Finite Group Databasebeta

← #44560306

Subgroups — #44560306

Order 32128.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22548689C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3#30632012C214yesmaximal, complemented
4#30632006C212yesmaximal, complemented
5#30632001C212yesmaximal, complemented
6#30631729C212yesmaximal, complemented
7#30631693C212yesmaximal, complemented
8#407×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10C22×C22⋊D251—21no—
11#22548689—21no—
12#22548689—21no—
13#22548689C2212yes—
14#22548689C2212yes—
15#22548674—21no—
16#22548674—21no—
17#22548674C2212yes—
18#22548674C2212yescomplemented
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic, complemented
21C2×C4⋊Dic251C2211yescharacteristic, complemented
22C2×C22⋊Dic251—21no—
23C2×(C2×D251)⋊C4—21no—
24C2×C4×Dic251—21no—
25#22548649C2214yescomplemented
26#22548647C2214yescomplemented
27#22548637C2214yescomplemented
28#22548637C2214yescomplemented
29C23×C4×C251C2211yescharacteristic
30C2×M16×C251C2212yes—
31C22⋊C8×C251C2212yescomplemented
32C22⋊C8×C251C2212yescomplemented
33C23×D251—21no—
34C2×D1004—21no—
35C2×D1004—24no—
36C2×D1004C2312yes—
37C2×C22⋊D251—22no—
38C2×C22⋊D251—22no—
39C2×C22⋊D251—44no—
40C22×Dic251—21no—
41C22×Dic251—21no—
42C22×Dic251—21no—
43C4⋊Dic251—21no—
44C4⋊Dic251—21no—
45C4⋊Dic251—21no—
46C4⋊Dic251C2312yes—
47C4⋊Dic251C2312yes—
48C4⋊Dic251C2312yes—
49C22⋊Dic251—22no—
50C22⋊Dic251—22no—
51(C2×D251)⋊C4—22no—
52(C2×D251)⋊C4—22no—
53C4×Dic251—22no—
54C4×Dic251—22no—
55C24×C251D411yescharacteristic
56C22×C4×C251—24no—
57C22×C4×C251D411yescharacteristic
58C22×C4×C251D411yescharacteristic
59C22×C4×C251D411yescharacteristic
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2311yescharacteristic
62C22×C4×C251C2311yescharacteristic
63M16×C251—24no—
64C2×C8×C251C2314yes—
65C22×D251—22no—
66C22×D251—42no—
67C22×D251—42no—
68C22×D251—42no—
69D1004—24no—
70D1004—42no—
71C22⋊D251—44no—
72C22⋊D251—44no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—42no—
77C2×Dic251—42no—
78C2×Dic251—42no—
79C23×C251—22no—
80C23×C251—22no—
81C23×C251—41no—
82C23×C251C2×D411yescharacteristic
83C23×C251C2×D411yescharacteristic
84C23×C251C2×D411yescharacteristic
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2×D411yescharacteristic
93C2×C4×C251#4114yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—24no—
96C22×C251—21no—
97C22×C251—21no—
98C22×C251—21no—
99C22×C251—21no—
100C22×C251—21no—
101C22×C251—21no—
102C22×C251—21no—
103C22×C251—44no—
104C22×C251#12312yes—
105C22×C251AGL(1,ℤ/8)12yes—
106C22×C251C22×D411yescharacteristic
107C2×D251—42no—
108C2×D251—42no—
109C2×D251—42no—
110C2×D251—84no—
111C4×C251—21no—
112C4×C251—21no—
113C4×C251#12312yes—
114C4×C251C2×#4112yes—
115Dic251—42no—
116Dic251—42no—
117Dic251—42no—
118C2×C251—22no—
119C2×C251—22no—
120C2×C251—41no—
121C2×C251C2×#12311yescharacteristic
122C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C251#57511yescharacteristic
124D251—82no—
125C251#309411yescharacteristic, complemented, Sylow
126#3094—2511nomaximal, Sylow
127C2×#121—2511no—
128C2×#117—2511no—
129#482—2514no—
130#481—2512no—
131#480—2512no—
132#428—2512no—
133#418—2512no—
134#407D25111yescharacteristic, complemented
135C22×D4—5021no—
136C22×D4—2511no—
137C23×C4C2×D25111yescharacteristic
138C2×M16C2×D25112yes—
139#121—5021no—
140#121—5021no—
141#121—2512no—
142#121—2512no—
143#117—5021no—
144#117—5021no—
145#117—2512no—
146#117—2512no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4—2511no—
149C2×C4⋊C4—2511no—
150C2×C22⋊C4—5021no—
151C2×C22⋊C4—5021no—
152C2×C42—5021no—
153#107—2514no—
154#106—2514no—
155#102—2514no—
156#102—2514no—
157C22⋊C8C2×D25112yescomplemented
158C22⋊C8C2×D25112yescomplemented
159C24—5021no—
160C24D100411yescharacteristic
161C2×D4—10044no—
162C2×D4—5022no—
163C2×D4—5022no—
164C2×D4—5021no—
165C2×D4—5024no—
166C2×D4—2512no—
167C22×C4—24no—
168C22×C4—5021no—
169C22×C4—5021no—
170C22×C4—5021no—
171C22×C4D100411yescharacteristic
172C22×C4D100411yescharacteristic
173C22×C4D100411yescharacteristic
174C22×C4C22×D25111yescharacteristic
175C22×C4C22×D25111yescharacteristic
176C22×C4C22×D25111yescharacteristic
177M16—24no—
178C2×C8C22×D25114yes—
179C4⋊C4—5021no—
180C4⋊C4—5021no—
181C4⋊C4—5021no—
182C4⋊C4—2512no—
183C4⋊C4—2512no—
184C4⋊C4—2512no—
185C22⋊C4—5022no—
186C22⋊C4—5022no—
187C22⋊C4—5022no—
188C22⋊C4—5022no—
189C42—5022no—
190C42—5022no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—10042no—
195C23—10042no—
196C23—10042no—
197C23—5022no—
198C23C2×D100411yescharacteristic
199C23C2×D100411yescharacteristic
200C23C2×D100411yescharacteristic
201D4—10044no—
202D4—10044no—
203D4—10042no—
204D4—5024no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—10042no—
211C2×C4—10042no—
212C2×C4—10042no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4C2×D100411yescharacteristic
217C2×C4C2×D100411yescharacteristic
218C2×C4C2×D100411yescharacteristic
219C2×C4#1686080914yes—
220C2×C4C23×D25111yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—20084no—
231C22—10042no—
232C22—10042no—
233C22—10042no—
234C22#2254869912yes—
235C22#2254874312yes—
236C22C22×D100411yescharacteristic
237C4—21no—
238C4—21no—
239C4—10042no—
240C4—10042no—
241C4—10042no—
242C4#2254869912yes—
243C4C2×#1686080912yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—20082no—
248C2C2×#2254869911yescharacteristic
249C2C2×#2254874311yescharacteristic
250C2#3063248711yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548689C2
hypercenter#407D251
hypocenterC251#3094
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
3. #30632012C2semidirect
4. #30632006C2semidirect
5. #30632001C2semidirect
6. #30631729C2semidirect
7. #30631693C2semidirect
8. #407×C251C2semidirect
18. #22548674C22semidirect
20. C2×C4⋊Dic251C22semidirect
21. C2×C4⋊Dic251C22semidirect
25. #22548649C22semidirect
26. #22548647C22semidirect
27. #22548637C22semidirect
28. #22548637C22semidirect
31. C22⋊C8×C251C22semidirect
32. C22⋊C8×C251C22semidirect
125. C251#3094semidirect
134. #407D251semidirect
157. C22⋊C8C2×D251semidirect
158. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
C2×#225486742
#306320122
#306320062
#306320012
#306317292
#306316932
#407×C2512
#3094251