Finite Group Databasebeta

← #35213494

Subgroups — #35213494

Order 20864.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#23782290C211yesmaximal, characteristic
2#23781823C211yesmaximal, characteristic, complemented
3#23781820C211yesmaximal, characteristic, complemented
4#23781578C211yesmaximal, characteristic
5#23781401C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic
7#328×C163C211yesmaximal, characteristic, complemented
8C22×Dic326C2211yescharacteristic
9C2×Dic652—21no—
10C2×#14579689—21no—
11#19104715—21no—
12#19104714C2211yescharacteristic
13#19104713—21no—
14#19104713—21no—
15#19104712—21no—
16#19104711—21no—
17C2×(C2×D163)⋊C4—21no—
18C2×(C2×D163)⋊C4C2211yescharacteristic
19C22⋊C4×D163—21no—
20#19104660—21no—
21#19104658—21no—
22C22⋊(C163⋊C8)C2211yescharacteristic
23C163⋊#95—21no—
24C22⋊Q8×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326—21no—
29C2×Dic326—21no—
30C2×Dic326—21no—
31C2×Dic326—42no—
32C2×Dic326D411yescharacteristic
33C22×Dic163—21no—
34C22×Dic163D411yescharacteristic
35C2×C4×D163—21no—
36C2×C4×D163—21no—
37Dic652—42no—
38#14579689—42no—
39C2×C163⋊C8—21no—
40C4⋊Dic163—21no—
41C4⋊Dic163D411yescharacteristic
42Dic163⋊C4—21no—
43Dic163⋊C4—21no—
44Dic163⋊C4—21no—
45C22⋊Dic163—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—21no—
48(C2×D163)⋊C4—21no—
49(C2×D163)⋊C4—21no—
50(C2×D163)⋊C4—42no—
51C2×Q8×C163D411yescharacteristic
52C22×C4×C163—21no—
53C22×C4×C163C2311yescharacteristic
54C2×C8×C163—21no—
55C4⋊C4×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58C22×D163—21no—
59C22×D163—21no—
60C22×D163—41no—
61C22×D163—41no—
62C22×D163Q1612yescomplemented
63Dic326—22no—
64Dic326—22no—
65Dic326—41no—
66Dic326—41no—
67Dic326—41no—
68Dic326—81no—
69C2×Dic163—22no—
70C2×Dic163—21no—
71C2×Dic163—21no—
72C2×Dic163—21no—
73C2×Dic163—41no—
74C2×Dic163—41no—
75C2×Dic163—41no—
76C4×D163—81no—
77C4×D163—81no—
78C163⋊C8—41no—
79C23×C163C2×D411yescharacteristic
80Q8×C163—41no—
81C2×C4×C163—21no—
82C2×C4×C163—21no—
83C2×C4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163—41no—
86C2×C4×C163—41no—
87C2×C4×C163C2×D411yescharacteristic
88C2×C4×C163C2×D411yescharacteristic
89C8×C163—41no—
90C22×C163—21no—
91C22×C163—21no—
92C22×C163#12011yescharacteristic
93C22×C163C2×Q1611yescharacteristic
94C22×C163#13711yescharacteristic
95C2×D163—22no—
96C2×D163—41no—
97C2×D163—41no—
98C2×D163—42no—
99C2×D163—81no—
100C4×C163—21no—
101C4×C163—21no—
102C4×C163—41no—
103C4×C163—41no—
104C4×C163—81no—
105Dic163—41no—
106Dic163—41no—
107Dic163—41no—
108Dic163—42no—
109C2×C163—21no—
110C2×C163—21no—
111C2×C163#45111yescharacteristic
112C2×C163#45511yescharacteristic
113C2×C163B(4,𝔽2)11yescharacteristic
114D163—42no—
115C163#160211yescharacteristic, complemented, Sylow
116#1602—1631nomaximal, Sylow
117#543—1631no—
118#451—1631no—
119#451—1631no—
120#393—1631no—
121#328—1631no—
122#328D16311yescharacteristic, complemented
123#323—1631no—
124C22×Q8—1631no—
125C2×Q16—3261no—
126C2×Q16—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8—3261no—
129C22⋊Q8—3261no—
130C22⋊Q8—3261no—
131C22⋊Q8—3261no—
132C22⋊Q8—1631no—
133C22⋊Q8C2×D16311yescharacteristic
134C2×C22⋊C4—3261no—
135C2×C22⋊C4—3261no—
136C2×C22⋊C4—1631no—
137#103—3261no—
138#103—3261no—
139C22⋊C8—1631no—
140C22⋊C8C2×D16311yescharacteristic
141#95—3261no—
142#95C2×D16311yescharacteristic
143C24—1631no—
144C2×Q8—3261no—
145C2×Q8—3261no—
146C2×Q8—3261no—
147C2×Q8—6522no—
148C2×Q8—1631no—
149C2×Q8D65211yescharacteristic
150C22×C4—21no—
151C22×C4—3261no—
152C22×C4—3261no—
153C22×C4—3261no—
154C22×C4—1631no—
155C22×C4C22×D16311yescharacteristic
156Q16—6522no—
157Q16—6522no—
158C2×C8—21no—
159C2×C8—3261no—
160C4⋊C4—21no—
161C4⋊C4—3261no—
162C4⋊C4—3261no—
163C4⋊C4—3261no—
164C4⋊C4—3261no—
165C4⋊C4—1631no—
166C4⋊C4D65211yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3261no—
169C22⋊C4—3261no—
170C22⋊C4—3261no—
171C22⋊C4—3261no—
172C22⋊C4—3261no—
173C22⋊C4—6522no—
174C23—3261no—
175C23—3261no—
176C23—6521no—
177C23—6521no—
178C23—1632no—
179C23D4×D16311yescharacteristic
180Q8—41no—
181Q8—13041no—
182Q8—3262no—
183Q8—3262no—
184Q8—6521no—
185Q8—6521no—
186Q8—6521no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—13041no—
194C2×C4—13041no—
195C2×C4—3262no—
196C2×C4—3261no—
197C2×C4—3261no—
198C2×C4—3261no—
199C2×C4—6521no—
200C2×C4—6521no—
201C2×C4—6521no—
202C2×C4D4×D16311yescharacteristic
203C2×C4C2×D65211yescharacteristic
204C8—41no—
205C8—6521no—
206C22—21no—
207C22—21no—
208C22—13041no—
209C22—3262no—
210C22—6522no—
211C22—6521no—
212C22—6521no—
213C22#1910470511yescharacteristic
214C22Q16×D16311yescharacteristic
215C22#1910476411yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—6521no—
222C4—6521no—
223C4—6521no—
224C4—6522no—
225C2—21no—
226C2—21no—
227C2—6522no—
228C2#2378182211yescharacteristic
229C2#2378184411yescharacteristic
230C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#328×C163C2
socleC22×C163#120
prime radicalC23×D163D4
hypercenter#328D163
hypocenterC163#1602
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #23781823C2semidirect
3. #23781820C2semidirect
5. #23781401C2semidirect
7. #328×C163C2semidirect
62. C22×D163Q16semidirect
115. C163#1602semidirect
122. #328D163semidirect

Maximal subgroups

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

subgroupindex
#237822902
#237818232
#237818202
#237815782
#237814012
#237813882
#328×C1632
#1602163