Finite Group Databasebeta

← #10596161

Subgroups — #10596161

Order 576.

684 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 193 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#64106C212yesmaximal, complemented
3#64105C212yesmaximal, complemented
4#64105C212yesmaximal, complemented
5#64102C211yesmaximal, characteristic
6#64102C211yesmaximal, characteristic
7C4⋊C4×C3⋊D3C212yesmaximal, complemented
8#64099C211yesmaximal, characteristic, complemented
9#64097C211yesmaximal, characteristic, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
11C4×#634C211yesmaximal, characteristic, complemented
12#5690—34nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4×C3⋊D3C2212yes—
15C2×C4×C3⋊D3C2212yes—
16C2×C4×C3⋊D3C2212yes—
17C2×#634C2211yescharacteristic
18C2×#634C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2212yes—
22C32×C4⋊C4C2212yes—
23C32×C42C2211yescharacteristic
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2212yes—
30#3750C2211yescharacteristic
31#3750C2211yescharacteristic
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2211yescharacteristic
36C3×#125D314yescomplemented
37#897—38no—
38#896—38no—
39#896—38no—
40#893—34no—
41#893—34no—
42D3×C4⋊C4—38no—
43#890—34no—
44#888—34no—
45C4×D12—34no—
46C4×Dic6—34no—
47C22×C3⋊D3C2312yes—
48C2×C32×C4C2311yescharacteristic
49C2×C32×C4C2312yes—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2311yescharacteristic
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3⋊Dic3C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C4⋊(C3⋊D3)Q814yescomplemented
59C4×C3⋊D3—24no—
60C4×C3⋊D3—42no—
61C4×C3⋊D3—42no—
62#634—22no—
63#634—42no—
64#554—91noSylow
65C2×D12—34no—
66C2×C4×D3—38no—
67C2×C4×D3—38no—
68C2×C4×D3—38no—
69C2×Dic6—34no—
70C2×Dic6—38no—
71C3×C4⋊C4C2×D314yes—
72C3×C4⋊C4C2×D314yes—
73C3×C4⋊C4C2×D318yes—
74C3×C4⋊C4C2×D318yes—
75C3×C42C2×D314yes—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—38no—
79C4⋊Dic3—34no—
80C4⋊Dic3—34no—
81C4⋊Dic3—38no—
82Dic3⋊C4—34no—
83Dic3⋊C4—34no—
84Dic3⋊C4—38no—
85Dic3⋊C4—38no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C22×C32C2411yescharacteristic
89C2×C3⋊D3—22no—
90C2×C3⋊D3—22no—
91C2×C3⋊D3C2×Q814yes—
92C32×C4—21no—
93C32×C4—22no—
94C32×C4—22no—
95C32×C4—21no—
96C32×C4C2×Q812yes—
97C3⋊Dic3—21no—
98C3⋊Dic3—21no—
99C3⋊Dic3—22no—
100C3⋊Dic3—21no—
101C3⋊Dic3#4112yes—
102C4⋊Q8—91no—
103#125—91no—
104#125C3⋊D311yescharacteristic, complemented
105C22⋊Q8—92no—
106C22⋊Q8—92no—
107C22⋊Q8—92no—
108C4×Q8—91no—
109C4×D4—91no—
110C4×D4—91no—
111C4×D4—91no—
112C2×C4⋊C4—92no—
113C22×D3—38no—
114C2×C3×C4C22×D314yes—
115C2×C3×C4C22×D318yes—
116C2×C3×C4C22×D318yes—
117C2×C3×C4C22×D314yes—
118C2×C3×C4C22×D314yes—
119C2×Dic3—34no—
120C2×Dic3—34no—
121C2×Dic3—34no—
122C2×Dic3—34no—
123C2×Dic3—38no—
124D12—316no—
125C4×D3—616no—
126C4×D3—128no—
127C4×D3—128no—
128Dic6—68no—
129Dic6—128no—
130C2×C32C22×Q811yescharacteristic
131C2×C32C2×#4111yescharacteristic
132C2×C32H(5,𝔽2)11yescharacteristic
133C3⋊D3—24no—
134C2×Q8—91no—
135C2×Q8—92no—
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—92no—
139C22×C4—92no—
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4C2×C3⋊D312yes—
148C4⋊C4C2×C3⋊D311yescharacteristic
149C4⋊C4C2×C3⋊D311yescharacteristic
150C4⋊C4C2×C3⋊D312yes—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C42—91no—
155C42—91no—
156C42C2×C3⋊D311yescharacteristic
157C22×C3C23×D314yes—
158C2×D3—316no—
159C2×D3—68no—
160C2×D3—68no—
161C3×C4—24no—
162C3×C4—28no—
163C3×C4—28no—
164C3×C4—24no—
165C3×C4D3×Q818yes—
166Dic3—38no—
167Dic3—64no—
168Dic3—64no—
169Dic3—68no—
170Dic3—64no—
171C32#55411yescharacteristic, complemented, Sylow
172C23—92no—
173Q8—182no—
174Q8—362no—
175D4—94no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—92no—
181C2×C4—184no—
182C2×C4—362no—
183C2×C4—362no—
184C2×C4C22×C3⋊D311yescharacteristic
185C2×C4C22×C3⋊D311yescharacteristic
186C2×C4C22×C3⋊D312yes—
187C2×C4C22×C3⋊D312yes—
188C2×C4C22×C3⋊D311yescharacteristic
189C2×C3C2×D3×Q814yes—
190C2×C3D3×#4114yes—
191C2×C3#100914yes—
192D3—616no—
193C22—94no—
194C22—182no—
195C22—182no—
196C22C23×C3⋊D311yescharacteristic
197C4—22no—
198C4—21no—
199C4—21no—
200C4—22no—
201C4—92no—
202C4—181no—
203C4—181no—
204C4—182no—
205C4—181no—
206C4Q8×C3⋊D312yes—
207C3#569014yescomplemented
208C2—184no—
209C2C2×Q8×C3⋊D311yescharacteristic
210C2#41×C3⋊D311yescharacteristic
211C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenter#125C3⋊D3
hypocenterC32#554
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. #64106C2semidirect
3. #64105C2semidirect
4. #64105C2semidirect
7. C4⋊C4×C3⋊D3C2semidirect
8. #64099C2semidirect
9. #64097C2semidirect
11. C4×#634C2semidirect
36. C3×#125D3semidirect
58. C4⋊(C3⋊D3)Q8semidirect
104. #125C3⋊D3semidirect
171. C32#554semidirect
207. C3#5690semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641062
#641052
#641022
C4⋊C4×C3⋊D32
#640992
#640972
C4×C4⋊(C3⋊D3)2
C4×#6342
#56903