Finite Group Databasebeta

← C22×(C3⋊D9)⋊C4 #11722525

Subgroups — C22×(C3⋊D9)⋊C4 #11722525

Order 864.

842 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 231 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
1C23×C3⋊D9C211yesmaximal, characteristic
2C22×C9×Dic3C211yesmaximal, characteristic, complemented
3C22×C3×Dic9C211yesmaximal, characteristic, complemented
4C2×(C3⋊D9)⋊C4C2112yesmaximal, direct product factor
5C22×#624—31nomaximal
6C22×C4×D9—31nomaximal
7C23×C3×C9C2211yescharacteristic
8C22×C3⋊D9C418yescomplemented
9C22×C3⋊D9C2216yes—
10C2×C9×Dic3C2216yescomplemented
11C2×C3×Dic9C2216yescomplemented
12(C3⋊D9)⋊C4C22116yesdirect product factor
13C23×C3⋊D3—31no—
14C22×C3×Dic3—31no—
15C22×C3×Dic3D311yescharacteristic
16C2×#624—312no—
17C23×D9—31no—
18C23×D9—61no—
19C22×C4×C9—31no—
20C22×Dic9D311yescharacteristic, complemented
21C2×C4×D9—312no—
22C22×C3×C9C2×C414yescomplemented
23C22×C3×C9C2313yes—
24C2×C3⋊D9C2×C4124yescomplemented
25C2×C3⋊D9C2314yes—
26C9×Dic3C2314yescomplemented
27C3×Dic9C2314yescomplemented
28C22×C4×D3—91no—
29C22×C4×D3—91no—
30C23×C32C2×D311yescharacteristic
31C22×C3⋊D3—38no—
32C22×C3⋊D3—36no—
33C2×C3×Dic3—36no—
34C2×C3×Dic3C2×D316yes—
35#624—316no—
36C23×C9—21no—
37C23×C9C2×D311yescharacteristic
38C22×D9—38no—
39C22×D9—36no—
40C22×D9—68no—
41C22×D9—66no—
42C2×C4×C9—36no—
43C2×Dic9C2×D316yescomplemented
44C4×D9—316no—
45C2×C3×C9C22×C416yescomplemented
46C2×C3×C9C2411yescharacteristic
47C3⋊D9C22×C418yescomplemented
48C23×D3—91no—
49C23×D3—91no—
50C23×D3—181no—
51C22×C3×C4—31no—
52C22×C3×C4—91no—
53C22×Dic3—31no—
54C22×Dic3D911yescharacteristic, complemented
55C2×C4×D3—912no—
56C2×C4×D3—912no—
57C22×C32C4×D314yes—
58C22×C32C22×D313yes—
59C2×C3⋊D3—324no—
60C2×C3⋊D3—34no—
61C3×Dic3—34no—
62C3×Dic3C22×D314yes—
63C22×C9—24no—
64C22×C9—23no—
65C22×C9C4×D314yescomplemented
66C22×C9C22×D313yes—
67C2×D9—324no—
68C2×D9—34no—
69C2×D9—624no—
70C2×D9—64no—
71C4×C9—34no—
72Dic9C22×D314yescomplemented
73C23×C4—271noSylow
74C3×C9C23×C411yescharacteristic, complemented, Sylow
75C23×C3—21no—
76C23×C3C2×D911yescharacteristic
77C23×C3D3211yescharacteristic
78C22×D3—98no—
79C22×D3—96no—
80C22×D3—98no—
81C22×D3—96no—
82C22×D3—188no—
83C22×D3—186no—
84C2×C3×C4—36no—
85C2×C3×C4—96no—
86C2×Dic3—36no—
87C2×Dic3C2×D916yescomplemented
88C4×D3—916no—
89C4×D3—916no—
90C2×C32C2×C4×D316yes—
91C2×C32C23×D311yescharacteristic
92C3⋊D3—38no—
93C2×C9—21no—
94C2×C9—26no—
95C2×C9C2×C4×D316yescomplemented
96C2×C9C23×D311yescharacteristic
97D9—38no—
98D9—68no—
99C24—271no—
100C22×C4—31no—
101C22×C4—91no—
102C22×C4—2712no—
103C22×C3—24no—
104C22×C3—23no—
105C22×C3C4×D914yescomplemented
106C22×C3C22×D913yes—
107C22×C3#62414yes—
108C22×C3C2×D3213yes—
109C2×D3—94no—
110C2×D3—924no—
111C2×D3—94no—
112C2×D3—924no—
113C2×D3—184no—
114C2×D3—1824no—
115C3×C4—34no—
116C3×C4—94no—
117Dic3—34no—
118Dic3C22×D914yescomplemented
119C32C22×C4×D311yescharacteristic
120C9—21no—
121C9C22×C4×D311yescharacteristic, complemented
122C23—278no—
123C23—276no—
124C23D3×D911yescharacteristic
125C2×C4—36no—
126C2×C4—96no—
127C2×C4—2716no—
128C2×C3—21no—
129C2×C3—26no—
130C2×C3C2×C4×D916yescomplemented
131C2×C3C23×D911yescharacteristic
132C2×C3C2×#62416yes—
133C2×C3C22×D3211yescharacteristic
134D3—98no—
135D3—98no—
136D3—188no—
137C22—274no—
138C22—2724no—
139C22(C3⋊D9)⋊C414yesdirect product factor
140C22C2×D3×D913yes—
141C4—34no—
142C4—94no—
143C3—21no—
144C3C22×C4×D911yescharacteristic, complemented
145C3C22×#62411yescharacteristic
146C2—278no—
147C2C2×(C3⋊D9)⋊C416yesdirect product factor
148C2C22×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC23D3×D9
commutator, hypocenterC3×C9C23×C4
FrattiniC2×C3C22×D32
FittingC23×C3×C9C22
socleC23×C32C2×D3
prime radicalC23×C3⋊D9C2
cosocleC2×C3×C9C24
LCSG ▷ C3×C9
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C9×Dic3C2semidirect
3. C22×C3×Dic9C2semidirect
4. C2×(C3⋊D9)⋊C4C2direct (N × H)
8. C22×C3⋊D9C4semidirect
10. C2×C9×Dic3C22semidirect
11. C2×C3×Dic9C22semidirect
12. (C3⋊D9)⋊C4C22direct (N × H)
20. C22×Dic9D3semidirect
22. C22×C3×C9C2×C4semidirect
24. C2×C3⋊D9C2×C4semidirect
26. C9×Dic3C23semidirect
27. C3×Dic9C23semidirect
43. C2×Dic9C2×D3semidirect
45. C2×C3×C9C22×C4semidirect
47. C3⋊D9C22×C4semidirect
54. C22×Dic3D9semidirect
65. C22×C9C4×D3semidirect
72. Dic9C22×D3semidirect
74. C3×C9C23×C4semidirect
87. C2×Dic3C2×D9semidirect
95. C2×C9C2×C4×D3semidirect
105. C22×C3C4×D9semidirect
118. Dic3C22×D9semidirect
121. C9C22×C4×D3semidirect
130. C2×C3C2×C4×D9semidirect
139. C22(C3⋊D9)⋊C4direct (N × H)
144. C3C22×C4×D9semidirect
147. C2C2×(C3⋊D9)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊D92
C22×C9×Dic32
C22×C3×Dic92
C2×(C3⋊D9)⋊C42
C22×#6243
C22×C4×D93