Finite Group Databasebeta

← #44562220

Subgroups — #44562220

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#30632387C211yesmaximal, characteristic, complemented
2#30632332C211yesmaximal, characteristic, complemented
3#30632042C211yesmaximal, characteristic, complemented
4#30632024C211yesmaximal, characteristic, complemented
5#30632014C211yesmaximal, characteristic, complemented
6#30631987C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631915C211yesmaximal, characteristic, complemented
9#30631902C211yesmaximal, characteristic, complemented
10#30631829C211yesmaximal, characteristic, complemented
11#30631819C211yesmaximal, characteristic, complemented
12C8×D1004C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15C4×D8×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860795C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic, complemented
22#22548708—21no—
23#22548707C2211yescharacteristic, complemented
24#22548703C2211yescharacteristic
25#22548699—21no—
26#22548698—21no—
27#22548695C2211yescharacteristic
28#22548690C2211yescharacteristic
29#22548689—21no—
30#22548688—21no—
31#22548686C2211yescharacteristic
32#22548686C2211yescharacteristic, complemented
33#22548685—21no—
34C4×D1004C2211yescharacteristic
35C4×C22⋊D251—21no—
36#22548673C2211yescharacteristic, complemented
37C4⋊C4×D251C2211yescharacteristic
38C2×(C2×D251)⋊C4—21no—
39#22548648C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41C4⋊(C251⋊C8)C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52C2×D8×C251C2211yescharacteristic
53C4×D4×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic, complemented
55#107×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic, complemented
58C4×C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860809—22no—
61#16860809—41no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—21no—
64C2×D1004D411yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70D4×D251—22no—
71D4×D251—41no—
72C22×Dic251—21no—
73C2×C4×D251—21no—
74C2×C4×D251C2311yescharacteristic
75C2×C4×D251C2311yescharacteristic
76#16860798—41no—
77#16860795—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C8×D251—41no—
82C4⋊Dic251—21no—
83C4⋊Dic251D411yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251—21no—
91(C2×D251)⋊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)⋊C4D411yescharacteristic
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C2×D4×C251C2311yescharacteristic
100C2×D4×C251C2311yescharacteristic
101C22×C4×C251—21no—
102C22×C4×C251—21no—
103D8×C251—22no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C22⋊C4×C251—21no—
109C22⋊C4×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116C22×D251C2×D411yescharacteristic
117Dic502—21no—
118Dic502—21no—
119D1004—22no—
120D1004—21no—
121D1004—21no—
122D1004—81no—
123C22⋊D251—42no—
124C22⋊D251—42no—
125C22⋊D251—42no—
126C22⋊D251—81no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—42no—
131C2×Dic251C2×D411yescharacteristic
132C2×Dic251C2×D411yescharacteristic
133C4×D251—21no—
134C4×D251—21no—
135C4×D251—21no—
136C4×D251—21no—
137C4×D251—81no—
138C251⋊C8—21no—
139C251⋊C8—21no—
140C23×C251—21no—
141C23×C251—21no—
142D4×C251—21no—
143D4×C251—21no—
144D4×C251—21no—
145D4×C251#4112yes—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—81no—
149C2×C4×C251—81no—
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—22no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
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—84no—
171C2×D251#13512yes—
172C4×C251—21no—
173C4×C251—21no—
174C4×C251—41no—
175C4×C251—41no—
176C4×C251C2×#4111yescharacteristic
177C4×C251H(5,𝔽2)11yescharacteristic
178Dic251—21no—
179Dic251—21no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—42no—
184C2×C251—81no—
185C2×C251#54611yescharacteristic
186C2×C251C2×#13511yescharacteristic
187C2×C251#57611yescharacteristic
188D251—22no—
189D251—41no—
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×D8D25111yescharacteristic, complemented
205C8×D4—2511no—
206#416—2511no—
207C2×#102—2511no—
208C2×#41—2511no—
209C22×D4—2511no—
210C2×SD16—2511no—
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#121—2511no—
226#120—5021no—
227C4×D4—5021no—
228C4×D4—2511no—
229C4×D4C2×D25111yescharacteristic
230C4×D4C2×D25111yescharacteristic, complemented
231#117—2511no—
232C2×C4⋊C4—2511no—
233C2×C22⋊C4—5021no—
234#107—2511no—
235#107C2×D25111yescharacteristic, complemented
236#106—2511no—
237C4⋊C8—2511no—
238#103—2511no—
239#103—2511no—
240#103—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102C2×D25111yescharacteristic
247#102C2×D25111yescharacteristic, complemented
248C22⋊C8—2511no—
249C22⋊C8—2511no—
250C4×C8C2×D25111yescharacteristic, complemented
251C24—5021no—
252#41—10041no—
253#41—5022no—
254C2×Q8—2511no—
255C2×D4—10041no—
256C2×D4—5021no—
257C2×D4—5021no—
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—21no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4—2511no—
271C22×C4—2511no—
272SD16—10041no—
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⋊C4C22×D25111yescharacteristic
289C4⋊C4C22×D25111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
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—
302C42C22×D25111yescharacteristic
303C23—21no—
304C23—21no—
305C23—20081no—
306C23—10042no—
307C23—10042no—
308C23—5021no—
309C23—2511no—
310C23—2511no—
311Q8—5021no—
312Q8—5021no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—20081no—
317D4—20081no—
318D4—10042no—
319D4—10042no—
320D4—10042no—
321D4—5022no—
322D4—5021no—
323D4—5021no—
324D4#1686081012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—20081no—
330C2×C4—10042no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—2511no—
339C2×C4—2511no—
340C2×C4D4×D25111yescharacteristic
341C2×C4D4×D25111yescharacteristic
342C2×C4C23×D25111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5021no—
346C8—5021no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—20084no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—5021no—
358C22—5021no—
359C22—5021no—
360C22—5021no—
361C22—5021no—
362C22—2512no—
363C22C2×D4×D25111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—10041no—
369C4—10041no—
370C4—10041no—
371C4—5021no—
372C4—5021no—
373C4C2×#1686081011yescharacteristic
374C4#2254876111yescharacteristic
375C2—42no—
376C2—81no—
377C2—20081no—
378C2—10041no—
379C2—5022no—
380C2#3063233311yescharacteristic
381C2#135×D25111yescharacteristic
382C2#3063248811yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×D8×C251C2
socleC22×C251C22×D4
hypercenterC4×D8D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #30632387C2semidirect
2. #30632332C2semidirect
3. #30632042C2semidirect
4. #30632024C2semidirect
5. #30632014C2semidirect
6. #30631987C2semidirect
7. #30631922C2semidirect
8. #30631915C2semidirect
9. #30631902C2semidirect
10. #30631829C2semidirect
11. #30631819C2semidirect
12. C8×D1004C2semidirect
13. #30631680C2semidirect
14. #102×D251C2semidirect
15. C4×D8×C251C2semidirect
18. C2×#16860798C22semidirect
21. C2×C8×D251C22semidirect
23. #22548707C22semidirect
32. #22548686C22semidirect
36. #22548673C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. C4⋊(C251⋊C8)C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548638C22semidirect
51. (C2×D251)⋊C8C22semidirect
54. C4×D4×C251C22semidirect
55. #107×C251C22semidirect
57. #102×C251C22semidirect
58. C4×C8×C251C22semidirect
191. C251#3297semidirect
204. C4×D8D251semidirect
230. C4×D4C2×D251semidirect
235. #107C2×D251semidirect
247. #102C2×D251semidirect
250. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323872
#306323322
#306320422
#306320242
#306320142
#306319872
#306319222
#306319152
#306319022
#306318292
#306318192
C8×D10042
#306316802
#102×D2512
C4×D8×C2512
#3297251