Finite Group Databasebeta

← #44560841

Subgroups — #44560841

Order 32128.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548748C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548737C211yesmaximal, characteristic, complemented
4C22×#16860799C211yesmaximal, characteristic, complemented
5#30632263C211yesmaximal, characteristic, complemented
6#30632256C211yesmaximal, characteristic, complemented
7#30632081C212yesmaximal, complemented
8#30632076C212yesmaximal, complemented
9#30632071C212yesmaximal, complemented
10#30632060C212yesmaximal, complemented
11#405×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14C22×Dic502C2211yescharacteristic
15C22×D1004C2211yescharacteristic
16#22548748—24no—
17#22548743—24no—
18#22548737—24no—
19C2×Dic1004C2211yescharacteristic, complemented
20C2×Dic1004C2212yescomplemented
21C2×#16860799—24no—
22C2×#16860799C2212yes—
23C2×#16860799C2212yes—
24C2×#16860799C2212yes—
25C2×#16860799C2211yescharacteristic, complemented
26C2×#16860799C2211yescharacteristic, complemented
27C2×#16860799C2212yescomplemented
28C2×D2008C2212yes—
29C2×D2008C2211yescharacteristic, complemented
30#22548715C2212yescomplemented
31#22548712C2212yes—
32#22548703C2212yes—
33#22548703C2212yescomplemented
34#22548699—22no—
35#22548695—22no—
36#22548690—22no—
37#22548685—22no—
38C2×M16×C251C2211yescharacteristic
39C22×C8×C251C2211yescharacteristic
40#117×C251C2211yescharacteristic
41#97×C251C2212yescomplemented
42C4×C8×C251C2212yescomplemented
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46#16860810—22no—
47#16860810—22no—
48C2×Dic502—21no—
49C2×Dic502—22no—
50C2×Dic502—22no—
51C2×Dic502C2312yes—
52C2×Dic502C2311yescharacteristic
53C2×Dic502C2311yescharacteristic
54C2×D1004—22no—
55C2×D1004—22no—
56C2×D1004—21no—
57C2×D1004—24no—
58C2×D1004C2312yes—
59C2×D1004C2311yescharacteristic
60C2×D1004C2311yescharacteristic
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C22×Dic251—21no—
64C2×C4×D251—21no—
65C2×C4×D251—21no—
66Dic1004—22no—
67Dic1004—24no—
68#16860799—22no—
69#16860799—22no—
70#16860799—24no—
71#16860799—24no—
72#16860799—24no—
73#16860799—24no—
74D2008—22no—
75D2008—24no—
76C4⋊Dic251—24no—
77(C2×D251)⋊C4—24no—
78(C2×D251)⋊C4—24no—
79C22×C4×C251C2311yescharacteristic
80M16×C251D414yescomplemented
81C2×C8×C251D414yescomplemented
82C2×C8×C251C2311yescharacteristic
83C2×C8×C251C2312yes—
84C2×C8×C251C2311yescharacteristic
85C4⋊C4×C251D412yes—
86C22⋊C4×C251D412yes—
87C42×C251C2312yes—
88C22×D251—21no—
89C22×D251—22no—
90C22×D251—21no—
91C22×D251—42no—
92C22×D251—82no—
93Dic502—22no—
94Dic502—21no—
95Dic502—22no—
96Dic502—21no—
97Dic502—21no—
98Dic502—21no—
99Dic502—42no—
100D1004—21no—
101D1004—22no—
102D1004—21no—
103D1004—21no—
104D1004—22no—
105D1004—21no—
106D1004—42no—
107D1004—84no—
108C22⋊D251—42no—
109C22⋊D251—42no—
110C2×Dic251—21no—
111C2×Dic251—22no—
112C2×Dic251—21no—
113C2×Dic251—42no—
114C4×D251—42no—
115C4×D251—42no—
116C23×C251C2×D411yescharacteristic
117C2×C4×C251C2×D414yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D412yes—
121C2×C4×C251C2411yescharacteristic
122C8×C251C2×D412yes—
123C8×C251C2×D412yes—
124C8×C251C2×D414yes—
125C22×C251—22no—
126C22×C251#12712yes—
127C22×C251C22×D411yescharacteristic
128C2×D251—41no—
129C2×D251—41no—
130C2×D251—42no—
131C2×D251—42no—
132C2×D251—44no—
133C2×D251—42no—
134C2×D251—84no—
135C4×C251—24no—
136C4×C251#12712yes—
137C4×C251C22×D411yescharacteristic
138C4×C251C22×D411yescharacteristic
139Dic251—41no—
140Dic251—41no—
141Dic251—42no—
142C2×C251—22no—
143C2×C251C2×#12711yescharacteristic
144C2×C251#57712yes—
145D251—81no—
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#536—2511no—
155#535—2511no—
156#497—2512no—
157#496—2512no—
158#495—2512no—
159#492—2512no—
160#405D25111yescharacteristic, complemented
161C2×#41—2511no—
162C2×#41—2511no—
163C22×Q8—2511no—
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—2511no—
175C2×SD16—2512no—
176C2×SD16—2512no—
177C2×D8—2511no—
178C2×D8—2512no—
179C2×M16C2×D25111yescharacteristic
180C22×C8C2×D25111yescharacteristic
181C4⋊Q8—2512no—
182#127—2512no—
183#124—2512no—
184#124—2512no—
185#123—5022no—
186C22⋊Q8—5022no—
187#121—5022no—
188#120—5022no—
189#117C2×D25111yescharacteristic
190#97C2×D25112yescomplemented
191C4×C8C2×D25112yescomplemented
192C24—5021no—
193#41—5022no—
194#41—5022no—
195#41—5022no—
196#41—5022no—
197C2×Q8—5022no—
198C2×Q8—5022no—
199C2×Q8—5021no—
200C2×Q8—2511no—
201C2×Q8—2512no—
202C2×Q8—2511no—
203C2×D4—5021no—
204C2×D4—5021no—
205C2×D4—5024no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—2511no—
210C2×D4—2512no—
211C2×D4—2511no—
212C22×C4—5021no—
213C22×C4—5021no—
214C22×C4—5021no—
215C22×C4C22×D25111yescharacteristic
216Q16—5022no—
217Q16—5024no—
218SD16—5024no—
219SD16—5024no—
220SD16—5022no—
221SD16—5022no—
222SD16—5024no—
223SD16—5024no—
224D8—5022no—
225D8—5024no—
226M16D100414yescomplemented
227C2×C8D100414yescomplemented
228C2×C8C22×D25111yescharacteristic
229C2×C8C22×D25112yes—
230C2×C8C22×D25111yescharacteristic
231C4⋊C4—5024no—
232C4⋊C4D100412yes—
233C22⋊C4—5024no—
234C22⋊C4—5024no—
235C22⋊C4D100412yes—
236C42C22×D25112yes—
237C23—20082no—
238C23—10042no—
239C23—5021no—
240C23—5022no—
241C23—5021no—
242C23C2×D100411yescharacteristic
243Q8—10042no—
244Q8—5022no—
245Q8—5021no—
246Q8—5022no—
247Q8—5021no—
248Q8—5021no—
249Q8—5021no—
250D4—20084no—
251D4—10042no—
252D4—10042no—
253D4—10042no—
254D4—5021no—
255D4—5022no—
256D4—5021no—
257D4—5021no—
258D4—5022no—
259D4—5021no—
260C2×C4—10042no—
261C2×C4—10042no—
262C2×C4—10042no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5021no—
266C2×C4C2×D100414yes—
267C2×C4C2×D100412yes—
268C2×C4C2×D100412yes—
269C2×C4C2×D100411yescharacteristic
270C2×C4C23×D25111yescharacteristic
271C8C2×D100412yes—
272C8C2×D100412yes—
273C8C2×D100414yes—
274C22—22no—
275C22—20084no—
276C22—10041no—
277C22—10041no—
278C22—10042no—
279C22—10042no—
280C22—10044no—
281C22—10042no—
282C22#2254871212yes—
283C22C22×D100411yescharacteristic
284C4—24no—
285C4—10041no—
286C4—10041no—
287C4—10042no—
288C4#2254871212yes—
289C4C22×D100411yescharacteristic
290C4C22×D100411yescharacteristic
291C2—22no—
292C2—20082no—
293C2—20081no—
294C2—20081no—
295C2C2×#2254871211yescharacteristic
296C2#3063249912yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#405×C251C2
socleC22×C251C22×D4
hypercenter#405D251
hypocenterC251#3154
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#22548748C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548737C2semidirect
4. C22×#16860799C2semidirect
5. #30632263C2semidirect
6. #30632256C2semidirect
7. #30632081C2semidirect
8. #30632076C2semidirect
9. #30632071C2semidirect
10. #30632060C2semidirect
11. #405×C251C2semidirect
19. C2×Dic1004C22semidirect
20. C2×Dic1004C22semidirect
25. C2×#16860799C22semidirect
26. C2×#16860799C22semidirect
27. C2×#16860799C22semidirect
29. C2×D2008C22semidirect
30. #22548715C22semidirect
33. #22548703C22semidirect
41. #97×C251C22semidirect
42. C4×C8×C251C22semidirect
80. M16×C251D4semidirect
81. C2×C8×C251D4semidirect
148. C251#3154semidirect
160. #405D251semidirect
190. #97C2×D251semidirect
191. C4×C8C2×D251semidirect
226. M16D1004semidirect
227. C2×C8D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×#225487432
C2×#225487372
C22×#168607992
#306322632
#306322562
#306320812
#306320762
#306320712
#306320602
#405×C2512
#3154251