Finite Group Databasebeta

← #44559214

Subgroups — #44559214

Order 32128.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D251C211yesmaximal, characteristic
2#30632159C212yesmaximal, complemented
3#30631800C211yesmaximal, characteristic, complemented
4#30631799C211yesmaximal, characteristic, complemented
5#30631798C211yesmaximal, characteristic, complemented
6C4×#16860795C212yesmaximal, complemented
7#30631767C212yesmaximal, complemented
8#30631722C211yesmaximal, characteristic, complemented
9#30631690C212yesmaximal, complemented
10#30631644C211yesmaximal, characteristic, complemented
11#442×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251C2212yes—
13C2×D2008C2211yescharacteristic
14C2×#16860795C2212yes—
15D8×D251C418yescomplemented
16C2×#16860793C2211yescharacteristic
17C2×C8×D251C2211yescharacteristic
18C4×D1004C2212yescomplemented
19D4×Dic251C2212yescomplemented
20#22548677C2212yescomplemented
21C4×C22⋊D251—22no—
22(C22⋊D251)⋊C4—22no—
23#22548673C2212yes—
24#22548672C2211yescharacteristic
25C2×(C2×D251)⋊C4—22no—
26C22⋊C4×D251—22no—
27#22548648C2212yescomplemented
28#22548647C2211yescharacteristic, complemented
29#22548637C2212yescomplemented
30#22548636C2212yescomplemented
31#22548635C2212yescomplemented
32#22548625C2211yescharacteristic, complemented
33C4×C251⋊C8C2211yescharacteristic, complemented
34C8×Dic251C2211yescharacteristic, complemented
35C2×D8×C251C2211yescharacteristic
36C4×D4×C251C2212yescomplemented
37#106×C251C2211yescharacteristic, complemented
38#102×C251C2212yescomplemented
39#97×C251C2211yescharacteristic, complemented
40C23×D251—22no—
41C2×D1004C2312yes—
42C2×C22⋊D251—22no—
43D4×D251—24no—
44D4×D251C2×C418yescomplemented
45C22×Dic251—22no—
46C2×C4×D251—22no—
47C2×C4×D251—22no—
48C2×C4×D251C2311yescharacteristic
49D2008C2×C414yescomplemented
50#16860795C2×C418yescomplemented
51#16860793—22no—
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C8×D251C2×C414yescomplemented
55C4⋊Dic251C2312yes—
56Dic251⋊C4D412yes—
57C22⋊Dic251—22no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4D412yes—
62C4×Dic251—22no—
63C4×Dic251C2311yescharacteristic
64C2×D4×C251C2312yes—
65C22×C4×C251—22no—
66D8×C251C2×C414yescomplemented
67C2×C8×C251C2311yescharacteristic
68C2×C8×C251C2311yescharacteristic
69C4⋊C4×C251C2312yes—
70C22⋊C4×C251—22no—
71C42×C251C2311yescharacteristic
72C22×D251—22no—
73C22×D251—28no—
74C22×D251—82no—
75C22×D251—82no—
76C22×D251C2×D411yescharacteristic
77D1004—22no—
78D1004C22×C414yes—
79C22⋊D251—28no—
80C2×Dic251—22no—
81C2×Dic251—82no—
82C2×Dic251C2×D412yes—
83C2×Dic251C2×D411yescharacteristic
84C4×D251—82no—
85C4×D251—82no—
86C4×D251C22×C412yes—
87C4×D251#4112yes—
88C251⋊C8—21no—
89C251⋊C8C22×C412yes—
90C23×C251—22no—
91D4×C251—22no—
92D4×C251C22×C414yes—
93C2×C4×C251—22no—
94C2×C4×C251—82no—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2411yescharacteristic
97C8×C251—21no—
98C8×C251C22×C412yes—
99C22×C251—24no—
100C22×C251—42no—
101C22×C251—42no—
102C22×C251C22×D411yescharacteristic
103C2×D251—22no—
104C2×D251—24no—
105C2×D251—42no—
106C2×D251—42no—
107C2×D251—88no—
108C2×D251C4×D412yes—
109C4×C251—22no—
110C4×C251—42no—
111C4×C251C23×C411yescharacteristic
112C4×C251C2×#4111yescharacteristic
113Dic251—22no—
114Dic251—42no—
115Dic251C4×D412yes—
116C2×C251—44no—
117C2×C251C2×C4×D411yescharacteristic
118C2×C251#57612yes—
119D251—22no—
120D251—44no—
121C251#294811yescharacteristic, complemented, Sylow
122#2948—2511nomaximal, Sylow
123C22×D8—2511no—
124#518—2512no—
125#442—2511no—
126#442—2511no—
127#442—2511no—
128#442D25111yescharacteristic, complemented
129C4×D8—2512no—
130C4×D8—2512no—
131#427—2511no—
132#418—2512no—
133#405—2511no—
134C22×D4—2512no—
135C2×D8—2512no—
136C2×D8—2518no—
137C2×D8—2511no—
138C2×D8C2×D25111yescharacteristic
139C2×M16—2511no—
140C22×C8—2511no—
141C4×D4—5022no—
142C4×D4—5022no—
143C4×D4—2512no—
144C4×D4—2512no—
145C4×D4—2512no—
146C4×D4C2×D25112yescomplemented
147#117—2512no—
148#117—2511no—
149C2×C22⋊C4—5022no—
150C2×C22⋊C4—5022no—
151#107—2512no—
152#106—2511no—
153#106C2×D25111yescharacteristic, complemented
154#102—2512no—
155#102—2512no—
156#102—2512no—
157#102C2×D25112yescomplemented
158#97—2511no—
159#97C2×D25111yescharacteristic, complemented
160C4×C8—2511no—
161C4×C8—2511no—
162C24—5022no—
163C2×D4—5024no—
164C2×D4—5022no—
165C2×D4—2518no—
166C2×D4—2512no—
167C2×D4C22×D25112yes—
168C22×C4—22no—
169C22×C4—5022no—
170C22×C4—5022no—
171C22×C4—5022no—
172C22×C4—2511no—
173D8—2518no—
174D8—2514no—
175D8C4×D25114yescomplemented
176M16—5022no—
177C2×C8—2511no—
178C2×C8—2511no—
179C2×C8—2514no—
180C2×C8C22×D25111yescharacteristic
181C2×C8C22×D25111yescharacteristic
182C4⋊C4—2512no—
183C4⋊C4—2512no—
184C4⋊C4C22×D25112yes—
185C22⋊C4—22no—
186C22⋊C4—5022no—
187C22⋊C4—5022no—
188C22⋊C4—5022no—
189C22⋊C4—5022no—
190C22⋊C4—2512no—
191C42—5022no—
192C42—2511no—
193C42C22×D25111yescharacteristic
194C23—22no—
195C23—20082no—
196C23—20082no—
197C23—5028no—
198C23—5022no—
199C23—2511no—
200D4—22no—
201D4—5028no—
202D4—5022no—
203D4—2514no—
204D4C2×C4×D25114yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—20082no—
208C2×C4—20082no—
209C2×C4—20082no—
210C2×C4—5022no—
211C2×C4—2512no—
212C2×C4—2511no—
213C2×C4—2512no—
214C2×C4—2512no—
215C2×C4D4×D25112yes—
216C2×C4C23×D25111yescharacteristic
217C8—21no—
218C8—5021no—
219C8—2512no—
220C8C2×C4×D25112yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—20088no—
225C22—10042no—
226C22—10042no—
227C22—5024no—
228C22—5022no—
229C22—2512no—
230C22C2×D4×D25111yescharacteristic
231C4—22no—
232C4—42no—
233C4—10042no—
234C4—5022no—
235C4—2512no—
236C4C22×C4×D25111yescharacteristic
237C4#41×D25111yescharacteristic
238C2—44no—
239C2—10044no—
240C2—5022no—
241C2C4×D4×D25111yescharacteristic
242C2#3063249112yes—
subgroupG/H
centerC22C2×D4×D251
commutatorC4×C251C23×C4
FrattiniC2×C4C23×D251
Fitting#442×C251C2
socleC22×C251C22×D4
prime radicalC2×D8×D251C2
cosocleC2×C4×C251C24
hypercenter#442D251
hypocenterC251#2948
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #30632159C2semidirect
3. #30631800C2semidirect
4. #30631799C2semidirect
5. #30631798C2semidirect
6. C4×#16860795C2semidirect
7. #30631767C2semidirect
8. #30631722C2semidirect
9. #30631690C2semidirect
10. #30631644C2semidirect
11. #442×C251C2semidirect
15. D8×D251C4semidirect
18. C4×D1004C22semidirect
19. D4×Dic251C22semidirect
20. #22548677C22semidirect
27. #22548648C22semidirect
28. #22548647C22semidirect
29. #22548637C22semidirect
30. #22548636C22semidirect
31. #22548635C22semidirect
32. #22548625C22semidirect
33. C4×C251⋊C8C22semidirect
34. C8×Dic251C22semidirect
36. C4×D4×C251C22semidirect
37. #106×C251C22semidirect
38. #102×C251C22semidirect
39. #97×C251C22semidirect
44. D4×D251C2×C4semidirect
49. D2008C2×C4semidirect
50. #16860795C2×C4semidirect
54. C8×D251C2×C4semidirect
66. D8×C251C2×C4semidirect
121. C251#2948semidirect
128. #442D251semidirect
146. C4×D4C2×D251semidirect
153. #106C2×D251semidirect
157. #102C2×D251semidirect
159. #97C2×D251semidirect
175. D8C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D2512
#306321592
#306318002
#306317992
#306317982
C4×#168607952
#306317672
#306317222
#306316902
#306316442
#442×C2512
#2948251