Finite Group Databasebeta

← #44560814

Subgroups — #44560814

Order 32128.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548742C212yesmaximal, complemented
2C2×#22548739C212yesmaximal, complemented
3#30632236C211yesmaximal, characteristic, complemented
4#127×D251C211yesmaximal, characteristic, complemented
5#30632077C211yesmaximal, characteristic, complemented
6#30632076C211yesmaximal, characteristic, complemented
7#30632074C211yesmaximal, characteristic, complemented
8#30632062C212yesmaximal, complemented
9#30632058C212yesmaximal, complemented
10#30631639C211yesmaximal, characteristic, complemented
11#496×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C2×#16860809C2211yescharacteristic
14C2×D4×D251—22no—
15C2×D4×D251C2212yes—
16#22548742—28no—
17#22548739—28no—
18C2×#16860799C2212yescomplemented
19C2×#16860798C2212yescomplemented
20C2×#16860797C2212yescomplemented
21C2×D2008C2212yescomplemented
22C2×#16860795C2212yescomplemented
23C2×#16860795C2212yescomplemented
24C2×#16860793C2212yes—
25#22548713C2211yescharacteristic, complemented
26#22548712C2211yescharacteristic
27#22548711—22no—
28#22548711C2212yes—
29#22548711C2211yescharacteristic, complemented
30#22548703C2211yescharacteristic, complemented
31#22548688—22no—
32(C22⋊D251)⋊C4—22no—
33C42×D251C2211yescharacteristic
34#22548626C2211yescharacteristic, complemented
35C8×Dic251C2212yescomplemented
36C2×SD16×C251C2212yescomplemented
37C2×D8×C251C2212yescomplemented
38#127×C251C2211yescharacteristic
39#124×C251C2211yescharacteristic, complemented
40#97×C251C2211yescharacteristic, complemented
41C23×D251—22no—
42#16860811—24no—
43#16860809—24no—
44C2×Dic502C2311yescharacteristic
45C2×D1004—21no—
46C2×D1004—22no—
47C2×D1004C2312yes—
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—21no—
50C2×C22⋊D251—24no—
51C2×C22⋊D251—22no—
52C2×C22⋊D251—22no—
53D4×D251—24no—
54D4×D251—24no—
55D4×D251—48no—
56C22×Dic251—21no—
57C2×C4×D251—21no—
58C2×C4×D251D412yes—
59C2×C4×D251C2311yescharacteristic
60#16860799—24no—
61#16860798—24no—
62#16860797—24no—
63D2008—24no—
64#16860795—24no—
65#16860795—24no—
66#16860793D418yescomplemented
67C2×C251⋊C8C2312yes—
68Dic251⋊C4—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251D412yes—
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251C2311yescharacteristic
73C2×D4×C251—22no—
74C2×D4×C251C2312yes—
75C2×D4×C251C2311yescharacteristic
76SD16×C251—24no—
77D8×C251—24no—
78C2×C8×C251C2312yes—
79C22⋊C4×C251—22no—
80C42×C251C2311yescharacteristic
81C22×D251—21no—
82C22×D251—22no—
83C22×D251—48no—
84C22×D251—44no—
85C22×D251C2×D411yescharacteristic
86Dic502—22no—
87D1004—22no—
88D1004—22no—
89D1004—22no—
90D1004—44no—
91D1004—42no—
92C22⋊D251—42no—
93C22⋊D251—44no—
94C22⋊D251—48no—
95C22⋊D251—44no—
96C2×Dic251—21no—
97C2×Dic251—42no—
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D412yes—
100C4×D251—24no—
101C4×D251—42no—
102C4×D251C2×D414yes—
103C251⋊C8C2×D414yescomplemented
104C23×C251—21no—
105C23×C251—22no—
106Q8×C251—22no—
107D4×C251—22no—
108D4×C251—22no—
109D4×C251—22no—
110D4×C251—44no—
111C2×C4×C251—21no—
112C2×C4×C251C2×D412yes—
113C2×C4×C251C2411yescharacteristic
114C8×C251C2×D414yescomplemented
115C22×C251—41no—
116C22×C251—42no—
117C22×C251—44no—
118C22×C251—42no—
119C22×C251C22×D411yescharacteristic
120C2×D251—22no—
121C2×D251—41no—
122C2×D251—42no—
123C2×D251—44no—
124C2×D251—42no—
125C2×D251—88no—
126C2×D251#12712yes—
127C4×C251—22no—
128C4×C251—41no—
129C4×C251C22×D412yes—
130Dic251—41no—
131Dic251#12712yes—
132Dic251AGL(1,ℤ/8)14yescomplemented
133C2×C251—82no—
134C2×C251—81no—
135C2×C251C2×#12711yescharacteristic
136C2×C251C2×AGL(1,ℤ/8)12yes—
137D251—22no—
138D251—81no—
139D251—82no—
140C251#315111yescharacteristic, complemented, Sylow
141#3151—2511nomaximal, Sylow
142C2×AGL(1,ℤ/8)—2512no—
143C2×AGL(1,ℤ/8)—2512no—
144#532—2511no—
145C2×#127—2511no—
146#496—2511no—
147#496—2511no—
148#496—2511no—
149#496D25111yescharacteristic, complemented
150#493—2512no—
151#492—2512no—
152C4×M16—2511no—
153C2×#41—2511no—
154C2×#41—2511no—
155C22×D4—5022no—
156C22×D4—2512no—
157AGL(1,ℤ/8)—5028no—
158AGL(1,ℤ/8)—5028no—
159C2×SD16—2512no—
160C2×SD16—2512no—
161C2×SD16—2512no—
162C2×SD16C2×D25112yescomplemented
163C2×D8—2512no—
164C2×D8—2512no—
165C2×D8—2512no—
166C2×D8C2×D25112yescomplemented
167C2×M16—2512no—
168C4⋊Q8—2511no—
169#127—5022no—
170#127—2512no—
171#127—2511no—
172#127—2511no—
173#127C2×D25111yescharacteristic
174#124—2511no—
175#124C2×D25111yescharacteristic, complemented
176#121—5022no—
177C4×D4—5022no—
178C2×C42—2511no—
179#97—2511no—
180#97C2×D25111yescharacteristic, complemented
181C4×C8—2512no—
182C24—5022no—
183#41—5024no—
184#41—5024no—
185C2×Q8—2511no—
186C2×Q8C22×D25111yescharacteristic
187C2×D4—22no—
188C2×D4—10048no—
189C2×D4—5021no—
190C2×D4—5021no—
191C2×D4—5024no—
192C2×D4—5024no—
193C2×D4—5022no—
194C2×D4—5022no—
195C2×D4—5024no—
196C2×D4—5022no—
197C2×D4—2511no—
198C2×D4—2512no—
199C2×D4C22×D25111yescharacteristic
200C2×D4C22×D25112yes—
201C22×C4—5021no—
202C22×C4—5021no—
203C22×C4—2512no—
204C22×C4—2511no—
205SD16—24no—
206SD16—5024no—
207SD16—5024no—
208SD16—5024no—
209D8—24no—
210D8—5024no—
211D8—5024no—
212D8—5024no—
213M16—2518no—
214C2×C8—2512no—
215C2×C8C22×D25112yes—
216C4⋊C4—5022no—
217C22⋊C4—22no—
218C22⋊C4—5022no—
219C42—2512no—
220C42—2511no—
221C42C22×D25111yescharacteristic
222C23—21no—
223C23—22no—
224C23—10048no—
225C23—10044no—
226C23—5021no—
227C23—5022no—
228C23—2511no—
229Q8—22no—
230Q8—5022no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—10044no—
236D4—10042no—
237D4—10042no—
238D4—10048no—
239D4—10044no—
240D4—10044no—
241D4—5022no—
242D4—5022no—
243D4—5022no—
244C2×C4—21no—
245C2×C4—10042no—
246C2×C4—10042no—
247C2×C4—5024no—
248C2×C4—5021no—
249C2×C4—2512no—
250C2×C4—2511no—
251C2×C4—2514no—
252C2×C4D4×D25112yes—
253C2×C4C23×D25111yescharacteristic
254C8—2514no—
255C8D4×D25114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—20088no—
261C22—10044no—
262C22—10042no—
263C22—10041no—
264C22—10042no—
265C22—5022no—
266C22—2512no—
267C22C2×D4×D25111yescharacteristic
268C4—22no—
269C4—41no—
270C4—10041no—
271C4—2514no—
272C4—2512no—
273C4C2×D4×D25112yes—
274C2—82no—
275C2—81no—
276C2—20082no—
277C2—20081no—
278C2—5022no—
279C2#127×D25111yescharacteristic
280C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#496×C251C2
socleC22×C251C22×D4
hypercenter#496D251
hypocenterC251#3151
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. C2×#22548739C2semidirect
3. #30632236C2semidirect
4. #127×D251C2semidirect
5. #30632077C2semidirect
6. #30632076C2semidirect
7. #30632074C2semidirect
8. #30632062C2semidirect
9. #30632058C2semidirect
10. #30631639C2semidirect
11. #496×C251C2semidirect
18. C2×#16860799C22semidirect
19. C2×#16860798C22semidirect
20. C2×#16860797C22semidirect
21. C2×D2008C22semidirect
22. C2×#16860795C22semidirect
23. C2×#16860795C22semidirect
25. #22548713C22semidirect
29. #22548711C22semidirect
30. #22548703C22semidirect
34. #22548626C22semidirect
35. C8×Dic251C22semidirect
36. C2×SD16×C251C22semidirect
37. C2×D8×C251C22semidirect
39. #124×C251C22semidirect
40. #97×C251C22semidirect
66. #16860793D4semidirect
103. C251⋊C8C2×D4semidirect
114. C8×C251C2×D4semidirect
132. Dic251AGL(1,ℤ/8)semidirect
140. C251#3151semidirect
149. #496D251semidirect
162. C2×SD16C2×D251semidirect
166. C2×D8C2×D251semidirect
175. #124C2×D251semidirect
180. #97C2×D251semidirect
255. C8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
C2×#225487392
#306322362
#127×D2512
#306320772
#306320762
#306320742
#306320622
#306320582
#306316392
#496×C2512
#3151251