Finite Group Databasebeta

← C3×#64106 #11723852

Subgroups — C3×#64106 #11723852

Order 864.

548 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#634C211yesmaximal, characteristic, complemented
3C33×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#3752C212yesmaximal
5C3×C4⋊(C3⋊Dic3)C212yesmaximal, complemented
6#64106C311yesmaximal, characteristic, direct product factor
7C3×#897—34nomaximal
8C22×C3×C3⋊D3C2211yescharacteristic
9C2×C33×C4C2212yes—
10C2×C33×C4C2211yescharacteristic
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊Dic3C2212yes—
13C3×C4×C3⋊D3—22no—
14C3×#634—22no—
15C2×C4×C3⋊D3C2×C311yescharacteristic
16C2×#634C2×C311yescharacteristic, complemented
17C2×C3×C4×D3—34no—
18C2×C3×Dic6—34no—
19C32×C4⋊C4—24no—
20C32×C4⋊C4D314yescomplemented
21C32×C4⋊C4C2×C311yescharacteristic, complemented
22#3752C2×C312yes—
23C4⋊(C3⋊Dic3)C2×C312yescomplemented
24C3×(C2×D3)⋊C4—38no—
25C3×C4⋊Dic3—38no—
26C22×C33C2311yescharacteristic
27C2×C3×C3⋊D3—21no—
28C2×C3×C3⋊D3—21no—
29C2×C3×C3⋊D3Q812yescomplemented
30C33×C4—22no—
31C33×C4D412yes—
32C3×C3⋊Dic3—22no—
33C3×C3⋊Dic3—21no—
34C3×C22⋊Q8—91no—
35#897—34no—
36C22×C3⋊D3C22×C311yescharacteristic
37C22×C3×D3—34no—
38C2×C32×C4—24no—
39C2×C32×C4—28no—
40C2×C32×C4C2×D318yes—
41C2×C32×C4C2×D314yes—
42C2×C32×C4C22×C312yes—
43C2×C32×C4C22×C311yescharacteristic
44C2×C3⋊Dic3C22×C311yescharacteristic
45C2×C3⋊Dic3C22×C312yes—
46C4×C3⋊D3—22no—
47#634—22no—
48C2×C3×Dic3—34no—
49C2×C3×Dic3—38no—
50C3×C4×D3—68no—
51C3×Dic6—68no—
52C2×C33C2×D411yescharacteristic
53C2×C33C2×Q811yescharacteristic
54C2×C33#4111yescharacteristic
55C3×C3⋊D3—22no—
56C2×C3×Q8—91no—
57C22×C3×C4—91no—
58C2×C4×D3—34no—
59C2×Dic6—34no—
60C3×C4⋊C4—24no—
61C3×C4⋊C4—92no—
62C3×C4⋊C4C3×D314yescomplemented
63C3×C4⋊C4C3⋊D311yescharacteristic, complemented
64C3×C22⋊C4—92no—
65(C2×D3)⋊C4—38no—
66C4⋊Dic3—38no—
67C22×C32—24no—
68C22×C32C22×D314yes—
69C22×C32C23×C311yescharacteristic
70C2×C3⋊D3—21no—
71C2×C3⋊D3—21no—
72C2×C3⋊D3C3×Q812yescomplemented
73C2×C3×D3—38no—
74C2×C3×D3—64no—
75C2×C3×D3—64no—
76C32×C4—22no—
77C32×C4—28no—
78C32×C4—28no—
79C32×C4—48no—
80C32×C4D1218yes—
81C32×C4C3×D412yes—
82C3⋊Dic3—22no—
83C3⋊Dic3—21no—
84C3×Dic3—68no—
85C3×Dic3—64no—
86C22⋊Q8—91noSylow
87C33C22⋊Q811yescharacteristic, complemented, Sylow
88C23×C3—91no—
89C22×D3—34no—
90C3×Q8—182no—
91C2×C3×C4—24no—
92C2×C3×C4—28no—
93C2×C3×C4—92no—
94C2×C3×C4—91no—
95C2×C3×C4—182no—
96C2×C3×C4C2×C3×D318yes—
97C2×C3×C4C2×C3×D314yes—
98C2×C3×C4C2×C3⋊D312yes—
99C2×C3×C4C2×C3⋊D311yescharacteristic
100C2×Dic3—34no—
101C2×Dic3—38no—
102C4×D3—68no—
103Dic6—68no—
104C2×C32—24no—
105C2×C32—24no—
106C2×C32—24no—
107C2×C32C2×D1214yes—
108C2×C32#23714yes—
109C2×C32D3×Q814yes—
110C2×C32C2×C3×D411yescharacteristic
111C2×C32C2×C3×Q811yescharacteristic
112C2×C32C3×#4111yescharacteristic
113C3⋊D3—22no—
114C3×D3—68no—
115C2×Q8—91no—
116C22×C4—91no—
117C4⋊C4—92no—
118C4⋊C4C3×C3⋊D311yescharacteristic, complemented
119C22⋊C4—92no—
120C22×C3—24no—
121C22×C3—92no—
122C22×C3—181no—
123C22×C3—181no—
124C22×C3C22×C3×D314yes—
125C22×C3C22×C3⋊D311yescharacteristic
126C2×D3—38no—
127C2×D3—64no—
128C2×D3—64no—
129C3×C4—28no—
130C3×C4—22no—
131C3×C4—28no—
132C3×C4—48no—
133C3×C4—181no—
134C3×C4—182no—
135C3×C4C3×D1218yes—
136C3×C4C4⋊(C3⋊D3)12yes—
137Dic3—68no—
138Dic3—64no—
139C32—24no—
140C32#89714yescomplemented
141C32C3×C22⋊Q811yescharacteristic, complemented
142C23—91no—
143Q8—182no—
144C2×C4—92no—
145C2×C4—91no—
146C2×C4—182no—
147C2×C4C2×C3×C3⋊D312yes—
148C2×C4C2×C3×C3⋊D311yescharacteristic
149C2×C3—24no—
150C2×C3—24no—
151C2×C3—24no—
152C2×C3—182no—
153C2×C3C2×C3×D1214yes—
154C2×C3C3×#23714yes—
155C2×C3C3×D3×Q814yes—
156C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
157C2×C3#383011yescharacteristic
158C2×C3Q8×C3⋊D311yescharacteristic
159D3—68no—
160C22—92no—
161C22—181no—
162C22—181no—
163C22C22×C3×C3⋊D311yescharacteristic
164C4—22no—
165C4—181no—
166C4—182no—
167C4C3×C4⋊(C3⋊D3)12yes—
168C3—24no—
169C3C3×#89714yescomplemented
170C3#6410611yescharacteristic, direct product factor
171C2—182no—
172C2C2×C3×C4⋊(C3⋊D3)11yescharacteristic
173C2C3×#383011yescharacteristic
174C2C3×Q8×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C22×C3⋊D3
commutator, cosocleC22×C32C23×C3
FrattiniC22C22×C3×C3⋊D3
FittingC33×C4⋊C4C2
socleC22×C33C23
prime radicalC22×C3×C3⋊D3C22
hypercenterC3×C4⋊C4C3⋊D3
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#634C2semidirect
3. C33×C4⋊C4C2semidirect
5. C3×C4⋊(C3⋊Dic3)C2semidirect
6. #64106C3direct (N × H)
16. C2×#634C2×C3semidirect
20. C32×C4⋊C4D3semidirect
21. C32×C4⋊C4C2×C3semidirect
23. C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×C3×C3⋊D3Q8semidirect
62. C3×C4⋊C4C3×D3semidirect
63. C3×C4⋊C4C3⋊D3semidirect
72. C2×C3⋊D3C3×Q8semidirect
87. C33C22⋊Q8semidirect
118. C4⋊C4C3×C3⋊D3semidirect
140. C32#897semidirect
141. C32C3×C22⋊Q8semidirect
169. C3C3×#897semidirect
170. C3#64106direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
C2×C3×#6342
C33×C4⋊C42
C3×#37522
C3×C4⋊(C3⋊Dic3)2
#641063
C3×#8973