Finite Group Databasebeta

← #45664759

Subgroups — #45664759

Order 32896.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#16861602C211yesmaximal, characteristic, complemented
2C2×#22551513C212yesmaximal, complemented
3#30656597C211yesmaximal, characteristic, complemented
4#30656499C212yesmaximal, complemented
5C2×#121×C257C211yesmaximal, characteristic
6C2×#16861602—24no—
7C2×#16861602—22no—
8C22×#14523339C2212yes—
9C2×C4⋊Dic257C2212yes—
10C2×C22⋊Dic257—21no—
11#22551513—24no—
12#22551499—21no—
13C22×D4×C257—21no—
14C22×D4×C257C2211yescharacteristic
15C23×C4×C257C2211yescharacteristic
16#121×C257—22no—
17#121×C257C414yescomplemented
18C2×C4⋊C4×C257C2211yescharacteristic
19C2×C22⋊C4×C257—21no—
20C22×Dic257—22no—
21#16861602—28no—
22C2×#14523339—22no—
23C2×#14523339—22no—
24C2×#14523339D414yescomplemented
25C4⋊Dic257—22no—
26C4⋊Dic257—22no—
27C22⋊Dic257—42no—
28C24×C257—21no—
29C24×C257D411yescharacteristic
30C2×D4×C257—24no—
31C2×D4×C257—22no—
32C2×D4×C257—21no—
33C2×D4×C257—22no—
34C2×D4×C257—44no—
35C2×D4×C257C2×C412yes—
36C22×C4×C257—22no—
37C22×C4×C257—22no—
38C22×C4×C257—21no—
39C22×C4×C257C2×C412yes—
40C22×C4×C257D411yescharacteristic
41C22×C4×C257D412yes—
42C22×C4×C257C2311yescharacteristic
43C4⋊C4×C257—21no—
44C4⋊C4×C257C2×C412yes—
45C22⋊C4×C257—24no—
46C2×Dic257—42no—
47C2×Dic257—42no—
48C2×Dic257—42no—
49#14523339—24no—
50C23×C257—22no—
51C23×C257—22no—
52C23×C257—21no—
53C23×C257—21no—
54C23×C257—22no—
55C23×C257—42no—
56C23×C257—42no—
57C23×C257—41no—
58C23×C257—41no—
59C23×C257C22⋊C412yes—
60C23×C257C2×D411yescharacteristic
61D4×C257—24no—
62D4×C257—48no—
63D4×C257—41no—
64D4×C257—41no—
65C2×C4×C257—22no—
66C2×C4×C257—24no—
67C2×C4×C257—22no—
68C2×C4×C257—22no—
69C2×C4×C257—22no—
70C2×C4×C257—42no—
71C2×C4×C257C22⋊C412yes—
72C2×C4×C257C22×C411yescharacteristic
73C2×C4×C257C2×D412yes—
74C2×C4×C257C2×D411yescharacteristic
75C2×C4×C257#4112yes—
76C22×C257—22no—
77C22×C257—22no—
78C22×C257—22no—
79C22×C257—42no—
80C22×C257—42no—
81C22×C257—42no—
82C22×C257—42no—
83C22×C257—42no—
84C22×C257—82no—
85C22×C257—82no—
86C22×C257C2×C22⋊C411yescharacteristic
87C22×C257#12112yes—
88C22×C257AGL(1,ℤ/8)11yescharacteristic
89C22×C257AGL(1,ℤ/8)12yes—
90C22×C257#13711yescharacteristic
91C4×C257—22no—
92C4×C257—42no—
93C4×C257C4×D412yes—
94C4×C257#12011yescharacteristic
95C4×C257#12311yescharacteristic
96Dic257—82no—
97C2×C257—42no—
98C2×C257—82no—
99C2×C257#38611yescharacteristic
100C2×C257#41711yescharacteristic
101C2×C257#41811yescharacteristic
102C2×C257#46812yes—
103C2×C257#46912yes—
104C257#189511yescharacteristic, complemented, Sylow
105#1895—2571nomaximal, Sylow
106C22×M16—2571no—
107C2×#121D25711yescharacteristic
108C2×#102—2572no—
109#380—2571no—
110#340—2572no—
111C22×D4—21no—
112C22×D4C2×D25711yescharacteristic
113C23×C4C2×D25711yescharacteristic
114C2×M16—5144no—
115C2×M16—5142no—
116C22×C8—2572no—
117#121—22no—
118#121Dic25714yescomplemented
119C2×C4⋊C4—2572no—
120C2×C4⋊C4C2×D25711yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—5141no—
123#102—5144no—
124#95—5141no—
125C24—21no—
126C24C22⋊D25711yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×Dic25712yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—5142no—
137C22×C4C2×Dic25712yes—
138C22×C4C22⋊D25711yescharacteristic
139C22×C4C22⋊D25712yes—
140C22×C4C22×D25711yescharacteristic
141M16—5148no—
142C2×C8—2574no—
143C2×C8—5142no—
144C2×C8—5142no—
145C4⋊C4—21no—
146C4⋊C4—5142no—
147C4⋊C4—5142no—
148C4⋊C4C2×Dic25712yes—
149C22⋊C4—24no—
150C22⋊C4—10282no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23C22⋊Dic25712yes—
161C23C2×C22⋊D25711yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—10282no—
173C2×C4—10282no—
174C2×C4—10282no—
175C2×C4C22⋊Dic25712yes—
176C2×C4C22×Dic25711yescharacteristic
177C2×C4D4×D25712yes—
178C2×C4C2×C22⋊D25711yescharacteristic
179C2×C4#1686161712yes—
180C8—5144no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊Dic25711yescharacteristic
192C22#2255156412yes—
193C22#2255161711yescharacteristic
194C22#2255161812yes—
195C22#2255162211yescharacteristic
196C4—22no—
197C4—42no—
198C4—20562no—
199C4D4×Dic25712yes—
200C4#2255156011yescharacteristic
201C4#2255157311yescharacteristic
202C2—42no—
203C2—82no—
204C2#3065661711yescharacteristic
205C2#3065673811yescharacteristic
206C2#3065674211yescharacteristic
207C2#3065698112yes—
208C2#3065698812yes—
subgroupG/H
centerC23C2×C22⋊D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting, prime radicalC2×#121×C257C2
socleC23×C257C2×D4
cosocleC22×C4×C257C23
hypercenterC2×#121D257
hypocenterC257#1895
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16861602C2semidirect
2. C2×#22551513C2semidirect
3. #30656597C2semidirect
4. #30656499C2semidirect
17. #121×C257C4semidirect
24. C2×#14523339D4semidirect
104. C257#1895semidirect
118. #121Dic257semidirect

Maximal subgroups

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

subgroupindex
C22×#168616022
C2×#225515132
#306565972
#306564992
C2×#121×C2572
#1895257