042
#306318192
D8×Dic2512
(C22⋊D251)⋊C82
#306316832
#102×D2512
#449×C2512
#3297251