Finite Group Databasebeta

← #35213744

Subgroups — #35213744

Order 20864.

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

normal subgroups only

This group has 208 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#23782376C211yesmaximal, characteristic
2#23782015C211yesmaximal, characteristic, complemented
3#23781951C211yesmaximal, characteristic, complemented
4#23781871C211yesmaximal, characteristic
5#23781395C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic
7#333×C163C211yesmaximal, characteristic, complemented
8C2×#14579688—21no—
9#19104735C2211yescharacteristic
10#19104726—21no—
11#19104715—21no—
12#19104711—22no—
13C4×Dic326C2211yescharacteristic
14C4×D652C2211yescharacteristic
15#19104697—21no—
16C4⋊C4×D163—21no—
17#19104667—21no—
18#19104660—21no—
19#19104660—21no—
20#19104659—22no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25C2×Dic326—21no—
26C2×Dic326D411yescharacteristic
27C2×D652D411yescharacteristic
28C2×C4×D163—22no—
29C2×C4×D163—21no—
30#14579688—42no—
31C2×C163⋊C8—21no—
32C4⋊Dic163—22no—
33C4⋊Dic163—21no—
34C4⋊Dic163—21no—
35C4⋊Dic163D411yescharacteristic
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—22no—
38Dic163⋊C4—21no—
39(C2×D163)⋊C4—22no—
40(C2×D163)⋊C4—21no—
41C4×Dic163—21no—
42C2×Q8×C163D412yes—
43C2×C8×C163—21no—
44C2×C8×C163—21no—
45C4⋊C4×C163—21no—
46C4⋊C4×C163—21no—
47C42×C163C2311yescharacteristic
48C22×D163—21no—
49Dic326—21no—
50Dic326—21no—
51Dic326—41no—
52Dic326—41no—
53D652—21no—
54D652SD1612yes—
55C2×Dic163—21no—
56C2×Dic163—21no—
57C2×Dic163—21no—
58C2×Dic163—41no—
59C4×D163—41no—
60C4×D163—41no—
61C4×D163—82no—
62C163⋊C8—41no—
63Q8×C163—42no—
64C2×C4×C163—22no—
65C2×C4×C163C2×D411yescharacteristic
66C2×C4×C163C2×D411yescharacteristic
67C2×C4×C163C2×D411yescharacteristic
68C8×C163—41no—
69C8×C163—41no—
70C22×C163#12011yescharacteristic
71C2×D163—22no—
72C2×D163—41no—
73C2×D163—41no—
74C4×C163—21no—
75C4×C163—42no—
76C4×C163C2×SD1611yescharacteristic
77C4×C163#13511yescharacteristic
78C4×C163#13711yescharacteristic
79C4×C163#13711yescharacteristic
80Dic163—41no—
81Dic163—41no—
82Dic163—41no—
83Dic163—81no—
84C2×C163#44811yescharacteristic
85C2×C163#45211yescharacteristic
86C2×C163#45511yescharacteristic
87D163—42no—
88C163#164311yescharacteristic, complemented, Sylow
89#1643—1631nomaximal, Sylow
90#554—1631no—
91#487—1631no—
92#475—1631no—
93#461—1631no—
94#333D16311yescharacteristic, complemented
95Q8⋊C8—1631no—
96D4⋊C8—1631no—
97C2×SD16—3261no—
98C4⋊Q8—1631no—
99C4⋊Q8C2×D16311yescharacteristic
100#125—3261no—
101C22⋊Q8—3261no—
102C22⋊Q8—3262no—
103C4×Q8—1631no—
104C4×D4—3261no—
105C4×D4—1631no—
106C2×C4⋊C4—3261no—
107#106—3261no—
108C4⋊C8C2×D16311yescharacteristic
109C4⋊C8C2×D16311yescharacteristic
110#103—3261no—
111#103—3262no—
112#103—3261no—
113C4×C8—1631no—
114C2×Q8—3261no—
115C2×Q8—1631no—
116C2×Q8D65212yes—
117C2×D4—1631no—
118C22×C4—3262no—
119C22×C4—3261no—
120SD16—6522no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3261no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3262no—
127C4⋊C4—3262no—
128C4⋊C4—3261no—
129C4⋊C4—3261no—
130C4⋊C4—3261no—
131C4⋊C4—1631no—
132C4⋊C4—1631no—
133C22⋊C4—3262no—
134C22⋊C4—3261no—
135C42—3261no—
136C42C22×D16311yescharacteristic
137C23—3261no—
138Q8—42no—
139Q8—3261no—
140Q8—3261no—
141Q8—6521no—
142Q8—6521no—
143D4—3261no—
144D4—1632no—
145C2×C4—22no—
146C2×C4—13042no—
147C2×C4—3261no—
148C2×C4—3261no—
149C2×C4—3261no—
150C2×C4—6521no—
151C2×C4—6521no—
152C2×C4—6521no—
153C2×C4D4×D16311yescharacteristic
154C2×C4D4×D16311yescharacteristic
155C2×C4C2×D65211yescharacteristic
156C8—41no—
157C8—41no—
158C8—6521no—
159C22—3262no—
160C22—6521no—
161C22—6521no—
162C22#1910470511yescharacteristic
163C4—21no—
164C4—42no—
165C4—13041no—
166C4—6521no—
167C4—6521no—
168C4—6521no—
169C4SD16×D16311yescharacteristic
170C4#1910475611yescharacteristic
171C4#1910476711yescharacteristic
172C4#1910476711yescharacteristic
173C2—6522no—
174C2#2378180511yescharacteristic
175C2#2378183011yescharacteristic
176C2#2378184511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#333×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#333D163
hypocenterC163#1643
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #23782015C2semidirect
3. #23781951C2semidirect
5. #23781395C2semidirect
7. #333×C163C2semidirect
88. C163#1643semidirect
94. #333D163semidirect

Maximal subgroups

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

subgroupindex
#237823762
#237820152
#237819512
#237818712
#237813952
#237813922
#333×C1632
#1643163