Finite Group Databasebeta

← #45663575

Subgroups — #45663575

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#30657460C211yesmaximal, characteristic
2#30657140C211yesmaximal, characteristic, complemented
3#30656939C211yesmaximal, characteristic
4#30656482C211yesmaximal, characteristic, complemented
5Q8⋊#14523339C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#475×C257C211yesmaximal, characteristic, complemented
8C2×#16861605—21no—
9#22551592C2211yescharacteristic
10#22551587—22no—
11#22551580—21no—
12#22551568—21no—
13C4×Dic514—21no—
14C4×D1028C2211yescharacteristic
15#22551550—21no—
16#22551525—22no—
17C4⋊#14523339C2211yescharacteristic
18#22551516—21no—
19C4×#14523339C2211yescharacteristic
20C4⋊Q8×C257C2211yescharacteristic
21C4×Q8×C257C2211yescharacteristic
22#106×C257—21no—
23C4⋊C8×C257C2211yescharacteristic
24#103×C257—21no—
25C2×Dic514—21no—
26C2×D1028D411yescharacteristic
27C2×C4×D257—21no—
28#16861605—42no—
29C2×#14523339—21no—
30C2×#14523339—21no—
31C4⋊Dic257—21no—
32C4⋊Dic257D411yescharacteristic
33C4⋊Dic257D412yes—
34#16861597—22no—
35#16861595—22no—
36#16861595—22no—
37#16861595—21no—
38C4×Dic257—22no—
39C2×Q8×C257—21no—
40C2×Q8×C257D411yescharacteristic
41C2×C8×C257—21no—
42C4⋊C4×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257D411yescharacteristic
46C42×C257—21no—
47C42×C257C2311yescharacteristic
48C22×D257—21no—
49Dic514—41no—
50Dic514—41no—
51D1028—21no—
52D1028—21no—
53C2×Dic257—21no—
54C2×Dic257—22no—
55C4×D257—41no—
56C4×D257—41no—
57#14523339—22no—
58#14523339—41no—
59Q8×C257—21no—
60Q8×C257—21no—
61Q8×C257—41no—
62Q8×C257—41no—
63C2×C4×C257—21no—
64C2×C4×C257—21no—
65C2×C4×C257—41no—
66C2×C4×C257C2×D411yescharacteristic
67C2×C4×C257C2×D411yescharacteristic
68C2×C4×C257C2×D411yescharacteristic
69C8×C257—41no—
70C22×C257#12011yescharacteristic
71C2×D257—41no—
72C2×D257—41no—
73C2×D257—41no—
74C4×C257—21no—
75C4×C257—41no—
76C4×C257—41no—
77C4×C257—81no—
78C4×C257#13511yescharacteristic
79C4×C257#13511yescharacteristic
80C4×C257AGL(1,ℤ/8)11yescharacteristic
81C4×C257#13711yescharacteristic
82Dic257—42no—
83Dic257—41no—
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#475D25711yescharacteristic, complemented
93#461—2571no—
94#332—2571no—
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—5141no—
104C4×Q8C2×D25711yescharacteristic
105C4×D4—2571no—
106#117—5141no—
107#107—5142no—
108#106—21no—
109C4⋊C8—2571no—
110C4⋊C8C2×D25711yescharacteristic
111#103—21no—
112#103—5141no—
113C4×C8—2571no—
114C2×Q8—21no—
115C2×Q8—5141no—
116C2×Q8C22⋊D25711yescharacteristic
117C2×D4—2571no—
118C22×C4—5141no—
119SD16—10282no—
120C2×C8—21no—
121C2×C8—5141no—
122C2×C8—5141no—
123C4⋊C4—2571no—
124C4⋊C4—2572no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—5142no—
129C4⋊C4—5141no—
130C4⋊C4C22⋊D25711yescharacteristic
131C22⋊C4—5142no—
132C22⋊C4—5142no—
133C22⋊C4—5141no—
134C42—21no—
135C42—5142no—
136C42C22×D25711yescharacteristic
137C23—5141no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—10281no—
143Q8—10281no—
144D4—5141no—
145D4—5141no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—5141no—
149C2×C4—5142no—
150C2×C4—41no—
151C2×C4—10281no—
152C2×C4—10281no—
153C2×C4D4×D25711yescharacteristic
154C2×C4D4×D25711yescharacteristic
155C2×C4C2×C22⋊D25711yescharacteristic
156C8—5142no—
157C8—41no—
158C8—10281no—
159C22—10281no—
160C22—10281no—
161C22—10281no—
162C22#2255156011yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—10282no—
167C4—10281no—
168C4—81no—
169C4#2255161211yescharacteristic
170C4#2255161311yescharacteristic
171C4#2255161911yescharacteristic
172C4#2255162311yescharacteristic
173C2—20561no—
174C2#3065688111yescharacteristic
175C2#3065689711yescharacteristic
176C2#3065691811yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#475×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#475D257
hypocenterC257#1650
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #30657140C2semidirect
4. #30656482C2semidirect
5. Q8⋊#14523339C2semidirect
7. #475×C257C2semidirect
88. C257#1650semidirect
92. #475D257semidirect

Maximal subgroups

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

subgroupindex
#306574602
#306571402
#306569392
#306564822
Q8⋊#145233392
#306564642
#475×C2572
#1650257