Finite Group Databasebeta

← #44560832

Subgroups — #44560832

Order 32128.

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

normal subgroups only

This group has 572 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×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4#30632233C211yesmaximal, characteristic, complemented
5C2×#22548704C211yesmaximal, characteristic, complemented
6#30632080C212yesmaximal, complemented
7#30632073C212yesmaximal, complemented
8#30632072C212yesmaximal, complemented
9#30632068C212yesmaximal, complemented
10M16×Dic251C211yesmaximal, characteristic, complemented
11C2×#137×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548746—24no—
15#22548743—24no—
16#22548740—24no—
17C2×#16860800C2212yescomplemented
18C2×#16860799C2212yes—
19C2×#16860797C2212yes—
20C2×#16860797C2212yescomplemented
21C2×D2008C2212yes—
22C2×#16860795C2212yes—
23C2×#16860794C2211yescharacteristic
24#22548713C2211yescharacteristic, complemented
25#22548711C2211yescharacteristic, complemented
26#22548704—22no—
27#22548704C2212yes—
28#22548704C2212yes—
29#22548704C2211yescharacteristic, complemented
30#22548702C2211yescharacteristic, complemented
31#22548687—22no—
32C4×C22⋊D251—22no—
33C2×(C2×D251)⋊C4—22no—
34C2×C4×Dic251C2211yescharacteristic
35#22548625C2212yescomplemented
36C8×Dic251C2212yescomplemented
37C2×#41×C251C2211yescharacteristic
38C22×Q8×C251C2211yescharacteristic
39#137×C251—24no—
40C2×Q16×C251C2212yescomplemented
41C2×SD16×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—22no—
53C2×C22⋊D251—21no—
54C22×Dic251D412yes—
55C2×C4×D251—21no—
56#16860800—24no—
57#16860799—24no—
58#16860797—24no—
59#16860797—24no—
60D2008—24no—
61#16860795—24no—
62#16860794D414yescomplemented
63C2×C251⋊C8C2312yes—
64Dic251⋊C4—22no—
65C22⋊Dic251—22no—
66(C2×D251)⋊C4—24no—
67(C2×D251)⋊C4—24no—
68(C2×D251)⋊C4—22no—
69C4×Dic251D412yes—
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72#41×C251—22no—
73#41×C251—22no—
74C2×Q8×C251—21no—
75C2×Q8×C251—22no—
76C2×Q8×C251—22no—
77C2×Q8×C251C2312yes—
78C2×Q8×C251C2311yescharacteristic
79C2×D4×C251—21no—
80C2×D4×C251C2311yescharacteristic
81C22×C4×C251—21no—
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84Q16×C251—24no—
85SD16×C251—24no—
86M16×C251D414yescomplemented
87C2×C8×C251C2312yes—
88C22×D251—22no—
89C22×D251—21no—
90C22×D251—44no—
91C22×D251—42no—
92Dic502—21no—
93Dic502—21no—
94D1004—22no—
95D1004—21no—
96D1004—22no—
97D1004—21no—
98D1004—42no—
99C22⋊D251—44no—
100C22⋊D251—42no—
101C2×Dic251—21no—
102C2×Dic251—42no—
103C2×Dic251C2×D412yes—
104C2×Dic251C2×D412yes—
105C4×D251—42no—
106C251⋊C8C2×D414yes—
107C23×C251—21no—
108C23×C251C2×D411yescharacteristic
109Q8×C251—22no—
110Q8×C251—21no—
111Q8×C251—22no—
112Q8×C251—21no—
113Q8×C251—42no—
114D4×C251—21no—
115D4×C251—21no—
116D4×C251—42no—
117C2×C4×C251—21no—
118C2×C4×C251—22no—
119C2×C4×C251—42no—
120C2×C4×C251—42no—
121C2×C4×C251C2×D411yescharacteristic
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2×D412yes—
124C2×C4×C251C2411yescharacteristic
125C8×C251C2×D414yes—
126C22×C251—22no—
127C22×C251—41no—
128C22×C251—42no—
129C22×C251#12712yes—
130C22×C251C22×D411yescharacteristic
131C2×D251—41no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—44no—
135C2×D251—84no—
136C4×C251—41no—
137C4×C251—42no—
138C4×C251#12712yes—
139C4×C251C22×D411yescharacteristic
140C4×C251C22×D411yescharacteristic
141Dic251—22no—
142Dic251—22no—
143Dic251—41no—
144C2×C251—22no—
145C2×C251—81no—
146C2×C251C2×#12711yescharacteristic
147C2×C251#57512yes—
148D251—81no—
149D251—82no—
150C251#315311yescharacteristic, complemented, Sylow
151#3153—2511nomaximal, Sylow
152C2×#137—2511no—
153C2×#137D25111yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2511no—
155C2×AGL(1,ℤ/8)—2511no—
156#532—2511no—
157C2×#124—2511no—
158#497—2512no—
159#496—2512no—
160#495—2512no—
161#495—2512no—
162C4×M16—2511no—
163C2×#41—2511no—
164C2×#41C2×D25111yescharacteristic
165C22×Q8C2×D25111yescharacteristic
166C22×D4—2511no—
167#137—24no—
168#137—5024no—
169AGL(1,ℤ/8)—5024no—
170AGL(1,ℤ/8)—5024no—
171C2×Q16—2512no—
172C2×Q16C2×D25112yescomplemented
173C2×SD16—2512no—
174C2×SD16—2512no—
175C2×SD16—2512no—
176C2×SD16C2×D25112yescomplemented
177C2×D8—2512no—
178C2×D8—2512no—
179C2×M16—2511no—
180C2×M16C2×D25111yescharacteristic
181C4⋊Q8—2511no—
182#127—2511no—
183#124—5022no—
184#124—2512no—
185#124—2512no—
186#124—2511no—
187#124—2511no—
188#121—5022no—
189C4×D4—5022no—
190C2×C22⋊C4—5022no—
191C2×C42—2511no—
192#97—2512no—
193C4×C8—2512no—
194C24—5021no—
195#41—22no—
196#41—22no—
197#41—5022no—
198#41—5022no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—2511no—
203C2×Q8C22×D25112yes—
204C2×Q8C22×D25111yescharacteristic
205C2×D4—21no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—5021no—
209C2×D4—5022no—
210C2×D4—5022no—
211C2×D4—2512no—
212C2×D4—2511no—
213C2×D4C22×D25111yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—5021no—
217C22×C4—2512no—
218C22×C4C22×D25111yescharacteristic
219Q16—24no—
220Q16—5024no—
221SD16—24no—
222SD16—5024no—
223SD16—5024no—
224SD16—5024no—
225D8—5024no—
226D8—5024no—
227M16—2514no—
228M16C22⋊D25114yescomplemented
229C2×C8—2512no—
230C2×C8C22×D25112yes—
231C4⋊C4—5022no—
232C22⋊C4—5024no—
233C22⋊C4—5024no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C42—2512no—
237C42—2511no—
238C42—2511no—
239C23—21no—
240C23—10044no—
241C23—10042no—
242C23—5022no—
243C23—5021no—
244C23D4×D25111yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—5021no—
251Q8—5021no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—10042no—
256D4—10044no—
257D4—10042no—
258D4—5022no—
259D4—5021no—
260D4—5021no—
261D4—5022no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—10042no—
267C2×C4—10042no—
268C2×C4—5021no—
269C2×C4—2512no—
270C2×C4—2512no—
271C2×C4D4×D25112yes—
272C2×C4D4×D25111yescharacteristic
273C2×C4C2×C22⋊D25112yes—
274C2×C4C23×D25111yescharacteristic
275C8—2514no—
276C8C2×C22⋊D25114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—20084no—
281C22—10041no—
282C22—10042no—
283C22—10042no—
284C22—10044no—
285C22#2254871112yes—
286C22C2×D4×D25111yescharacteristic
287C4—41no—
288C4—42no—
289C4—10041no—
290C4—5022no—
291C4—5022no—
292C4#2254871112yes—
293C4C2×D4×D25111yescharacteristic
294C4C22×C22⋊D25111yescharacteristic
295C2—22no—
296C2—81no—
297C2—20082no—
298C2—20081no—
299C2C2×#2254871111yescharacteristic
300C2#3063248612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
hypercenterC2×#137D251
hypocenterC251#3153
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548740C2semidirect
4. #30632233C2semidirect
5. C2×#22548704C2semidirect
6. #30632080C2semidirect
7. #30632073C2semidirect
8. #30632072C2semidirect
9. #30632068C2semidirect
10. M16×Dic251C2semidirect
11. C2×#137×C251C2semidirect
17. C2×#16860800C22semidirect
20. C2×#16860797C22semidirect
24. #22548713C22semidirect
25. #22548711C22semidirect
29. #22548704C22semidirect
30. #22548702C22semidirect
35. #22548625C22semidirect
36. C8×Dic251C22semidirect
40. C2×Q16×C251C22semidirect
41. C2×SD16×C251C22semidirect
62. #16860794D4semidirect
86. M16×C251D4semidirect
150. C251#3153semidirect
153. C2×#137D251semidirect
172. C2×Q16C2×D251semidirect
176. C2×SD16C2×D251semidirect
228. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487432
C2×#225487402
#306322332
C2×#225487042
#306320802
#306320732
#306320722
#306320682
M16×Dic2512
C2×#137×C2512
#3153251