Finite Group Databasebeta

← #46781919

Subgroups — #46781919

Order 33664.

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

normal subgroups only

This group has 194 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#30682290C211yesmaximal, characteristic, complemented
2#30682233C211yesmaximal, characteristic, complemented
3#30682228C211yesmaximal, characteristic
4#30681684C211yesmaximal, characteristic, complemented
5Q8⋊(C263⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C263⋊C8)C211yesmaximal, characteristic
7#556×C263C211yesmaximal, characteristic
8#22554470C2211yescharacteristic
9#22554403—21no—
10#22554401—21no—
11C4⋊(C263⋊C8)C2211yescharacteristic
12C4⋊(C263⋊C8)C2211yescharacteristic
13#22554394—21no—
14#22554393—22no—
15#22554391—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—
26C2×C263⋊C8—21no—
27C2×C263⋊C8—21no—
28C2×C263⋊C8—21no—
29C4⋊Dic263—21no—
30C4⋊Dic263—21no—
31C2×Q8×C263—21no—
32C2×Q8×C263D411yescharacteristic
33C2×D4×C263D411yescharacteristic
34C22×C4×C263—21no—
35C4⋊C4×C263—22no—
36C4⋊C4×C263—21no—
37C4⋊C4×C263—21no—
38C4⋊C4×C263D412yes—
39C4⋊C4×C263D411yescharacteristic
40C4⋊C4×C263D411yescharacteristic
41C22⋊C4×C263—22no—
42C22⋊C4×C263—22no—
43C22⋊C4×C263—21no—
44C42×C263—22no—
45C42×C263—21no—
46C42×C263C2311yescharacteristic
47Dic526—42no—
48C2×Dic263—41no—
49C263⋊C8—41no—
50C263⋊C8—41no—
51C263⋊C8—41no—
52C23×C263—21no—
53Q8×C263—21no—
54Q8×C263—21no—
55Q8×C263—41no—
56Q8×C263—41no—
57D4×C263—21no—
58D4×C263—21no—
59C2×C4×C263—21no—
60C2×C4×C263—21no—
61C2×C4×C263—21no—
62C2×C4×C263—22no—
63C2×C4×C263—41no—
64C2×C4×C263—41no—
65C2×C4×C263C2×D411yescharacteristic
66C2×C4×C263C2×D411yescharacteristic
67C2×C4×C263C2×D411yescharacteristic
68C22×C263—41no—
69C22×C263—41no—
70C22×C263—41no—
71C22×C263#12011yescharacteristic
72C4×C263—21no—
73C4×C263—41no—
74C4×C263—41no—
75C4×C263—42no—
76C4×C263—41no—
77C4×C263#13511yescharacteristic
78C4×C263#13511yescharacteristic
79C4×C263#13711yescharacteristic
80C4×C263#13711yescharacteristic
81Dic263—81no—
82C2×C263—81no—
83C2×C263#45211yescharacteristic
84C2×C263#45211yescharacteristic
85C2×C263#45411yescharacteristic
86C263#163811yescharacteristic, complemented, Sylow
87#1638—2631nomaximal, Sylow
88#556D26311yescharacteristic
89#487—2631no—
90#477—2631no—
91#476—2631no—
92#332—2631no—
93Q8⋊C8—2631no—
94D4⋊C8—2631no—
95C4⋊Q8—2631no—
96C4⋊Q8C2×D26311yescharacteristic
97#126—22no—
98#124—21no—
99C22⋊Q8—21no—
100C4×Q8—21no—
101C4×Q8C2×D26311yescharacteristic
102C4×D4C2×D26311yescharacteristic
103#117—21no—
104#107—5261no—
105#106—5261no—
106C4⋊C8—2631no—
107C4⋊C8—2631no—
108#103—5261no—
109#103—5262no—
110#102—5261no—
111C4×C8—2631no—
112C2×Q8—21no—
113C2×Q8—5261no—
114C2×Q8C22⋊D26311yescharacteristic
115C2×D4C22⋊D26311yescharacteristic
116C22×C4—21no—
117C2×C8—5261no—
118C2×C8—5261no—
119C2×C8—5261no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—5261no—
124C4⋊C4—5261no—
125C4⋊C4C22⋊D26312yes—
126C4⋊C4C22⋊D26311yescharacteristic
127C4⋊C4C22⋊D26311yescharacteristic
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—21no—
131C42—22no—
132C42—21no—
133C42C22×D26311yescharacteristic
134C23—21no—
135Q8—21no—
136Q8—21no—
137Q8—41no—
138Q8—41no—
139Q8—10522no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—10521no—
149C2×C4C2×C22⋊D26311yescharacteristic
150C2×C4C2×C22⋊D26311yescharacteristic
151C2×C4C2×C22⋊D26311yescharacteristic
152C8—10521no—
153C8—10521no—
154C8—10521no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#2255443911yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—42no—
163C4—41no—
164C4—21041no—
165C4#2255449011yescharacteristic
166C4#2255449011yescharacteristic
167C4#2255450011yescharacteristic
168C4#2255450011yescharacteristic
169C2—81no—
170C2#3068210011yescharacteristic
171C2#3068210011yescharacteristic
172C2#3068211111yescharacteristic
subgroupG/H
centerC22#22554439
commutator, cosocleC42×C263C23
FrattiniC42C22×D263
Fitting#556×C263C2
socleC22×C263#120
prime radicalC2×D4×C263D4
hypercenter#556D263
hypocenterC263#1638
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. #30682290C2semidirect
2. #30682233C2semidirect
4. #30681684C2semidirect
5. Q8⋊(C263⋊C8)C2semidirect
86. C263#1638semidirect

Maximal subgroups

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

subgroupindex
#306822902
#306822332
#306822282
#306816842
Q8⋊(C263⋊C8)2
D4⋊(C263⋊C8)2
#556×C2632
#1638263