Finite Group Databasebeta

← #44559768

Subgroups — #44559768

Order 32128.

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

normal subgroups only

This group has 772 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#30632256C212yesmaximal, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30631875C218yesmaximal, complemented
5#30631660C212yesmaximal, complemented
6#410×C251C211yesmaximal, characteristic, complemented
7#22548763—21no—
8#22548762—22no—
9#22548761—22no—
10#22548760—21no—
11#22548760C2214yes—
12C2×#16860810—22no—
13C2×#16860810C2212yes—
14#41×D251—24no—
15C22×D1004C2211yescharacteristic, complemented
16#22548712—22no—
17#22548711—21no—
18#22548704—21no—
19#22548703—22no—
20#22548690—24no—
21#22548689—22no—
22#22548685C2218yescomplemented
23#22548683C2214yescomplemented
24#22548674C2211yescharacteristic, complemented
25#22548629C2218yescomplemented
26C2×#41×C251C2211yescharacteristic
27#117×C251C2212yescomplemented
28#99×C251C2214yescomplemented
29C23×D251C2312yes—
30#16860811—24no—
31#16860810—24no—
32#16860810—24no—
33#16860810—22no—
34#16860810—22no—
35#16860810—24no—
36#16860810—22no—
37#16860810—22no—
38#16860809—24no—
39#16860809—28no—
40C2×Dic502—22no—
41C2×Dic502D412yescomplemented
42Q8×D251—24no—
43C2×D1004—22no—
44C2×D1004—28no—
45C2×D1004—21no—
46C2×D1004—22no—
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004D412yescomplemented
50C2×C22⋊D251—24no—
51C2×C22⋊D251—24no—
52C2×C22⋊D251C2314yes—
53D4×D251—24no—
54D4×D251—28no—
55C2×C4×D251—24no—
56C2×C4×D251D414yescomplemented
57C4⋊Dic251—21no—
58C22⋊Dic251C2312yes—
59(C2×D251)⋊C4—28no—
60(C2×D251)⋊C4—24no—
61C4×Dic251—21no—
62#41×C251—22no—
63#41×C251—22no—
64C2×Q8×C251D411yescharacteristic, complemented
65C2×D4×C251D411yescharacteristic, complemented
66C2×D4×C251C2312yes—
67C22×C4×C251D412yescomplemented
68C22×C4×C251C2311yescharacteristic
69C4⋊C4×C251—22no—
70C22⋊C4×C251C2314yes—
71C42×C251—22no—
72C22×D251—24no—
73C22×D251—22no—
74C22×D251—24no—
75C22×D251—82no—
76C22×D251C2×D414yes—
77Dic502—24no—
78Dic502—22no—
79Dic502—22no—
80D1004—24no—
81D1004—22no—
82D1004—22no—
83D1004—41no—
84D1004—41no—
85D1004—42no—
86D1004—88no—
87C22⋊D251—28no—
88C22⋊D251—24no—
89C22⋊D251—24no—
90C22⋊D251—84no—
91C2×Dic251—24no—
92C2×Dic251—22no—
93C2×Dic251C2×D414yes—
94C4×D251—28no—
95C4×D251—24no—
96C4×D251—24no—
97C23×C251C2×D412yes—
98C23×C251C2411yescharacteristic
99Q8×C251—22no—
100D4×C251—24no—
101D4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—24no—
105C2×C4×C251C2×D412yes—
106C2×C4×C251C2×D412yes—
107C2×C4×C251C2×D412yes—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×D411yescharacteristic
110C22×C251—22no—
111C22×C251—41no—
112C22×C251—42no—
113C22×C251C22×D412yes—
114C22×C251C22×D411yescharacteristic
115C2×D251—24no—
116C2×D251—44no—
117C2×D251—44no—
118C2×D251—48no—
119C2×D251—42no—
120C2×D251—84no—
121C4×C251—22no—
122C4×C251—44no—
123C4×C251#12011yescharacteristic
124C4×C251#12011yescharacteristic
125C4×C251#12012yes—
126Dic251—24no—
127Dic251—42no—
128C2×C251—21no—
129C2×C251—22no—
130C2×C251—42no—
131C2×C251C2×#12011yescharacteristic
132D251—44no—
133D251—82no—
134C251#302711yescharacteristic, complemented, Sylow
135#3027—2511nomaximal, Sylow
136#585—2511no—
137#535—2511no—
138#535—2512no—
139B(4,𝔽2)—2518no—
140#410—2512no—
141#410D25111yescharacteristic, complemented
1422−1+4—5022no—
1432−1+4—5021no—
144H(5,𝔽2)—5022no—
145H(5,𝔽2)—5021no—
146H(5,𝔽2)—2514no—
147C2×#41—5024no—
148C2×#41—5022no—
149C2×#41—2512no—
150C2×#41C2×D25111yescharacteristic
151C22×D4—2511no—
152#127—5022no—
153#127—5021no—
154#124—5022no—
155#124—5021no—
156#121—5024no—
157#121—5022no—
158#120—2514no—
159#120—2518no—
160#117—2511no—
161#117C2×D25112yescomplemented
162#99—2518no—
163#99C2×D25114yescomplemented
164C24—2512no—
165#41—22no—
166#41—22no—
167#41—5024no—
168#41—5024no—
169#41—5024no—
170#41—5024no—
171#41—5028no—
172#41—5022no—
173#41—5022no—
174#41—5022no—
175#41—5024no—
176#41—5022no—
177C2×Q8—5022no—
178C2×Q8—5024no—
179C2×Q8—2512no—
180C2×Q8D100411yescharacteristic, complemented
181C2×D4—5028no—
182C2×D4—5022no—
183C2×D4—5024no—
184C2×D4—5028no—
185C2×D4—5024no—
186C2×D4—5024no—
187C2×D4—5022no—
188C2×D4—5022no—
189C2×D4—5021no—
190C2×D4—5021no—
191C2×D4—2514no—
192C2×D4—2512no—
193C2×D4D100411yescharacteristic, complemented
194C2×D4C22×D25112yes—
195C22×C4—5024no—
196C22×C4—2514no—
197C22×C4D100412yescomplemented
198C22×C4C22×D25111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—5021no—
201C22⋊C4—5028no—
202C22⋊C4—5024no—
203C22⋊C4—2512no—
204C22⋊C4C22×D25114yes—
205C42—22no—
206C42—5021no—
207C23—20082no—
208C23—5024no—
209C23—5024no—
210C23—5022no—
211C23—2514no—
212C23C2×D100412yes—
213C23C23×D25111yescharacteristic
214Q8—22no—
215Q8—5024no—
216Q8—5022no—
217Q8—5022no—
218D4—24no—
219D4—22no—
220D4—20088no—
221D4—20084no—
222D4—10041no—
223D4—10041no—
224D4—10042no—
225D4—5024no—
226D4—5028no—
227D4—5024no—
228D4—5024no—
229D4—5022no—
230D4—5022no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—5028no—
235C2×C4—5022no—
236C2×C4—5024no—
237C2×C4—5024no—
238C2×C4—5024no—
239C2×C4—2514no—
240C2×C4D4×D25112yes—
241C2×C4D4×D25112yes—
242C2×C4C2×D100412yes—
243C2×C4C2×D100411yescharacteristic
244C2×C4C2×D100411yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—20084no—
249C22—10048no—
250C22—10044no—
251C22—10044no—
252C22—10042no—
253C22—5024no—
254C22C2×D4×D25112yes—
255C22C22×D100411yescharacteristic
256C4—22no—
257C4—44no—
258C4—10042no—
259C4—5024no—
260C4#2254868511yescharacteristic
261C4#2254868511yescharacteristic
262C4#2254868512yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—20082no—
267C2—10044no—
268C2C2×#2254868511yescharacteristic
subgroupG/H
centerC2C2×#22548685
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
Fitting#410×C251C2
socleC2×C251C2×#120
hypercenter#410D251
hypocenterC251#3027
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. #30632256C2semidirect
3. #30632251C2semidirect
4. #30631875C2semidirect
5. #30631660C2semidirect
6. #410×C251C2semidirect
15. C22×D1004C22semidirect
22. #22548685C22semidirect
23. #22548683C22semidirect
24. #22548674C22semidirect
25. #22548629C22semidirect
27. #117×C251C22semidirect
28. #99×C251C22semidirect
41. C2×Dic502D4semidirect
49. C2×D1004D4semidirect
56. C2×C4×D251D4semidirect
64. C2×Q8×C251D4semidirect
65. C2×D4×C251D4semidirect
67. C22×C4×C251D4semidirect
134. C251#3027semidirect
141. #410D251semidirect
161. #117C2×D251semidirect
163. #99C2×D251semidirect
180. C2×Q8D1004semidirect
193. C2×D4D1004semidirect
197. C22×C4D1004semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306322562
#306322512
#306318752
#306316602
#410×C2512
#3027251