Finite Group Databasebeta

← #45663578

Subgroups — #45663578

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#30657457C211yesmaximal, characteristic
2#30657142C211yesmaximal, characteristic, complemented
3#30657023C211yesmaximal, characteristic, complemented
4#30656942C211yesmaximal, characteristic
5#30656482C211yesmaximal, characteristic, complemented
6#30656469C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic
8C2×#16861606—21no—
9#22551592C2211yescharacteristic
10#22551592C2211yescharacteristic
11#22551585—22no—
12#22551579—21no—
13#22551571—21no—
14C4×Dic514C2211yescharacteristic
15#22551557—21no—
16#22551551—21no—
17#22551526—22no—
18#22551524—21no—
19C4⋊#14523339C2211yescharacteristic
20#22551517—21no—
21#22551515—21no—
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28#16861606—42no—
29C2×#14523339—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32C4⋊Dic257—21no—
33C4⋊Dic257D411yescharacteristic
34C4⋊Dic257D412yes—
35#16861597—22no—
36#16861597—21no—
37C22⋊Dic257—22no—
38C22⋊Dic257—22no—
39C4×Dic257—22no—
40C4×Dic257—21no—
41C2×D4×C257D411yescharacteristic
42C22×C4×C257—21no—
43C2×C8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257D411yescharacteristic
46C22⋊C4×C257—21no—
47C42×C257C2311yescharacteristic
48Dic514—21no—
49Dic514—21no—
50Dic514—41no—
51Dic514—41no—
52Dic514—41no—
53Dic514—41no—
54C2×Dic257—21no—
55C2×Dic257—21no—
56C2×Dic257—22no—
57C2×Dic257—41no—
58#14523339—41no—
59C23×C257—21no—
60D4×C257—21no—
61D4×C257—21no—
62C2×C4×C257—21no—
63C2×C4×C257—41no—
64C2×C4×C257—41no—
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C2×C4×C257C2×D411yescharacteristic
68C8×C257—22no—
69C8×C257—41no—
70C22×C257—41no—
71C22×C257—41no—
72C22×C257—41no—
73C22×C257#12011yescharacteristic
74C4×C257—21no—
75C4×C257—41no—
76C4×C257#13511yescharacteristic
77C4×C257#13511yescharacteristic
78C4×C257AGL(1,ℤ/8)11yescharacteristic
79C4×C257#13711yescharacteristic
80Dic257—42no—
81Dic257—41no—
82Dic257—41no—
83Dic257—81no—
84C2×C257—81no—
85C2×C257#44911yescharacteristic
86C2×C257#45211yescharacteristic
87C2×C257#45611yescharacteristic
88C257#165011yescharacteristic, complemented, Sylow
89#1650—2571nomaximal, Sylow
90#556—2571no—
91#500—2571no—
92#475—2571no—
93#461—2571no—
94#332—2571no—
95Q8⋊C8—2571no—
96D4⋊C8D25711yescharacteristic
97C2×SD16—5141no—
98C4⋊Q8—2571no—
99C4⋊Q8—2571no—
100#126—5142no—
101#124—5141no—
102C22⋊Q8—5141no—
103C4×Q8—2571no—
104C4×Q8—5141no—
105C4×D4C2×D25711yescharacteristic
106#117—5141no—
107#107—5142no—
108#106—5141no—
109C4⋊C8—2571no—
110C4⋊C8C2×D25711yescharacteristic
111#103—5141no—
112#103—5141no—
113C4×C8C2×D25711yescharacteristic
114C2×Q8—2571no—
115C2×Q8—5141no—
116C2×Q8—5141no—
117C2×D4D102811yescharacteristic
118C22×C4—21no—
119SD16—10282no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5141no—
123C4⋊C4—2571no—
124C4⋊C4—2572no—
125C4⋊C4—5142no—
126C4⋊C4—5141no—
127C4⋊C4—5141no—
128C4⋊C4—5141no—
129C4⋊C4—5141no—
130C4⋊C4D102811yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—5142no—
133C22⋊C4—5142no—
134C42—5142no—
135C42—5141no—
136C42C22×D25711yescharacteristic
137C23—21no—
138Q8—5141no—
139Q8—5141no—
140Q8—10281no—
141Q8—10281no—
142Q8—10281no—
143Q8—10281no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—5141no—
148C2×C4—5142no—
149C2×C4—5141no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—10281no—
153C2×C4D4×D25711yescharacteristic
154C2×C4D4×D25711yescharacteristic
155C2×C4C2×D102811yescharacteristic
156C8—22no—
157C8—41no—
158C8—10281no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2255156211yescharacteristic
163C4—21no—
164C4—41no—
165C4—10282no—
166C4—10281no—
167C4—10281no—
168C4—20561no—
169C4#2255161111yescharacteristic
170C4#2255161411yescharacteristic
171C4#2255162011yescharacteristic
172C4#2255162111yescharacteristic
173C2—81no—
174C2#3065688411yescharacteristic
175C2#3065690011yescharacteristic
176C2#3065692111yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenterD4⋊C8D257
hypocenterC257#1650
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30657142C2semidirect
3. #30657023C2semidirect
5. #30656482C2semidirect
6. #30656469C2semidirect
88. C257#1650semidirect

Maximal subgroups

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

subgroupindex
#306574572
#306571422
#306570232
#306569422
#306564822
#306564692
D4⋊C8×C2572
#1650257