Finite Group Databasebeta

← #37549047

Subgroups — #37549047

Order 23168.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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#26042298C211yesmaximal, characteristic, complemented
2#26042283C211yesmaximal, characteristic, complemented
3#26042127C212yesmaximal, complemented
4D4⋊#13089685C212yesmaximal, complemented
5C4×D4×D181C211yesmaximal, characteristic
6C2×#15668459—21no—
7C4⋊#13089685—21no—
8C4⋊#13089685—21no—
9C4⋊#13089685C2212yescomplemented
10#19153873—21no—
11C4×#13089685C2212yescomplemented
12C2×D4×D181C411yescharacteristic
13C22×C4×D181—21no—
14C4×D724C411yescharacteristic
15D4×Dic181C2211yescharacteristic
16#19153778C2211yescharacteristic
17C4×C22⋊D181—21no—
18#19153776—21no—
19C4⋊C4×D181C411yescharacteristic
20C22⋊C4×D181—21no—
21C42×D181C2211yescharacteristic
22C4×D4×C181C411yescharacteristic
23#15668459—42no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C23×D181—21no—
27C2×D724C2×C411yescharacteristic
28C2×C22⋊D181—21no—
29D4×D181—22no—
30D4×D181—22no—
31C22×Dic181—21no—
32C2×C4×D181—21no—
33C2×C4×D181—21no—
34C2×C4×D181—21no—
35C2×C4×D181—42no—
36C2×C4×D181—42no—
37C2×C4×D181C2×C411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C2×C4×D181D411yescharacteristic
40C4⋊Dic181C2×C411yescharacteristic
41#15668428—21no—
42C22⋊Dic181—21no—
43#15668426—21no—
44#15668426—21no—
45C4×Dic181D411yescharacteristic
46C4×Dic181D411yescharacteristic
47C4×Dic181C2311yescharacteristic
48C2×D4×C181C2×C411yescharacteristic
49C22×C4×C181—21no—
50C4⋊C4×C181C2×C411yescharacteristic
51C22⋊C4×C181—21no—
52C42×C181C2×C411yescharacteristic
53#13089685—42no—
54#13089685—42no—
55C22×D181—24no—
56C22×D181—21no—
57C22×D181—42no—
58C22×D181—42no—
59C22×D181C22⋊C411yescharacteristic
60D724—21no—
61D724M1612yescomplemented
62C22⋊D181—24no—
63C2×Dic181—21no—
64C2×Dic181—41no—
65C2×Dic181—41no—
66C2×Dic181C22⋊C411yescharacteristic
67C2×Dic181C2×D411yescharacteristic
68C2×Dic181C2×D411yescharacteristic
69C4×D181—21no—
70C4×D181—21no—
71C4×D181—41no—
72C4×D181—42no—
73C4×D181—42no—
74C4×D181—41no—
75C4×D181—41no—
76C4×D181D812yes—
77C4×D181SD1612yes—
78C23×C181—21no—
79D4×C181—21no—
80D4×C181M1612yescomplemented
81C2×C4×C181—21no—
82C2×C4×C181—42no—
83C2×C4×C181C22⋊C411yescharacteristic
84C2×C4×C181C22⋊C411yescharacteristic
85C2×C4×C181C22×C411yescharacteristic
86C22×C181—22no—
87C22×C181—41no—
88C22×C181—41no—
89C22×C181C2×C22⋊C411yescharacteristic
90C2×D181—21no—
91C2×D181—21no—
92C2×D181—22no—
93C2×D181—48no—
94C2×D181—41no—
95C2×D181—41no—
96C2×D181#10212yes—
97C4×C181—21no—
98C4×C181—41no—
99C4×C181#10212yes—
100C4×C181C2×M1611yescharacteristic
101C4×C181C2×M1611yescharacteristic
102Dic181—21no—
103Dic181—21no—
104Dic181—41no—
105Dic181C2×D811yescharacteristic
106Dic181C2×SD1611yescharacteristic
107C2×C181—42no—
108C2×C181#40711yescharacteristic
109C2×C181C2×#10211yescharacteristic
110C2×C181#42111yescharacteristic
111D181—22no—
112D181—42no—
113C181#148611yescharacteristic, complemented, Sylow
114#1486—1811nomaximal, Sylow
115C2×C4×D4—1811no—
116#431—1811no—
117#423—1811no—
118D4⋊C8—1812no—
119D4⋊C8—1812no—
120C22×D4—1811no—
121C23×C4—3621no—
122C2×M16—3621no—
123C4×D4—3621no—
124C4×D4—3621no—
125C4×D4—1811no—
126C4×D4—1811no—
127C4×D4—1811no—
128C4×D4#1062438211yescharacteristic
129C2×C4⋊C4—1811no—
130C2×C22⋊C4—3621no—
131C2×C42—1811no—
132C4⋊C8—3621no—
133C4⋊C8—3621no—
134C4⋊C8—1812no—
135C22⋊C8—3621no—
136C4×C8—1812no—
137C24—3621no—
138C2×D4—3622no—
139C2×D4—3622no—
140C2×D4—3621no—
141C2×D4—1811no—
142C2×D4C2×#1062438211yescharacteristic
143C22×C4—21no—
144C22×C4—3621no—
145C22×C4—3621no—
146C22×C4—3621no—
147C22×C4—3621no—
148C22×C4—1811no—
149C22×C4—1811no—
150C22×C4—1811no—
151C22×C4—7242no—
152C22×C4—7242no—
153M16—7242no—
154C2×C8—3622no—
155C2×C8—3622no—
156C4⋊C4—3621no—
157C4⋊C4—1811no—
158C4⋊C4C2×#1062438211yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3621no—
161C22⋊C4—3621no—
162C22⋊C4—3621no—
163C42—1811no—
164C42—1811no—
165C42—1811no—
166C42C2×#1062438211yescharacteristic
167C23—21no—
168C23—3624no—
169C23—3621no—
170C23—1811no—
171C23—7242no—
172C23—7242no—
173D4—21no—
174D4—3624no—
175D4—3621no—
176D4—1812no—
177D4#1566845912yescomplemented
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—3621no—
181C2×C4—3621no—
182C2×C4—3621no—
183C2×C4—1811no—
184C2×C4—1811no—
185C2×C4—1811no—
186C2×C4—1812no—
187C2×C4—1812no—
188C2×C4—7241no—
189C2×C4—7241no—
190C2×C4—7241no—
191C2×C4—7242no—
192C2×C4—7241no—
193C2×C4—7241no—
194C2×C4—7242no—
195C2×C4C22⋊#1062438211yescharacteristic
196C2×C4C22⋊#1062438211yescharacteristic
197C2×C4C22×#1062438211yescharacteristic
198C8—7242no—
199C8—7242no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—3621no—
204C22—3621no—
205C22—3622no—
206C22—1812no—
207C22—7248no—
208C22—7241no—
209C22—7241no—
210C22C2×C22⋊#1062438211yescharacteristic
211C4—21no—
212C4—41no—
213C4—3621no—
214C4—3621no—
215C4—1811no—
216C4—1811no—
217C4—7241no—
218C4#1915387812yes—
219C4D181⋊M1611yescharacteristic
220C4C2×#1566845911yescharacteristic
221C2—42no—
222C2—3622no—
223C2—7242no—
224C2#2604225011yescharacteristic
225C2C2×#1915387811yescharacteristic
226C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4×D4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC2×D4×D181C4
cosocleC4×Dic181C23
hypercenterC4×D4#10624382
hypocenterC181#1486
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26042298C2semidirect
2. #26042283C2semidirect
3. #26042127C2semidirect
4. D4⋊#13089685C2semidirect
9. C4⋊#13089685C22semidirect
11. C4×#13089685C22semidirect
61. D724M16semidirect
80. D4×C181M16semidirect
113. C181#1486semidirect
177. D4#15668459semidirect

Maximal subgroups

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

subgroupindex
#260422982
#260422832
#260421272
D4⋊#130896852
C4×D4×D1812
#1486181