Finite Group Databasebeta

← #35213868

Subgroups — #35213868

Order 20864.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#23782419C211yesmaximal, characteristic
2#23782033C211yesmaximal, characteristic, complemented
3#23782027C211yesmaximal, characteristic
4#23781407C211yesmaximal, characteristic, complemented
5#23781406C211yesmaximal, characteristic, complemented
6#23781404C211yesmaximal, characteristic
7#501×C163C211yesmaximal, characteristic, complemented
8#19104734—21no—
9#19104730—21no—
10#19104730—21no—
11#19104727C2211yescharacteristic
12#19104723C2211yescharacteristic
13#19104720—21no—
14#19104715—21no—
15#19104713—21no—
16#19104659—21no—
17#19104659—21no—
18#19104658—21no—
19#19104655—21no—
20#19104646C2211yescharacteristic
21#19104646C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23#125×C163C2211yescharacteristic
24#107×C163—21no—
25#106×C163—21no—
26#97×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326D411yescharacteristic
29C2×D652D411yescharacteristic
30C2×C4×D163—21no—
31C2×C163⋊C8—21no—
32C2×C163⋊C8—21no—
33C4⋊Dic163—21no—
34C4⋊Dic163D411yescharacteristic
35C4⋊Dic163D411yescharacteristic
36Dic163⋊C4—21no—
37Dic163⋊C4—21no—
38Dic163⋊C4—21no—
39(C2×D163)⋊C4—21no—
40(C2×D163)⋊C4—21no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44C4×Dic163—21no—
45C2×Q8×C163—21no—
46C2×C8×C163—21no—
47C4⋊C4×C163—21no—
48C4⋊C4×C163—21no—
49C4⋊C4×C163—21no—
50C4⋊C4×C163—21no—
51C4⋊C4×C163D411yescharacteristic
52C4⋊C4×C163D411yescharacteristic
53C42×C163C2311yescharacteristic
54C22×D163—21no—
55Dic326—41no—
56Dic326—81no—
57D652—41no—
58C2×Dic163—21no—
59C2×Dic163—21no—
60C2×Dic163—21no—
61C4×D163—81no—
62C163⋊C8—41no—
63C163⋊C8—41no—
64Q8×C163—42no—
65C2×C4×C163—21no—
66C2×C4×C163—21no—
67C2×C4×C163—41no—
68C2×C4×C163C2×D411yescharacteristic
69C2×C4×C163C2×D411yescharacteristic
70C2×C4×C163C2×D411yescharacteristic
71C8×C163—22no—
72C22×C163#12011yescharacteristic
73C2×D163—41no—
74C2×D163—41no—
75C2×D163—41no—
76C4×C163—21no—
77C4×C163—21no—
78C4×C163—21no—
79C4×C163—41no—
80C4×C163—41no—
81C4×C163—81no—
82Dic163—41no—
83Dic163—41no—
84Dic163—41no—
85C2×C163#45411yescharacteristic
86C2×C163#45511yescharacteristic
87C2×C163#45611yescharacteristic
88D163—81no—
89C163#166411yescharacteristic, complemented, Sylow
90#1664—1631nomaximal, Sylow
91#563—1631no—
92#501D16311yescharacteristic, complemented
93#491—1631no—
94#489—1631no—
95#330—1631no—
96#330—1631no—
97#329—1631no—
98C4⋊Q8—3261no—
99C4⋊Q8C2×D16311yescharacteristic
100#126—3261no—
101#126—3261no—
102#125—1631no—
103#125C2×D16311yescharacteristic
104#124—1631no—
105#123—3261no—
106C22⋊Q8—3261no—
107C22⋊Q8—3261no—
108#107—21no—
109#106—21no—
110#103—3261no—
111#103—3261no—
112#103—3261no—
113#102—3261no—
114#97—1631no—
115#97—1631no—
116#97C2×D16311yescharacteristic
117C2×Q8—21no—
118C2×Q8—3261no—
119C2×Q8—1631no—
120C2×D4—1631no—
121C22×C4—3261no—
122C2×C8—21no—
123C2×C8—3261no—
124C2×C8—3261no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—3261no—
130C4⋊C4—3261no—
131C4⋊C4—3261no—
132C4⋊C4—3261no—
133C4⋊C4—1631no—
134C4⋊C4—1631no—
135C4⋊C4C22⋊D16311yescharacteristic
136C4⋊C4C22⋊D16311yescharacteristic
137C22⋊C4—3261no—
138C22⋊C4—3261no—
139C22⋊C4—3261no—
140C22⋊C4—3261no—
141C22⋊C4—3261no—
142C42—3261no—
143C42C22×D16311yescharacteristic
144C23—3261no—
145Q8—42no—
146Q8—13041no—
147Q8—6521no—
148D4—6521no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—13041no—
153C2×C4—3261no—
154C2×C4—3261no—
155C2×C4—3261no—
156C2×C4D4×D16311yescharacteristic
157C2×C4D4×D16311yescharacteristic
158C2×C4C2×C22⋊D16311yescharacteristic
159C8—22no—
160C8—6521no—
161C8—6521no—
162C22—6521no—
163C22—6521no—
164C22—6521no—
165C22#1910470311yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—6521no—
173C4—6521no—
174C4—6521no—
175C2—13041no—
176C2#2378183611yescharacteristic
177C2#2378184111yescharacteristic
178C2#2378184711yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#501×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#501D163
hypocenterC163#1664
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #23782033C2semidirect
4. #23781407C2semidirect
5. #23781406C2semidirect
7. #501×C163C2semidirect
89. C163#1664semidirect
92. #501D163semidirect

Maximal subgroups

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

subgroupindex
#237824192
#237820332
#237820272
#237814072
#237814062
#237814042
#501×C1632
#1664163