Finite Group Databasebeta

← #35213587

Subgroups — #35213587

Order 20864.

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

normal subgroups only

This group has 240 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#23782255C211yesmaximal, characteristic
2#23781998C211yesmaximal, characteristic, complemented
3#23781984C211yesmaximal, characteristic, complemented
4#23781602C211yesmaximal, characteristic, complemented
5#23781398C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic, complemented
7#328×C163C211yesmaximal, characteristic, complemented
8#19104730—21no—
9#19104724—21no—
10#19104720—21no—
11#19104710—21no—
12#19104709C2211yescharacteristic
13#19104705—21no—
14(C22⋊D163)⋊C4—21no—
15C2×C4⋊Dic163—21no—
16C2×C4⋊Dic163C2211yescharacteristic
17C2×(C2×D163)⋊C4—21no—
18C2×(C2×D163)⋊C4C2211yescharacteristic
19#19104668—21no—
20#19104667—21no—
21#19104659—21no—
22#19104657—21no—
23C22⋊(C163⋊C8)C2211yescharacteristic
24C22⋊Q8×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×D652—21no—
29C2×D652—21no—
30C2×D652D411yescharacteristic
31C2×C22⋊D163—21no—
32C22×Dic163—21no—
33C22×Dic163D411yescharacteristic
34C2×C4×D163—21no—
35C2×C163⋊C8—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163—21no—
38C4⋊Dic163—42no—
39C4⋊Dic163D411yescharacteristic
40Dic163⋊C4—21no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—42no—
48C4×Dic163—21no—
49C2×Q8×C163D411yescharacteristic
50C22×C4×C163—21no—
51C22×C4×C163C2311yescharacteristic
52C2×C8×C163—21no—
53C4⋊C4×C163—21no—
54C4⋊C4×C163D411yescharacteristic
55C22⋊C4×C163—21no—
56C22×D163—21no—
57C22×D163—21no—
58C22×D163—21no—
59C22×D163—21no—
60C22×D163—41no—
61C22×D163—41no—
62D652—41no—
63D652—81no—
64D652—81no—
65C22⋊D163—41no—
66C22⋊D163—41no—
67C2×Dic163—21no—
68C2×Dic163—21no—
69C2×Dic163—41no—
70C2×Dic163—41no—
71C2×Dic163—41no—
72C2×Dic163—41no—
73C4×D163—81no—
74C163⋊C8—41no—
75C23×C163C2×D411yescharacteristic
76Q8×C163—41no—
77C2×C4×C163—22no—
78C2×C4×C163—21no—
79C2×C4×C163—21no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163—41no—
83C2×C4×C163C2×D411yescharacteristic
84C2×C4×C163C2×D411yescharacteristic
85C8×C163—41no—
86C22×C163—21no—
87C22×C163—21no—
88C22×C163#12011yescharacteristic
89C22×C163#13511yescharacteristic
90C22×C163#13711yescharacteristic
91C2×D163—42no—
92C2×D163—41no—
93C2×D163—41no—
94C2×D163—41no—
95C2×D163—41no—
96C2×D163—41no—
97C2×D163—41no—
98C2×D163—81no—
99C4×C163—21no—
100C4×C163—21no—
101C4×C163—42no—
102C4×C163—41no—
103C4×C163—41no—
104Dic163—41no—
105Dic163—41no—
106Dic163—81no—
107C2×C163—21no—
108C2×C163—21no—
109C2×C163#45211yescharacteristic
110C2×C163#45311yescharacteristic
111C2×C163#45811yescharacteristic
112D163—81no—
113D163—81no—
114C163#161811yescharacteristic, complemented, Sylow
115#1618—1631nomaximal, Sylow
116#538—1631no—
117#484—1631no—
118#481—1631no—
119#399—1631no—
120#328D16311yescharacteristic, complemented
121#327—1631no—
122#323—1631no—
123#126—3261no—
124#124—3261no—
125#123—3261no—
126C22⋊Q8C2×D16311yescharacteristic
127#121—3261no—
128#121—1631no—
129#120—3261no—
130C4×D4—3261no—
131C2×C4⋊C4—3261no—
132C2×C4⋊C4—1631no—
133C2×C22⋊C4—3261no—
134C2×C22⋊C4—1631no—
135#107—3261no—
136#106—3261no—
137#103—3261no—
138#102—3261no—
139C22⋊C8—1631no—
140C22⋊C8C2×D16311yescharacteristic
141#95C2×D16311yescharacteristic
142C24—1631no—
143C2×Q8D65211yescharacteristic
144C2×D4—3261no—
145C2×D4—3261no—
146C2×D4—3261no—
147C2×D4—1631no—
148C22×C4—21no—
149C22×C4—3261no—
150C22×C4—3261no—
151C22×C4—1631no—
152C22×C4C22×D16311yescharacteristic
153C2×C8—21no—
154C2×C8—3261no—
155C4⋊C4—21no—
156C4⋊C4—3261no—
157C4⋊C4—3261no—
158C4⋊C4—3261no—
159C4⋊C4—6522no—
160C4⋊C4—1631no—
161C4⋊C4D65211yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3261no—
164C22⋊C4—3261no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C22⋊C4—6522no—
170C42—3261no—
171C23—3261no—
172C23—3261no—
173C23—3261no—
174C23—3261no—
175C23—6521no—
176C23—6521no—
177C23D4×D16311yescharacteristic
178Q8—41no—
179D4—13041no—
180D4—13041no—
181D4—6521no—
182D4—6521no—
183D4—6521no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—13041no—
191C2×C4—3261no—
192C2×C4—3261no—
193C2×C4—6521no—
194C2×C4—6521no—
195C2×C4—6521no—
196C2×C4—6521no—
197C2×C4D4×D16311yescharacteristic
198C2×C4C2×D65211yescharacteristic
199C8—41no—
200C8—6521no—
201C22—21no—
202C22—21no—
203C22—13041no—
204C22—6522no—
205C22—6521no—
206C22—6521no—
207C22—6521no—
208C22—6521no—
209C22—6521no—
210C22—6521no—
211C22#1910470511yescharacteristic
212C22#1910475611yescharacteristic
213C22#1910476711yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—13041no—
220C4—6521no—
221C4—6521no—
222C2—21no—
223C2—21no—
224C2—13041no—
225C2—13041no—
226C2#2378183011yescharacteristic
227C2#2378183411yescharacteristic
228C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#328×C163C2
socleC22×C163#120
prime radical#23782255C2
hypercenter#328D163
hypocenterC163#1618
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. #23781998C2semidirect
3. #23781984C2semidirect
4. #23781602C2semidirect
5. #23781398C2semidirect
6. #23781388C2semidirect
7. #328×C163C2semidirect
114. C163#1618semidirect
120. #328D163semidirect

Maximal subgroups

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

subgroupindex
#237822552
#237819982
#237819842
#237816022
#237813982
#237813882
#328×C1632
#1618163