Finite Group Databasebeta

← #44562213

Subgroups — #44562213

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#30632385C211yesmaximal, characteristic, complemented
2#30632328C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632014C211yesmaximal, characteristic, complemented
5#30631990C211yesmaximal, characteristic, complemented
6#30631924C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631905C211yesmaximal, characteristic, complemented
9#30631835C211yesmaximal, characteristic, complemented
10#30631823C211yesmaximal, characteristic, complemented
11#30631769C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14C2×#22548635C211yesmaximal, characteristic, complemented
15#484×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860797C2211yescharacteristic, complemented
19C2×D2008C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic, complemented
23#22548710—21no—
24#22548706C2211yescharacteristic, complemented
25#22548704C2211yescharacteristic
26#22548700—21no—
27#22548700—21no—
28#22548690—21no—
29#22548690C2211yescharacteristic
30#22548690C2211yescharacteristic, complemented
31#22548689—21no—
32#22548689C2211yescharacteristic
33#22548685—21no—
34#22548677C2211yescharacteristic
35#22548677C2211yescharacteristic, complemented
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38#22548673C2211yescharacteristic, complemented
39C22⋊C4×D251—21no—
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22⋊Q8×C251C2211yescharacteristic
53C2×C4⋊C4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#107×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C23×D251—21no—
60#16860811—22no—
61#16860811—41no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—21no—
67C2×D1004—41no—
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251D411yescharacteristic
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251C2311yescharacteristic
77#16860797—41no—
78D2008—22no—
79D2008—41no—
80C2×C251⋊C8—41no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C8×D251—41no—
84C4⋊Dic251—21no—
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4D411yescharacteristic
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)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C4×Dic251C2311yescharacteristic
100C2×Q8×C251C2311yescharacteristic
101C22×C4×C251—21no—
102C22×C4×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—81no—
117C22×D251C2×D411yescharacteristic
118D1004—21no—
119D1004—21no—
120D1004—21no—
121D1004—21no—
122D1004—21no—
123D1004—42no—
124D1004—42no—
125D1004—81no—
126D1004—81no—
127D1004#4112yes—
128C22⋊D251—22no—
129C22⋊D251—42no—
130C2×Dic251—21no—
131C2×Dic251C2×D411yescharacteristic
132C2×Dic251C2×D411yescharacteristic
133C2×Dic251C2×D411yescharacteristic
134C4×D251—21no—
135C4×D251—21no—
136C4×D251—42no—
137C4×D251—81no—
138C4×D251—81no—
139C251⋊C8—21no—
140C251⋊C8—21no—
141C23×C251C2×D411yescharacteristic
142Q8×C251—21no—
143Q8×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—81no—
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251#13512yes—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—22no—
161C2×D251—21no—
162C2×D251—21no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—84no—
172C4×C251—21no—
173C4×C251—41no—
174C4×C251—41no—
175C4×C251—41no—
176C4×C251—41no—
177C4×C251C2×#4111yescharacteristic
178C4×C251H(5,𝔽2)11yescharacteristic
179Dic251—21no—
180Dic251—21no—
181Dic251—21no—
182Dic251—41no—
183C2×C251—22no—
184C2×C251#54611yescharacteristic
185C2×C251C2×#13511yescharacteristic
186C2×C251#57611yescharacteristic
187D251—41no—
188D251—42no—
189D251—81no—
190D251—81no—
191C251#329711yescharacteristic, complemented, Sylow
192#3297—2511nomaximal, Sylow
193#553—2511no—
194#546—2511no—
195#488—2511no—
196#484D25111yescharacteristic, complemented
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#416—2511no—
207C2×#102—2511no—
208C2×#41—2511no—
209C22×D4—2511no—
210C2×SD16—2511no—
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8—2511no—
214C22×C8—2511no—
215#126—5021no—
216#125—2511no—
217#124—2511no—
218#123—5021no—
219#123—5021no—
220C22⋊Q8C2×D25111yescharacteristic
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#117—2511no—
232C2×C4⋊C4C2×D25111yescharacteristic
233C2×C22⋊C4—5021no—
234#107C2×D25111yescharacteristic, complemented
235#107C2×D25111yescharacteristic, complemented
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#102—2511no—
247#102—2511no—
248C22⋊C8—2511no—
249C22⋊C8C2×D25111yescharacteristic
250C4×C8—2511no—
251C24—5021no—
252#41—10041no—
253#41—5022no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—10041no—
256C2×D4—5021no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5022no—
260C2×D4—5021no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4—2511no—
264C2×D4—2511no—
265C2×D4—2511no—
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—10041no—
276C2×C8—10041no—
277C2×C8—2511no—
278C2×C8—2511no—
279C2×C8C22×D25111yescharacteristic
280C2×C8C22×D25111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—5021no—
284C4⋊C4—5021no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4C22×D25111yescharacteristic
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—5021no—
302C42—2511no—
303C23—20081no—
304C23—10042no—
305C23—10042no—
306C23—5021no—
307C23—5021no—
308C23—5021no—
309C23—2511no—
310C23D4×D25111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—20081no—
314D4—20081no—
315D4—10042no—
316D4—10042no—
317D4—10042no—
318D4—5022no—
319D4—5021no—
320D4—5021no—
321D4—5021no—
322D4—5021no—
323D4—5021no—
324D4—2512no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—20081no—
333C2×C4—20081no—
334C2×C4—10042no—
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—20084no—
350C22—10041no—
351C22—10041no—
352C22—10041no—
353C22—10041no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22—5021no—
359C22—5021no—
360C22—5022no—
361C22—5021no—
362C22#2254873812yes—
363C22C2×D4×D25111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—10041no—
370C4—5021no—
371C4—5021no—
372C4—5021no—
373C4#41×D25111yescharacteristic
374C4#2254876111yescharacteristic
375C2—22no—
376C2—20081no—
377C2—20081no—
378C2—10042no—
379C2—10041no—
380C2#3063232911yescharacteristic
381C2C2×#2254873811yescharacteristic
382C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
hypercenter#484D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #30632385C2semidirect
2. #30632328C2semidirect
3. #30632043C2semidirect
4. #30632014C2semidirect
5. #30631990C2semidirect
6. #30631924C2semidirect
7. #30631918C2semidirect
8. #30631905C2semidirect
9. #30631835C2semidirect
10. #30631823C2semidirect
11. #30631769C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631682C2semidirect
14. C2×#22548635C2semidirect
15. #484×C251C2semidirect
18. C2×#16860797C22semidirect
22. C2×C8×D251C22semidirect
24. #22548706C22semidirect
30. #22548690C22semidirect
35. #22548677C22semidirect
38. #22548673C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548639C22semidirect
49. #22548635C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
54. #107×C251C22semidirect
55. #107×C251C22semidirect
56. #103×C251C22semidirect
57. #103×C251C22semidirect
191. C251#3297semidirect
196. #484D251semidirect
234. #107C2×D251semidirect
235. #107C2×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
#306323852
#306323282
#306320432
#306320142
#306319902
#306319242
#306319182
#306319052
#306318352
#306318232
#306317692
(C22⋊D251)⋊C82
#306316822
C2×#225486352
#484×C2512
#3297251