Finite Group Databasebeta

← #35213558

Subgroups — #35213558

Order 20864.

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

normal subgroups only

This group has 288 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#23782227C211yesmaximal, characteristic
2#23781984C212yesmaximal, complemented
3#23781601C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#323×C163C211yesmaximal, characteristic, complemented
6C22×C22⋊D163—21no—
7#19104719—22no—
8#19104709—21no—
9#19104709C2212yes—
10#19104705—22no—
11C2×C4⋊Dic163C2211yescharacteristic, complemented
12C2×Dic163⋊C4—21no—
13C2×C22⋊Dic163—21no—
14#19104667—22no—
15#19104657—22no—
16C163⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yescomplemented
19C23×D163—21no—
20C2×D652—21no—
21C2×D652D412yes—
22C2×C22⋊D163—22no—
23C2×C22⋊D163—22no—
24C2×C22⋊D163—42no—
25C2×C22⋊D163—42no—
26C22×Dic163—21no—
27C22×Dic163—21no—
28C22×Dic163—21no—
29C4⋊Dic163—21no—
30C4⋊Dic163—22no—
31C4⋊Dic163D412yes—
32Dic163⋊C4—42no—
33C22⋊Dic163—42no—
34(C2×D163)⋊C4—22no—
35(C2×D163)⋊C4—22no—
36C24×C163D411yescharacteristic
37C22×C4×C163D411yescharacteristic
38C22×C4×C163C2311yescharacteristic
39C2×C8×C163—22no—
40C22⋊C4×C163—22no—
41C22×D163—22no—
42C22×D163—42no—
43C22×D163—81no—
44C22×D163—81no—
45D652—42no—
46D652—81no—
47C22⋊D163—44no—
48C22⋊D163—42no—
49C22⋊D163—42no—
50C2×Dic163—22no—
51C2×Dic163—22no—
52C2×Dic163—41no—
53C2×Dic163—42no—
54C2×Dic163—41no—
55C2×Dic163—41no—
56C2×Dic163—41no—
57C2×Dic163—41no—
58C23×C163—21no—
59C23×C163—21no—
60C23×C163—41no—
61C23×C163—41no—
62C23×C163SD1612yes—
63C23×C163C2×D411yescharacteristic
64C2×C4×C163—21no—
65C2×C4×C163—41no—
66C2×C4×C163—41no—
67C2×C4×C163C2×D412yes—
68C8×C163—42no—
69C22×C163—21no—
70C22×C163—21no—
71C22×C163—22no—
72C22×C163—42no—
73C22×C163—41no—
74C22×C163—41no—
75C22×C163—81no—
76C22×C163#12011yescharacteristic
77C22×C163C2×SD1611yescharacteristic
78C22×C163AGL(1,ℤ/8)11yescharacteristic
79C2×D163—42no—
80C2×D163—42no—
81C2×D163—42no—
82C2×D163—84no—
83C4×C163—22no—
84C4×C163—41no—
85Dic163—42no—
86Dic163—42no—
87Dic163—81no—
88C2×C163—21no—
89C2×C163—21no—
90C2×C163—42no—
91C2×C163#45011yescharacteristic
92C2×C163#45311yescharacteristic
93C2×C163#45811yescharacteristic
94D163—82no—
95C163#161311yescharacteristic, complemented, Sylow
96#1613—1631nomaximal, Sylow
97#534—1631no—
98#481—1632no—
99#399—1631no—
100#327—1632no—
101#323D16311yescharacteristic, complemented
102C22×D4—3261no—
103#123—3262no—
104#121—3261no—
105#121—1632no—
106#120—3262no—
107C2×C4⋊C4—3261no—
108C2×C4⋊C4—1631no—
109C2×C22⋊C4—3261no—
110C2×C22⋊C4C2×D16311yescharacteristic
111#106—3262no—
112#102—3262no—
113C22⋊C8C2×D16312yescomplemented
114#95—1631no—
115C24—3261no—
116C24D65211yescharacteristic
117C2×D4—3261no—
118C2×D4—3262no—
119C2×D4—3262no—
120C2×D4—6522no—
121C2×D4—6522no—
122C2×D4—1632no—
123C22×C4—3261no—
124C22×C4—3261no—
125C22×C4—3261no—
126C22×C4D65211yescharacteristic
127C22×C4C22×D16311yescharacteristic
128C2×C8—22no—
129C4⋊C4—3261no—
130C4⋊C4—3262no—
131C4⋊C4—6522no—
132C4⋊C4—1632no—
133C22⋊C4—22no—
134C22⋊C4—3262no—
135C22⋊C4—3262no—
136C22⋊C4—6522no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—13041no—
142C23—13041no—
143C23—3262no—
144C23—6522no—
145C23#1457968812yes—
146C23C2×D65211yescharacteristic
147D4—13041no—
148D4—6522no—
149D4—6524no—
150D4—6522no—
151D4—6522no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—3262no—
156C2×C4—3262no—
157C2×C4—6521no—
158C2×C4—6521no—
159C2×C4—6521no—
160C2×C4—6522no—
161C2×C4—6521no—
162C2×C4—6521no—
163C2×C4D4×D16312yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—13044no—
173C22—6522no—
174C22—6522no—
175C22—6522no—
176C22#1910470511yescharacteristic
177C22C2×#1457968811yescharacteristic
178C22#1910476311yescharacteristic
179C4—22no—
180C4—41no—
181C4—13041no—
182C4—6522no—
183C4—6522no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—13042no—
188C2#2378181811yescharacteristic
189C2#2378183511yescharacteristic
190C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#323×C163C2
socleC22×C163#120
prime radical#23782227C2
hypercenter#323D163
hypocenterC163#1613
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #23781984C2semidirect
3. #23781601C2semidirect
4. #23781400C2semidirect
5. #323×C163C2semidirect
11. C2×C4⋊Dic163C22semidirect
16. C163⋊#95C22semidirect
18. C22⋊C8×C163C22semidirect
95. C163#1613semidirect
101. #323D163semidirect
113. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822272
#237819842
#237816012
#237814002
#323×C1632
#1613163