Finite Group Databasebeta

← #44562202

Subgroups — #44562202

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#30632377C211yesmaximal, characteristic, complemented
2#30632333C211yesmaximal, characteristic
3#30632042C211yesmaximal, characteristic, complemented
4#30632018C211yesmaximal, characteristic, complemented
5#30632012C211yesmaximal, characteristic, complemented
6#30631988C211yesmaximal, characteristic, complemented
7#30631928C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631910C211yesmaximal, characteristic, complemented
10#30631851C211yesmaximal, characteristic, complemented
11#30631840C211yesmaximal, characteristic, complemented
12C4×#16860799C211yesmaximal, characteristic, complemented
13#30631684C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15C8×D4×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×Dic1004C2211yescharacteristic, complemented
19C2×#16860799C2211yescharacteristic
20C2×#16860799C2211yescharacteristic
21#22548708—21no—
22#22548707C2211yescharacteristic, complemented
23#22548705—21no—
24#22548703C2211yescharacteristic
25#22548700—21no—
26#22548696—21no—
27#22548694C2211yescharacteristic
28#22548694C2211yescharacteristic
29#22548694C2211yescharacteristic, complemented
30#22548689C2211yescharacteristic
31#22548688—21no—
32#22548683—21no—
33C4×Dic502C2211yescharacteristic, complemented
34C4×D1004C2211yescharacteristic
35#22548674C2211yescharacteristic, complemented
36#22548671—21no—
37C2×C4⋊Dic251C2211yescharacteristic
38C22⋊C4×D251—21no—
39#22548649C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41#22548647C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548640C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548637C2211yescharacteristic
49#22548637C2211yescharacteristic
50#22548637C2211yescharacteristic
51#22548637C2211yescharacteristic
52C22×C8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic, complemented
56C22⋊C8×C251C2211yescharacteristic
57C22⋊C8×C251C2211yescharacteristic, complemented
58C4×C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62C2×Dic502C2311yescharacteristic
63C2×Dic502C2311yescharacteristic
64C2×D1004—22no—
65C2×D1004—21no—
66C2×D1004—41no—
67C2×D1004C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74Dic1004—41no—
75#16860799—22no—
76#16860799—41no—
77C4⋊Dic251—21no—
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96C4×Dic251—21no—
97C4×Dic251—21no—
98C2×D4×C251D411yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C2×C8×C251—41no—
102C2×C8×C251—41no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—81no—
116Dic502—21no—
117Dic502—21no—
118Dic502—21no—
119Dic502—21no—
120D1004—21no—
121D1004—21no—
122D1004—21no—
123D1004—42no—
124D1004#4112yes—
125C22⋊D251—42no—
126C22⋊D251—81no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—42no—
134C4×D251—42no—
135C4×D251—81no—
136C23×C251C2×D411yescharacteristic
137C23×C251C2×D411yescharacteristic
138D4×C251—22no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—21no—
149C8×C251—21no—
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251#13512yes—
158C22×C251C22×D411yescharacteristic
159C2×D251—22no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—84no—
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—21no—
169C4×C251—21no—
170C4×C251C2×#4111yescharacteristic
171C4×C251H(5,𝔽2)11yescharacteristic
172Dic251—41no—
173Dic251—41no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—22no—
179C2×C251—41no—
180C2×C251#54611yescharacteristic
181C2×C251C2×#13511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—42no—
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×SD16—2511no—
199C8×D4D25111yescharacteristic, complemented
200#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4—2511no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C22×C8C2×D25111yescharacteristic
208C22×C8C2×D25111yescharacteristic, complemented
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×Q8—2511no—
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#106—2511no—
231C4⋊C8C2×D25111yescharacteristic, complemented
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102—2511no—
242C22⋊C8C2×D25111yescharacteristic
243C22⋊C8C2×D25111yescharacteristic, complemented
244C4×C8C2×D25111yescharacteristic, complemented
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8—2511no—
250C2×D4—10041no—
251C2×D4—5022no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4D100411yescharacteristic
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4C22×D25111yescharacteristic
262C22×C4C22×D25111yescharacteristic
263Q16—10041no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D25111yescharacteristic
269C2×C8C22×D25111yescharacteristic
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—5021no—
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⋊C4D100411yescharacteristic
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4D100411yescharacteristic
293C22⋊C4D100411yescharacteristic
294C42—5021no—
295C42—5021no—
296C42C22×D25111yescharacteristic
297C23—20081no—
298C23—10042no—
299C23—10042no—
300C23—5021no—
301C23—5021no—
302C23C2×D100411yescharacteristic
303C23C2×D100411yescharacteristic
304Q8—5021no—
305Q8—5021no—
306Q8—5021no—
307Q8—5021no—
308D4—22no—
309D4—20081no—
310D4—10042no—
311D4—10042no—
312D4—5021no—
313D4—5021no—
314D4—5021no—
315D4—2512no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—20081no—
321C2×C4—10042no—
322C2×C4—10042no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4C2×D100411yescharacteristic
330C2×C4C2×D100411yescharacteristic
331C2×C4C2×D100411yescharacteristic
332C2×C4C2×D100411yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—20084no—
344C22—10041no—
345C22—10041no—
346C22—10041no—
347C22—10041no—
348C22—10041no—
349C22—5022no—
350C22#2254873712yes—
351C22C22×D100411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—10041no—
357C4—10041no—
358C4—10041no—
359C4—10041no—
360C4—10041no—
361C4—10041no—
362C4#41×D25111yescharacteristic
363C4#2254876011yescharacteristic
364C2—22no—
365C2—41no—
366C2—20081no—
367C2—10042no—
368C2#3063233211yescharacteristic
369C2C2×#2254873711yescharacteristic
370C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
prime radical#30632333C2
hypercenterC8×D4D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #30632377C2semidirect
3. #30632042C2semidirect
4. #30632018C2semidirect
5. #30632012C2semidirect
6. #30631988C2semidirect
7. #30631928C2semidirect
8. #30631916C2semidirect
9. #30631910C2semidirect
10. #30631851C2semidirect
11. #30631840C2semidirect
12. C4×#16860799C2semidirect
13. #30631684C2semidirect
14. C2×#22548637C2semidirect
15. C8×D4×C251C2semidirect
18. C2×Dic1004C22semidirect
22. #22548707C22semidirect
29. #22548694C22semidirect
33. C4×Dic502C22semidirect
35. #22548674C22semidirect
39. #22548649C22semidirect
40. #22548647C22semidirect
41. #22548647C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548640C22semidirect
45. #22548640C22semidirect
53. C22×C8×C251C22semidirect
55. C4⋊C8×C251C22semidirect
57. C22⋊C8×C251C22semidirect
58. C4×C8×C251C22semidirect
185. C251#3296semidirect
199. C8×D4D251semidirect
208. C22×C8C2×D251semidirect
231. C4⋊C8C2×D251semidirect
243. C22⋊C8C2×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
#306323772
#306323332
#306320422
#306320182
#306320122
#306319882
#306319282
#306319162
#306319102
#306318512
#306318402
C4×#168607992
#306316842
C2×#225486372
C8×D4×C2512
#3296251