Finite Group Databasebeta

← #44558824

Subgroups — #44558824

Order 32128.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548761C211yesmaximal, characteristic
2#30632161C212yesmaximal, complemented
3#30631661C211yesmaximal, characteristic, complemented
4#30631660C212yesmaximal, complemented
5#99×D251C218yesmaximal, complemented
6#410×C251C211yesmaximal, characteristic, complemented
7#22548761—24no—
8#22548761—24no—
9C2×#16860811C412yescomplemented
10C2×#16860810C412yescomplemented
11C2×#16860810C2211yescharacteristic
12C22×D1004C412yescomplemented
13C22×D1004C2211yescharacteristic
14C2×D4×D251C412yescomplemented
15C2×D4×D251C2214yes—
16C4×D1004—24no—
17#22548677—24no—
18#22548674C2212yescomplemented
19C22⋊C4×D251C2218yescomplemented
20#22548630C2214yescomplemented
21#22548629C2218yescomplemented
22C2×#41×C251C2211yescharacteristic
23#117×C251C2212yescomplemented
24#99×C251C2214yescomplemented
25C23×D251C2×C414yescomplemented
26C23×D251C2312yes—
27#16860811—28no—
28#16860810—24no—
29#16860810—24no—
30#16860810—22no—
31#16860810—22no—
32C2×Dic502C2×C411yescharacteristic, complemented
33C2×D1004—24no—
34C2×D1004—24no—
35C2×D1004C2×C412yes—
36C2×D1004C2×C412yes—
37C2×D1004C2×C414yescomplemented
38C2×D1004C2×C411yescharacteristic, complemented
39C2×D1004D414yes—
40C2×D1004D414yes—
41C2×C22⋊D251C2×C414yescomplemented
42C2×C22⋊D251C2312yes—
43D4×D251—216no—
44D4×D251—28no—
45C2×C4×D251—28no—
46C2×C4×D251C2×C412yescomplemented
47C2×C4×D251C2×C414yescomplemented
48C4⋊Dic251—22no—
49C22⋊Dic251C2314yes—
50(C2×D251)⋊C4—28no—
51C4×Dic251—22no—
52#41×C251—22no—
53#41×C251—22no—
54C2×Q8×C251C2×C411yescharacteristic, complemented
55C2×D4×C251C2×C411yescharacteristic, complemented
56C2×D4×C251C2312yes—
57C22×C4×C251C2×C412yescomplemented
58C22×C4×C251C2311yescharacteristic
59C4⋊C4×C251—22no—
60C22⋊C4×C251C2314yes—
61C42×C251—22no—
62C22×D251—28no—
63C22×D251—82no—
64C22×D251—84no—
65C22×D251C22×C412yes—
66C22×D251C22×C414yes—
67C22×D251C2×D418yes—
68Dic502—21no—
69Dic502—21no—
70D1004—28no—
71D1004—21no—
72D1004—21no—
73D1004C22⋊C418yes—
74D1004C22⋊C414yes—
75D1004C22⋊C414yes—
76C22⋊D251—24no—
77C22⋊D251—28no—
78C2×Dic251—24no—
79C2×Dic251C22×C412yes—
80C4×D251—28no—
81C4×D251—24no—
82C4×D251—88no—
83C23×C251C22×C412yes—
84C23×C251C2411yescharacteristic
85Q8×C251—22no—
86D4×C251—24no—
87D4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—24no—
91C2×C4×C251C22×C412yes—
92C2×C4×C251C22×C411yescharacteristic
93C2×C4×C251C22×C411yescharacteristic
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D412yes—
96C22×C251—22no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251C23×C411yescharacteristic
100C22×C251C22×D412yes—
101C2×D251—22no—
102C2×D251—48no—
103C2×D251—44no—
104C2×D251—42no—
105C2×D251—88no—
106C2×D251C2×C22⋊C418yes—
107C4×C251—22no—
108C4×C251—44no—
109C4×C251C2×C22⋊C411yescharacteristic
110C4×C251C2×C22⋊C411yescharacteristic
111C4×C251C2×C22⋊C412yes—
112Dic251—22no—
113Dic251—44no—
114C2×C251—22no—
115C2×C251—21no—
116C2×C251—42no—
117C2×C251C22×C22⋊C411yescharacteristic
118D251—28no—
119D251—42no—
120C251#288311yescharacteristic, complemented, Sylow
121#2883—2511nomaximal, Sylow
122C2×H(5,𝔽2)—2511no—
123#518—2512no—
124#410—2511no—
125#410—2512no—
126#410D25111yescharacteristic, complemented
127C2×#99—2518no—
128H(5,𝔽2)—5024no—
129H(5,𝔽2)—5024no—
130C2×#41—2512no—
131C2×#41—2512no—
132C2×#41—2511no—
133C2×#41C2×D25111yescharacteristic
134C22×D4—2512no—
135C22×D4—2514no—
136C22×D4—2512no—
137C22×D4—2511no—
138C4×D4—5024no—
139C4×D4—5024no—
140#117—2512no—
141#117C2×D25112yescomplemented
142C2×C22⋊C4—2518no—
143#99—2514no—
144#99—2518no—
145#99C2×D25114yescomplemented
146C24—2514no—
147C24—2512no—
148#41—22no—
149#41—22no—
150#41—5024no—
151#41—5024no—
152#41—5028no—
153#41—5022no—
154#41—5022no—
155C2×Q8—2511no—
156C2×Q8C4×D25111yescharacteristic, complemented
157C2×D4—5024no—
158C2×D4—5024no—
159C2×D4—50216no—
160C2×D4—5028no—
161C2×D4—2514no—
162C2×D4—2512no—
163C2×D4—2514no—
164C2×D4—2511no—
165C2×D4—2512no—
166C2×D4—2512no—
167C2×D4—2514no—
168C2×D4—2514no—
169C2×D4C4×D25111yescharacteristic, complemented
170C2×D4C22×D25112yes—
171C22×C4—5028no—
172C22×C4—2514no—
173C22×C4—2512no—
174C22×C4C4×D25112yescomplemented
175C22×C4C22×D25111yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—5022no—
178C22⋊C4—5028no—
179C22⋊C4—2514no—
180C22⋊C4C22×D25114yes—
181C42—22no—
182C42—5022no—
183C23—20084no—
184C23—20082no—
185C23—5028no—
186C23—2512no—
187C23—2514no—
188C23—2518no—
189C23C2×C4×D25112yes—
190C23C23×D25111yescharacteristic
191Q8—22no—
192Q8—5021no—
193Q8—5021no—
194D4—24no—
195D4—22no—
196D4—5028no—
197D4—5028no—
198D4—5024no—
199D4—5021no—
200D4—5021no—
201D4—2514no—
202D4—2514no—
203D4—2518no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—20088no—
208C2×C4—5028no—
209C2×C4—5024no—
210C2×C4—5024no—
211C2×C4—2512no—
212C2×C4C2×C4×D25112yes—
213C2×C4C2×C4×D25111yescharacteristic
214C2×C4C2×C4×D25111yescharacteristic
215C2×C4D4×D25112yes—
216C2×C4D4×D25112yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—20088no—
221C22—10042no—
222C22—10048no—
223C22—10044no—
224C22—5022no—
225C22—2518no—
226C22C22×C4×D25111yescharacteristic
227C22C2×D4×D25112yes—
228C4—22no—
229C4—44no—
230C4—10044no—
231C4—5022no—
232C4C22⋊C4×D25111yescharacteristic
233C4C22⋊C4×D25111yescharacteristic
234C4C22⋊C4×D25112yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—10042no—
239C2—5028no—
240C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC23C23×D251
Fitting#410×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548761C2
cosocleC23×C251C24
hypercenter#410D251
hypocenterC251#2883
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30632161C2semidirect
3. #30631661C2semidirect
4. #30631660C2semidirect
5. #99×D251C2semidirect
6. #410×C251C2semidirect
9. C2×#16860811C4semidirect
10. C2×#16860810C4semidirect
12. C22×D1004C4semidirect
14. C2×D4×D251C4semidirect
18. #22548674C22semidirect
19. C22⋊C4×D251C22semidirect
20. #22548630C22semidirect
21. #22548629C22semidirect
23. #117×C251C22semidirect
24. #99×C251C22semidirect
25. C23×D251C2×C4semidirect
32. C2×Dic502C2×C4semidirect
37. C2×D1004C2×C4semidirect
38. C2×D1004C2×C4semidirect
41. C2×C22⋊D251C2×C4semidirect
46. C2×C4×D251C2×C4semidirect
47. C2×C4×D251C2×C4semidirect
54. C2×Q8×C251C2×C4semidirect
55. C2×D4×C251C2×C4semidirect
57. C22×C4×C251C2×C4semidirect
120. C251#2883semidirect
126. #410D251semidirect
141. #117C2×D251semidirect
145. #99C2×D251semidirect
156. C2×Q8C4×D251semidirect
169. C2×D4C4×D251semidirect
174. C22×C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306321612
#306316612
#306316602
#99×D2512
#410×C2512
#2883251