Finite Group Databasebeta

← #44562207

Subgroups — #44562207

Order 32128.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632378C211yesmaximal, characteristic, complemented
2#30632330C211yesmaximal, characteristic
3#30632041C211yesmaximal, characteristic, complemented
4#30632016C211yesmaximal, characteristic, complemented
5#30632009C211yesmaximal, characteristic, complemented
6#30631988C211yesmaximal, characteristic, complemented
7#30631925C211yesmaximal, characteristic, complemented
8#30631913C211yesmaximal, characteristic, complemented
9#30631848C211yesmaximal, characteristic, complemented
10#30631836C211yesmaximal, characteristic, complemented
11C4×#16860797C211yesmaximal, characteristic, complemented
12#30631752C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#464×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic, complemented
21#22548709—21no—
22#22548707C2211yescharacteristic, complemented
23#22548705—21no—
24#22548700—21no—
25#22548697—21no—
26#22548691C2211yescharacteristic
27#22548691C2211yescharacteristic
28#22548691C2211yescharacteristic, complemented
29#22548688—21no—
30#22548686C2211yescharacteristic
31#22548685—21no—
32C4×D1004C2211yescharacteristic
33C4×D1004C2211yescharacteristic
34#22548673C2211yescharacteristic, complemented
35#22548672—21no—
36C4⋊C4×D251C2211yescharacteristic
37C22⋊C4×D251—21no—
38#22548649C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C4×C251⋊C8C2211yescharacteristic, complemented
52C2×Q16×C251C2211yescharacteristic, complemented
53C2×SD16×C251C2211yescharacteristic
54#124×C251C2211yescharacteristic
55C4×Q8×C251C2211yescharacteristic, complemented
56C4⋊C8×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60#16860811—22no—
61#16860811—41no—
62C2×D1004—21no—
63C2×D1004D411yescharacteristic
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67D4×D251—22no—
68D4×D251—41no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73C2×C4×D251C2311yescharacteristic
74#16860797—22no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—41no—
78C8×D251—41no—
79C4⋊Dic251D411yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89C22⋊Dic251—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
96(C2×D251)⋊C4D411yescharacteristic
97C4×Dic251—21no—
98C2×Q8×C251C2311yescharacteristic
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251C2311yescharacteristic
101Q16×C251—41no—
102SD16×C251—41no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22⋊C4×C251—21no—
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116C22×D251C2×D411yescharacteristic
117D1004—22no—
118D1004—21no—
119D1004—42no—
120D1004#4112yes—
121C22⋊D251—42no—
122C22⋊D251—81no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C4×D251—21no—
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—42no—
133C4×D251—42no—
134C4×D251—81no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251—21no—
138Q8×C251—21no—
139Q8×C251—21no—
140Q8×C251—21no—
141Q8×C251—21no—
142D4×C251—21no—
143D4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—22no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—84no—
165C2×D251#13512yes—
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251—41no—
171C4×C251C2×#4111yescharacteristic
172C4×C251H(5,𝔽2)11yescharacteristic
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—81no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—22no—
183D251—41no—
184D251—42no—
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546—2511no—
189#488—2511no—
190#483—2511no—
191#482—2511no—
192#478—2511no—
193#467—2511no—
194#465—2511no—
195#464D25111yescharacteristic, complemented
196#452—2511no—
197#450—2511no—
198C4×SD16—2511no—
199C8×D4—2511no—
200#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4—2511no—
204C2×Q16C2×D25111yescharacteristic, complemented
205C2×SD16—2511no—
206C2×SD16C2×D25111yescharacteristic
207C22×C8—2511no—
208C22×C8—2511no—
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124C2×D25111yescharacteristic
213#123—5021no—
214#123—5021no—
215C22⋊Q8—2511no—
216C22⋊Q8—2511no—
217C22⋊Q8—2511no—
218#121—5021no—
219#121—2511no—
220#120—5021no—
221C4×Q8C2×D25111yescharacteristic, complemented
222C4×D4—2511no—
223C4×D4—2511no—
224#117—5021no—
225#117—2511no—
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—5021no—
228#107—2511no—
229#106—2511no—
230#106—2511no—
231C4⋊C8C2×D25111yescharacteristic, complemented
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103C2×D25111yescharacteristic, complemented
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102C2×D25111yescharacteristic
242C22⋊C8—2511no—
243C22⋊C8—2511no—
244C4×C8—2511no—
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8C22×D25111yescharacteristic
249C2×Q8C22×D25111yescharacteristic
250C2×D4—10041no—
251C2×D4—5021no—
252C2×D4—5022no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4—2511no—
263Q16—41no—
264SD16—41no—
265SD16—5022no—
266C2×C8—10041no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4—2511no—
282C4⋊C4—2511no—
283C4⋊C4C22×D25111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—2511no—
293C22⋊C4—2511no—
294C42—21no—
295C42—5021no—
296C42C22×D25111yescharacteristic
297C23—21no—
298C23—20081no—
299C23—10042no—
300C23—10042no—
301C23—5021no—
302C23—2511no—
303C23—2511no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—20081no—
311D4—10042no—
312D4—10042no—
313D4—5022no—
314D4—5021no—
315D4—2512no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—20081no—
320C2×C4—10042no—
321C2×C4—10042no—
322C2×C4—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4D4×D25111yescharacteristic
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5021no—
337C8—5021no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—20084no—
342C22—10041no—
343C22—10041no—
344C22—5021no—
345C22—5021no—
346C22—5022no—
347C22—5021no—
348C22—5021no—
349C22—5021no—
350C22—2512no—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—10041no—
360C4—5021no—
361C4—5021no—
362C4#41×D25111yescharacteristic
363C4#2254876011yescharacteristic
364C2—81no—
365C2—10042no—
366C2—10041no—
367C2—5022no—
368C2#3063233411yescharacteristic
369C2#135×D25111yescharacteristic
370C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#464×C251C2
socleC22×C251C22×D4
prime radical#30632330C2
hypercenter#464D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #30632378C2semidirect
3. #30632041C2semidirect
4. #30632016C2semidirect
5. #30632009C2semidirect
6. #30631988C2semidirect
7. #30631925C2semidirect
8. #30631913C2semidirect
9. #30631848C2semidirect
10. #30631836C2semidirect
11. C4×#16860797C2semidirect
12. #30631752C2semidirect
13. #30631682C2semidirect
14. #102×D251C2semidirect
15. #464×C251C2semidirect
20. C2×C8×D251C22semidirect
22. #22548707C22semidirect
28. #22548691C22semidirect
34. #22548673C22semidirect
38. #22548649C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
43. #22548639C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C4×C251⋊C8C22semidirect
52. C2×Q16×C251C22semidirect
55. C4×Q8×C251C22semidirect
56. C4⋊C8×C251C22semidirect
57. #103×C251C22semidirect
185. C251#3296semidirect
195. #464D251semidirect
204. C2×Q16C2×D251semidirect
221. C4×Q8C2×D251semidirect
231. C4⋊C8C2×D251semidirect
235. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323782
#306323302
#306320412
#306320162
#306320092
#306319882
#306319252
#306319132
#306318482
#306318362
C4×#168607972
#306317522
#306316822
#102×D2512
#464×C2512
#3296251