Finite Group Databasebeta

← #45664758

Subgroups — #45664758

Order 32896.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#22551566C211yesmaximal, characteristic
2C2×#22551514C212yesmaximal, complemented
3#30656597C211yesmaximal, characteristic, complemented
4#30656500C212yesmaximal, complemented
5C22×M16×C257C211yesmaximal, characteristic, complemented
6C22×D1028C2211yescharacteristic
7C22×C22⋊D257—21no—
8#22551566—22no—
9#22551566C414yescomplemented
10C2×C4⋊Dic257C2212yes—
11C2×C4⋊Dic257C2211yescharacteristic
12C2×C22⋊Dic257—21no—
13C2×#16861595—21no—
14#22551514—24no—
15#22551499—21no—
16C23×C4×C257C2211yescharacteristic
17C2×M16×C257—22no—
18C2×M16×C257—24no—
19C22×C8×C257C2212yes—
20C23×D257—21no—
21C2×D1028—22no—
22C2×D1028—21no—
23C2×D1028—22no—
24C2×D1028C2×C412yes—
25C2×C22⋊D257—24no—
26C2×C22⋊D257—44no—
27C22×Dic257—21no—
28C22×Dic257—22no—
29C4⋊Dic257—21no—
30C4⋊Dic257—22no—
31C4⋊Dic257—22no—
32C4⋊Dic257C2×C412yes—
33C22⋊Dic257—42no—
34#16861595—24no—
35C24×C257D411yescharacteristic
36C22×C4×C257—22no—
37C22×C4×C257—22no—
38C22×C4×C257C2×C412yes—
39C22×C4×C257D412yes—
40C22×C4×C257D411yescharacteristic
41C22×C4×C257C2311yescharacteristic
42M16×C257—28no—
43C2×C8×C257—22no—
44C2×C8×C257—22no—
45C2×C8×C257D414yescomplemented
46C22×D257—22no—
47C22×D257—42no—
48C22×D257—42no—
49C22×D257—41no—
50C22×D257—41no—
51D1028—24no—
52D1028—41no—
53D1028—41no—
54C22⋊D257—48no—
55C2×Dic257—22no—
56C2×Dic257—42no—
57C2×Dic257—42no—
58C2×Dic257—42no—
59C2×Dic257—42no—
60C23×C257—22no—
61C23×C257—21no—
62C23×C257—22no—
63C23×C257—21no—
64C23×C257C22⋊C412yes—
65C23×C257C2×D411yescharacteristic
66C2×C4×C257—22no—
67C2×C4×C257—24no—
68C2×C4×C257—22no—
69C2×C4×C257—22no—
70C2×C4×C257C22⋊C412yes—
71C2×C4×C257C22×C411yescharacteristic
72C2×C4×C257C2×D412yes—
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257#4112yes—
75C8×C257—24no—
76C22×C257—22no—
77C22×C257—22no—
78C22×C257—22no—
79C22×C257—42no—
80C22×C257—42no—
81C22×C257C2×C22⋊C411yescharacteristic
82C22×C257#12112yes—
83C22×C257AGL(1,ℤ/8)11yescharacteristic
84C22×C257AGL(1,ℤ/8)12yes—
85C22×C257#13711yescharacteristic
86C2×D257—42no—
87C2×D257—42no—
88C2×D257—42no—
89C2×D257—82no—
90C2×D257—82no—
91C4×C257—22no—
92C4×C257C4×D412yes—
93C4×C257#12011yescharacteristic
94C4×C257#12311yescharacteristic
95Dic257—42no—
96Dic257—82no—
97C2×C257—42no—
98C2×C257#38611yescharacteristic
99C2×C257#41711yescharacteristic
100C2×C257#41811yescharacteristic
101C2×C257#46812yes—
102C2×C257#46912yes—
103D257—82no—
104C257#189511yescharacteristic, complemented, Sylow
105#1895—2571nomaximal, Sylow
106C22×M16D25711yescharacteristic, complemented
107C2×#121—2571no—
108C2×#102—2572no—
109#380—2571no—
110#340—2572no—
111C22×D4—2571no—
112C22×D4—5141no—
113C23×C4C2×D25711yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D25712yes—
117#121—2574no—
118#121—5142no—
119C2×C4⋊C4—2571no—
120C2×C4⋊C4—2572no—
121C2×C22⋊C4—5141no—
122C2×C22⋊C4—5141no—
123#102—5144no—
124#95—5141no—
125C24—5141no—
126C24D102811yescharacteristic
127C2×D4—2572no—
128C2×D4—5144no—
129C2×D4—5142no—
130C2×D4—5141no—
131C2×D4—5142no—
132C2×D4—10284no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—5141no—
136C22×C4—5142no—
137C22×C4C4×D25712yes—
138C22×C4C22⋊D25712yes—
139C22×C4D102811yescharacteristic
140C22×C4C22×D25711yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D25714yescomplemented
145C4⋊C4—2572no—
146C4⋊C4—5141no—
147C4⋊C4—5142no—
148C4⋊C4—5142no—
149C22⋊C4—5144no—
150C22⋊C4—10282no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—5142no—
156C23—10282no—
157C23—10282no—
158C23—10281no—
159C23—10281no—
160C23#1686159512yes—
161C23C2×D102811yescharacteristic
162D4—5144no—
163D4—10288no—
164D4—10281no—
165D4—10281no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—5142no—
171C2×C4—10282no—
172C2×C4—10282no—
173C2×C4—10282no—
174C2×C4—10282no—
175C2×C4#1686159512yes—
176C2×C4C2×C4×D25711yescharacteristic
177C2×C4C2×C22⋊D25712yes—
178C2×C4C2×C22⋊D25711yescharacteristic
179C2×C4#1686161812yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—10282no—
187C22—10282no—
188C22—10282no—
189C22—20562no—
190C22—20562no—
191C22C2×#1686159511yescharacteristic
192C22#2255156612yes—
193C22#2255162012yes—
194C22#2255162011yescharacteristic
195C22#2255162511yescharacteristic
196C4—22no—
197C4—10282no—
198C4—20562no—
199C4C4×C22⋊D25712yes—
200C4#2255156111yescharacteristic
201C4#2255157511yescharacteristic
202C2—42no—
203C2—20562no—
204C2#3065661611yescharacteristic
205C2#3065673911yescharacteristic
206C2#3065674311yescharacteristic
207C2#3065698312yes—
208C2#3065699012yes—
subgroupG/H
centerC23C2×D1028
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC22×M16×C257C2
socleC23×C257C2×D4
prime radicalC2×#22551566C2
cosocleC22×C4×C257C23
hypercenterC22×M16D257
hypocenterC257#1895
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#22551514C2semidirect
3. #30656597C2semidirect
4. #30656500C2semidirect
5. C22×M16×C257C2semidirect
9. #22551566C4semidirect
45. C2×C8×C257D4semidirect
104. C257#1895semidirect
106. C22×M16D257semidirect
144. C2×C8C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515662
C2×#225515142
#306565972
#306565002
C22×M16×C2572
#1895257