Finite Group Databasebeta

← #44562205

Subgroups — #44562205

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#30632332C211yesmaximal, characteristic
3#30632042C211yesmaximal, characteristic, complemented
4#30632019C211yesmaximal, characteristic, complemented
5#30632013C211yesmaximal, characteristic, complemented
6#30631987C211yesmaximal, characteristic, complemented
7#30631927C211yesmaximal, characteristic, complemented
8#30631919C211yesmaximal, characteristic, complemented
9#30631909C211yesmaximal, characteristic, complemented
10#30631846C211yesmaximal, characteristic, complemented
11#30631836C211yesmaximal, characteristic, complemented
12C8×D1004C211yesmaximal, characteristic, complemented
13#30631680C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15C4×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860800C2211yescharacteristic, complemented
19C2×#16860797C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic, complemented
22#22548709—21no—
23#22548707C2211yescharacteristic, complemented
24#22548705—21no—
25#22548703C2211yescharacteristic
26#22548700—21no—
27#22548699—21no—
28#22548695C2211yescharacteristic
29#22548691C2211yescharacteristic
30#22548691C2211yescharacteristic, complemented
31#22548690C2211yescharacteristic
32#22548689—21no—
33#22548685—21no—
34C4×D1004C2211yescharacteristic
35#22548673C2211yescharacteristic, complemented
36#22548672—21no—
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×SD16×C251C2211yescharacteristic
53C4×Q8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55#106×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
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—
68D4×D251—22no—
69D4×D251—41no—
70C22×Dic251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73C2×C4×D251C2311yescharacteristic
74#16860800—41no—
75#16860797—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C8×D251—41no—
79C8×D251—41no—
80C4⋊Dic251—21no—
81C4⋊Dic251D411yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊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)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251C2311yescharacteristic
101C22×C4×C251—21no—
102SD16×C251—22no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
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
117Dic502—21no—
118Dic502—21no—
119D1004—22no—
120D1004—21no—
121D1004—21no—
122D1004—81no—
123C22⋊D251—42no—
124C22⋊D251—42no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—42no—
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—21no—
134C4×D251—21no—
135C4×D251—42no—
136C251⋊C8—21no—
137C251⋊C8—21no—
138C23×C251—21no—
139Q8×C251—21no—
140Q8×C251—21no—
141D4×C251—21no—
142D4×C251#4112yes—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—81no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—22no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—41no—
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—42no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—22no—
183D251—41no—
184D251—81no—
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#464—2511no—
196#452—2511no—
197#450—2511no—
198C4×SD16D25111yescharacteristic, complemented
199C8×D4—2511no—
200#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4—2511no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16C2×D25111yescharacteristic
207C22×C8—2511no—
208C22×C8—2511no—
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124—2511no—
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×D4C2×D25111yescharacteristic
224#117—5021no—
225#117—2511no—
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—5021no—
228#107—2511no—
229#106—2511no—
230#106C2×D25111yescharacteristic, complemented
231C4⋊C8—2511no—
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×C8C2×D25111yescharacteristic, complemented
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—10041no—
251C2×D4—5021no—
252C2×D4—5021no—
253C2×D4—5022no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4—2511no—
263Q16—10041no—
264SD16—22no—
265SD16—10041no—
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⋊C4C22×D25111yescharacteristic
283C4⋊C4C22×D25111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—5021no—
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—5021no—
307Q8—5021no—
308D4—21no—
309D4—20081no—
310D4—10042no—
311D4—10042no—
312D4—5022no—
313D4—5021no—
314D4—5021no—
315D4#1686081012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
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—22no—
339C22—41no—
340C22—41no—
341C22—20084no—
342C22—10041no—
343C22—10041no—
344C22—10041no—
345C22—5021no—
346C22—5021no—
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—
362C4C2×#1686081011yescharacteristic
363C4#2254876111yescharacteristic
364C2—42no—
365C2—20081no—
366C2—10041no—
367C2—5022no—
368C2#3063233311yescharacteristic
369C2#135×D25111yescharacteristic
370C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
prime radical#30632332C2
hypercenterC4×SD16D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #30632378C2semidirect
3. #30632042C2semidirect
4. #30632019C2semidirect
5. #30632013C2semidirect
6. #30631987C2semidirect
7. #30631927C2semidirect
8. #30631919C2semidirect
9. #30631909C2semidirect
10. #30631846C2semidirect
11. #30631836C2semidirect
12. C8×D1004C2semidirect
13. #30631680C2semidirect
14. #102×D251C2semidirect
15. C4×SD16×C251C2semidirect
18. C2×#16860800C22semidirect
21. C2×C8×D251C22semidirect
23. #22548707C22semidirect
30. #22548691C22semidirect
35. #22548673C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. C4⋊(C251⋊C8)C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548638C22semidirect
51. (C2×D251)⋊C8C22semidirect
53. C4×Q8×C251C22semidirect
55. #106×C251C22semidirect
56. #103×C251C22semidirect
58. C4×C8×C251C22semidirect
185. C251#3296semidirect
198. C4×SD16D251semidirect
221. C4×Q8C2×D251semidirect
230. #106C2×D251semidirect
235. #103C2×D251semidirect
244. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323782
#306323322
#306320422
#306320192
#306320132
#306319872
#306319272
#306319192
#306319092
#306318462
#306318362
C8×D10042
#306316802
#102×D2512
C4×SD16×C2512
#3296251