Finite Group Databasebeta

← #45663577

Subgroups — #45663577

Order 32896.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#30657461C211yesmaximal, characteristic
2#30657140C211yesmaximal, characteristic, complemented
3#30657023C211yesmaximal, characteristic, complemented
4#30656943C211yesmaximal, characteristic
5#30656467C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#332×C257C211yesmaximal, characteristic, complemented
8C2×#16861607—21no—
9#22551592C2211yescharacteristic
10#22551587—22no—
11#22551581—21no—
12#22551572—21no—
13Q8×Dic257—21no—
14C4×Dic514C2211yescharacteristic
15C4×D1028C2211yescharacteristic
16#22551550—21no—
17#22551525—22no—
18#22551524—21no—
19#22551517—21no—
20#22551517—21no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514D411yescharacteristic
27C2×D1028D411yescharacteristic
28C2×C4×D257—21no—
29#16861607—42no—
30C2×#14523339—21no—
31C4⋊Dic257—22no—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257D411yescharacteristic
35C4⋊Dic257D411yescharacteristic
36#16861597—21no—
37#16861595—22no—
38#16861595—22no—
39#16861595—21no—
40C4×Dic257—22no—
41C4×Dic257—21no—
42C2×Q8×C257—21no—
43C2×C8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257D412yes—
47C42×C257C2311yescharacteristic
48C22×D257—21no—
49Dic514—21no—
50Dic514—21no—
51Dic514—41no—
52Dic514—41no—
53D1028—21no—
54D1028—21no—
55C2×Dic257—21no—
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—41no—
59C4×D257—41no—
60C4×D257—41no—
61#14523339—22no—
62Q8×C257—41no—
63Q8×C257—41no—
64C2×C4×C257—22no—
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C2×C4×C257C2×D411yescharacteristic
68C8×C257—41no—
69C8×C257—41no—
70C22×C257#12011yescharacteristic
71C2×D257—41no—
72C2×D257—41no—
73C2×D257—41no—
74C4×C257—21no—
75C4×C257—42no—
76C4×C257#13511yescharacteristic
77C4×C257#13511yescharacteristic
78C4×C257AGL(1,ℤ/8)11yescharacteristic
79C4×C257#13711yescharacteristic
80Dic257—41no—
81Dic257—41no—
82Dic257—41no—
83Dic257—81no—
84C2×C257#44911yescharacteristic
85C2×C257#45211yescharacteristic
86C2×C257#45611yescharacteristic
87D257—81no—
88C257#165011yescharacteristic, complemented, Sylow
89#1650—2571nomaximal, Sylow
90#556—2571no—
91#500—2571no—
92#475—2571no—
93#461—2571no—
94#332D25711yescharacteristic, complemented
95Q8⋊C8—2571no—
96D4⋊C8—2571no—
97C2×SD16—5141no—
98C4⋊Q8—2571no—
99C4⋊Q8C2×D25711yescharacteristic
100#126—5142no—
101#124—5141no—
102C22⋊Q8—5141no—
103C4×Q8—2571no—
104C4×Q8—5141no—
105C4×D4—2571no—
106#117—5141no—
107#107—5142no—
108#106—5141no—
109C4⋊C8C2×D25711yescharacteristic
110C4⋊C8C2×D25711yescharacteristic
111#103—5141no—
112#103—5141no—
113C4×C8—2571no—
114C2×Q8—2571no—
115C2×Q8—21no—
116C2×Q8—5141no—
117C2×D4—2571no—
118C22×C4—5141no—
119SD16—10282no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5141no—
123C4⋊C4—2571no—
124C4⋊C4—2571no—
125C4⋊C4—21no—
126C4⋊C4—5142no—
127C4⋊C4—5141no—
128C4⋊C4—5141no—
129C4⋊C4—5141no—
130C4⋊C4D102812yes—
131C22⋊C4—5142no—
132C22⋊C4—5142no—
133C22⋊C4—5141no—
134C42—5142no—
135C42—5141no—
136C42C22×D25711yescharacteristic
137C23—5141no—
138Q8—5141no—
139Q8—5141no—
140Q8—41no—
141Q8—41no—
142Q8—10281no—
143Q8—10281no—
144D4—5141no—
145D4—5141no—
146C2×C4—22no—
147C2×C4—5141no—
148C2×C4—5141no—
149C2×C4—5141no—
150C2×C4—10281no—
151C2×C4—10281no—
152C2×C4—10281no—
153C2×C4D4×D25711yescharacteristic
154C2×C4D4×D25711yescharacteristic
155C2×C4C2×D102811yescharacteristic
156C8—5142no—
157C8—41no—
158C8—41no—
159C22—10281no—
160C22—10281no—
161C22—10281no—
162C22#2255156211yescharacteristic
163C4—21no—
164C4—42no—
165C4—10281no—
166C4—10281no—
167C4—10281no—
168C4—20561no—
169C4#2255161311yescharacteristic
170C4#2255161311yescharacteristic
171C4#2255161611yescharacteristic
172C4#2255162411yescharacteristic
173C2—20561no—
174C2#3065688311yescharacteristic
175C2#3065690211yescharacteristic
176C2#3065692211yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#332D257
hypocenterC257#1650
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #30657140C2semidirect
3. #30657023C2semidirect
5. #30656467C2semidirect
7. #332×C257C2semidirect
88. C257#1650semidirect
94. #332D257semidirect

Maximal subgroups

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

subgroupindex
#306574612
#306571402
#306570232
#306569432
#306564672
#306564642
#332×C2572
#1650257