Finite Group Databasebeta

← #44562380

Subgroups — #44562380

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
1D4×D1004C211yesmaximal, characteristic
2#30632053C211yesmaximal, characteristic, complemented
3#30632011C211yesmaximal, characteristic, complemented
4#30632000C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631935C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631887C211yesmaximal, characteristic, complemented
9#30631837C211yesmaximal, characteristic, complemented
10#30631837C211yesmaximal, characteristic, complemented
11#30631785C211yesmaximal, characteristic, complemented
12#30631759C211yesmaximal, characteristic, complemented
13#30631691C211yesmaximal, characteristic, complemented
14C2×#22548635C211yesmaximal, characteristic, complemented
15#552×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×D4×D251—21no—
19C2×#16860797C2211yescharacteristic
20C2×#16860797C2211yescharacteristic, complemented
21C2×#16860797C2211yescharacteristic, complemented
22C2×#16860794C2211yescharacteristic, complemented
23C22×C251⋊C8C2211yescharacteristic, complemented
24#22548712C2211yescharacteristic
25#22548690—21no—
26#22548689C2211yescharacteristic
27#22548689C2211yescharacteristic
28#22548685—21no—
29#22548685—21no—
30C4×D1004C2211yescharacteristic
31#22548648C2211yescharacteristic, complemented
32#22548648C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548639C2211yescharacteristic, complemented
36#22548635C2211yescharacteristic
37#22548635C2211yescharacteristic
38#22548635C2211yescharacteristic
39#22548635C2211yescharacteristic, complemented
40#22548635C2211yescharacteristic, complemented
41#22548635C2211yescharacteristic, complemented
42#22548635C2211yescharacteristic, complemented
43#22548635C2211yescharacteristic, complemented
44#22548635C2211yescharacteristic, complemented
45C22⋊(C251⋊C8)C2211yescharacteristic, complemented
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47#22548626C2211yescharacteristic, complemented
48#126×C251—21no—
49#125×C251—21no—
50#125×C251C2211yescharacteristic, complemented
51#123×C251—21no—
52C22⋊Q8×C251C2211yescharacteristic, complemented
53C22⋊Q8×C251C2211yescharacteristic, complemented
54C4×Q8×C251C2211yescharacteristic, complemented
55C4×D4×C251C2211yescharacteristic
56#117×C251—21no—
57#117×C251C2211yescharacteristic, complemented
58C2×C4⋊C4×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C23×D251—21no—
61C2×D1004—22no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—21no—
67C2×D1004—41no—
68C2×D1004—41no—
69C2×D1004C2311yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×D1004C2311yescharacteristic
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74D4×D251—44no—
75C2×C4×D251—21no—
76#16860797—22no—
77#16860797—41no—
78#16860797—41no—
79#16860794—41no—
80C2×C251⋊C8—41no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C2×C251⋊C8C2311yescharacteristic
84C2×C251⋊C8C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89C2×Q8×C251C2311yescharacteristic
90C2×D4×C251D411yescharacteristic
91C22×C4×C251—21no—
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—21no—
95C4⋊C4×C251—21no—
96C4⋊C4×C251—21no—
97C4⋊C4×C251—21no—
98C4⋊C4×C251—21no—
99C4⋊C4×C251—21no—
100C4⋊C4×C251D411yescharacteristic
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C42×C251—21no—
110C42×C251—21no—
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—42no—
118C22×D251—42no—
119C22×D251—81no—
120C22×D251—81no—
121D1004—21no—
122D1004—21no—
123D1004—21no—
124D1004—21no—
125D1004—21no—
126D1004—42no—
127D1004—42no—
128D1004—42no—
129D1004—81no—
130D1004#4112yes—
131C22⋊D251—42no—
132C22⋊D251—42no—
133C2×Dic251—21no—
134C4×D251—42no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C251⋊C8—21no—
138C251⋊C8—21no—
139C23×C251C2×D411yescharacteristic
140C23×C251C2×D411yescharacteristic
141Q8×C251—21no—
142Q8×C251—21no—
143D4×C251—22no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—42no—
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2×D411yescharacteristic
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2411yescharacteristic
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—21no—
163C22×C251AGL(1,ℤ/8)12yes—
164C22×C251C22×D411yescharacteristic
165C2×D251—22no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—84no—
175C2×D251—84no—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—21no—
179C4×C251—21no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251—41no—
183C4×C251—41no—
184C4×C251C2×#4111yescharacteristic
185C4×C251H(5,𝔽2)11yescharacteristic
186Dic251—41no—
187C2×C251—22no—
188C2×C251—41no—
189C2×C251#54611yescharacteristic
190C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C251#57711yescharacteristic
192D251—42no—
193D251—81no—
194D251—81no—
195C251#331011yescharacteristic, complemented, Sylow
196#3310—2511nomaximal, Sylow
197#552D25111yescharacteristic, complemented
198D42—2511no—
199#490—2511no—
200#482—2511no—
201#480—2511no—
202#478—2511no—
203#468—2511no—
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×SD16—2511no—
218C2×M16—2511no—
219C22×C8—2511no—
220#127—2511no—
221#126—21no—
222#125—21no—
223#125C2×D25111yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D25111yescharacteristic, complemented
226C22⋊Q8C2×D25111yescharacteristic, complemented
227#121—5021no—
228#121—2511no—
229#121—2511no—
230#120—5021no—
231#120—5021no—
232C4×Q8C2×D25111yescharacteristic, complemented
233C4×D4—2511no—
234C4×D4C2×D25111yescharacteristic
235#117—21no—
236#117C2×D25111yescharacteristic, complemented
237C2×C4⋊C4C2×D25111yescharacteristic, complemented
238#107—2511no—
239#107—2511no—
240#106—2511no—
241C4⋊C8—2511no—
242#103—2511no—
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#102—2511no—
252C22⋊C8—2511no—
253C22⋊C8—2511no—
254#97—2511no—
255C24—5021no—
256C24—5021no—
257C2×Q8C22×D25111yescharacteristic
258C2×D4—10041no—
259C2×D4—10041no—
260C2×D4—10044no—
261C2×D4—5022no—
262C2×D4—5022no—
263C2×D4—5021no—
264C2×D4—5021no—
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—21no—
274C22×C4—5021no—
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277SD16—10041no—
278SD16—10041no—
279SD16—5022no—
280M16—10041no—
281C2×C8—10041no—
282C2×C8—2511no—
283C2×C8—2511no—
284C2×C8—2511no—
285C2×C8—2511no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—2511no—
293C4⋊C4C22⋊D25111yescharacteristic
294C4⋊C4C22×D25111yescharacteristic
295C4⋊C4C22×D25111yescharacteristic
296C4⋊C4C22×D25111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4C22⋊D25111yescharacteristic
304C22⋊C4C22⋊D25111yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D25111yescharacteristic
308C23—20081no—
309C23—20081no—
310C23—10042no—
311C23—10042no—
312C23—10042no—
313C23—10042no—
314C23—5021no—
315C23—5021no—
316C23—5021no—
317C23C2×C22⋊D25111yescharacteristic
318C23C2×C22⋊D25111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—20081no—
323D4—10042no—
324D4—10042no—
325D4—10042no—
326D4—10042no—
327D4—10042no—
328D4—5021no—
329D4—5021no—
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—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—10042no—
344C2×C4—5021no—
345C2×C4C2×C22⋊D25111yescharacteristic
346C2×C4C2×C22⋊D25111yescharacteristic
347C2×C4C2×C22⋊D25111yescharacteristic
348C2×C4C2×C22⋊D25111yescharacteristic
349C2×C4C23×D25111yescharacteristic
350C8—5021no—
351C8—5021no—
352C8—5021no—
353C8—5021no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—20084no—
360C22—20084no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22—10041no—
369C22—5022no—
370C22#2254874012yes—
371C22C22×C22⋊D25111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—10041no—
381C4#41×D25111yescharacteristic
382C4#2254876111yescharacteristic
383C2—22no—
384C2—41no—
385C2—20081no—
386C2—20081no—
387C2—10042no—
388C2#3063232511yescharacteristic
389C2C2×#2254874011yescharacteristic
390C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#552×C251C2
socleC22×C251C22×D4
prime radicalD4×D1004C2
hypercenter#552D251
hypocenterC251#3310
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #30632053C2semidirect
3. #30632011C2semidirect
4. #30632000C2semidirect
5. #30631984C2semidirect
6. #30631935C2semidirect
7. #30631918C2semidirect
8. #30631887C2semidirect
9. #30631837C2semidirect
10. #30631837C2semidirect
11. #30631785C2semidirect
12. #30631759C2semidirect
13. #30631691C2semidirect
14. C2×#22548635C2semidirect
15. #552×C251C2semidirect
20. C2×#16860797C22semidirect
21. C2×#16860797C22semidirect
22. C2×#16860794C22semidirect
23. C22×C251⋊C8C22semidirect
31. #22548648C22semidirect
32. #22548648C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548639C22semidirect
39. #22548635C22semidirect
40. #22548635C22semidirect
41. #22548635C22semidirect
42. #22548635C22semidirect
43. #22548635C22semidirect
44. #22548635C22semidirect
45. C22⋊(C251⋊C8)C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. #22548626C22semidirect
50. #125×C251C22semidirect
52. C22⋊Q8×C251C22semidirect
53. C22⋊Q8×C251C22semidirect
54. C4×Q8×C251C22semidirect
57. #117×C251C22semidirect
58. C2×C4⋊C4×C251C22semidirect
195. C251#3310semidirect
197. #552D251semidirect
223. #125C2×D251semidirect
225. C22⋊Q8C2×D251semidirect
226. C22⋊Q8C2×D251semidirect
232. C4×Q8C2×D251semidirect
236. #117C2×D251semidirect
237. C2×C4⋊C4C2×D251semidirect

Maximal subgroups

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

subgroupindex
D4×D10042
#306320532
#306320112
#306320002
#306319842
#306319352
#306319182
#306318872
#306318372
#306317852
#306317592
#306316912
C2×#225486352
#552×C2512
#3310251