Finite Group Databasebeta

← #44560737

Subgroups — #44560737

Order 32128.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632240C211yesmaximal, characteristic, complemented
2C2×#22548711C211yesmaximal, characteristic
3#30632050C214yesmaximal, complemented
4#30632032C214yesmaximal, complemented
5#30631693C211yesmaximal, characteristic, complemented
6#30631692C211yesmaximal, characteristic, complemented
7#30631691C211yesmaximal, characteristic, complemented
8M16×Dic251C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251—22no—
12C2×#16860794C2211yescharacteristic
13#22548714C2212yescomplemented
14#22548711—22no—
15#22548711C2212yes—
16#22548711C2212yes—
17#22548706C2212yescomplemented
18#22548692—22no—
19C4×Dic502—22no—
20#22548674C2211yescharacteristic
21C2×C4×Dic251C2211yescharacteristic
22#22548637C2214yescomplemented
23#22548636C2214yescomplemented
24#22548635C2214yescomplemented
25#22548625C2212yescomplemented
26C8×Dic251C2212yescomplemented
27C22×D4×C251C2211yescharacteristic
28C2×M16×C251C2211yescharacteristic
29#117×C251C2211yescharacteristic
30#102×C251C2214yescomplemented
31C23×D251—21no—
32C2×Dic502—22no—
33C2×D1004—21no—
34C2×D1004—22no—
35C2×D1004—22no—
36C2×D1004C2312yes—
37C2×C22⋊D251—24no—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—48no—
40C22×Dic251D412yes—
41#16860794—22no—
42C2×C251⋊C8C2312yes—
43C4⋊Dic251—22no—
44C4⋊Dic251C2312yes—
45Dic251⋊C4—22no—
46Dic251⋊C4—22no—
47C22⋊Dic251—21no—
48C4×Dic251—21no—
49C4×Dic251D412yes—
50C4×Dic251C2311yescharacteristic
51C4×Dic251C2311yescharacteristic
52C24×C251—21no—
53C2×D4×C251—21no—
54C2×D4×C251—22no—
55C2×D4×C251—22no—
56C2×D4×C251C2312yes—
57C22×C4×C251C2311yescharacteristic
58M16×C251—22no—
59C2×C8×C251C2312yes—
60C4⋊C4×C251C2312yes—
61C22⋊C4×C251—21no—
62C42×C251—21no—
63C22×D251—22no—
64C22×D251—44no—
65C22×D251—42no—
66Dic502—44no—
67D1004—22no—
68D1004—22no—
69D1004—42no—
70C22⋊D251—48no—
71C22⋊D251—48no—
72C2×Dic251—24no—
73C2×Dic251—22no—
74C2×Dic251C2×D412yes—
75C2×Dic251C2×D412yes—
76C251⋊C8—22no—
77C23×C251—22no—
78C23×C251—44no—
79C23×C251—42no—
80C23×C251C2×D411yescharacteristic
81D4×C251—22no—
82D4×C251—22no—
83D4×C251—42no—
84C2×C4×C251—22no—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251#4112yes—
87C2×C4×C251#4112yes—
88C2×C4×C251C2411yescharacteristic
89C8×C251—22no—
90C22×C251—22no—
91C22×C251—42no—
92C22×C251—44no—
93C22×C251—84no—
94C22×C251#12412yes—
95C22×C251C22×D411yescharacteristic
96C2×D251—42no—
97C2×D251—44no—
98C2×D251—84no—
99C4×C251—42no—
100C4×C251#12412yes—
101C4×C251C2×#4111yescharacteristic
102C4×C251C2×#4111yescharacteristic
103Dic251—22no—
104Dic251—42no—
105Dic251AGL(1,ℤ/8)14yes—
106C2×C251—22no—
107C2×C251—82no—
108C2×C251C2×#12411yescharacteristic
109C2×C251C2×AGL(1,ℤ/8)12yes—
110D251—82no—
111C251#313811yescharacteristic, complemented, Sylow
112#3138—2511nomaximal, Sylow
113#533—2511no—
114C2×#127—2511no—
115#490—2514no—
116#486—2514no—
117#418—2511no—
118#418—2511no—
119#418—2511no—
120#418D25111yescharacteristic, complemented
121C4×M16—2511no—
122C22×D4—5022no—
123C22×D4—2511no—
124C22×D4C2×D25111yescharacteristic
125C2×M16—2511no—
126C2×M16C2×D25111yescharacteristic
127C4⋊Q8—2512no—
128#127—5022no—
129#127—2512no—
130#127—2512no—
131#125—2512no—
132C22⋊Q8—5022no—
133C4×Q8—5022no—
134#117—2511no—
135#117C2×D25111yescharacteristic
136C2×C42—2511no—
137#102—2514no—
138#102—2514no—
139#102—2514no—
140#102C2×D25114yescomplemented
141#97—2512no—
142C4×C8—2512no—
143C24—21no—
144C24—5021no—
145C2×Q8—5022no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—10048no—
150C2×D4—5024no—
151C2×D4—5024no—
152C2×D4—5021no—
153C2×D4—5022no—
154C2×D4—5022no—
155C2×D4—2512no—
156C2×D4C22×D25112yes—
157C22×C4—2512no—
158C22×C4C22×D25111yescharacteristic
159M16—22no—
160M16—5022no—
161C2×C8—2512no—
162C2×C8C22×D25112yes—
163C4⋊C4—5022no—
164C4⋊C4—5022no—
165C4⋊C4—5022no—
166C4⋊C4—2512no—
167C4⋊C4C22×D25112yes—
168C22⋊C4—21no—
169C22⋊C4—5021no—
170C42—21no—
171C42—5021no—
172C42—2512no—
173C42—2511no—
174C42—2511no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—10044no—
179C23—10042no—
180C23—5022no—
181C23D4×D25111yescharacteristic
182Q8—10044no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—10048no—
187D4—10048no—
188D4—10042no—
189D4—5022no—
190D4—5022no—
191C2×C4—22no—
192C2×C4—5022no—
193C2×C4—5024no—
194C2×C4—2512no—
195C2×C4—2512no—
196C2×C4D4×D25111yescharacteristic
197C2×C4#1686080912yes—
198C2×C4#1686081012yes—
199C2×C4C23×D25111yescharacteristic
200C8—22no—
201C8—5022no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—20084no—
207C22—10042no—
208C22—10044no—
209C22#2254870112yes—
210C22C2×D4×D25111yescharacteristic
211C4—42no—
212C4—10042no—
213C4—5022no—
214C4—2514no—
215C4#2254870112yes—
216C4C2×#1686080911yescharacteristic
217C4C2×#1686081011yescharacteristic
218C2—22no—
219C2—82no—
220C2—20082no—
221C2C2×#2254870111yescharacteristic
222C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548711C2
hypercenter#418D251
hypocenterC251#3138
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #30632240C2semidirect
3. #30632050C2semidirect
4. #30632032C2semidirect
5. #30631693C2semidirect
6. #30631692C2semidirect
7. #30631691C2semidirect
8. M16×Dic251C2semidirect
9. #418×C251C2semidirect
13. #22548714C22semidirect
17. #22548706C22semidirect
22. #22548637C22semidirect
23. #22548636C22semidirect
24. #22548635C22semidirect
25. #22548625C22semidirect
26. C8×Dic251C22semidirect
30. #102×C251C22semidirect
111. C251#3138semidirect
120. #418D251semidirect
140. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322402
C2×#225487112
#306320502
#306320322
#306316932
#306316922
#306316912
M16×Dic2512
#418×C2512
#3138251