Finite Group Databasebeta

← C23×C4×C3⋊D9 #13164972

Subgroups — C23×C4×C3⋊D9 #13164972

Order 1728.

6810 conjugacy classes of subgroups; only the 1743 normal subgroups are shown.

normal subgroups only

This group has 6810 conjugacy classes of subgroups — too many to draw. Only its 1743 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
1C24×C3⋊D9C211yesmaximal, characteristic
2C23×C3×C4×C9C211yesmaximal, characteristic, complemented
3C23×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C22×C4×C3⋊D9C2128yesmaximal, direct product factor
5C23×C4×C3⋊D3—31nomaximal
6C23×C4×D9—33nomaximal
7C24×C3×C9C2211yescharacteristic
8C23×C3⋊D9C4116yesdirect product factor
9C23×C3⋊D9C22114yes—
10C22×C3×C4×C9C22114yescomplemented
11C22×C3⋊Dic9C22114yescomplemented
12C2×C4×C3⋊D9C221112yesdirect product factor
13C24×C3⋊D3—31no—
14C23×C32×C4D311yescharacteristic
15C23×C3⋊Dic3—31no—
16C22×C4×C3⋊D3—328no—
17C24×D9—33no—
18C23×C4×C9D313yescomplemented
19C23×Dic9—33no—
20C22×C4×D9—384no—
21C23×C3×C9C2×C418yescomplemented
22C23×C3×C9C2317yes—
23C22×C3⋊D9C2×C41112yesdirect product factor
24C22×C3⋊D9C23128yes—
25C2×C3×C4×C9C23128yescomplemented
26C2×C3⋊Dic9C23128yescomplemented
27C4×C3⋊D9C23164yesdirect product factor
28C23×C4×D3—93no—
29C23×C4×D3—91no—
30C24×C32C2×D311yescharacteristic
31C23×C3⋊D3—316no—
32C23×C3⋊D3—314no—
33C22×C32×C4C2×D3114yes—
34C22×C3⋊Dic3—314no—
35C2×C4×C3⋊D3—3112no—
36C24×C9C2×D313yes—
37C23×D9—348no—
38C23×D9—342no—
39C22×C4×C9C2×D3142yescomplemented
40C22×Dic9—342no—
41C2×C4×D9—3336no—
42C22×C3×C9C22×C4128yescomplemented
43C22×C3×C9C2417yes—
44C2×C3⋊D9C22×C41112yesdirect product factor
45C2×C3⋊D9C2418yes—
46C3×C4×C9C2418yescomplemented
47C3⋊Dic9C2418yescomplemented
48C24×D3—93no—
49C24×D3—91no—
50C23×C3×C4D913yescomplemented
51C23×C3×C4C3⋊D311yescharacteristic
52C23×Dic3—93no—
53C23×Dic3—91no—
54C22×C4×D3—984no—
55C22×C4×D3—928no—
56C23×C32C4×D318yes—
57C23×C32C22×D317yes—
58C22×C3⋊D3—3112no—
59C22×C3⋊D3—328no—
60C2×C32×C4C22×D3128yes—
61C2×C3⋊Dic3—328no—
62C4×C3⋊D3—364no—
63C23×C9C4×D3124yescomplemented
64C23×C9C22×D3121yes—
65C22×D9—3336no—
66C22×D9—384no—
67C2×C4×C9C22×D3184yescomplemented
68C2×Dic9—384no—
69C4×D9—3192no—
70C24×C4—271noSylow
71C2×C3×C9C23×C4114yescomplemented
72C2×C3×C9C2511yescharacteristic
73C3⋊D9C23×C4116yesdirect product factor
74C24×C3C2×D913yes—
75C24×C3C2×C3⋊D311yescharacteristic
76C23×D3—948no—
77C23×D3—942no—
78C23×D3—916no—
79C23×D3—914no—
80C22×C3×C4C2×D9142yescomplemented
81C22×C3×C4C2×C3⋊D3114yes—
82C22×Dic3—942no—
83C22×Dic3—914no—
84C2×C4×D3—9336no—
85C2×C4×D3—9112no—
86C22×C32C2×C4×D3128yes—
87C22×C32C23×D317yes—
88C2×C3⋊D3—3112no—
89C2×C3⋊D3—38no—
90C32×C4C23×D318yes—
91C3⋊Dic3—38no—
92C22×C9C2×C4×D3184yescomplemented
93C22×C9C23×D3121yes—
94C2×D9—3336no—
95C2×D9—324no—
96C4×C9C23×D3124yescomplemented
97Dic9—324no—
98C25—271no—
99C23×C4—271no—
100C23×C4—2728no—
101C23×C4C3⋊D911yescharacteristic, direct product factor
102C3×C9C24×C411yescharacteristic, complemented, Sylow
103C23×C3C4×D9124yescomplemented
104C23×C3C22×D9121yes—
105C23×C3C4×C3⋊D318yes—
106C23×C3C22×C3⋊D317yes—
107C22×D3—928no—
108C22×D3—9336no—
109C22×D3—984no—
110C22×D3—9112no—
111C2×C3×C4C22×D9184yescomplemented
112C2×C3×C4C22×C3⋊D3128yes—
113C2×Dic3—928no—
114C2×Dic3—984no—
115C4×D3—964no—
116C4×D3—9192no—
117C2×C32C22×C4×D3114yes—
118C2×C32C24×D311yescharacteristic
119C3⋊D3—316no—
120C2×C9C22×C4×D3142yescomplemented
121C2×C9C24×D313yes—
122D9—348no—
123C24—2716no—
124C24—2714no—
125C24C2×C3⋊D911yescharacteristic
126C22×C4—2714no—
127C22×C4—27112no—
128C22×C4C2×C3⋊D9114yesdirect product factor
129C22×C3C2×C4×D9184yescomplemented
130C22×C3C23×D9121yes—
131C22×C3C2×C4×C3⋊D3128yes—
132C22×C3C23×C3⋊D317yes—
133C2×D3—9112no—
134C2×D3—98no—
135C2×D3—924no—
136C2×D3—9336no—
137C3×C4C23×D9124yescomplemented
138C3×C4C23×C3⋊D318yes—
139Dic3—98no—
140Dic3—924no—
141C32C23×C4×D311yescharacteristic
142C9C23×C4×D313yescomplemented
143C23—2728no—
144C23—27112no—
145C23C4×C3⋊D918yesdirect product factor
146C23C22×C3⋊D917yes—
147C2×C4—2728no—
148C2×C4—2764no—
149C2×C4C22×C3⋊D9128yesdirect product factor
150C2×C3C22×C4×D9142yescomplemented
151C2×C3C24×D913yes—
152C2×C3C22×C4×C3⋊D3114yes—
153C2×C3C24×C3⋊D311yescharacteristic
154D3—948no—
155D3—916no—
156C22—278no—
157C22—27112no—
158C22C2×C4×C3⋊D9128yesdirect product factor
159C22C23×C3⋊D917yes—
160C4—278no—
161C4C23×C3⋊D918yesdirect product factor
162C3C23×C4×D913yescomplemented
163C3C23×C4×C3⋊D311yescharacteristic
164C2—2716no—
165C2C22×C4×C3⋊D9114yesdirect product factor
166C2C24×C3⋊D911yescharacteristic
subgroupG/H
center, hypercenterC23×C4C3⋊D9
commutator, hypocenterC3×C9C24×C4
FrattiniC2×C3C24×C3⋊D3
FittingC23×C3×C4×C9C2
socleC24×C32C2×D3
prime radicalC24×C3⋊D9C2
cosocleC2×C3×C9C25
LCSG ▷ C3×C9
UCSC1 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4×C9C2semidirect
3. C23×C3⋊Dic9C2semidirect
4. C22×C4×C3⋊D9C2direct (N × H)
8. C23×C3⋊D9C4direct (N × H)
10. C22×C3×C4×C9C22semidirect
11. C22×C3⋊Dic9C22semidirect
12. C2×C4×C3⋊D9C22direct (N × H)
18. C23×C4×C9D3semidirect
21. C23×C3×C9C2×C4semidirect
23. C22×C3⋊D9C2×C4direct (N × H)
25. C2×C3×C4×C9C23semidirect
26. C2×C3⋊Dic9C23semidirect
27. C4×C3⋊D9C23direct (N × H)
39. C22×C4×C9C2×D3semidirect
42. C22×C3×C9C22×C4semidirect
44. C2×C3⋊D9C22×C4direct (N × H)
46. C3×C4×C9C24semidirect
47. C3⋊Dic9C24semidirect
50. C23×C3×C4D9semidirect
63. C23×C9C4×D3semidirect
67. C2×C4×C9C22×D3semidirect
71. C2×C3×C9C23×C4semidirect
73. C3⋊D9C23×C4direct (N × H)
80. C22×C3×C4C2×D9semidirect
92. C22×C9C2×C4×D3semidirect
96. C4×C9C23×D3semidirect
101. C23×C4C3⋊D9direct (N × H)
102. C3×C9C24×C4semidirect
103. C23×C3C4×D9semidirect
111. C2×C3×C4C22×D9semidirect
120. C2×C9C22×C4×D3semidirect
128. C22×C4C2×C3⋊D9direct (N × H)
129. C22×C3C2×C4×D9semidirect
137. C3×C4C23×D9semidirect
142. C9C23×C4×D3semidirect
145. C23C4×C3⋊D9direct (N × H)
149. C2×C4C22×C3⋊D9direct (N × H)
150. C2×C3C22×C4×D9semidirect
158. C22C2×C4×C3⋊D9direct (N × H)
161. C4C23×C3⋊D9direct (N × H)
162. C3C23×C4×D9semidirect
165. C2C22×C4×C3⋊D9direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3⋊D92
C23×C3×C4×C92
C23×C3⋊Dic92
C22×C4×C3⋊D92
C23×C4×C3⋊D33
C23×C4×D93