Finite Group Databasebeta

← #10596152

Subgroups — #10596152

Order 576.

708 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 185 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#64156C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#64149C211yesmaximal, characteristic
5D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#64095C212yesmaximal, complemented
7#64094C212yesmaximal, complemented
8#64092C212yesmaximal, complemented
9#64090C212yesmaximal, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C4×#634C211yesmaximal, characteristic, complemented
12#5681—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×#638C2212yes—
17C22×C3⋊Dic3C2212yes—
18C2×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊D3C2212yes—
20C2×#634C2211yescharacteristic
21C32×C22⋊C4C2212yes—
22C32×C22⋊C4C2212yes—
23C32×C42C2211yescharacteristic
24#3757C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2212yes—
31#3750C2211yescharacteristic
32#3750C2211yescharacteristic
33C4×C3⋊Dic3C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×#124D314yescomplemented
37#947—34no—
38Q8×Dic3—34no—
39#940—34no—
40D4×Dic3—34no—
41#886—38no—
42#885—38no—
43#883—38no—
44#67⋊C4—38no—
45C4×D12—34no—
46C4×Dic6—34no—
47C23×C32C2312yes—
48C22×C3⋊D3C2312yes—
49C32×Q8—22no—
50C32×D4—22no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55#638—24no—
56#638—42no—
57C2×C3⋊Dic3—42no—
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62C2×C3⋊Dic3C2311yescharacteristic
63C4⋊(C3⋊D3)—22no—
64C4×C3⋊D3—42no—
65#634—22no—
66#559—91noSylow
67C2×C3×Q8C2×D314yes—
68C2×C3×D4C2×D314yes—
69C2×#67—38no—
70C2×#67—38no—
71C22×Dic3—38no—
72C2×D12—34no—
73C2×C4×D3—38no—
74C2×Dic6—34no—
75C3×C22⋊C4C2×D318yes—
76C3×C22⋊C4C2×D318yes—
77C3×C42C2×D314yes—
78#217—38no—
79(C2×D3)⋊C4—38no—
80(C2×D3)⋊C4—38no—
81(C2×D3)⋊C4—38no—
82C4⋊Dic3—34no—
83C4⋊Dic3—34no—
84C4⋊Dic3—38no—
85Dic3⋊C4—34no—
86Dic3⋊C4—34no—
87C4×Dic3—34no—
88C4×Dic3—34no—
89C4×Dic3—38no—
90C22×C32—22no—
91C22×C32—22no—
92C22×C32—22no—
93C22×C32C2411yescharacteristic
94C2×C3⋊D3—22no—
95C2×C3⋊D3—22no—
96C2×C3⋊D3—22no—
97C32×C4—21no—
98C32×C4—22no—
99C32×C4—21no—
100C32×C4#4112yes—
101C3⋊Dic3—21no—
102C3⋊Dic3—22no—
103C3⋊Dic3—21no—
104C3⋊Dic3—21no—
105C3⋊Dic3#4112yes—
106#127—91no—
107#124—91no—
108#124—92no—
109#124C3⋊D311yescharacteristic, complemented
110#123—92no—
111#123—92no—
112C4×Q8—91no—
113C4×Q8—91no—
114C4×D4—91no—
115C4×D4—92no—
116C4×D4—91no—
117C23×C3C22×D318yes—
118C22×D3—38no—
119C3×Q8—28no—
120C3×D4—28no—
121C2×C3×C4C22×D318yes—
122C2×C3×C4C22×D314yes—
123C2×C3×C4C22×D314yes—
124C2×C3×C4C22×D314yes—
125#67—616no—
126#67—128no—
127C2×Dic3—34no—
128C2×Dic3—34no—
129C2×Dic3—34no—
130C2×Dic3—34no—
131C2×Dic3—38no—
132C2×Dic3—128no—
133D12—68no—
134C4×D3—128no—
135Dic6—68no—
136C2×C32—42no—
137C2×C32C2×#4111yescharacteristic
138C2×C32C2×#4111yescharacteristic
139C2×C32H(5,𝔽2)11yescharacteristic
140C3⋊D3—42no—
141C2×Q8—91no—
142C2×Q8C2×C3⋊D311yescharacteristic
143C2×D4—92no—
144C2×D4—92no—
145C2×D4—91no—
146C2×D4C2×C3⋊D311yescharacteristic
147C22×C4—92no—
148C22×C4—92no—
149C4⋊C4—91no—
150C4⋊C4—91no—
151C4⋊C4—91no—
152C4⋊C4—92no—
153C4⋊C4—91no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C22⋊C4—92no—
157C22⋊C4—92no—
158C22⋊C4C2×C3⋊D312yes—
159C22⋊C4C2×C3⋊D312yes—
160C42—92no—
161C42—91no—
162C42—91no—
163C42C2×C3⋊D311yescharacteristic
164C22×C3—28no—
165C22×C3—28no—
166C22×C3—28no—
167C22×C3C23×D314yes—
168C2×D3—68no—
169C2×D3—68no—
170C2×D3—68no—
171C3×C4—28no—
172C3×C4—24no—
173C3×C4—24no—
174C3×C4#23718yes—
175Dic3—38no—
176Dic3—64no—
177Dic3—68no—
178Dic3—64no—
179Dic3—64no—
180C32#55911yescharacteristic, complemented, Sylow
181C23—92no—
182C23C22×C3⋊D312yes—
183Q8—22no—
184Q8—182no—
185D4—22no—
186D4—184no—
187D4—182no—
188D4—362no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—92no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—362no—
195C2×C4—362no—
196C2×C4C22×C3⋊D312yes—
197C2×C4C22×C3⋊D311yescharacteristic
198C2×C4C22×C3⋊D311yescharacteristic
199C2×C4C22×C3⋊D311yescharacteristic
200C2×C3—48no—
201C2×C3C2×#23714yes—
202C2×C3D3×#4114yes—
203C2×C3#100914yes—
204D3—128no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—182no—
209C22—182no—
210C22—182no—
211C22C23×C3⋊D311yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—92no—
216C4—181no—
217C4—182no—
218C4—181no—
219C4—181no—
220C4#383012yes—
221C3#568114yescomplemented
222C2—42no—
223C2—362no—
224C2C2×#383011yescharacteristic
225C2#41×C3⋊D311yescharacteristic
226C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
prime radical#64149C2
hypercenter#124C3⋊D3
hypocenterC32#559
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. #64156C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. D4×C3⋊Dic3C2semidirect
6. #64095C2semidirect
7. #64094C2semidirect
8. #64092C2semidirect
9. #64090C2semidirect
10. C4×C4⋊(C3⋊D3)C2semidirect
11. C4×#634C2semidirect
36. C3×#124D3semidirect
109. #124C3⋊D3semidirect
180. C32#559semidirect
221. C3#5681semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641562
Q8×C3⋊Dic32
#641492
D4×C3⋊Dic32
#640952
#640942
#640922
#640902
C4×C4⋊(C3⋊D3)2
C4×#6342
#56813