Finite Group Databasebeta

← #44562216

Subgroups — #44562216

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#30632381C211yesmaximal, characteristic, complemented
2#30632325C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632021C211yesmaximal, characteristic, complemented
5#30632011C211yesmaximal, characteristic, complemented
6#30631985C211yesmaximal, characteristic, complemented
7#30631923C211yesmaximal, characteristic, complemented
8#30631919C211yesmaximal, characteristic, complemented
9#30631905C211yesmaximal, characteristic, complemented
10#30631832C211yesmaximal, characteristic, complemented
11#30631822C211yesmaximal, characteristic, complemented
12#30631767C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#416×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860797C2211yescharacteristic, complemented
18C2×D2008C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548709—21no—
22#22548706C2211yescharacteristic, complemented
23#22548704C2211yescharacteristic
24#22548698—21no—
25#22548697—21no—
26#22548691C2211yescharacteristic
27#22548690C2211yescharacteristic
28#22548689C2211yescharacteristic
29#22548689C2211yescharacteristic, complemented
30#22548688—21no—
31#22548687—21no—
32#22548683—21no—
33C4×D1004—21no—
34D4×Dic251C2211yescharacteristic
35#22548677C2211yescharacteristic, complemented
36C4×C22⋊D251C2211yescharacteristic
37C4⋊C4×D251C2211yescharacteristic
38C2×(C2×D251)⋊C4—21no—
39#22548649C2211yescharacteristic, complemented
40#22548648C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic, complemented
52C2×#41×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic, complemented
54#117×C251C2211yescharacteristic, complemented
55#103×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60C2×D1004—21no—
61C2×D1004C2311yescharacteristic
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251D411yescharacteristic
67D4×D251—22no—
68D4×D251—41no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73#16860797—41no—
74D2008—41no—
75#16860795—22no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C8×D251—41no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4D411yescharacteristic
86C22⋊Dic251—21no—
87C22⋊Dic251D411yescharacteristic
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104C2×C8×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—81no—
116C22×D251C2×D411yescharacteristic
117D1004—21no—
118D1004—21no—
119D1004—21no—
120D1004—21no—
121D1004—42no—
122C22⋊D251—22no—
123C22⋊D251—42no—
124C22⋊D251—81no—
125C22⋊D251—81no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—81no—
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C2×Dic251C2×D411yescharacteristic
132C4×D251—21no—
133C4×D251—21no—
134C4×D251—81no—
135C4×D251—81no—
136C251⋊C8—21no—
137C251⋊C8—21no—
138C23×C251—21no—
139C23×C251C2×D411yescharacteristic
140Q8×C251—21no—
141Q8×C251—21no—
142D4×C251—21no—
143D4×C251—42no—
144D4×C251#4112yes—
145C2×C4×C251—21no—
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×C251C22×D411yescharacteristic
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
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×C251—41no—
176C4×C251C2×#4111yescharacteristic
177C4×C251H(5,𝔽2)11yescharacteristic
178Dic251—21no—
179Dic251—21no—
180Dic251—21no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—42no—
184C2×C251—41no—
185C2×C251#54611yescharacteristic
186C2×C251C2×#13511yescharacteristic
187C2×C251#57611yescharacteristic
188D251—22no—
189D251—81no—
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#449—2511no—
203#447—2511no—
204C4×D8—2511no—
205C8×D4—2511no—
206#416D25111yescharacteristic, complemented
207C2×#102—2511no—
208C2×#41C2×D25111yescharacteristic
209C22×D4—2511no—
210C2×SD16—2511no—
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8—2511no—
214C22×C8C2×D25111yescharacteristic, complemented
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#121—2511no—
226#120—5021no—
227C4×D4—5021no—
228C4×D4—2511no—
229C4×D4—2511no—
230C4×D4—2511no—
231#117C2×D25111yescharacteristic, complemented
232C2×C4⋊C4—2511no—
233C2×C22⋊C4—5021no—
234#107—2511no—
235#107—2511no—
236#106—2511no—
237C4⋊C8—2511no—
238#103—2511no—
239#103C2×D25111yescharacteristic, complemented
240#103C2×D25111yescharacteristic, complemented
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102C2×D25111yescharacteristic
247#102C2×D25111yescharacteristic
248C22⋊C8—2511no—
249C22⋊C8—2511no—
250C4×C8—2511no—
251C24—5021no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—21no—
256C2×D4—10041no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—5022no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4—2511no—
264C2×D4—2511no—
265C2×D4C22×D25111yescharacteristic
266C22×C4—21no—
267C22×C4—5021no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4—2511no—
271C22×C4C22×D25111yescharacteristic
272SD16—10041no—
273D8—10041no—
274D8—5022no—
275C2×C8—41no—
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⋊C4C22×D25111yescharacteristic
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—21no—
302C42—2511no—
303C23—21no—
304C23—20081no—
305C23—10042no—
306C23—10042no—
307C23—5021no—
308C23—5021no—
309C23—2511no—
310C23D4×D25111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—20081no—
316D4—20081no—
317D4—10042no—
318D4—10042no—
319D4—5022no—
320D4—5021no—
321D4—5021no—
322D4—5021no—
323D4—5021no—
324D4#1686081012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—20081no—
332C2×C4—20081no—
333C2×C4—20081no—
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—20084no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22—10041no—
359C22—10041no—
360C22—5021no—
361C22—5021no—
362C22—2512no—
363C22C2×D4×D25111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—10041no—
369C4—10041no—
370C4—5021no—
371C4—5021no—
372C4—5021no—
373C4C2×#1686081011yescharacteristic
374C4#2254876011yescharacteristic
375C2—41no—
376C2—42no—
377C2—20081no—
378C2—20081no—
379C2—5022no—
380C2#3063232711yescharacteristic
381C2#135×D25111yescharacteristic
382C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #30632381C2semidirect
2. #30632325C2semidirect
3. #30632043C2semidirect
4. #30632021C2semidirect
5. #30632011C2semidirect
6. #30631985C2semidirect
7. #30631923C2semidirect
8. #30631919C2semidirect
9. #30631905C2semidirect
10. #30631832C2semidirect
11. #30631822C2semidirect
12. #30631767C2semidirect
13. C8×C22⋊D251C2semidirect
14. #102×D251C2semidirect
15. #416×C251C2semidirect
17. C2×#16860797C22semidirect
22. #22548706C22semidirect
29. #22548689C22semidirect
35. #22548677C22semidirect
39. #22548649C22semidirect
40. #22548648C22semidirect
41. #22548646C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548639C22semidirect
45. #22548637C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
51. C8×Dic251C22semidirect
53. C22×C8×C251C22semidirect
54. #117×C251C22semidirect
55. #103×C251C22semidirect
56. #103×C251C22semidirect
191. C251#3297semidirect
206. #416D251semidirect
214. C22×C8C2×D251semidirect
231. #117C2×D251semidirect
239. #103C2×D251semidirect
240. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323812
#306323252
#306320432
#306320212
#306320112
#306319852
#306319232
#306319192
#306319052
#306318322
#306318222
#306317672
C8×C22⋊D2512
#102×D2512
#416×C2512
#3297251