Finite Group Databasebeta

← #44559713

Subgroups — #44559713

Order 32128.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#30632523C211yesmaximal, characteristic, complemented
2#30632485C212yesmaximal, complemented
3#30631869C212yesmaximal, complemented
4#30631864C212yesmaximal, complemented
5#30631859C212yesmaximal, complemented
6#30631854C212yesmaximal, complemented
7C2×#22548643C212yesmaximal, complemented
8#30631671C211yesmaximal, characteristic, complemented
9#532×C251C211yesmaximal, characteristic, complemented
10#22548763C2212yes—
11#22548762—22no—
12#22548761—22no—
13#22548760—22no—
14#22548760C2212yes—
15C2×#16860810—22no—
16C2×#16860810C2212yes—
17#41×D251—24no—
18#22548746C2214yescomplemented
19#22548745—22no—
20#22548741C2214yescomplemented
21#22548740—22no—
22#22548735—24no—
23C2×#16860794C2212yescomplemented
24#22548643C2214yescomplemented
25#22548643C2214yescomplemented
26#22548634C2212yescomplemented
27#22548632C2212yescomplemented
28C2×#41×C251C2211yescharacteristic
29C2×#41×C251C2211yescharacteristic, complemented
30C4⋊Q8×C251C2211yescharacteristic, complemented
31#127×C251C2211yescharacteristic, complemented
32#124×C251C2211yescharacteristic, complemented
33#124×C251C2211yescharacteristic, complemented
34#121×C251—22no—
35C4×D4×C251—22no—
36C2×C42×C251C2211yescharacteristic, complemented
37#16860811—24no—
38#16860810—24no—
39#16860810—24no—
40#16860810—22no—
41#16860810—24no—
42#16860810—22no—
43#16860810D414yescomplemented
44#16860810C2314yes—
45#16860809—28no—
46#16860809—24no—
47C2×Dic502—22no—
48C2×Dic502D412yescomplemented
49Q8×D251—24no—
50C2×D1004—22no—
51C2×D1004D412yescomplemented
52C2×C22⋊D251—24no—
53C2×C22⋊D251—22no—
54D4×D251—28no—
55D4×D251—24no—
56C2×C4×D251—24no—
57C2×C4×D251—22no—
58#16860800—24no—
59#16860798—24no—
60#16860797—24no—
61#16860795—24no—
62#16860794—22no—
63#16860794C2314yes—
64C2×C251⋊C8—22no—
65#41×C251—22no—
66#41×C251—22no—
67#41×C251—22no—
68#41×C251—22no—
69C2×Q8×C251C2311yescharacteristic
70C2×Q8×C251C2311yescharacteristic
71C2×D4×C251—22no—
72C2×D4×C251—21no—
73C2×D4×C251D412yes—
74C2×D4×C251C2311yescharacteristic
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251—21no—
78C22×C4×C251D412yes—
79C22×C4×C251C2311yescharacteristic
80C4⋊C4×C251—22no—
81C22⋊C4×C251—22no—
82C22⋊C4×C251—22no—
83C42×C251—21no—
84C42×C251C2311yescharacteristic
85C42×C251C2311yescharacteristic
86C22×D251—24no—
87C22×D251—22no—
88Dic502—24no—
89Dic502—22no—
90Dic502C2×D414yes—
91D1004—24no—
92D1004—22no—
93D1004C2×D414yes—
94C22⋊D251—28no—
95C22⋊D251—24no—
96C22⋊D251—24no—
97C2×Dic251—24no—
98C2×Dic251—22no—
99C4×D251—28no—
100C4×D251—24no—
101C4×D251—24no—
102C251⋊C8—24no—
103C23×C251—21no—
104C23×C251C2×D412yes—
105C23×C251C2×D411yescharacteristic
106Q8×C251—22no—
107Q8×C251—22no—
108D4×C251—24no—
109D4×C251—22no—
110D4×C251—22no—
111D4×C251—44no—
112D4×C251—42no—
113C2×C4×C251—21no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—21no—
117C2×C4×C251—21no—
118C2×C4×C251—42no—
119C2×C4×C251—41no—
120C2×C4×C251—41no—
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2×D412yes—
124C2×C4×C251C2×D411yescharacteristic
125C2×C4×C251C2411yescharacteristic
126C22×C251—22no—
127C22×C251—41no—
128C22×C251—42no—
129C22×C251—42no—
130C22×C251—41no—
131C22×C251#12012yes—
132C22×C251C22×D411yescharacteristic
133C2×D251—24no—
134C2×D251—82no—
135C2×D251—84no—
136C4×C251—22no—
137C4×C251—22no—
138C4×C251—41no—
139C4×C251—41no—
140C4×C251#12011yescharacteristic
141C4×C251#12011yescharacteristic
142C4×C251C22×D412yes—
143Dic251—24no—
144C2×C251—21no—
145C2×C251—22no—
146C2×C251—42no—
147C2×C251—81no—
148C2×C251C2×#12011yescharacteristic
149D251—44no—
150C251#301811yescharacteristic, complemented, Sylow
151#3018—2511nomaximal, Sylow
152#585—2511no—
153#575—2512no—
154#532D25111yescharacteristic, complemented
155#456—2512no—
156#455—2512no—
157#454—2512no—
158#453—2512no—
159C2×#104—2512no—
160#413—2511no—
1612−1+4—5022no—
1622−1+4—2512no—
163H(5,𝔽2)—5022no—
164H(5,𝔽2)—5022no—
165H(5,𝔽2)—2512no—
166C2×#41—5024no—
167C2×#41—5022no—
168C2×#41—2512no—
169C2×#41C2×D25111yescharacteristic
170C2×#41C2×D25111yescharacteristic, complemented
171#137—5022no—
172#137—2514no—
173AGL(1,ℤ/8)—5022no—
174AGL(1,ℤ/8)—2514no—
175#135—5024no—
176C2×M16—2512no—
177C4⋊Q8C2×D25111yescharacteristic, complemented
178#127C2×D25111yescharacteristic, complemented
179#124C2×D25111yescharacteristic, complemented
180#124C2×D25111yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D25111yescharacteristic, complemented
184#104—2514no—
185#104—2514no—
186#101—2512no—
187#100—2512no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—5024no—
193#41—5024no—
194#41—5028no—
195#41—5024no—
196#41—5024no—
197#41—5022no—
198#41—5022no—
199#41—5024no—
200#41—2514no—
201#41—2514no—
202C2×Q8—5022no—
203C2×Q8—5024no—
204C2×Q8—2512no—
205C2×Q8C22×D25111yescharacteristic
206C2×Q8C22×D25111yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—5022no—
210C2×D4—5028no—
211C2×D4—5024no—
212C2×D4—5024no—
213C2×D4—5022no—
214C2×D4—2512no—
215C2×D4C22⋊D25112yes—
216C2×D4C22×D25111yescharacteristic
217C2×D4C22×D25111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—5024no—
221C22×C4—5022no—
222C22×C4C22⋊D25112yes—
223C22×C4C22×D25111yescharacteristic
224Q16—5024no—
225SD16—5024no—
226SD16—5024no—
227D8—5024no—
228M16—5022no—
229M16—2514no—
230C2×C8—5022no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D25111yescharacteristic
236C42C22×D25111yescharacteristic
237C23—21no—
238C23—5024no—
239C23—5022no—
240C23C2×C22⋊D25112yes—
241C23C2×C22⋊D25111yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—5024no—
245Q8—5022no—
246Q8—2514no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—5024no—
253D4—5028no—
254D4—5024no—
255D4—5024no—
256D4—5022no—
257D4—2514no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—5028no—
267C2×C4—5024no—
268C2×C4—5022no—
269C2×C4—5024no—
270C2×C4—5024no—
271C2×C4D4×D25112yes—
272C2×C4D4×D25112yes—
273C2×C4C2×C22⋊D25112yes—
274C2×C4C2×C22⋊D25111yescharacteristic
275C2×C4C23×D25111yescharacteristic
276C8—5024no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—20082no—
283C22—20084no—
284C22—5024no—
285C22#2254868312yes—
286C22C22×C22⋊D25111yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—5024no—
292C4#2254868311yescharacteristic
293C4#2254868311yescharacteristic
294C4C2×D4×D25112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—10044no—
300C2C2×#2254868311yescharacteristic
subgroupG/H
centerC4#22548683
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#532×C251C2
socleC2×C251C2×#120
hypercenter#532D251
hypocenterC251#3018
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. #30632485C2semidirect
3. #30631869C2semidirect
4. #30631864C2semidirect
5. #30631859C2semidirect
6. #30631854C2semidirect
7. C2×#22548643C2semidirect
8. #30631671C2semidirect
9. #532×C251C2semidirect
18. #22548746C22semidirect
20. #22548741C22semidirect
23. C2×#16860794C22semidirect
24. #22548643C22semidirect
25. #22548643C22semidirect
26. #22548634C22semidirect
27. #22548632C22semidirect
29. C2×#41×C251C22semidirect
30. C4⋊Q8×C251C22semidirect
31. #127×C251C22semidirect
32. #124×C251C22semidirect
33. #124×C251C22semidirect
36. C2×C42×C251C22semidirect
43. #16860810D4semidirect
48. C2×Dic502D4semidirect
51. C2×D1004D4semidirect
150. C251#3018semidirect
154. #532D251semidirect
170. C2×#41C2×D251semidirect
177. C4⋊Q8C2×D251semidirect
178. #127C2×D251semidirect
179. #124C2×D251semidirect
180. #124C2×D251semidirect
183. C2×C42C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306324852
#306318692
#306318642
#306318592
#306318542
C2×#225486432
#306316712
#532×C2512
#3018251