Finite Group Databasebeta

← #35213500

Subgroups — #35213500

Order 20864.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782260C211yesmaximal, characteristic
2#23781823C211yesmaximal, characteristic, complemented
3#23781803C211yesmaximal, characteristic
4#23781586C211yesmaximal, characteristic, complemented
5#23781401C211yesmaximal, characteristic, complemented
6#23781390C211yesmaximal, characteristic, complemented
7#327×C163C211yesmaximal, characteristic
8C22×Dic326C2211yescharacteristic
9C2×Dic652—21no—
10C2×#14579687—21no—
11#19104728—21no—
12#19104725—21no—
13#19104716—21no—
14#19104716—21no—
15#19104714C2211yescharacteristic
16#19104712—21no—
17D4×Dic163—21no—
18C2×C4⋊Dic163—21no—
19C2×Dic163⋊C4C2211yescharacteristic
20#19104660—21no—
21#19104658—21no—
22C22⋊(C163⋊C8)C2211yescharacteristic
23C163⋊#95—21no—
24#121×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326—21no—
29C2×Dic326—42no—
30C2×Dic326D411yescharacteristic
31C22×Dic163—21no—
32C22×Dic163—21no—
33C22×Dic163D411yescharacteristic
34C22×Dic163D411yescharacteristic
35Dic652—42no—
36#14579687—42no—
37C2×C163⋊C8—21no—
38C4⋊Dic163—21no—
39C4⋊Dic163—42no—
40C4⋊Dic163D411yescharacteristic
41Dic163⋊C4—21no—
42Dic163⋊C4—21no—
43Dic163⋊C4—21no—
44Dic163⋊C4—21no—
45Dic163⋊C4—21no—
46C22⋊Dic163—21no—
47C22⋊Dic163—21no—
48C22⋊Dic163—21no—
49C22⋊Dic163—21no—
50C4×Dic163—21no—
51C2×D4×C163—21no—
52C2×D4×C163D411yescharacteristic
53C22×C4×C163—21no—
54C22×C4×C163C2311yescharacteristic
55C2×C8×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58Dic326—22no—
59Dic326—22no—
60Dic326—41no—
61Dic326—41no—
62Dic326—41no—
63C2×Dic163—22no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C2×Dic163—21no—
67C2×Dic163—21no—
68C2×Dic163—21no—
69C2×Dic163—21no—
70C2×Dic163—41no—
71C2×Dic163—41no—
72C2×Dic163—41no—
73C2×Dic163—81no—
74C163⋊C8—41no—
75C23×C163—21no—
76C23×C163C2×D411yescharacteristic
77D4×C163—41no—
78D4×C163—41no—
79D4×C163—41no—
80C2×C4×C163—22no—
81C2×C4×C163—21no—
82C2×C4×C163—41no—
83C2×C4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163C2×D411yescharacteristic
86C2×C4×C163C2×D411yescharacteristic
87C8×C163—41no—
88C22×C163—21no—
89C22×C163—21no—
90C22×C163—41no—
91C22×C163—41no—
92C22×C163—41no—
93C22×C163#12011yescharacteristic
94C22×C163#13511yescharacteristic
95C22×C163#13711yescharacteristic
96C4×C163—21no—
97C4×C163—21no—
98C4×C163—42no—
99C4×C163—41no—
100Dic163—41no—
101Dic163—41no—
102Dic163—42no—
103Dic163—41no—
104Dic163—41no—
105C2×C163—21no—
106C2×C163—21no—
107C2×C163—81no—
108C2×C163#45211yescharacteristic
109C2×C163#45611yescharacteristic
110C2×C163B(4,𝔽2)11yescharacteristic
111C163#160311yescharacteristic, complemented, Sylow
112#1603—1631nomaximal, Sylow
113#539—1631no—
114#451—1631no—
115#448—1631no—
116#395—1631no—
117#328—1631no—
118#327D16311yescharacteristic
119#324—1631no—
120C22×Q8—1631no—
121C2×Q16—3261no—
122C2×SD16—3261no—
123#126—3261no—
124#125—3261no—
125#123—3261no—
126#123—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8—1631no—
129#121C2×D16311yescharacteristic
130C4×D4—3261no—
131C2×C4⋊C4—3261no—
132C2×C4⋊C4—1631no—
133#103—3261no—
134#103—3261no—
135C22⋊C8—1631no—
136C22⋊C8C2×D16311yescharacteristic
137#95—3261no—
138#95C2×D16311yescharacteristic
139C2×Q8—3261no—
140C2×Q8—3261no—
141C2×Q8—6522no—
142C2×Q8—1631no—
143C2×D4—21no—
144C2×D4D65211yescharacteristic
145C22×C4—21no—
146C22×C4—3261no—
147C22×C4—3261no—
148C22×C4—1631no—
149C22×C4—1631no—
150C22×C4C22×D16311yescharacteristic
151Q16—6522no—
152SD16—6522no—
153C2×C8—21no—
154C2×C8—3261no—
155C4⋊C4—3261no—
156C4⋊C4—3261no—
157C4⋊C4—3261no—
158C4⋊C4—3261no—
159C4⋊C4—3261no—
160C4⋊C4—3261no—
161C4⋊C4—6522no—
162C4⋊C4—1631no—
163C4⋊C4D65211yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C42—3261no—
170C23—21no—
171C23D4×D16311yescharacteristic
172Q8—3262no—
173Q8—3262no—
174Q8—6521no—
175Q8—6521no—
176Q8—6521no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—13041no—
186C2×C4—3262no—
187C2×C4—3261no—
188C2×C4—3261no—
189C2×C4—3261no—
190C2×C4—3261no—
191C2×C4—3261no—
192C2×C4—3261no—
193C2×C4—6521no—
194C2×C4—6521no—
195C2×C4—6521no—
196C2×C4D4×D16311yescharacteristic
197C2×C4C2×D65211yescharacteristic
198C8—41no—
199C8—6521no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1910470511yescharacteristic
206C22#1910475411yescharacteristic
207C22#1910476411yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—6521no—
213C4—6521no—
214C4—6521no—
215C4—6521no—
216C4—6522no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2378182811yescharacteristic
221C2#2378184911yescharacteristic
222C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#327×C163C2
socleC22×C163#120
prime radical#121×C163C22
hypercenter#327D163
hypocenterC163#1603
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #23781823C2semidirect
4. #23781586C2semidirect
5. #23781401C2semidirect
6. #23781390C2semidirect
111. C163#1603semidirect

Maximal subgroups

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

subgroupindex
#237822602
#237818232
#237818032
#237815862
#237814012
#237813902
#327×C1632
#1603163