Finite Group Databasebeta

← C3×D4×#157 #11724503

Subgroups — C3×D4×#157 #11724503

Order 864.

328 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C22×C3×#157C212yesmaximal, complemented
2C3×D4×C3⋊D3C211yesmaximal, characteristic
3C3×C22⋊#157C212yesmaximal, complemented
4C3×C4⋊#157C211yesmaximal, characteristic, complemented
5C3×C4×#157C211yesmaximal, characteristic, complemented
6D4×#157C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊D3C2212yes—
8C2×C3×#157—24no—
9C2×C3×#157C2212yes—
10C2×C3×#157C2211yescharacteristic
11C2×C3×#157C2211yescharacteristic
12C33×D4C411yescharacteristic, complemented
13C3×#638C412yescomplemented
14C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
15C3×C4×C3⋊D3C2211yescharacteristic
16C22×#157C2×C312yescomplemented
17D4×C3⋊D3C2×C311yescharacteristic
18C3×D3×D4—62no—
19C22⋊#157C2×C312yescomplemented
20C4⋊#157C2×C311yescharacteristic, complemented
21C4×#157C2×C311yescharacteristic, complemented
22C22×C33C2×C412yescomplemented
23C2×C3×C3⋊D3—22no—
24C2×C3×C3⋊D3—22no—
25C2×C3×C3⋊D3C2×C412yescomplemented
26C2×C3×C3⋊D3C2311yescharacteristic
27C3×#157—22no—
28C3×#157—21no—
29C3×#157D412yesdirect product factor
30C33×C4C2×C411yescharacteristic, complemented
31C3×C3⋊Dic3C2×C411yescharacteristic, complemented
32C3×C4×D4—91nomaximal
33C22×C3⋊D3C22×C312yes—
34C22×C3×D3—64no—
35C2×#157—24no—
36C2×#157C22×C312yes—
37C2×#157C22×C311yescharacteristic
38C2×#157C22×C311yescharacteristic
39C32×D4—22no—
40C32×D4—42no—
41C32×D4C3×C411yescharacteristic, complemented
42#638C3×C412yescomplemented
43C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
44C4×C3⋊D3C22×C311yescharacteristic
45C3×#67—64no—
46C3×D12—62no—
47C3×C4×D3—62no—
48C2×C33—22no—
49C2×C33C22×C411yescharacteristic
50C3×C3⋊D3—22no—
51C3×C3⋊D3C2×D411yescharacteristic
52C3×C3⋊D3#4111yescharacteristic
53C2×C3×D4—91no—
54C22×C3×C4—92no—
55D3×D4—62no—
56C3×C4⋊C4—91no—
57C3×C22⋊C4—92no—
58C3×C42—91no—
59C22×C32—24no—
60C22×C32—44no—
61C22×C32C2×C3×C412yescomplemented
62C2×C3⋊D3—22no—
63C2×C3⋊D3—22no—
64C2×C3⋊D3C2×C3×C412yescomplemented
65C2×C3⋊D3C23×C311yescharacteristic
66C2×C3×D3—62no—
67C2×C3×D3—64no—
68C2×C3×D3—124no—
69C2×C3×D3—124no—
70#157—22no—
71#157—21no—
72#157C3×D412yesdirect product factor
73C32×C4—22no—
74C32×C4—42no—
75C32×C4C2×C3×C411yescharacteristic, complemented
76C3⋊Dic3C2×C3×C411yescharacteristic, complemented
77C3×Dic3—62no—
78C4×D4—91noSylow
79C33C4×D411yescharacteristic, complemented, Sylow
80C23×C3—92no—
81C22×D3—64no—
82C3×D4—22no—
83C3×D4—42no—
84C3×D4—92no—
85C3×D4—91no—
86C3×D4#15711yescharacteristic, direct product factor
87C2×C3×C4—91no—
88C2×C3×C4—91no—
89C2×C3×C4—91no—
90C2×C3×C4—92no—
91C2×C3×C4—184no—
92#67—64no—
93D12—62no—
94C4×D3—62no—
95C2×C32—22no—
96C2×C32—22no—
97C2×C32—42no—
98C2×C32—44no—
99C2×C32—84no—
100C2×C32C22×C3×C411yescharacteristic
101C3⋊D3—22no—
102C3⋊D3C2×C3×D411yescharacteristic
103C3⋊D3C3×#4111yescharacteristic
104C3×D3—62no—
105C3×D3—62no—
106C3×D3—124no—
107C2×D4—91no—
108C22×C4—92no—
109C4⋊C4—91no—
110C22⋊C4—92no—
111C42—91no—
112C22×C3—24no—
113C22×C3—44no—
114C22×C3—91no—
115C22×C3—92no—
116C22×C3—182no—
117C22×C3—182no—
118C22×C3C2×#15712yescomplemented
119C2×D3—62no—
120C2×D3—64no—
121C2×D3—124no—
122C2×D3—124no—
123C3×C4—22no—
124C3×C4—42no—
125C3×C4—92no—
126C3×C4—91no—
127C3×C4—182no—
128C3×C4—181no—
129C3×C4C2×#15711yescharacteristic, complemented
130Dic3—62no—
131C32—22no—
132C32—42no—
133C32C3×C4×D411yescharacteristic, complemented
134C23—92no—
135D4—92no—
136D4—91no—
137D4C3×#15711yescharacteristic, direct product factor
138C2×C4—91no—
139C2×C4—91no—
140C2×C4—91no—
141C2×C4—92no—
142C2×C4—184no—
143C2×C3—22no—
144C2×C3—22no—
145C2×C3—44no—
146C2×C3—42no—
147C2×C3—84no—
148C2×C3—91no—
149C2×C3—91no—
150C2×C3—182no—
151C2×C3C22×#15711yescharacteristic
152D3—62no—
153D3—62no—
154D3—124no—
155C22—91no—
156C22—92no—
157C22—182no—
158C22—182no—
159C22C2×C3×#15712yescomplemented
160C4—92no—
161C4—91no—
162C4—182no—
163C4—181no—
164C4C2×C3×#15711yescharacteristic, complemented
165C3—22no—
166C3—42no—
167C3D4×#15711yescharacteristic, direct product factor
168C2—22no—
169C2—91no—
170C2—91no—
171C2—182no—
172C2C22×C3×#15711yescharacteristic
subgroupG/H
centerC2×C3C22×#157
commutatorC2×C32C22×C3×C4
FrattiniC2C22×C3×#157
FittingC33×D4C4
socleC2×C33C22×C4
prime radicalC3×D4×C3⋊D3C2
cosocleC2×C3⋊D3C23×C3
hypercenterC3×D4#157
hypocenterC32C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#157C2semidirect
3. C3×C22⋊#157C2semidirect
4. C3×C4⋊#157C2semidirect
5. C3×C4×#157C2semidirect
6. D4×#157C3direct (N × H)
12. C33×D4C4semidirect
13. C3×#638C4semidirect
14. C3×C4⋊(C3⋊D3)C4semidirect
16. C22×#157C2×C3semidirect
19. C22⋊#157C2×C3semidirect
20. C4⋊#157C2×C3semidirect
21. C4×#157C2×C3semidirect
22. C22×C33C2×C4semidirect
25. C2×C3×C3⋊D3C2×C4semidirect
29. C3×#157D4direct (N × H)
30. C33×C4C2×C4semidirect
31. C3×C3⋊Dic3C2×C4semidirect
41. C32×D4C3×C4semidirect
42. #638C3×C4semidirect
43. C4⋊(C3⋊D3)C3×C4semidirect
61. C22×C32C2×C3×C4semidirect
64. C2×C3⋊D3C2×C3×C4semidirect
72. #157C3×D4direct (N × H)
75. C32×C4C2×C3×C4semidirect
76. C3⋊Dic3C2×C3×C4semidirect
79. C33C4×D4semidirect
86. C3×D4#157direct (N × H)
118. C22×C3C2×#157semidirect
129. C3×C4C2×#157semidirect
133. C32C3×C4×D4semidirect
137. D4C3×#157direct (N × H)
159. C22C2×C3×#157semidirect
164. C4C2×C3×#157semidirect
167. C3D4×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1572
C3×D4×C3⋊D32
C3×C22⋊#1572
C3×C4⋊#1572
C3×C4×#1572
D4×#1573
C3×C4×D49