Finite Group Databasebeta

← #46782003

Subgroups — #46782003

Order 33664.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#30682341C211yesmaximal, characteristic, complemented
2#30682223C211yesmaximal, characteristic, complemented
3#30682143C211yesmaximal, characteristic
4#30681684C211yesmaximal, characteristic, complemented
5Q8⋊(C263⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C263⋊C8)C211yesmaximal, characteristic
7#556×C263C211yesmaximal, characteristic
8C2×#16862364—21no—
9#22554470C2211yescharacteristic
10#22554403—22no—
11#22554401—21no—
12C4⋊(C263⋊C8)C2211yescharacteristic
13C4⋊(C263⋊C8)C2211yescharacteristic
14#22554394—21no—
15#22554393—21no—
16C4×C263⋊C8C2211yescharacteristic
17C4⋊Q8×C263C2211yescharacteristic
18#126×C263—22no—
19#124×C263—21no—
20C22⋊Q8×C263—21no—
21C4×Q8×C263—21no—
22C4×Q8×C263C2211yescharacteristic
23C4×D4×C263C2211yescharacteristic
24#117×C263—21no—
25C2×Dic526—21no—
26#16862364—42no—
27C2×C263⋊C8—21no—
28C2×C263⋊C8—21no—
29C2×C263⋊C8—21no—
30C4⋊Dic263—21no—
31C4⋊Dic263—21no—
32C2×Q8×C263—21no—
33C2×Q8×C263D411yescharacteristic
34C2×D4×C263D411yescharacteristic
35C22×C4×C263—21no—
36C4⋊C4×C263—22no—
37C4⋊C4×C263—21no—
38C4⋊C4×C263—21no—
39C4⋊C4×C263D411yescharacteristic
40C4⋊C4×C263D411yescharacteristic
41C4⋊C4×C263D412yes—
42C22⋊C4×C263—22no—
43C22⋊C4×C263—22no—
44C22⋊C4×C263—21no—
45C42×C263—22no—
46C42×C263—21no—
47C42×C263C2311yescharacteristic
48Dic526—41no—
49Dic526—41no—
50C2×Dic263—41no—
51C263⋊C8—22no—
52C263⋊C8—41no—
53C263⋊C8—41no—
54C23×C263—21no—
55Q8×C263—21no—
56Q8×C263—21no—
57Q8×C263—41no—
58Q8×C263—41no—
59D4×C263—21no—
60D4×C263—21no—
61C2×C4×C263—21no—
62C2×C4×C263—21no—
63C2×C4×C263—21no—
64C2×C4×C263—22no—
65C2×C4×C263—41no—
66C2×C4×C263—41no—
67C2×C4×C263C2×D411yescharacteristic
68C2×C4×C263C2×D411yescharacteristic
69C2×C4×C263C2×D411yescharacteristic
70C22×C263—41no—
71C22×C263—41no—
72C22×C263—41no—
73C22×C263#12011yescharacteristic
74C4×C263—21no—
75C4×C263—42no—
76C4×C263—41no—
77C4×C263—41no—
78C4×C263—41no—
79C4×C263#13511yescharacteristic
80C4×C263#13511yescharacteristic
81C4×C263AGL(1,ℤ/8)11yescharacteristic
82C4×C263#13711yescharacteristic
83Dic263—81no—
84C2×C263—81no—
85C2×C263#44911yescharacteristic
86C2×C263#45211yescharacteristic
87C2×C263#45611yescharacteristic
88C263#165011yescharacteristic, complemented, Sylow
89#1650—2631nomaximal, Sylow
90#556D26311yescharacteristic
91#500—2631no—
92#475—2631no—
93#461—2631no—
94#332—2631no—
95Q8⋊C8—2631no—
96D4⋊C8—2631no—
97C2×SD16—5261no—
98C4⋊Q8—2631no—
99C4⋊Q8C2×D26311yescharacteristic
100#126—22no—
101#124—21no—
102C22⋊Q8—21no—
103C4×Q8—21no—
104C4×Q8C2×D26311yescharacteristic
105C4×D4C2×D26311yescharacteristic
106#117—21no—
107#107—5262no—
108#106—5261no—
109C4⋊C8—2631no—
110C4⋊C8—2631no—
111#103—5261no—
112#103—5261no—
113C4×C8—2631no—
114C2×Q8—21no—
115C2×Q8—5261no—
116C2×Q8C22⋊D26311yescharacteristic
117C2×D4C22⋊D26311yescharacteristic
118C22×C4—21no—
119SD16—10522no—
120C2×C8—5261no—
121C2×C8—5261no—
122C2×C8—5261no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—5261no—
127C4⋊C4—5261no—
128C4⋊C4C22⋊D26311yescharacteristic
129C4⋊C4C22⋊D26311yescharacteristic
130C4⋊C4C22⋊D26312yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—22no—
135C42—21no—
136C42C22×D26311yescharacteristic
137C23—21no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—10521no—
143Q8—10521no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—10521no—
153C2×C4C2×C22⋊D26311yescharacteristic
154C2×C4C2×C22⋊D26311yescharacteristic
155C2×C4C2×C22⋊D26311yescharacteristic
156C8—5262no—
157C8—10521no—
158C8—10521no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2255443911yescharacteristic
163C4—21no—
164C4—41no—
165C4—42no—
166C4—41no—
167C4—41no—
168C4—21041no—
169C4#2255449011yescharacteristic
170C4#2255449011yescharacteristic
171C4#2255449511yescharacteristic
172C4#2255450011yescharacteristic
173C2—81no—
174C2#3068208311yescharacteristic
175C2#3068210011yescharacteristic
176C2#3068212111yescharacteristic
subgroupG/H
centerC22#22554439
commutator, cosocleC42×C263C23
FrattiniC42C22×D263
Fitting#556×C263C2
socleC22×C263#120
prime radicalC2×D4×C263D4
hypercenter#556D263
hypocenterC263#1650
LCSG ▷ C42×C263 ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #30682341C2semidirect
2. #30682223C2semidirect
4. #30681684C2semidirect
5. Q8⋊(C263⋊C8)C2semidirect
88. C263#1650semidirect

Maximal subgroups

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

subgroupindex
#306823412
#306822232
#306821432
#306816842
Q8⋊(C263⋊C8)2
D4⋊(C263⋊C8)2
#556×C2632
#1650263