Finite Group Databasebeta

← #44562381

Subgroups — #44562381

Order 32128.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632371C211yesmaximal, characteristic, complemented
2#30632321C211yesmaximal, characteristic
3#30632050C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30632002C211yesmaximal, characteristic, complemented
6#30631990C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631885C211yesmaximal, characteristic, complemented
9#30631837C211yesmaximal, characteristic, complemented
10#30631836C211yesmaximal, characteristic, complemented
11#30631790C211yesmaximal, characteristic, complemented
12#30631756C211yesmaximal, characteristic, complemented
13#30631693C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#468×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C2×D4×D251—21no—
18C2×D4×D251C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×#16860797C2211yescharacteristic, complemented
21C2×C8×D251C2211yescharacteristic, complemented
22#22548711C2211yescharacteristic
23#22548708—21no—
24#22548706C2211yescharacteristic, complemented
25#22548705—21no—
26#22548697—21no—
27#22548691C2211yescharacteristic, complemented
28#22548690C2211yescharacteristic
29#22548688—21no—
30#22548685—21no—
31#22548683—21no—
32Q8×Dic251C2211yescharacteristic, complemented
33#22548677C2211yescharacteristic
34C4×C22⋊D251C2211yescharacteristic
35#22548674C2211yescharacteristic, complemented
36#22548673—21no—
37C4⋊C4×D251C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548646C2211yescharacteristic, complemented
40Dic251⋊C8C2211yescharacteristic, complemented
41#22548637C2211yescharacteristic
42#22548637C2211yescharacteristic
43#22548637C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic, complemented
53C2×M16×C251C2211yescharacteristic, complemented
54C22⋊Q8×C251C2211yescharacteristic, complemented
55#121×C251C2211yescharacteristic
56#107×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C23×D251—21no—
61C2×D1004—22no—
62C2×D1004—21no—
63C2×D1004—21no—
64C2×D1004—41no—
65C2×D1004C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251D411yescharacteristic
71D4×D251—22no—
72D4×D251—41no—
73D4×D251—44no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251C2311yescharacteristic
77#16860797—22no—
78#16860797—41no—
79C2×C251⋊C8C2311yescharacteristic
80C2×C251⋊C8C2311yescharacteristic
81C8×D251—41no—
82C4⋊Dic251—21no—
83C4⋊Dic251—21no—
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91C22⋊Dic251D411yescharacteristic
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251—21no—
97C4×Dic251—21no—
98C4×Dic251C2311yescharacteristic
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103SD16×C251—41no—
104M16×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22⋊C4×C251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—42no—
118C22×D251—81no—
119C22×D251—81no—
120C22×D251C2×D411yescharacteristic
121D1004—21no—
122D1004—21no—
123D1004—21no—
124D1004—42no—
125D1004—42no—
126D1004#4112yes—
127C22⋊D251—22no—
128C22⋊D251—42no—
129C22⋊D251—42no—
130C22⋊D251—81no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251C2×D411yescharacteristic
134C2×Dic251C2×D411yescharacteristic
135C2×Dic251C2×D411yescharacteristic
136C4×D251—21no—
137C4×D251—21no—
138C4×D251—42no—
139C4×D251—42no—
140C251⋊C8—21no—
141C251⋊C8—21no—
142C23×C251—21no—
143C23×C251C2×D411yescharacteristic
144Q8×C251—21no—
145Q8×C251—21no—
146D4×C251—21no—
147D4×C251—21no—
148D4×C251—42no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2411yescharacteristic
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251C22×D411yescharacteristic
165C2×D251—21no—
166C2×D251—21no—
167C2×D251—22no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—84no—
174C2×D251—84no—
175C2×D251AGL(1,ℤ/8)12yes—
176C4×C251—21no—
177C4×C251—41no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251C2×#4111yescharacteristic
181C4×C251H(5,𝔽2)11yescharacteristic
182Dic251—21no—
183Dic251—21no—
184Dic251—21no—
185Dic251—41no—
186Dic251—41no—
187C2×C251—41no—
188C2×C251—81no—
189C2×C251#54611yescharacteristic
190C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C251#57711yescharacteristic
192D251—22no—
193D251—42no—
194D251—81no—
195C251#331011yescharacteristic, complemented, Sylow
196#3310—2511nomaximal, Sylow
197#552—2511no—
198D42—2511no—
199#490—2511no—
200#482—2511no—
201#480—2511no—
202#478—2511no—
203#468D25111yescharacteristic, complemented
204#465—2511no—
205#460—2511no—
206#450—2511no—
207#450—2511no—
208#440—2511no—
209#435—2511no—
210#418—2511no—
211C2×#102—2511no—
212C22×D4—5021no—
213C22×D4—2511no—
214C22×D4—2511no—
215C2×SD16—2511no—
216C2×SD16—2511no—
217C2×SD16C2×D25111yescharacteristic, complemented
218C2×M16C2×D25111yescharacteristic, complemented
219C22×C8—2511no—
220#127—2511no—
221#126—5021no—
222#125—5021no—
223#125—2511no—
224#123—5021no—
225C22⋊Q8—2511no—
226C22⋊Q8C2×D25111yescharacteristic, complemented
227#121—5021no—
228#121—2511no—
229#121C2×D25111yescharacteristic
230#120—5021no—
231#120—5021no—
232C4×Q8—2511no—
233C4×D4—2511no—
234C4×D4—2511no—
235#117—5021no—
236#117—2511no—
237C2×C4⋊C4—2511no—
238#107—2511no—
239#107C2×D25111yescharacteristic, complemented
240#106—2511no—
241C4⋊C8—2511no—
242#103C2×D25111yescharacteristic, complemented
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102—2511no—
251#102C2×D25111yescharacteristic, complemented
252C22⋊C8—2511no—
253C22⋊C8—2511no—
254#97—2511no—
255C24—5021no—
256C24—5021no—
257C2×Q8C22×D25111yescharacteristic
258C2×D4—21no—
259C2×D4—10041no—
260C2×D4—10044no—
261C2×D4—10041no—
262C2×D4—5021no—
263C2×D4—5022no—
264C2×D4—5022no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—5021no—
268C2×D4—5021no—
269C2×D4—2511no—
270C2×D4—2511no—
271C2×D4—2511no—
272C2×D4C22×D25111yescharacteristic
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—2511no—
276C22×C4C22×D25111yescharacteristic
277SD16—41no—
278SD16—10041no—
279SD16—5022no—
280M16—41no—
281C2×C8—10041no—
282C2×C8—2511no—
283C2×C8—2511no—
284C2×C8C22×D25111yescharacteristic
285C2×C8C22×D25111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—5021no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—5021no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4—2511no—
295C4⋊C4C22×D25111yescharacteristic
296C4⋊C4C22×D25111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—2511no—
304C22⋊C4—2511no—
305C42—5021no—
306C42—5021no—
307C42—2511no—
308C23—21no—
309C23—20081no—
310C23—20081no—
311C23—10042no—
312C23—10042no—
313C23—10042no—
314C23—10042no—
315C23—5021no—
316C23—5021no—
317C23—2511no—
318C23D4×D25111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—20081no—
325D4—10042no—
326D4—10042no—
327D4—10042no—
328D4—10042no—
329D4—5022no—
330D4—5021no—
331D4—5021no—
332D4—5021no—
333D4—2512no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—10042no—
340C2×C4—10042no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—2511no—
346C2×C4—2511no—
347C2×C4—2511no—
348C2×C4D4×D25111yescharacteristic
349C2×C4C23×D25111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5021no—
353C8—5021no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—20084no—
361C22—20084no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—5022no—
368C22—5021no—
369C22—5021no—
370C22—2512no—
371C22C2×D4×D25111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—10041no—
377C4—10041no—
378C4—5021no—
379C4—5021no—
380C4—5021no—
381C4#41×D25111yescharacteristic
382C4#2254876011yescharacteristic
383C2—41no—
384C2—81no—
385C2—20081no—
386C2—10042no—
387C2—5022no—
388C2#3063232611yescharacteristic
389C2AGL(1,ℤ/8)×D25111yescharacteristic
390C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#468×C251C2
socleC22×C251C22×D4
prime radical#30632321C2
hypercenter#468D251
hypocenterC251#3310
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #30632371C2semidirect
3. #30632050C2semidirect
4. #30632008C2semidirect
5. #30632002C2semidirect
6. #30631990C2semidirect
7. #30631919C2semidirect
8. #30631885C2semidirect
9. #30631837C2semidirect
10. #30631836C2semidirect
11. #30631790C2semidirect
12. #30631756C2semidirect
13. #30631693C2semidirect
14. #102×D251C2semidirect
15. #468×C251C2semidirect
20. C2×#16860797C22semidirect
21. C2×C8×D251C22semidirect
24. #22548706C22semidirect
27. #22548691C22semidirect
32. Q8×Dic251C22semidirect
35. #22548674C22semidirect
37. C4⋊C4×D251C22semidirect
38. #22548648C22semidirect
39. #22548646C22semidirect
40. Dic251⋊C8C22semidirect
43. #22548637C22semidirect
44. #22548637C22semidirect
45. #22548636C22semidirect
46. #22548636C22semidirect
48. #22548635C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. #22548625C22semidirect
52. C2×SD16×C251C22semidirect
53. C2×M16×C251C22semidirect
54. C22⋊Q8×C251C22semidirect
56. #107×C251C22semidirect
57. #103×C251C22semidirect
58. #102×C251C22semidirect
195. C251#3310semidirect
203. #468D251semidirect
217. C2×SD16C2×D251semidirect
218. C2×M16C2×D251semidirect
226. C22⋊Q8C2×D251semidirect
239. #107C2×D251semidirect
242. #103C2×D251semidirect
251. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323712
#306323212
#306320502
#306320082
#306320022
#306319902
#306319192
#306318852
#306318372
#306318362
#306317902
#306317562
#306316932
#102×D2512
#468×C2512
#3310251