Finite Group Databasebeta

← #35213821

Subgroups — #35213821

Order 20864.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#23782318C211yesmaximal, characteristic
2#23782061C211yesmaximal, characteristic, complemented
3#23781946C211yesmaximal, characteristic, complemented
4#23781871C211yesmaximal, characteristic, complemented
5#23781392C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic, complemented
7#332×C163C211yesmaximal, characteristic, complemented
8C2×#14579700—21no—
9C2×#14579688—21no—
10#19104735C2211yescharacteristic
11#19104720—22no—
12#19104711—21no—
13#19104710—21no—
14C4×D652C2211yescharacteristic
15C4×D652C2211yescharacteristic
16C4⋊C4×D163—21no—
17#19104669—21no—
18#19104666—22no—
19#19104660—21no—
20#19104657—21no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25#14579700—42no—
26#14579700—42no—
27C2×Dic326—21no—
28C2×D652—22no—
29C2×D652D411yescharacteristic
30C2×D652D411yescharacteristic
31C2×C4×D163—22no—
32C2×C4×D163—21no—
33C2×C4×D163—21no—
34#14579688—42no—
35C2×C163⋊C8—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163—21no—
38C4⋊Dic163D411yescharacteristic
39C4⋊Dic163D411yescharacteristic
40Dic163⋊C4—22no—
41(C2×D163)⋊C4—22no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44C2×Q8×C163—21no—
45C2×C8×C163—21no—
46C2×C8×C163—21no—
47C4⋊C4×C163—21no—
48C4⋊C4×C163D412yes—
49C42×C163C2311yescharacteristic
50C22×D163—21no—
51C22×D163—21no—
52Dic326—41no—
53Dic326—41no—
54D652—21no—
55D652—21no—
56D652—21no—
57D652—44no—
58D652SD1612yescomplemented
59C2×Dic163—21no—
60C2×Dic163—21no—
61C2×Dic163—41no—
62C4×D163—44no—
63C4×D163—41no—
64C4×D163—41no—
65C4×D163—41no—
66C4×D163—41no—
67C163⋊C8—22no—
68Q8×C163—41no—
69Q8×C163—41no—
70C2×C4×C163—22no—
71C2×C4×C163C2×D411yescharacteristic
72C2×C4×C163C2×D411yescharacteristic
73C2×C4×C163C2×D411yescharacteristic
74C8×C163—41no—
75C8×C163—41no—
76C22×C163#12011yescharacteristic
77C2×D163—22no—
78C2×D163—41no—
79C2×D163—41no—
80C2×D163—41no—
81C2×D163—41no—
82C2×D163—41no—
83C4×C163—21no—
84C4×C163—42no—
85C4×C163C2×SD1611yescharacteristic
86C4×C163#13511yescharacteristic
87C4×C163AGL(1,ℤ/8)11yescharacteristic
88C4×C163AGL(1,ℤ/8)11yescharacteristic
89Dic163—41no—
90Dic163—41no—
91Dic163—81no—
92C2×C163#44911yescharacteristic
93C2×C163#45011yescharacteristic
94C2×C163#45611yescharacteristic
95D163—42no—
96D163—81no—
97C163#165411yescharacteristic, complemented, Sylow
98#1654—1631nomaximal, Sylow
99#547—1631no—
100#498—1631no—
101#474—1631no—
102#461—1631no—
103#332D16311yescharacteristic, complemented
104D4⋊C8—1631no—
105D4⋊C8—1631no—
106C2×#41—3261no—
107C2×SD16—3261no—
108C4⋊Q8—1631no—
109C4⋊Q8C2×D16311yescharacteristic
110#123—3262no—
111C22⋊Q8—3261no—
112#121—3261no—
113C4×D4—1631no—
114C4×D4—1631no—
115C2×C4⋊C4—3261no—
116#107—3261no—
117#106—3262no—
118C4⋊C8C2×D16311yescharacteristic
119C4⋊C8C2×D16311yescharacteristic
120#103—3261no—
121#102—3261no—
122C4×C8—1631no—
123#41—6522no—
124#41—6522no—
125C2×Q8—21no—
126C2×Q8—3261no—
127C2×D4—3262no—
128C2×D4—1631no—
129C2×D4—1631no—
130C22×C4—3262no—
131C22×C4—3261no—
132C22×C4—3261no—
133SD16—6522no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—3261no—
137C4⋊C4—21no—
138C4⋊C4—3262no—
139C4⋊C4—3261no—
140C4⋊C4—3261no—
141C4⋊C4—1631no—
142C4⋊C4—1631no—
143C4⋊C4D65212yes—
144C22⋊C4—3262no—
145C22⋊C4—3261no—
146C22⋊C4—3261no—
147C42C22×D16311yescharacteristic
148C23—3261no—
149C23—3261no—
150Q8—41no—
151Q8—41no—
152Q8—6521no—
153Q8—6521no—
154D4—3261no—
155D4—3261no—
156D4—3261no—
157D4—6524no—
158D4—1632no—
159C2×C4—22no—
160C2×C4—3261no—
161C2×C4—3261no—
162C2×C4—6524no—
163C2×C4—6521no—
164C2×C4—6521no—
165C2×C4—6521no—
166C2×C4—6521no—
167C2×C4—6521no—
168C2×C4D4×D16311yescharacteristic
169C2×C4D4×D16311yescharacteristic
170C2×C4C2×D65211yescharacteristic
171C8—41no—
172C8—41no—
173C8—3262no—
174C22—3262no—
175C22—6521no—
176C22—6521no—
177C22—6521no—
178C22—6521no—
179C22—6521no—
180C22#1910470511yescharacteristic
181C4—21no—
182C4—42no—
183C4—13041no—
184C4—6521no—
185C4—6521no—
186C4SD16×D16311yescharacteristic
187C4#1910475811yescharacteristic
188C4#1910475911yescharacteristic
189C4#1910476211yescharacteristic
190C2—13041no—
191C2—6522no—
192C2#2378181311yescharacteristic
193C2#2378181711yescharacteristic
194C2#2378185011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#332×C163C2
socleC22×C163#120
prime radical#23782318C2
hypercenter#332D163
hypocenterC163#1654
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #23782061C2semidirect
3. #23781946C2semidirect
4. #23781871C2semidirect
5. #23781392C2semidirect
6. #23781392C2semidirect
7. #332×C163C2semidirect
58. D652SD16semidirect
97. C163#1654semidirect
103. #332D163semidirect

Maximal subgroups

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

subgroupindex
#237823182
#237820612
#237819462
#237818712
#237813922
#332×C1632
#1654163