Finite Group Databasebeta

← #35213507

Subgroups — #35213507

Order 20864.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#23782246C211yesmaximal, characteristic
2#23781806C212yesmaximal, complemented
3#23781585C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#324×C163C211yesmaximal, characteristic, complemented
6C2×#14579699—21no—
7C22×Dic326C2211yescharacteristic
8C2×#14579688—22no—
9#19104715—22no—
10#19104710—22no—
11#19104709—21no—
12#19104709C2212yes—
13C2×Dic163⋊C4—21no—
14#19104660—22no—
15C163⋊#95—21no—
16C163⋊#95C2211yescharacteristic
17C2×C4⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yes—
19#14579699—42no—
20#14579699—42no—
21C2×Dic326—21no—
22C2×Dic326—22no—
23C2×Dic326—42no—
24C2×D652—21no—
25C2×D652—21no—
26C2×D652D412yes—
27C2×C22⋊D163—22no—
28C22×Dic163—21no—
29C22×Dic163—21no—
30C2×C4×D163—22no—
31#14579688—44no—
32C4⋊Dic163—21no—
33C4⋊Dic163D412yes—
34Dic163⋊C4—42no—
35(C2×D163)⋊C4—22no—
36(C2×D163)⋊C4—22no—
37C22×C4×C163D411yescharacteristic
38C22×C4×C163D411yescharacteristic
39C22×C4×C163C2311yescharacteristic
40C2×C8×C163—22no—
41C4⋊C4×C163—22no—
42C22×D163—22no—
43Dic326—24no—
44Dic326—42no—
45Dic326—42no—
46D652—42no—
47D652—42no—
48D652—81no—
49C22⋊D163—42no—
50C22⋊D163—42no—
51C2×Dic163—22no—
52C2×Dic163—22no—
53C2×Dic163—22no—
54C2×Dic163—41no—
55C2×Dic163—41no—
56C2×Dic163—41no—
57C2×Dic163—41no—
58C4×D163—44no—
59C23×C163C2×D411yescharacteristic
60C2×C4×C163—21no—
61C2×C4×C163—21no—
62C2×C4×C163—21no—
63C2×C4×C163—41no—
64C2×C4×C163—41no—
65C2×C4×C163SD1612yescomplemented
66C2×C4×C163C2×D412yes—
67C8×C163—42no—
68C22×C163—21no—
69C22×C163—21no—
70C22×C163#12011yescharacteristic
71C22×C163C2×SD1611yescharacteristic
72C22×C163AGL(1,ℤ/8)11yescharacteristic
73C2×D163—42no—
74C2×D163—42no—
75C2×D163—42no—
76C4×C163—22no—
77C4×C163—22no—
78C4×C163—41no—
79Dic163—42no—
80Dic163—42no—
81Dic163—42no—
82C2×C163—21no—
83C2×C163—21no—
84C2×C163#45011yescharacteristic
85C2×C163#45611yescharacteristic
86C2×C163B(4,𝔽2)11yescharacteristic
87D163—82no—
88C163#160411yescharacteristic, complemented, Sylow
89#1604—1631nomaximal, Sylow
90#537—1631no—
91#448—1632no—
92#395—1631no—
93#327—1632no—
94#324D16311yescharacteristic, complemented
95C2×#41—3261no—
96C22×Q8—1631no—
97C2×SD16—3262no—
98C22⋊Q8—3262no—
99#121—3262no—
100#121—3261no—
101#121—1632no—
102C2×C4⋊C4—3261no—
103C2×C4⋊C4C2×D16311yescharacteristic
104#103—3262no—
105C22⋊C8C2×D16312yes—
106#95—3261no—
107#95—1631no—
108#41—6522no—
109#41—6522no—
110C2×Q8—3261no—
111C2×Q8—3262no—
112C2×Q8—6522no—
113C2×D4—3261no—
114C2×D4—3261no—
115C2×D4—3262no—
116C2×D4—1632no—
117C22×C4—3262no—
118C22×C4—3261no—
119C22×C4—3261no—
120C22×C4D65211yescharacteristic
121C22×C4D65211yescharacteristic
122C22×C4C22×D16311yescharacteristic
123SD16—6524no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—3261no—
127C4⋊C4—6522no—
128C4⋊C4—1632no—
129C22⋊C4—3262no—
130C22⋊C4—3262no—
131C23—3262no—
132C23C2×D65211yescharacteristic
133Q8—3264no—
134Q8—6522no—
135Q8—6522no—
136D4—13041no—
137D4—6522no—
138D4—6522no—
139D4—6522no—
140D4—6522no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3262no—
147C2×C4—3262no—
148C2×C4—3262no—
149C2×C4—6524no—
150C2×C4—6521no—
151C2×C4—6521no—
152C2×C4—6521no—
153C2×C4—6521no—
154C2×C4#1457968812yescomplemented
155C2×C4D4×D16312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—6522no—
160C22—6522no—
161C22—6522no—
162C22#1910470511yescharacteristic
163C22C2×#1457968811yescharacteristic
164C22#1910476311yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—6522no—
169C4—6522no—
170C4—6522no—
171C2—21no—
172C2—21no—
173C2—13042no—
174C2#2378181811yescharacteristic
175C2#2378185011yescharacteristic
176C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
prime radical#23782246C2
hypercenter#324D163
hypocenterC163#1604
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #23781806C2semidirect
3. #23781585C2semidirect
4. #23781400C2semidirect
5. #324×C163C2semidirect
65. C2×C4×C163SD16semidirect
88. C163#1604semidirect
94. #324D163semidirect
154. C2×C4#14579688semidirect

Maximal subgroups

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

subgroupindex
#237822462
#237818062
#237815852
#237814002
#324×C1632
#1604163