Finite Group Databasebeta

← #35213667

Subgroups — #35213667

Order 20864.

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

normal subgroups only

This group has 214 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#23781957C211yesmaximal, characteristic, complemented
3#23781952C211yesmaximal, characteristic
4#23781414C211yesmaximal, characteristic, complemented
5#23781397C211yesmaximal, characteristic, complemented
6#23781394C211yesmaximal, characteristic
7#494×C163C211yesmaximal, characteristic, complemented
8#19104726—21no—
9#19104715—21no—
10#19104711—22no—
11C4×Dic326C2211yescharacteristic
12C4×D652C2211yescharacteristic
13#19104697—21no—
14C4⋊C4×D163—21no—
15#19104668—21no—
16#19104666—21no—
17C4⋊(C163⋊C8)C2211yescharacteristic
18C4⋊(C163⋊C8)C2211yescharacteristic
19#19104658—21no—
20#19104655—21no—
21C2×Q16×C163—22no—
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊Q8×C163C2211yescharacteristic
24C4×C8×C163C2211yescharacteristic
25C2×Dic326D411yescharacteristic
26C2×D652D411yescharacteristic
27C2×C4×D163—22no—
28C2×C4×D163—21no—
29C2×C163⋊C8—21no—
30C2×C163⋊C8—21no—
31C4⋊Dic163—22no—
32C4⋊Dic163D411yescharacteristic
33C4⋊Dic163D411yescharacteristic
34Dic163⋊C4—22no—
35Dic163⋊C4—21no—
36(C2×D163)⋊C4—22no—
37(C2×D163)⋊C4—21no—
38C4×Dic163—21no—
39C2×Q8×C163—21no—
40C2×Q8×C163D412yes—
41Q16×C163—44no—
42C2×C8×C163—21no—
43C4⋊C4×C163—21no—
44C4⋊C4×C163—21no—
45C4⋊C4×C163—21no—
46C4⋊C4×C163—21no—
47C42×C163C2311yescharacteristic
48C22×D163—21no—
49Dic326—21no—
50Dic326—21no—
51D652—21no—
52D652Q1612yescomplemented
53C2×Dic163—21no—
54C2×Dic163—21no—
55C2×Dic163—21no—
56C4×D163—41no—
57C4×D163—41no—
58C4×D163—82no—
59C163⋊C8—41no—
60C163⋊C8—41no—
61Q8×C163—42no—
62Q8×C163—42no—
63C2×C4×C163—22no—
64C2×C4×C163—41no—
65C2×C4×C163C2×D411yescharacteristic
66C2×C4×C163C2×D411yescharacteristic
67C2×C4×C163C2×D411yescharacteristic
68C8×C163—22no—
69C22×C163#12011yescharacteristic
70C2×D163—22no—
71C2×D163—41no—
72C2×D163—41no—
73C4×C163—21no—
74C4×C163—42no—
75C4×C163—81no—
76C4×C163C2×Q1611yescharacteristic
77C4×C163#13511yescharacteristic
78C4×C163#13711yescharacteristic
79C4×C163#13711yescharacteristic
80Dic163—41no—
81Dic163—41no—
82Dic163—41no—
83C2×C163#45111yescharacteristic
84C2×C163#45211yescharacteristic
85C2×C163#45311yescharacteristic
86D163—42no—
87C163#163211yescharacteristic, complemented, Sylow
88#1632—1631nomaximal, Sylow
89#554—1631no—
90#494D16311yescharacteristic, complemented
91#477—1631no—
92#476—1631no—
93#333—1631no—
94Q8⋊C8—1631no—
95D4⋊C8—1631no—
96C2×Q16—22no—
97C4⋊Q8C2×D16311yescharacteristic
98C4⋊Q8C2×D16311yescharacteristic
99#125—3261no—
100C22⋊Q8—3261no—
101C22⋊Q8—3262no—
102C4×Q8—1631no—
103C4×D4—3261no—
104C4×D4—1631no—
105C2×C4⋊C4—3261no—
106#107—3261no—
107#106—3261no—
108C4⋊C8—1631no—
109C4⋊C8—1631no—
110#103—3261no—
111#102—3261no—
112C4×C8C2×D16311yescharacteristic
113C2×Q8—21no—
114C2×Q8—1631no—
115C2×Q8C22⋊D16312yes—
116C2×D4—1631no—
117C22×C4—3262no—
118C22×C4—3261no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—3261no—
122C2×C8—3261no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3262no—
128C4⋊C4—3262no—
129C4⋊C4—3261no—
130C4⋊C4—1631no—
131C4⋊C4—1631no—
132C22⋊C4—3262no—
133C22⋊C4—3261no—
134C42—3261no—
135C42C22×D16311yescharacteristic
136C23—3261no—
137Q8—42no—
138Q8—42no—
139Q8—3261no—
140Q8—3261no—
141D4—3261no—
142D4—1632no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—13042no—
146C2×C4—3261no—
147C2×C4—3261no—
148C2×C4—3261no—
149C2×C4—6521no—
150C2×C4—6521no—
151C2×C4D4×D16311yescharacteristic
152C2×C4D4×D16311yescharacteristic
153C2×C4C2×C22⋊D16311yescharacteristic
154C8—22no—
155C8—6521no—
156C8—6521no—
157C22—3262no—
158C22—6521no—
159C22—6521no—
160C22#1910470311yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—6521no—
165C4—6521no—
166C4—6521no—
167C4Q16×D16311yescharacteristic
168C4#1910475811yescharacteristic
169C4#1910476711yescharacteristic
170C4#1910476711yescharacteristic
171C2—6522no—
172C2#2378181911yescharacteristic
173C2#2378182611yescharacteristic
174C2#2378183211yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#494×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#494D163
hypocenterC163#1632
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #23781957C2semidirect
4. #23781414C2semidirect
5. #23781397C2semidirect
7. #494×C163C2semidirect
52. D652Q16semidirect
87. C163#1632semidirect
90. #494D163semidirect

Maximal subgroups

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

subgroupindex
#237823762
#237819572
#237819522
#237814142
#237813972
#237813942
#494×C1632
#1632163