Finite Group Databasebeta

← #44560813

Subgroups — #44560813

Order 32128.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548741C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4#30632233C211yesmaximal, characteristic, complemented
5C2×#22548711C211yesmaximal, characteristic, complemented
6#30632075C212yesmaximal, complemented
7#30632073C212yesmaximal, complemented
8#30632062C212yesmaximal, complemented
9#30632058C212yesmaximal, complemented
10M16×Dic251C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14C22×C22⋊D251—22no—
15#22548743—24no—
16#22548741—24no—
17#22548740—24no—
18C2×#16860799C2212yescomplemented
19C2×#16860798C2212yescomplemented
20C2×#16860797C2212yescomplemented
21C2×D2008C2212yescomplemented
22C2×#16860795C2212yescomplemented
23C2×#16860795C2212yescomplemented
24C2×#16860794C2211yescharacteristic
25#22548713C2211yescharacteristic, complemented
26#22548711—22no—
27#22548711C2212yes—
28#22548711C2212yes—
29#22548711C2211yescharacteristic, complemented
30#22548704C2211yescharacteristic, complemented
31#22548702C2211yescharacteristic, complemented
32#22548687—22no—
33C4×C22⋊D251—22no—
34C2×C4×Dic251C2211yescharacteristic
35#22548625C2212yescomplemented
36C8×Dic251C2212yescomplemented
37C2×#41×C251C2211yescharacteristic
38C22×D4×C251C2211yescharacteristic
39AGL(1,ℤ/8)×C251—24no—
40C2×SD16×C251C2212yescomplemented
41C2×D8×C251C2212yescomplemented
42C2×M16×C251C2211yescharacteristic
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004C2312yes—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—24no—
53C2×C22⋊D251—24no—
54C2×C22⋊D251—22no—
55C2×C22⋊D251—21no—
56C2×C22⋊D251—48no—
57C22×Dic251D412yes—
58C2×C4×D251—21no—
59#16860799—24no—
60#16860798—24no—
61#16860797—24no—
62D2008—24no—
63#16860795—24no—
64#16860795—24no—
65#16860794D414yescomplemented
66C2×C251⋊C8C2312yes—
67Dic251⋊C4—22no—
68C22⋊Dic251—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251D412yes—
71C4×Dic251C2311yescharacteristic
72C4×Dic251C2311yescharacteristic
73C24×C251—21no—
74#41×C251—22no—
75#41×C251—22no—
76C2×Q8×C251C2311yescharacteristic
77C2×D4×C251—21no—
78C2×D4×C251—22no—
79C2×D4×C251—22no—
80C2×D4×C251—21no—
81C2×D4×C251C2311yescharacteristic
82C2×D4×C251C2312yes—
83C22×C4×C251—21no—
84C22×C4×C251C2311yescharacteristic
85SD16×C251—24no—
86D8×C251—24no—
87M16×C251D414yescomplemented
88C2×C8×C251C2312yes—
89C22×D251—21no—
90C22×D251—22no—
91C22×D251—44no—
92C22×D251—42no—
93Dic502—21no—
94Dic502—21no—
95D1004—21no—
96D1004—22no—
97D1004—21no—
98D1004—22no—
99D1004—42no—
100C22⋊D251—48no—
101C22⋊D251—42no—
102C22⋊D251—44no—
103C22⋊D251—48no—
104C2×Dic251—24no—
105C2×Dic251—21no—
106C2×Dic251C2×D412yes—
107C2×Dic251C2×D412yes—
108C4×D251—42no—
109C251⋊C8C2×D414yescomplemented
110C23×C251—21no—
111C23×C251—22no—
112C23×C251—44no—
113C23×C251—42no—
114C23×C251C2×D411yescharacteristic
115Q8×C251—21no—
116Q8×C251—21no—
117D4×C251—21no—
118D4×C251—22no—
119D4×C251—22no—
120D4×C251—21no—
121D4×C251—42no—
122D4×C251—42no—
123C2×C4×C251—21no—
124C2×C4×C251—42no—
125C2×C4×C251C2×D411yescharacteristic
126C2×C4×C251C2×D412yes—
127C2×C4×C251C2×D412yes—
128C2×C4×C251C2411yescharacteristic
129C8×C251C2×D414yescomplemented
130C22×C251—22no—
131C22×C251—41no—
132C22×C251—42no—
133C22×C251—44no—
134C22×C251—42no—
135C22×C251—84no—
136C22×C251#12712yes—
137C22×C251C22×D411yescharacteristic
138C2×D251—41no—
139C2×D251—42no—
140C2×D251—44no—
141C2×D251—42no—
142C2×D251—84no—
143C4×C251—41no—
144C4×C251#12712yes—
145C4×C251C22×D411yescharacteristic
146C4×C251C22×D411yescharacteristic
147Dic251—22no—
148Dic251—41no—
149Dic251AGL(1,ℤ/8)14yescomplemented
150C2×C251—22no—
151C2×C251—81no—
152C2×C251—82no—
153C2×C251C2×#12711yescharacteristic
154C2×C251C2×AGL(1,ℤ/8)12yes—
155D251—81no—
156D251—82no—
157C251#315111yescharacteristic, complemented, Sylow
158#3151—2511nomaximal, Sylow
159C2×AGL(1,ℤ/8)—2511no—
160C2×AGL(1,ℤ/8)—2511no—
161C2×AGL(1,ℤ/8)—2511no—
162C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
163#532—2511no—
164C2×#127—2511no—
165#496—2512no—
166#496—2512no—
167#493—2512no—
168#492—2512no—
169C4×M16—2511no—
170C2×#41—2511no—
171C2×#41C2×D25111yescharacteristic
172C22×D4—5022no—
173C22×D4—2511no—
174C22×D4C2×D25111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5024no—
177AGL(1,ℤ/8)—5024no—
178AGL(1,ℤ/8)—5024no—
179C2×SD16—2512no—
180C2×SD16—2512no—
181C2×SD16—2512no—
182C2×SD16C2×D25112yescomplemented
183C2×D8—2512no—
184C2×D8—2512no—
185C2×D8—2512no—
186C2×D8C2×D25112yescomplemented
187C2×M16—2511no—
188C2×M16C2×D25111yescharacteristic
189C4⋊Q8—2511no—
190#127—5022no—
191#127—2512no—
192#127—2512no—
193#127—2511no—
194#124—2511no—
195#124—2511no—
196#121—5022no—
197C4×D4—5022no—
198C2×C42—2511no—
199#97—2512no—
200C4×C8—2512no—
201C24—21no—
202C24—5021no—
203#41—22no—
204#41—22no—
205#41—5022no—
206#41—5022no—
207C2×Q8—2511no—
208C2×Q8C22×D25111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—10048no—
214C2×D4—5021no—
215C2×D4—5022no—
216C2×D4—5024no—
217C2×D4—5024no—
218C2×D4—5021no—
219C2×D4—5022no—
220C2×D4—5022no—
221C2×D4—2511no—
222C2×D4—2512no—
223C2×D4C22×D25111yescharacteristic
224C2×D4C22×D25112yes—
225C22×C4—21no—
226C22×C4—5021no—
227C22×C4—2512no—
228C22×C4C22×D25111yescharacteristic
229SD16—24no—
230SD16—5024no—
231SD16—5024no—
232SD16—5024no—
233D8—24no—
234D8—5024no—
235D8—5024no—
236D8—5024no—
237M16—2514no—
238M16C22⋊D25114yescomplemented
239C2×C8—2512no—
240C2×C8C22×D25112yes—
241C4⋊C4—5022no—
242C22⋊C4—5022no—
243C22⋊C4—5022no—
244C42—2512no—
245C42—2511no—
246C42—2511no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—10044no—
252C23—10042no—
253C23—5021no—
254C23—5022no—
255C23D4×D25111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—5021no—
259Q8—5021no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—10042no—
267D4—10048no—
268D4—10048no—
269D4—10044no—
270D4—10042no—
271D4—5021no—
272D4—5022no—
273D4—5021no—
274D4—5022no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—10042no—
278C2×C4—5021no—
279C2×C4—5024no—
280C2×C4—2512no—
281C2×C4—2512no—
282C2×C4D4×D25112yes—
283C2×C4D4×D25111yescharacteristic
284C2×C4C2×C22⋊D25112yes—
285C2×C4C23×D25111yescharacteristic
286C8—2514no—
287C8C2×C22⋊D25114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—20084no—
295C22—10044no—
296C22—10042no—
297C22—10041no—
298C22—10042no—
299C22#2254871112yes—
300C22C2×D4×D25111yescharacteristic
301C4—41no—
302C4—10041no—
303C4—5022no—
304C4—2514no—
305C4#2254871112yes—
306C4C2×D4×D25111yescharacteristic
307C4C22×C22⋊D25111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—20082no—
312C2—20081no—
313C2C2×#2254871111yescharacteristic
314C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3151
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548741C2semidirect
3. C2×#22548740C2semidirect
4. #30632233C2semidirect
5. C2×#22548711C2semidirect
6. #30632075C2semidirect
7. #30632073C2semidirect
8. #30632062C2semidirect
9. #30632058C2semidirect
10. M16×Dic251C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
18. C2×#16860799C22semidirect
19. C2×#16860798C22semidirect
20. C2×#16860797C22semidirect
21. C2×D2008C22semidirect
22. C2×#16860795C22semidirect
23. C2×#16860795C22semidirect
25. #22548713C22semidirect
29. #22548711C22semidirect
30. #22548704C22semidirect
31. #22548702C22semidirect
35. #22548625C22semidirect
36. C8×Dic251C22semidirect
40. C2×SD16×C251C22semidirect
41. C2×D8×C251C22semidirect
65. #16860794D4semidirect
87. M16×C251D4semidirect
109. C251⋊C8C2×D4semidirect
129. C8×C251C2×D4semidirect
149. Dic251AGL(1,ℤ/8)semidirect
157. C251#3151semidirect
162. C2×AGL(1,ℤ/8)D251semidirect
182. C2×SD16C2×D251semidirect
186. C2×D8C2×D251semidirect
238. M16C22⋊D251semidirect
287. C8C2×C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487412
C2×#225487402
#306322332
C2×#225487112
#306320752
#306320732
#306320622
#306320582
M16×Dic2512
C2×AGL(1,ℤ/8)×C2512
#3151251