Finite Group Databasebeta

← #44560843

Subgroups — #44560843

Order 32128.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548746C211yesmaximal, characteristic, complemented
2C2×#22548740C211yesmaximal, characteristic, complemented
3C2×#22548735C211yesmaximal, characteristic, complemented
4C22×#16860797C211yesmaximal, characteristic, complemented
5#30632256C211yesmaximal, characteristic, complemented
6#30632079C212yesmaximal, complemented
7#30632077C212yesmaximal, complemented
8#30632069C212yesmaximal, complemented
9#30632061C212yesmaximal, complemented
10#30631643C211yesmaximal, characteristic, complemented
11#536×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548746—24no—
15#22548740—24no—
16#22548735—24no—
17C2×#16860800C2211yescharacteristic, complemented
18C2×#16860800C2212yescomplemented
19C2×#16860798C2211yescharacteristic, complemented
20C2×#16860797—24no—
21C2×#16860797C2212yes—
22C2×#16860797C2212yes—
23C2×#16860797C2212yes—
24C2×#16860797C2211yescharacteristic, complemented
25C2×#16860797C2212yescomplemented
26C2×#16860795C2212yes—
27C2×#16860795C2211yescharacteristic, complemented
28C2×#16860794C2211yescharacteristic
29C22×C251⋊C8C2211yescharacteristic
30#22548712C2212yes—
31#22548703C2212yes—
32#22548690—22no—
33#22548685—22no—
34#22548626C2212yescomplemented
35C4×C251⋊C8C2212yescomplemented
36C2×#41×C251C2211yescharacteristic
37C22×Q8×C251C2211yescharacteristic
38C4⋊Q8×C251C2212yescomplemented
39#124×C251C2212yescomplemented
40#123×C251—22no—
41C22⋊Q8×C251—22no—
42#117×C251C2211yescharacteristic
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004—24no—
51C2×D1004C2312yes—
52C2×D1004C2311yescharacteristic
53C2×C22⋊D251—21no—
54C2×C4×D251—21no—
55#16860800—24no—
56#16860800—22no—
57#16860798—22no—
58#16860797—24no—
59#16860797—24no—
60#16860797—24no—
61#16860797—24no—
62#16860797—22no—
63#16860795—24no—
64#16860795—22no—
65#16860794D414yescomplemented
66C2×C251⋊C8D414yescomplemented
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2312yes—
69C2×C251⋊C8C2311yescharacteristic
70(C2×D251)⋊C4—24no—
71#41×C251—22no—
72#41×C251—22no—
73C2×Q8×C251—21no—
74C2×Q8×C251—22no—
75C2×Q8×C251—22no—
76C2×Q8×C251C2312yes—
77C2×Q8×C251C2311yescharacteristic
78C2×D4×C251—21no—
79C2×D4×C251C2311yescharacteristic
80C22×C4×C251—21no—
81C22×C4×C251—21no—
82C22×C4×C251C2311yescharacteristic
83C4⋊C4×C251—24no—
84C4⋊C4×C251D412yes—
85C22⋊C4×C251—24no—
86C22⋊C4×C251D412yes—
87C42×C251C2312yes—
88C22×D251—22no—
89C22×D251—21no—
90C22×D251—42no—
91C22×D251—82no—
92Dic502—21no—
93Dic502—21no—
94D1004—22no—
95D1004—21no—
96D1004—22no—
97D1004—21no—
98D1004—42no—
99D1004—84no—
100C22⋊D251—42no—
101C2×Dic251—21no—
102C4×D251—42no—
103C251⋊C8C2×D412yes—
104C251⋊C8C2×D412yes—
105C251⋊C8C2×D414yes—
106C23×C251—21no—
107C23×C251C2×D411yescharacteristic
108Q8×C251—22no—
109Q8×C251—22no—
110Q8×C251—21no—
111Q8×C251—21no—
112Q8×C251—42no—
113D4×C251—21no—
114D4×C251—21no—
115D4×C251—42no—
116C2×C4×C251—22no—
117C2×C4×C251—21no—
118C2×C4×C251—42no—
119C2×C4×C251—42no—
120C2×C4×C251C2×D412yes—
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D414yes—
123C2×C4×C251C2×D411yescharacteristic
124C2×C4×C251C2411yescharacteristic
125C22×C251—22no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251#12712yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—41no—
131C2×D251—42no—
132C2×D251—42no—
133C2×D251—44no—
134C2×D251—84no—
135C4×C251—24no—
136C4×C251—41no—
137C4×C251—42no—
138C4×C251#12712yes—
139C4×C251C22×D411yescharacteristic
140C4×C251C22×D411yescharacteristic
141Dic251—41no—
142C2×C251—22no—
143C2×C251—81no—
144C2×C251C2×#12711yescharacteristic
145C2×C251#57712yes—
146D251—81no—
147D251—82no—
148C251#315411yescharacteristic, complemented, Sylow
149#3154—2511nomaximal, Sylow
150C2×#137—2511no—
151C2×AGL(1,ℤ/8)—2511no—
152C2×#135—2511no—
153C22×SD16—2511no—
154#536D25111yescharacteristic, complemented
155#535—2511no—
156#497—2512no—
157#496—2512no—
158#495—2512no—
159#492—2512no—
160#405—2511no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8C2×D25111yescharacteristic
164C22×D4—2511no—
165#137—5024no—
166AGL(1,ℤ/8)—5024no—
167#135—5024no—
168C2×Q16—2511no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2512no—
172C2×SD16—2512no—
173C2×SD16—2511no—
174C2×SD16—2512no—
175C2×SD16—2512no—
176C2×SD16—2511no—
177C2×D8—2511no—
178C2×D8—2512no—
179C2×M16—2511no—
180C22×C8—2511no—
181C4⋊Q8C2×D25112yescomplemented
182#127—2512no—
183#124—2512no—
184#124C2×D25112yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—5022no—
188#120—5022no—
189#117C2×D25111yescharacteristic
190#97—2512no—
191C4×C8—2512no—
192C24—5021no—
193#41—22no—
194#41—22no—
195#41—5022no—
196#41—5022no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—2511no—
201C2×Q8C22×D25112yes—
202C2×Q8C22×D25111yescharacteristic
203C2×D4—21no—
204C2×D4—5021no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5024no—
208C2×D4—5021no—
209C2×D4—2512no—
210C2×D4—2511no—
211C2×D4C22×D25111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5021no—
215C22×C4C22×D25111yescharacteristic
216Q16—5022no—
217Q16—5024no—
218SD16—5022no—
219SD16—5024no—
220SD16—5024no—
221SD16—5022no—
222SD16—5024no—
223SD16—5024no—
224D8—5022no—
225D8—5024no—
226M16—2514no—
227C2×C8—2514no—
228C2×C8—2511no—
229C2×C8—2512no—
230C2×C8—2511no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D25112yes—
233C22⋊C4—24no—
234C22⋊C4—5024no—
235C22⋊C4C22⋊D25112yes—
236C42C22×D25112yes—
237C23—21no—
238C23—20082no—
239C23—10042no—
240C23—5022no—
241C23—5021no—
242C23C2×C22⋊D25111yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—5021no—
249Q8—5021no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—20084no—
254D4—10042no—
255D4—10042no—
256D4—5022no—
257D4—5021no—
258D4—5022no—
259D4—5021no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—10042no—
265C2×C4—5021no—
266C2×C4D4×D25112yes—
267C2×C4D4×D25112yes—
268C2×C4C2×C22⋊D25114yes—
269C2×C4C2×C22⋊D25111yescharacteristic
270C2×C4C23×D25111yescharacteristic
271C8—2512no—
272C8—2512no—
273C8—2514no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—20084no—
278C22—10041no—
279C22—10042no—
280C22—10042no—
281C22—10044no—
282C22#2254871112yes—
283C22C22×C22⋊D25111yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—10041no—
288C4#2254871112yes—
289C4C2×D4×D25111yescharacteristic
290C4C2×D4×D25111yescharacteristic
291C2—22no—
292C2—81no—
293C2—20082no—
294C2—20081no—
295C2C2×#2254871111yescharacteristic
296C2#3063249512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#536×C251C2
socleC22×C251C22×D4
hypercenter#536D251
hypocenterC251#3154
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548740C2semidirect
3. C2×#22548735C2semidirect
4. C22×#16860797C2semidirect
5. #30632256C2semidirect
6. #30632079C2semidirect
7. #30632077C2semidirect
8. #30632069C2semidirect
9. #30632061C2semidirect
10. #30631643C2semidirect
11. #536×C251C2semidirect
17. C2×#16860800C22semidirect
18. C2×#16860800C22semidirect
19. C2×#16860798C22semidirect
24. C2×#16860797C22semidirect
25. C2×#16860797C22semidirect
27. C2×#16860795C22semidirect
34. #22548626C22semidirect
35. C4×C251⋊C8C22semidirect
38. C4⋊Q8×C251C22semidirect
39. #124×C251C22semidirect
65. #16860794D4semidirect
66. C2×C251⋊C8D4semidirect
148. C251#3154semidirect
154. #536D251semidirect
181. C4⋊Q8C2×D251semidirect
184. #124C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487402
C2×#225487352
C22×#168607972
#306322562
#306320792
#306320772
#306320692
#306320612
#306316432
#536×C2512
#3154251