Finite Group Databasebeta

← #10596165

Subgroups — #10596165

Order 576.

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

normal subgroups only

This group has 642 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#64106C211yesmaximal, characteristic, complemented
4#64106C211yesmaximal, characteristic, complemented
5#64103C212yesmaximal
6#64101C212yesmaximal, complemented
7#64101C211yesmaximal, characteristic, complemented
8C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
9#64099C212yesmaximal, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
11C4×#634C211yesmaximal, characteristic, complemented
12#5694—34nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊D3C2212yes—
17C2×#634C2211yescharacteristic
18C32×C4⋊C4C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2212yes—
22C32×C42C2211yescharacteristic
23#3752C2211yescharacteristic
24#3752C2211yescharacteristic
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2212yes—
31C4⋊(C3⋊Dic3)C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2212yes—
35C4×C3⋊Dic3C2212yes—
36C3×#125D314yescomplemented
37#898—38no—
38#897—34no—
39#897—34no—
40#894—38no—
41#892—38no—
42#892—34no—
43D3×C4⋊C4—34no—
44#890—38no—
45C4×D12—34no—
46C4×Dic6—34no—
47C22×C3⋊D3C2311yescharacteristic
48C22×C3⋊D3C2311yescharacteristic
49C2×C32×C4C2311yescharacteristic
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3C2312yes—
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C4⋊(C3⋊D3)—22no—
59C4×C3⋊D3—24no—
60C4×C3⋊D3—22no—
61C4×C3⋊D3—22no—
62#634—22no—
63#552—91noSylow
64C2×D12—34no—
65C2×C4×D3—34no—
66C2×C4×D3—34no—
67C2×C4×D3—38no—
68C2×Dic6—34no—
69C3×C4⋊C4C2×D318yes—
70C3×C4⋊C4C2×D314yes—
71C3×C4⋊C4C2×D314yes—
72C3×C4⋊C4C2×D318yes—
73C3×C42C2×D314yes—
74(C2×D3)⋊C4—34no—
75(C2×D3)⋊C4—34no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—38no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81C4⋊Dic3—38no—
82C4⋊Dic3—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85C4×Dic3—38no—
86C4×Dic3—38no—
87C22×C32C2411yescharacteristic
88C2×C3⋊D3—21no—
89C2×C3⋊D3—21no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3#4112yes—
94C32×C4—22no—
95C32×C4—21no—
96C32×C4—21no—
97C32×C4—22no—
98C32×C4#4112yes—
99C3⋊Dic3—22no—
100C3⋊Dic3—21no—
101C3⋊Dic3—21no—
102C3⋊Dic3—22no—
103#126—92no—
104#125—92no—
105#125C3⋊D311yescharacteristic, complemented
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8—91no—
109C4×Q8—91no—
110C4×D4—91no—
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C22×D3—34no—
115C22×D3—34no—
116C2×C3×C4C22×D314yes—
117C2×C3×C4C22×D318yes—
118C2×C3×C4C22×D314yes—
119C2×C3×C4C22×D318yes—
120C2×C3×C4C22×D314yes—
121C2×Dic3—38no—
122C2×Dic3—34no—
123C2×Dic3—38no—
124C2×Dic3—34no—
125D12—68no—
126C4×D3—616no—
127C4×D3—68no—
128C4×D3—68no—
129Dic6—68no—
130C2×C32C2×#4111yescharacteristic
131C2×C32C2×#4111yescharacteristic
132C2×C322−1+411yescharacteristic
133C3⋊D3—22no—
134C3⋊D3—41no—
135C2×Q8—91no—
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—91no—
139C22×C4—91no—
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—91no—
144C4⋊C4—92no—
145C4⋊C4—92no—
146C4⋊C4C2×C3⋊D311yescharacteristic
147C4⋊C4C2×C3⋊D311yescharacteristic
148C4⋊C4C2×C3⋊D312yes—
149C4⋊C4C2×C3⋊D312yes—
150C22⋊C4—91no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—91no—
154C22⋊C4—92no—
155C42—92no—
156C42—92no—
157C42C2×C3⋊D311yescharacteristic
158C22×C3C23×D314yes—
159C2×D3—38no—
160C2×D3—64no—
161C2×D3—64no—
162C2×D3—64no—
163C2×D3—64no—
164C2×D3—64no—
165C3×C4—28no—
166C3×C4—24no—
167C3×C4—24no—
168C3×C4—28no—
169C3×C4#23918yes—
170Dic3—68no—
171Dic3—64no—
172Dic3—64no—
173Dic3—68no—
174C32#55211yescharacteristic, complemented, Sylow
175C23—91no—
176C23—91no—
177Q8—182no—
178D4—182no—
179C2×C4—91no—
180C2×C4—92no—
181C2×C4—92no—
182C2×C4—91no—
183C2×C4—184no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4C22×C3⋊D311yescharacteristic
187C2×C4C22×C3⋊D312yes—
188C2×C4C22×C3⋊D311yescharacteristic
189C2×C4C22×C3⋊D312yes—
190C2×C4C22×C3⋊D311yescharacteristic
191C2×C3C2×#23914yes—
192C2×C3D3×#4114yes—
193C2×C3#101014yes—
194D3—68no—
195D3—124no—
196C22—92no—
197C22—181no—
198C22—181no—
199C22—181no—
200C22—181no—
201C22—181no—
202C22C23×C3⋊D311yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—22no—
207C4—181no—
208C4—181no—
209C4—182no—
210C4—182no—
211C4#383212yes—
212C3#569414yescomplemented
213C2—182no—
214C2—361no—
215C2C2×#383211yescharacteristic
216C2#41×C3⋊D311yescharacteristic
217C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenter#125C3⋊D3
hypocenterC32#552
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. #64106C2semidirect
4. #64106C2semidirect
6. #64101C2semidirect
7. #64101C2semidirect
8. C4⋊C4×C3⋊D3C2semidirect
9. #64099C2semidirect
11. C4×#634C2semidirect
36. C3×#125D3semidirect
105. #125C3⋊D3semidirect
174. C32#552semidirect
212. C3#5694semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641072
#641062
#641032
#641012
C4⋊C4×C3⋊D32
#640992
C4×C4⋊(C3⋊D3)2
C4×#6342
#56943