Finite Group Databasebeta

← #35213698

Subgroups — #35213698

Order 20864.

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

normal subgroups only

This group has 228 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#23782330C211yesmaximal, characteristic
2#23781957C211yesmaximal, characteristic, complemented
3#23781952C211yesmaximal, characteristic
4#23781411C211yesmaximal, characteristic, complemented
5#23781397C211yesmaximal, characteristic, complemented
6#23781394C211yesmaximal, characteristic
7#487×C163C211yesmaximal, characteristic, complemented
8C2×Q8×D163—21no—
9#19104734—21no—
10#19104713—22no—
11#19104711—21no—
12C4×Dic326C2211yescharacteristic
13C4×D652C2211yescharacteristic
14#19104697—21no—
15#19104668—21no—
16#19104666—21no—
17C4⋊(C163⋊C8)C2211yescharacteristic
18C4⋊(C163⋊C8)C2211yescharacteristic
19#19104658—21no—
20#19104655—21no—
21C4⋊Q8×C163C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23#103×C163—22no—
24C4×C8×C163C2211yescharacteristic
25C2×Dic326—22no—
26C2×Dic326D411yescharacteristic
27Q8×D163—42no—
28Q8×D163—42no—
29C2×D652D411yescharacteristic
30C2×C4×D163—22no—
31C2×C4×D163—21no—
32C2×C163⋊C8—21no—
33C2×C163⋊C8—21no—
34C4⋊Dic163D411yescharacteristic
35C4⋊Dic163D411yescharacteristic
36Dic163⋊C4—22no—
37Dic163⋊C4—21no—
38(C2×D163)⋊C4—22no—
39(C2×D163)⋊C4—21no—
40C4×Dic163—21no—
41C2×Q8×C163—21no—
42C2×Q8×C163—21no—
43C2×C8×C163—21no—
44C4⋊C4×C163—21no—
45C4⋊C4×C163—21no—
46C4⋊C4×C163—21no—
47C4⋊C4×C163D412yes—
48C42×C163C2311yescharacteristic
49C22×D163—21no—
50Dic326—21no—
51Dic326—44no—
52Dic326SD1612yes—
53D652—21no—
54D652Q1612yes—
55C2×Dic163—21no—
56C2×Dic163—21no—
57C2×Dic163—21no—
58C4×D163—44no—
59C4×D163—41no—
60C4×D163—41no—
61C163⋊C8—41no—
62C163⋊C8—41no—
63Q8×C163—42no—
64Q8×C163—41no—
65Q8×C163—41no—
66C2×C4×C163—22no—
67C2×C4×C163—41no—
68C2×C4×C163C2×D411yescharacteristic
69C2×C4×C163C2×D411yescharacteristic
70C2×C4×C163C2×D411yescharacteristic
71C8×C163—41no—
72C22×C163#12011yescharacteristic
73C2×D163—22no—
74C2×D163—41no—
75C2×D163—41no—
76C4×C163—21no—
77C4×C163—42no—
78C4×C163—81no—
79C4×C163C2×SD1611yescharacteristic
80C4×C163C2×Q1611yescharacteristic
81C4×C163AGL(1,ℤ/8)11yescharacteristic
82C4×C163#13711yescharacteristic
83Dic163—22no—
84Dic163—41no—
85Dic163—41no—
86C2×C163#45011yescharacteristic
87C2×C163#45111yescharacteristic
88C2×C163#45411yescharacteristic
89D163—42no—
90C163#163711yescharacteristic, complemented, Sylow
91#1637—1631nomaximal, Sylow
92D4×Q8—1631no—
93#487D16311yescharacteristic, complemented
94#477—1631no—
95#476—1631no—
96#332—1631no—
97Q8⋊C8—1631no—
98D4⋊C8—1631no—
99C22×Q8—3261no—
100C4⋊Q8—3261no—
101C4⋊Q8C2×D16311yescharacteristic
102C4⋊Q8C2×D16311yescharacteristic
103C22⋊Q8—3261no—
104C22⋊Q8—3262no—
105C4×Q8—1631no—
106C4×D4—3261no—
107C4×D4—1631no—
108#107—3261no—
109#106—3261no—
110C4⋊C8—1631no—
111C4⋊C8—1631no—
112#103—22no—
113#103—3261no—
114#102—3261no—
115C4×C8C2×D16311yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—3262no—
119C2×Q8—6522no—
120C2×Q8—6522no—
121C2×Q8—1631no—
122C2×D4—1631no—
123C22×C4—3262no—
124C22×C4—3261no—
125C2×C8—21no—
126C2×C8—3261no—
127C2×C8—3261no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3262no—
132C4⋊C4—3261no—
133C4⋊C4—1631no—
134C4⋊C4—1631no—
135C4⋊C4C22⋊D16312yes—
136C22⋊C4—3262no—
137C22⋊C4—3261no—
138C42—3261no—
139C42C22×D16311yescharacteristic
140C23—3261no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—3261no—
145Q8—6524no—
146Q8—1632no—
147D4—3261no—
148D4—1632no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—3261no—
152C2×C4—3261no—
153C2×C4—3261no—
154C2×C4—6524no—
155C2×C4—6521no—
156C2×C4—6521no—
157C2×C4D4×D16311yescharacteristic
158C2×C4D4×D16311yescharacteristic
159C2×C4C2×C22⋊D16311yescharacteristic
160C8—41no—
161C8—6521no—
162C8—6521no—
163C22—3262no—
164C22—6521no—
165C22—6521no—
166C22#1910470311yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—3262no—
171C4—6521no—
172C4—6521no—
173C4SD16×D16311yescharacteristic
174C4Q16×D16311yescharacteristic
175C4#1910476211yescharacteristic
176C4#1910476711yescharacteristic
177C2—6522no—
178C2#2378181411yescharacteristic
179C2#2378181911yescharacteristic
180C2#2378183711yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#487×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#487D163
hypocenterC163#1637
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #23781957C2semidirect
4. #23781411C2semidirect
5. #23781397C2semidirect
7. #487×C163C2semidirect
90. C163#1637semidirect
93. #487D163semidirect

Maximal subgroups

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

subgroupindex
#237823302
#237819572
#237819522
#237814112
#237813972
#237813942
#487×C1632
#1637163