Finite Group Databasebeta

← #10596166

Subgroups — #10596166

Order 576.

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

normal subgroups only

This group has 714 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×#125C211yesmaximal, characteristic, complemented
2#64107C212yesmaximal, complemented
3#64104C212yesmaximal, complemented
4#64104C211yesmaximal, characteristic, complemented
5#64104C211yesmaximal, characteristic, complemented
6#64103C212yesmaximal, complemented
7#64102C212yesmaximal, complemented
8#64101C211yesmaximal, characteristic, complemented
9C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
12#5695—34nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2212yes—
16C2×C4⋊(C3⋊D3)C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2212yes—
20C2×C4×C3⋊D3C2212yes—
21C32×C4⋊C4C2211yescharacteristic
22C32×C4⋊C4C2211yescharacteristic
23C32×C4⋊C4C2212yes—
24C32×C4⋊C4C2212yes—
25C32×C42C2211yescharacteristic
26#3752C2211yescharacteristic
27#3752C2211yescharacteristic
28#3752C2212yes—
29#3752C2212yes—
30#3752C2212yes—
31#3752C2212yes—
32C4⋊(C3⋊Dic3)C2211yescharacteristic
33C4⋊(C3⋊Dic3)C2211yescharacteristic
34#3750C2212yes—
35C4×C3⋊Dic3C2212yes—
36C3×#125D314yescomplemented
37#898—38no—
38#895—38no—
39#895—34no—
40#895—34no—
41#894—38no—
42#893—38no—
43#892—34no—
44D3×C4⋊C4—34no—
45C4×D12—34no—
46C4×D12—34no—
47C22×C3⋊D3C2311yescharacteristic
48C22×C3⋊D3C2312yes—
49C22×C3⋊D3C2311yescharacteristic
50C2×C32×C4C2312yes—
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C4⋊(C3⋊D3)—22no—
59C4⋊(C3⋊D3)—24no—
60C4⋊(C3⋊D3)—22no—
61C4⋊(C3⋊D3)—42no—
62C4×C3⋊D3—22no—
63C4×C3⋊D3—22no—
64C4×C3⋊D3—42no—
65C4×C3⋊D3—42no—
66#553—91noSylow
67C2×D12—34no—
68C2×D12—34no—
69C2×D12—38no—
70C2×D12—38no—
71C2×C4×D3—34no—
72C2×C4×D3—34no—
73C2×C4×D3—38no—
74C2×C4×D3—38no—
75C3×C4⋊C4C2×D314yes—
76C3×C4⋊C4C2×D314yes—
77C3×C4⋊C4C2×D318yes—
78C3×C4⋊C4C2×D318yes—
79C3×C42C2×D314yes—
80(C2×D3)⋊C4—34no—
81(C2×D3)⋊C4—34no—
82(C2×D3)⋊C4—38no—
83(C2×D3)⋊C4—38no—
84(C2×D3)⋊C4—38no—
85(C2×D3)⋊C4—38no—
86C4⋊Dic3—34no—
87C4⋊Dic3—34no—
88Dic3⋊C4—38no—
89C4×Dic3—38no—
90C22×C32C2411yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—22no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3—22no—
97C2×C3⋊D3—22no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3#4112yes—
100C32×C4—22no—
101C32×C4—21no—
102C32×C4—21no—
103C32×C4—22no—
104C32×C4#4112yes—
105C3⋊Dic3—21no—
106C3⋊Dic3—21no—
107C3⋊Dic3—22no—
108#126—92no—
109#125C3⋊D311yescharacteristic, complemented
110#123—92no—
111#121—92no—
112#121—91no—
113#121—91no—
114C4×D4—92no—
115C4×D4—91no—
116C4×D4—91no—
117#117—91no—
118C2×C4⋊C4—91no—
119C22×D3—34no—
120C22×D3—34no—
121C22×D3—38no—
122C2×C3×C4C22×D318yes—
123C2×C3×C4C22×D318yes—
124C2×C3×C4C22×D314yes—
125C2×C3×C4C22×D314yes—
126C2×C3×C4C22×D314yes—
127C2×Dic3—34no—
128C2×Dic3—38no—
129C2×Dic3—34no—
130D12—68no—
131D12—616no—
132D12—68no—
133D12—128no—
134C4×D3—68no—
135C4×D3—68no—
136C4×D3—128no—
137C4×D3—128no—
138C2×C32C2×#4111yescharacteristic
139C2×C32C2×#4111yescharacteristic
140C2×C32H(5,𝔽2)11yescharacteristic
141C3⋊D3—22no—
142C3⋊D3—41no—
143C3⋊D3—42no—
144C2×D4—92no—
145C2×D4—91no—
146C2×D4—91no—
147C2×D4—92no—
148C22×C4—91no—
149C22×C4—91no—
150C22×C4—92no—
151C22×C4—92no—
152C4⋊C4—91no—
153C4⋊C4—92no—
154C4⋊C4—91no—
155C4⋊C4C2×C3⋊D311yescharacteristic
156C4⋊C4C2×C3⋊D311yescharacteristic
157C4⋊C4C2×C3⋊D312yes—
158C4⋊C4C2×C3⋊D312yes—
159C22⋊C4—92no—
160C22⋊C4—91no—
161C22⋊C4—92no—
162C22⋊C4—92no—
163C22⋊C4—91no—
164C22⋊C4—92no—
165C42—92no—
166C42C2×C3⋊D311yescharacteristic
167C22×C3C23×D314yes—
168C2×D3—38no—
169C2×D3—64no—
170C2×D3—64no—
171C2×D3—64no—
172C2×D3—68no—
173C2×D3—64no—
174C2×D3—68no—
175C2×D3—68no—
176C2×D3—64no—
177C3×C4—28no—
178C3×C4—24no—
179C3×C4—24no—
180C3×C4—28no—
181C3×C4#23918yes—
182Dic3—64no—
183Dic3—64no—
184Dic3—68no—
185C32#55311yescharacteristic, complemented, Sylow
186C23—91no—
187C23—91no—
188C23—92no—
189D4—184no—
190D4—182no—
191D4—182no—
192D4—362no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—91no—
196C2×C4—182no—
197C2×C4—182no—
198C2×C4—362no—
199C2×C4—362no—
200C2×C4C22×C3⋊D312yes—
201C2×C4C22×C3⋊D311yescharacteristic
202C2×C4C22×C3⋊D311yescharacteristic
203C2×C4C22×C3⋊D312yes—
204C2×C4C22×C3⋊D311yescharacteristic
205C2×C3C2×#23914yes—
206C2×C3D3×#4114yes—
207C2×C3#100914yes—
208D3—68no—
209D3—124no—
210D3—128no—
211C22—92no—
212C22—181no—
213C22—181no—
214C22—181no—
215C22—182no—
216C22—181no—
217C22—182no—
218C22—182no—
219C22—181no—
220C22C23×C3⋊D311yescharacteristic
221C4—22no—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—181no—
226C4—181no—
227C4—182no—
228C4#383212yes—
229C3#569514yescomplemented
230C2—182no—
231C2—361no—
232C2—362no—
233C2C2×#383211yescharacteristic
234C2#41×C3⋊D311yescharacteristic
235C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
hypercenter#125C3⋊D3
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C32×#125C2semidirect
2. #64107C2semidirect
3. #64104C2semidirect
4. #64104C2semidirect
5. #64104C2semidirect
6. #64103C2semidirect
7. #64102C2semidirect
8. #64101C2semidirect
9. C4⋊C4×C3⋊D3C2semidirect
10. C4×C4⋊(C3⋊D3)C2semidirect
11. C4×C4⋊(C3⋊D3)C2semidirect
36. C3×#125D3semidirect
109. #125C3⋊D3semidirect
185. C32#553semidirect
229. C3#5695semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641072
#641042
#641032
#641022
#641012
C4⋊C4×C3⋊D32
C4×C4⋊(C3⋊D3)2
#56953