Finite Group Databasebeta

← #44562221

Subgroups — #44562221

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#30632389C211yesmaximal, characteristic, complemented
2#30632328C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632020C211yesmaximal, characteristic, complemented
5#30632013C211yesmaximal, characteristic, complemented
6#30631986C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631831C211yesmaximal, characteristic, complemented
10#30631820C211yesmaximal, characteristic, complemented
11#30631767C211yesmaximal, characteristic, complemented
12C8×C22⋊D251C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15#465×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic, complemented
21#22548708—21no—
22#22548706C2211yescharacteristic, complemented
23#22548704C2211yescharacteristic
24#22548700—21no—
25#22548699—21no—
26#22548690—21no—
27#22548690C2211yescharacteristic
28#22548687—21no—
29#22548686C2211yescharacteristic, complemented
30#22548685—21no—
31D4×Dic251—21no—
32D4×Dic251C2211yescharacteristic, complemented
33#22548677C2211yescharacteristic
34C4×C22⋊D251C2211yescharacteristic
35#22548673C2211yescharacteristic, complemented
36C2×C4⋊Dic251C2211yescharacteristic
37C22⋊C4×D251—21no—
38#22548648C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40Dic251⋊C8C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic, complemented
53C22×C8×C251C2211yescharacteristic
54C22⋊Q8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860811—22no—
61#16860811—41no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—41no—
67C2×D1004C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251D411yescharacteristic
71C22×Dic251—21no—
72C22×Dic251—21no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251C2311yescharacteristic
76#16860795—22no—
77#16860795—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C4⋊Dic251—21no—
82C4⋊Dic251—21no—
83C4⋊Dic251C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251D411yescharacteristic
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×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98C2×Q8×C251C2311yescharacteristic
99C2×D4×C251—21no—
100C2×D4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102SD16×C251—41no—
103C2×C8×C251—41no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C22⋊C4×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—42no—
121D1004—42no—
122D1004—81no—
123D1004#4112yes—
124C22⋊D251—22no—
125C22⋊D251—81no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—81no—
129C2×Dic251—81no—
130C2×Dic251C2×D411yescharacteristic
131C2×Dic251C2×D411yescharacteristic
132C2×Dic251C2×D411yescharacteristic
133C4×D251—21no—
134C4×D251—21no—
135C4×D251—42no—
136C4×D251—81no—
137C251⋊C8—21no—
138C251⋊C8—21no—
139C23×C251—21no—
140C23×C251C2×D411yescharacteristic
141Q8×C251—21no—
142Q8×C251—21no—
143D4×C251—21no—
144D4×C251—21no—
145D4×C251—42no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251#13512yes—
161C22×C251C22×D411yescharacteristic
162C2×D251—21no—
163C2×D251—22no—
164C2×D251—21no—
165C2×D251—21no—
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×C251C2×#4111yescharacteristic
177C4×C251H(5,𝔽2)11yescharacteristic
178Dic251—21no—
179Dic251—21no—
180Dic251—21no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—22no—
184C2×C251—81no—
185C2×C251#54611yescharacteristic
186C2×C251C2×#13511yescharacteristic
187C2×C251#57611yescharacteristic
188D251—41no—
189D251—42no—
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#465D25111yescharacteristic, complemented
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×SD16C2×D25111yescharacteristic, complemented
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8—2511no—
214C22×C8C2×D25111yescharacteristic
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#121C2×D25111yescharacteristic
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#106C2×D25111yescharacteristic, complemented
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, complemented
248C22⋊C8—2511no—
249C22⋊C8—2511no—
250C4×C8—2511no—
251C24—5021no—
252#41—10041no—
253#41—5022no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—21no—
256C2×D4—10041no—
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×D4C22×D25111yescharacteristic
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4—2511no—
271C22×C4C22×D25111yescharacteristic
272SD16—41no—
273D8—10041no—
274D8—5022no—
275C2×C8—41no—
276C2×C8—10041no—
277C2×C8—2511no—
278C2×C8—2511no—
279C2×C8C22×D25111yescharacteristic
280C2×C8C22×D25111yescharacteristic
281C4⋊C4—21no—
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—
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—21no—
315D4—42no—
316D4—20081no—
317D4—20081no—
318D4—10042no—
319D4—10042no—
320D4—5022no—
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—20081no—
331C2×C4—20081no—
332C2×C4—20081no—
333C2×C4—10042no—
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—41no—
350C22—41no—
351C22—41no—
352C22—20084no—
353C22—10041no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22—5021no—
359C22—5021no—
360C22—5022no—
361C22—5021no—
362C22#2254873612yes—
363C22C2×D4×D25111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—10041no—
369C4—10041no—
370C4—5021no—
371C4—5021no—
372C4—5021no—
373C4#41×D25111yescharacteristic
374C4#2254876011yescharacteristic
375C2—22no—
376C2—81no—
377C2—20081no—
378C2—10042no—
379C2—10041no—
380C2#3063232611yescharacteristic
381C2C2×#2254873611yescharacteristic
382C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#465×C251C2
socleC22×C251C22×D4
hypercenter#465D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #30632389C2semidirect
2. #30632328C2semidirect
3. #30632043C2semidirect
4. #30632020C2semidirect
5. #30632013C2semidirect
6. #30631986C2semidirect
7. #30631922C2semidirect
8. #30631904C2semidirect
9. #30631831C2semidirect
10. #30631820C2semidirect
11. #30631767C2semidirect
12. C8×C22⋊D251C2semidirect
13. #30631682C2semidirect
14. C2×#22548637C2semidirect
15. #465×C251C2semidirect
20. C2×C8×D251C22semidirect
22. #22548706C22semidirect
29. #22548686C22semidirect
32. D4×Dic251C22semidirect
35. #22548673C22semidirect
38. #22548648C22semidirect
39. #22548648C22semidirect
40. Dic251⋊C8C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
52. C2×SD16×C251C22semidirect
56. #106×C251C22semidirect
57. #103×C251C22semidirect
58. #102×C251C22semidirect
191. C251#3297semidirect
200. #465D251semidirect
210. C2×SD16C2×D251semidirect
236. #106C2×D251semidirect
240. #103C2×D251semidirect
247. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323892
#306323282
#306320432
#306320202
#306320132
#306319862
#306319222
#306319042
#306318312
#306318202
#306317672
C8×C22⋊D2512
#306316822
C2×#225486372
#465×C2512
#3297251