Finite Group Databasebeta

← #10596151

Subgroups — #10596151

Order 576.

702 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C32×#124C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#64147C211yesmaximal, characteristic
5D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#64092C212yesmaximal, complemented
7#64090C212yesmaximal, complemented
8#64088C212yesmaximal, complemented
9#64086C212yesmaximal, complemented
10C42×C3⋊D3C211yesmaximal, characteristic, complemented
11#64079C211yesmaximal, characteristic, complemented
12#5680—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C22×C3⋊Dic3C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C32×C22⋊C4C2212yes—
21C32×C22⋊C4C2212yes—
22C32×C42C2211yescharacteristic
23#3757C2212yes—
24#3757C2212yes—
25#3752C2212yes—
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2212yes—
29#3750C2212yes—
30#3750C2212yes—
31#3750C2212yes—
32C4×C3⋊Dic3C2211yescharacteristic
33C4×C3⋊Dic3C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×#124D314yescomplemented
37#946—34no—
38Q8×Dic3—34no—
39#938—34no—
40D4×Dic3—34no—
41#883—38no—
42#67⋊C4—38no—
43#879—38no—
44#877—38no—
45C42×D3—34no—
46#870—34no—
47C23×C32C2312yes—
48C22×C3⋊D3C2311yescharacteristic
49C32×Q8—22no—
50C32×D4—22no—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C32×C4C2311yescharacteristic
55#638—24no—
56C2×C3⋊Dic3—24no—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2311yescharacteristic
62C4×C3⋊D3—22no—
63C4×C3⋊D3—22no—
64C4×C3⋊D3—22no—
65#529—91noSylow
66C2×C3×Q8C2×D314yes—
67C2×C3×D4C2×D314yes—
68C2×#67—38no—
69C22×Dic3—38no—
70C2×C4×D3—34no—
71C2×C4×D3—34no—
72C2×C4×D3—34no—
73C3×C22⋊C4C2×D318yes—
74C3×C22⋊C4C2×D318yes—
75C3×C42C2×D314yes—
76#217—38no—
77#217—38no—
78(C2×D3)⋊C4—38no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81C4⋊Dic3—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85C4×Dic3—34no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C4×Dic3—38no—
89C22×C32—22no—
90C22×C32—22no—
91C22×C32—22no—
92C22×C32C2411yescharacteristic
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C2×C3⋊D3#4112yes—
96C32×C4—21no—
97C32×C4—21no—
98C32×C4—22no—
99C32×C4#4112yes—
100C3⋊Dic3—21no—
101C3⋊Dic3—21no—
102C3⋊Dic3—22no—
103C3⋊Dic3—22no—
104C3⋊Dic3#4112yes—
105#126—92no—
106#125—91no—
107#124C3⋊D311yescharacteristic, complemented
108#123—92no—
109C22⋊Q8—91no—
110#121—91no—
111C4×Q8—91no—
112C4×D4—91no—
113C4×D4—92no—
114#117—92no—
115C2×C42—91no—
116C23×C3C22×D318yes—
117C22×D3—34no—
118C3×Q8—28no—
119C3×D4—28no—
120C2×C3×C4C22×D314yes—
121C2×C3×C4C22×D314yes—
122C2×C3×C4C22×D318yes—
123C2×C3×C4C22×D314yes—
124#67—616no—
125C2×Dic3—34no—
126C2×Dic3—34no—
127C2×Dic3—34no—
128C2×Dic3—38no—
129C2×Dic3—38no—
130C2×Dic3—616no—
131C4×D3—68no—
132C4×D3—68no—
133C4×D3—68no—
134C2×C32—42no—
135C2×C32C2×#4111yescharacteristic
136C2×C32C2×#4111yescharacteristic
137C2×C32C2×#4111yescharacteristic
138C3⋊D3—22no—
139C2×Q8C2×C3⋊D311yescharacteristic
140C2×D4—92no—
141C2×D4C2×C3⋊D311yescharacteristic
142C22×C4—92no—
143C22×C4—91no—
144C22×C4—91no—
145C22×C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—91no—
149C4⋊C4—92no—
150C4⋊C4—91no—
151C4⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4—92no—
155C22⋊C4C2×C3⋊D312yes—
156C22⋊C4C2×C3⋊D312yes—
157C42—91no—
158C42—91no—
159C42—91no—
160C42—92no—
161C42C2×C3⋊D311yescharacteristic
162C22×C3—28no—
163C22×C3—28no—
164C22×C3—28no—
165C22×C3C23×D314yes—
166C2×D3—38no—
167C2×D3—64no—
168C2×D3—64no—
169C3×C4—28no—
170C3×C4—24no—
171C3×C4—24no—
172C3×C4#23718yes—
173Dic3—38no—
174Dic3—64no—
175Dic3—64no—
176Dic3—68no—
177Dic3—68no—
178C32#52911yescharacteristic, complemented, Sylow
179C23—91no—
180C23C22×C3⋊D312yes—
181Q8—22no—
182D4—22no—
183D4—184no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—92no—
189C2×C4—182no—
190C2×C4—182no—
191C2×C4—184no—
192C2×C4—182no—
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C4C22×C3⋊D311yescharacteristic
195C2×C4C22×C3⋊D312yes—
196C2×C4C22×C3⋊D311yescharacteristic
197C2×C3—48no—
198C2×C3C2×#23714yes—
199C2×C3D3×#4114yes—
200C2×C3D3×#4114yes—
201D3—68no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—92no—
206C22—181no—
207C22—181no—
208C22C23×C3⋊D311yescharacteristic
209C4—22no—
210C4—21no—
211C4—21no—
212C4—92no—
213C4—181no—
214C4—182no—
215C4—181no—
216C4—182no—
217C4#383012yes—
218C3#568014yescomplemented
219C2—42no—
220C2—182no—
221C2C2×#383011yescharacteristic
222C2#41×C3⋊D311yescharacteristic
223C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
prime radical#64147C2
hypercenter#124C3⋊D3
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C32×#124C2semidirect
2. #64155C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. D4×C3⋊Dic3C2semidirect
6. #64092C2semidirect
7. #64090C2semidirect
8. #64088C2semidirect
9. #64086C2semidirect
10. C42×C3⋊D3C2semidirect
11. #64079C2semidirect
36. C3×#124D3semidirect
107. #124C3⋊D3semidirect
178. C32#529semidirect
218. C3#5680semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641552
Q8×C3⋊Dic32
#641472
D4×C3⋊Dic32
#640922
#640902
#640882
#640862
C42×C3⋊D32
#640792
#56803