Finite Group Databasebeta

← #44560475

Subgroups — #44560475

Order 32128.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632223C211yesmaximal, characteristic, complemented
2#41×Dic251C211yesmaximal, characteristic, complemented
3#30632025C211yesmaximal, characteristic, complemented
4#30632012C211yesmaximal, characteristic, complemented
5#30631996C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631911C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic
10#30631796C211yesmaximal, characteristic, complemented
11#30631794C211yesmaximal, characteristic, complemented
12#30631790C211yesmaximal, characteristic, complemented
13#30631788C211yesmaximal, characteristic, complemented
14#30631741C211yesmaximal, characteristic, complemented
15#452×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18#22548710—21no—
19#22548706C2211yescharacteristic
20#22548704C2211yescharacteristic
21#22548699—21no—
22#22548689C2211yescharacteristic
23Q8×Dic251C2211yescharacteristic
24Q8×Dic251C2211yescharacteristic
25Q8×Dic251C2211yescharacteristic
26D4×Dic251—21no—
27D4×Dic251—21no—
28D4×Dic251C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30#22548677C2211yescharacteristic
31(C22⋊D251)⋊C4—21no—
32#22548674—21no—
33#22548674C2211yescharacteristic
34#22548673—21no—
35C2×C4×Dic251—21no—
36C2×C4×Dic251C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548647C2211yescharacteristic
40#22548646C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic
51#22548625C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C2×Q16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C22×Dic251—21no—
62C22×Dic251—21no—
63C22×Dic251D411yescharacteristic
64C22×Dic251D411yescharacteristic
65C2×C4×D251—21no—
66#16860797—22no—
67#16860795—22no—
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C4⋊Dic251—21no—
71C4⋊Dic251—21no—
72C4⋊Dic251—21no—
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78C22⋊Dic251—21no—
79C22⋊Dic251—21no—
80C22⋊Dic251—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83(C2×D251)⋊C4—21no—
84(C2×D251)⋊C4—21no—
85C4×Dic251—21no—
86C4×Dic251—21no—
87C4×Dic251—21no—
88C4×Dic251—21no—
89C4×Dic251D411yescharacteristic
90C4×Dic251D411yescharacteristic
91C4×Dic251C2311yescharacteristic
92C4×Dic251C2311yescharacteristic
93#41×C251—22no—
94#41×C251—22no—
95C2×Q8×C251C2311yescharacteristic
96C2×Q8×C251C2311yescharacteristic
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100C22×C4×C251C2311yescharacteristic
101Q16×C251—22no—
102SD16×C251—22no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22×D251—21no—
109D1004—21no—
110D1004—21no—
111C22⋊D251—42no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—41no—
116C2×Dic251—41no—
117C2×Dic251—42no—
118C2×Dic251—42no—
119C2×Dic251C2×D411yescharacteristic
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C4×D251—42no—
124C251⋊C8—21no—
125C251⋊C8—21no—
126C23×C251—21no—
127C23×C251C2×D411yescharacteristic
128Q8×C251—21no—
129Q8×C251—21no—
130Q8×C251—21no—
131Q8×C251#4112yes—
132D4×C251—24no—
133D4×C251—21no—
134D4×C251#4112yes—
135C2×C4×C251—24no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—21no—
146C22×C251—22no—
147C22×C251—21no—
148C22×C251—21no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—41no—
153C2×D251—41no—
154C2×D251—41no—
155C4×C251—21no—
156C4×C251—22no—
157C4×C251—41no—
158C4×C251—41no—
159C4×C251C2×#4111yescharacteristic
160C4×C251C2×#4111yescharacteristic
161Dic251—21no—
162Dic251—21no—
163Dic251—21no—
164Dic251—21no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—41no—
169C2×C251—42no—
170C2×C251#52511yescharacteristic
171C2×C251#57511yescharacteristic
172C2×C251#57511yescharacteristic
173D251—81no—
174C251#311111yescharacteristic, complemented, Sylow
175#3111—2511nomaximal, Sylow
176#529—2511no—
177C4×#41—2511no—
178#485—2511no—
179#482—2511no—
180#479—2511no—
181#478—2511no—
182#464—2511no—
183#463—2511no—
184#452D25111yescharacteristic, complemented
185#449—2511no—
186#442—2511no—
187#441—2511no—
188#440—2511no—
189#440—2511no—
190#432—2511no—
191C2×#41C2×D25111yescharacteristic
192C2×Q16C2×D25111yescharacteristic
193C2×SD16—2511no—
194C2×SD16C2×D25111yescharacteristic
195C2×D8—2511no—
196#126—5021no—
197#125—2511no—
198#124—2511no—
199#123—5021no—
200C22⋊Q8C2×D25111yescharacteristic
201#121—2511no—
202C4×Q8—2511no—
203C4×Q8—2511no—
204C4×Q8—2511no—
205C4×D4—5021no—
206C4×D4—5021no—
207C4×D4—5021no—
208C4×D4—2511no—
209C4×D4—2511no—
210C4×D4—2511no—
211#117—5021no—
212#117—5021no—
213#117—2511no—
214C2×C42—5021no—
215C2×C42—2511no—
216#107—2511no—
217#107—2511no—
218#106—2511no—
219#106—2511no—
220C4⋊C8—2511no—
221C4⋊C8—2511no—
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8C2×D25111yescharacteristic
232#97—2511no—
233#97—2511no—
234#41—22no—
235#41—22no—
236C2×Q8C22×D25111yescharacteristic
237C2×Q8C22×D25111yescharacteristic
238C2×D4—21no—
239C2×D4—5021no—
240C2×D4—2511no—
241C2×D4C22×D25111yescharacteristic
242C22×C4—21no—
243C22×C4—5021no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—2511no—
247C22×C4—2511no—
248C22×C4C22×D25111yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—5022no—
252D8—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4C22×D25111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5021no—
269C22⋊C4—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C42—5021no—
276C42—5021no—
277C42—5021no—
278C42—5021no—
279C42—2511no—
280C42—2511no—
281C42—2511no—
282C42—2511no—
283C23—21no—
284C23—5021no—
285C23D4×D25111yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8#1686080912yes—
290D4—24no—
291D4—21no—
292D4—10042no—
293D4—5021no—
294D4—5021no—
295D4#1686080912yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—10042no—
303C2×C4—10042no—
304C2×C4—10042no—
305C2×C4—10041no—
306C2×C4—10041no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4—2511no—
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—10041no—
327C22—10041no—
328C22—10041no—
329C22C2×D4×D25111yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—5021no—
338C4—5021no—
339C4—5021no—
340C4—5021no—
341C4C2×#1686080911yescharacteristic
342C4C2×#1686080911yescharacteristic
343C2—41no—
344C2—42no—
345C2—20081no—
346C2#3063219311yescharacteristic
347C2#3063248411yescharacteristic
348C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#452×C251C2
socleC22×C251C22×D4
prime radical#30631832C2
hypercenter#452D251
hypocenterC251#3111
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #30632223C2semidirect
2. #41×Dic251C2semidirect
3. #30632025C2semidirect
4. #30632012C2semidirect
5. #30631996C2semidirect
6. #30631989C2semidirect
7. #30631911C2semidirect
8. #30631904C2semidirect
10. #30631796C2semidirect
11. #30631794C2semidirect
12. #30631790C2semidirect
13. #30631788C2semidirect
14. #30631741C2semidirect
15. #452×C251C2semidirect
174. C251#3111semidirect
184. #452D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
#41×Dic2512
#306320252
#306320122
#306319962
#306319892
#306319112
#306319042
#306318322
#306317962
#306317942
#306317902
#306317882
#306317412
#452×C2512
#3111251