Finite Group Databasebeta

← #44560613

Subgroups — #44560613

Order 32128.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632297C211yesmaximal, characteristic, complemented
2#30632161C211yesmaximal, characteristic
3#30632028C211yesmaximal, characteristic, complemented
4#30631990C211yesmaximal, characteristic, complemented
5#30631986C211yesmaximal, characteristic, complemented
6#30631891C211yesmaximal, characteristic, complemented
7#30631891C211yesmaximal, characteristic, complemented
8#30631840C211yesmaximal, characteristic
9#30631837C211yesmaximal, characteristic
10#30631790C211yesmaximal, characteristic, complemented
11#30631786C211yesmaximal, characteristic, complemented
12#30631774C211yesmaximal, characteristic, complemented
13#30631772C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#485×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C2×#16860799C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×#16860797C2211yescharacteristic
21#22548713C2211yescharacteristic, complemented
22#22548708—21no—
23#22548708—21no—
24#22548706—21no—
25#22548706C2211yescharacteristic, complemented
26#22548705—21no—
27#22548694C2211yescharacteristic
28Q8×Dic251C2211yescharacteristic, complemented
29#22548680C2211yescharacteristic, complemented
30C4×D1004—21no—
31C4×D1004—21no—
32#22548677C2211yescharacteristic
33#22548677C2211yescharacteristic
34#22548677C2211yescharacteristic
35#22548677C2211yescharacteristic
36#22548674C2211yescharacteristic
37C22⋊C4×D251—21no—
38C22⋊C4×D251—21no—
39#22548648C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22⋊Q8×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C23×D251—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004—22no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C4×D251—21no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70#16860799—22no—
71#16860799—22no—
72#16860797—22no—
73#16860797—22no—
74C2×C251⋊C8C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C4⋊Dic251—21no—
77C4⋊Dic251—21no—
78C4⋊Dic251D411yescharacteristic
79C4⋊Dic251D411yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251D411yescharacteristic
88C22⋊Dic251D411yescharacteristic
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—24no—
111C22×D251—81no—
112C22×D251—81no—
113Dic502—21no—
114Dic502—21no—
115D1004—24no—
116D1004—21no—
117D1004—21no—
118D1004#4112yes—
119D1004#4112yes—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—81no—
127C4×D251—81no—
128C4×D251—81no—
129C4×D251—81no—
130C251⋊C8—21no—
131C251⋊C8—21no—
132C23×C251C2×D411yescharacteristic
133Q8×C251—21no—
134Q8×C251—21no—
135C2×C4×C251—21no—
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—21no—
147C22×C251—21no—
148C22×C251C22×D411yescharacteristic
149C2×D251—22no—
150C2×D251—22no—
151C2×D251—41no—
152C2×D251—41no—
153C2×D251—41no—
154C2×D251—41no—
155C2×D251—84no—
156C4×C251—21no—
157C4×C251—41no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251C2×#4111yescharacteristic
162C4×C251C2×#4111yescharacteristic
163Dic251—21no—
164Dic251—21no—
165Dic251—21no—
166Dic251—21no—
167Dic251—41no—
168Dic251—41no—
169C2×C251—41no—
170C2×C251#52511yescharacteristic
171C2×C251#57711yescharacteristic
172C2×C251#57711yescharacteristic
173D251—42no—
174D251—42no—
175C251#312311yescharacteristic, complemented, Sylow
176#3123—2511nomaximal, Sylow
177#541—2511no—
178#518—2511no—
179#485—2511no—
180#485D25111yescharacteristic, complemented
181#478—2511no—
182#478—2511no—
183#461—2511no—
184#461—2511no—
185#450—2511no—
186#450—2511no—
187#440—2511no—
188#440—2511no—
189C4×SD16—2511no—
190C4×SD16—2511no—
191#433—2511no—
192C22×D4—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×SD16—2511no—
196C2×SD16—2511no—
197C4⋊Q8—2511no—
198#126—5021no—
199#126—5021no—
200#125—5021no—
201#125—5021no—
202#125—2511no—
203C22⋊Q8—2511no—
204C22⋊Q8C2×D25111yescharacteristic
205C4×Q8—2511no—
206C4×Q8—2511no—
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4—2511no—
212C4×D4—2511no—
213#117—2511no—
214#117C2×D25111yescharacteristic
215C2×C22⋊C4—5021no—
216C2×C22⋊C4—5021no—
217#107—2511no—
218#107C2×D25111yescharacteristic, complemented
219#106—2511no—
220#106C2×D25111yescharacteristic, complemented
221C4⋊C8—2511no—
222C4⋊C8—2511no—
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic, complemented
226#103C2×D25111yescharacteristic, complemented
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102—2511no—
231C22⋊C8—2511no—
232C22⋊C8C2×D25111yescharacteristic
233#97—2511no—
234C4×C8—2511no—
235C24—5021no—
236C2×Q8—2511no—
237C2×Q8C22×D25111yescharacteristic
238C2×D4—5022no—
239C2×D4—5022no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4—2511no—
243C22×C4—5021no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4C22×D25111yescharacteristic
248SD16—5022no—
249SD16—5022no—
250SD16—5022no—
251SD16—5022no—
252C2×C8—2511no—
253C2×C8—2511no—
254C2×C8C22×D25111yescharacteristic
255C2×C8C22×D25111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—5021no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4C22×D25111yescharacteristic
268C4⋊C4C22×D25111yescharacteristic
269C4⋊C4C22×D25111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4—5021no—
277C22⋊C4—2511no—
278C22⋊C4—2511no—
279C42—21no—
280C42—5021no—
281C42—5021no—
282C42—2511no—
283C42—2511no—
284C23—20081no—
285C23—20081no—
286C23—5024no—
287C23—5021no—
288C23—5021no—
289C23D4×D25111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5021no—
293Q8—5021no—
294D4—5024no—
295D4—5021no—
296D4—5021no—
297D4—2512no—
298D4—2512no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—20081no—
306C2×C4—20081no—
307C2×C4—20081no—
308C2×C4—20081no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4—2511no—
314C2×C4—2511no—
315C2×C4D4×D25111yescharacteristic
316C2×C4C23×D25111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5021no—
320C8—5021no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—20084no—
325C22—10041no—
326C22—10041no—
327C22—10041no—
328C22—10041no—
329C22—5022no—
330C22—5022no—
331C22C2×D4×D25111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—10041no—
338C4—10041no—
339C4—5021no—
340C4—5021no—
341C4—5021no—
342C4—5021no—
343C4#41×D25111yescharacteristic
344C4#41×D25111yescharacteristic
345C2—41no—
346C2—10042no—
347C2—10042no—
348C2#3063219511yescharacteristic
349C2#3063249411yescharacteristic
350C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radicalC22×D1004C22
hypercenter#485D251
hypocenterC251#3123
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #30632297C2semidirect
3. #30632028C2semidirect
4. #30631990C2semidirect
5. #30631986C2semidirect
6. #30631891C2semidirect
7. #30631891C2semidirect
10. #30631790C2semidirect
11. #30631786C2semidirect
12. #30631774C2semidirect
13. #30631772C2semidirect
14. #30631745C2semidirect
15. #485×C251C2semidirect
21. #22548713C22semidirect
25. #22548706C22semidirect
28. Q8×Dic251C22semidirect
29. #22548680C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548638C22semidirect
44. #22548638C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. #103×C251C22semidirect
57. #103×C251C22semidirect
175. C251#3123semidirect
180. #485D251semidirect
218. #107C2×D251semidirect
220. #106C2×D251semidirect
225. #103C2×D251semidirect
226. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322972
#306321612
#306320282
#306319902
#306319862
#306318912
#306318402
#306318372
#306317902
#306317862
#306317742
#306317722
#306317452
#485×C2512
#3123251