Finite Group Databasebeta

← #45663417

Subgroups — #45663417

Order 32896.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657462C211yesmaximal, characteristic, complemented
2#30657085C211yesmaximal, characteristic
3#30656938C211yesmaximal, characteristic
4#30656938C211yesmaximal, characteristic
5#30656467C211yesmaximal, characteristic, complemented
6#30656467C211yesmaximal, characteristic, complemented
7#332×C257C211yesmaximal, characteristic, complemented
8C2×#16861607—21no—
9C2×#16861607—21no—
10#22551590—21no—
11#22551589C2211yescharacteristic
12#22551583—21no—
13#22551582—22no—
14Q8×Dic257—21no—
15C4×Dic514C2211yescharacteristic
16C4×Dic514C2211yescharacteristic
17#22551557—21no—
18#22551514—21no—
19#22551514—21no—
20#22551512—22no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514D411yescharacteristic
26C2×Dic514D411yescharacteristic
27C2×D1028—21no—
28C2×D1028—21no—
29C2×D1028—21no—
30#16861607—42no—
31#16861607—42no—
32C2×#14523339—21no—
33C4⋊Dic257—22no—
34C4⋊Dic257D411yescharacteristic
35C4⋊Dic257D411yescharacteristic
36#16861597—22no—
37#16861597—22no—
38#16861597—21no—
39#16861597—21no—
40C4×Dic257—22no—
41C4×Dic257—21no—
42C4×Dic257—21no—
43C2×Q8×C257—21no—
44C2×C8×C257—21no—
45C2×C8×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257D412yes—
48C42×C257C2311yescharacteristic
49C22×D257—41no—
50Dic514—21no—
51Dic514—21no—
52Dic514—21no—
53Dic514SD1612yescomplemented
54D1028—42no—
55D1028—41no—
56D1028—41no—
57D1028—41no—
58D1028—41no—
59C2×Dic257—21no—
60C2×Dic257—21no—
61C2×Dic257—21no—
62C2×Dic257—21no—
63#14523339—41no—
64Q8×C257—41no—
65Q8×C257—41no—
66C2×C4×C257—22no—
67C2×C4×C257C2×D411yescharacteristic
68C2×C4×C257C2×D411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C8×C257—41no—
71C8×C257—41no—
72C22×C257#12011yescharacteristic
73C2×D257—81no—
74C2×D257—81no—
75C2×D257—81no—
76C4×C257—21no—
77C4×C257—42no—
78C4×C257C2×SD1611yescharacteristic
79C4×C257#13511yescharacteristic
80C4×C257AGL(1,ℤ/8)11yescharacteristic
81C4×C257#13711yescharacteristic
82Dic257—22no—
83Dic257—41no—
84Dic257—41no—
85Dic257—41no—
86C2×C257#44811yescharacteristic
87C2×C257#44911yescharacteristic
88C2×C257#45411yescharacteristic
89D257—161no—
90C257#162611yescharacteristic, complemented, Sylow
91#1626—2571nomaximal, Sylow
92#557—2571no—
93#486—2571no—
94#460—2571no—
95#460—2571no—
96#332D25711yescharacteristic, complemented
97Q8⋊C8—2571no—
98Q8⋊C8—2571no—
99C2×SD16—5141no—
100C2×SD16—5141no—
101C4⋊Q8—5141no—
102C4⋊Q8C2×D25711yescharacteristic
103#127—2571no—
104#125—5141no—
105#125—5142no—
106C4×Q8—2571no—
107C4×Q8—2571no—
108C4×Q8—5141no—
109C4×Q8—5141no—
110C4⋊C8C2×D25711yescharacteristic
111C4⋊C8C2×D25711yescharacteristic
112#102—5141no—
113#102—5141no—
114#102—5142no—
115C4×C8—2571no—
116C2×Q8—2571no—
117C2×Q8—2571no—
118C2×Q8—21no—
119C2×D4—5141no—
120C2×D4—5141no—
121C2×D4—5141no—
122SD16—10282no—
123SD16—10282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—5141no—
127C4⋊C4—2571no—
128C4⋊C4—2571no—
129C4⋊C4—21no—
130C4⋊C4—5142no—
131C4⋊C4—5142no—
132C4⋊C4—5142no—
133C4⋊C4—5141no—
134C4⋊C4—5141no—
135C4⋊C4D102812yes—
136C42—5142no—
137C42—5141no—
138C42—5141no—
139C42C22×D25711yescharacteristic
140C23—10281no—
141Q8—2572no—
142Q8—5141no—
143Q8—5141no—
144Q8—5141no—
145Q8—41no—
146Q8—41no—
147D4—10282no—
148D4—10281no—
149D4—10281no—
150D4—10281no—
151D4—10281no—
152C2×C4—22no—
153C2×C4—5141no—
154C2×C4—5141no—
155C2×C4—5141no—
156C2×C4—5141no—
157C2×C4D4×D25711yescharacteristic
158C2×C4D4×D25711yescharacteristic
159C2×C4C2×D102811yescharacteristic
160C8—41no—
161C8—41no—
162C8—10281no—
163C22—20561no—
164C22—20561no—
165C22—20561no—
166C22#2255156211yescharacteristic
167C4—21no—
168C4—5142no—
169C4—42no—
170C4—10281no—
171C4—10281no—
172C4—10281no—
173C4SD16×D25711yescharacteristic
174C4#2255161311yescharacteristic
175C4#2255161611yescharacteristic
176C4#2255162411yescharacteristic
177C2—41121no—
178C2#3065687711yescharacteristic
179C2#3065688311yescharacteristic
180C2#3065691211yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257#120
prime radical#22551589C22
hypercenter#332D257
hypocenterC257#1626
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #30657462C2semidirect
5. #30656467C2semidirect
6. #30656467C2semidirect
7. #332×C257C2semidirect
53. Dic514SD16semidirect
90. C257#1626semidirect
96. #332D257semidirect

Maximal subgroups

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

subgroupindex
#306574622
#306570852
#306569382
#306564672
#332×C2572
#1626257