Finite Group Databasebeta

← C3×D4×AGL(1,𝔽9) #13180929

Subgroups — C3×D4×AGL(1,𝔽9) #13180929

Order 1728.

338 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 102 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×AGL(1,𝔽9)C212yesmaximal, complemented
2C3×D4×#157C211yesmaximal, characteristic
3C3×C22⋊AGL(1,𝔽9)C212yesmaximal, complemented
4C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
5C3×C4×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
6D4×AGL(1,𝔽9)C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C2×C3×AGL(1,𝔽9)—24no—
9C2×C3×AGL(1,𝔽9)C2212yes—
10C2×C3×AGL(1,𝔽9)C2211yescharacteristic
11C2×C3×AGL(1,𝔽9)C2211yescharacteristic
12C3×D4×C3⋊D3C411yescharacteristic
13C3×C22⋊#157C412yes—
14C3×C4⋊#157C411yescharacteristic
15C3×C4×#157C2211yescharacteristic
16C22×AGL(1,𝔽9)C2×C312yescomplemented
17D4×#157C2×C311yescharacteristic
18C22⋊AGL(1,𝔽9)C2×C312yescomplemented
19C4⋊AGL(1,𝔽9)C2×C311yescharacteristic, complemented
20C4×AGL(1,𝔽9)C2×C311yescharacteristic, complemented
21C22×C3×C3⋊D3C2×C412yes—
22C2×C3×#157—22no—
23C2×C3×#157—22no—
24C2×C3×#157C2×C412yes—
25C2×C3×#157C2×C411yescharacteristic
26C2×C3×#157C2311yescharacteristic
27C3×AGL(1,𝔽9)—22no—
28C3×AGL(1,𝔽9)—21no—
29C3×AGL(1,𝔽9)D412yesdirect product factor
30C33×D4C811yescharacteristic, complemented
31C3×#638C812yescomplemented
32C3×C4⋊(C3⋊D3)C811yescharacteristic, complemented
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C8×D4—91nomaximal
35C22×#157C22×C312yes—
36C2×AGL(1,𝔽9)—24no—
37C2×AGL(1,𝔽9)C22×C312yes—
38C2×AGL(1,𝔽9)C22×C311yescharacteristic
39C2×AGL(1,𝔽9)C22×C311yescharacteristic
40D4×C3⋊D3C3×C411yescharacteristic
41C3×D3×D4—121no—
42C22⋊#157C3×C412yes—
43C4⋊#157C3×C411yescharacteristic
44C4×#157C22×C311yescharacteristic
45C22×C33C2×C812yescomplemented
46C2×C3×C3⋊D3—22no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3C2×C812yescomplemented
49C2×C3×C3⋊D3C22×C411yescharacteristic
50C3×#157—22no—
51C3×#157—21no—
52C3×#157C2×D411yescharacteristic
53C3×#157#4111yescharacteristic
54C33×C4C2×C811yescharacteristic, complemented
55C3×C3⋊Dic3C2×C811yescharacteristic, complemented
56C22×C3×C8—92no—
57C3×C4×D4—91no—
58C3×C4⋊C8—91no—
59C3×C22⋊C8—92no—
60C3×C4×C8—91no—
61C22×C3⋊D3C2×C3×C412yes—
62C22×C3×D3—122no—
63C2×#157—22no—
64C2×#157—22no—
65C2×#157C2×C3×C412yes—
66C2×#157C2×C3×C411yescharacteristic
67C2×#157C23×C311yescharacteristic
68AGL(1,𝔽9)—22no—
69AGL(1,𝔽9)—21no—
70AGL(1,𝔽9)C3×D412yesdirect product factor
71C32×D4—41no—
72C32×D4—81no—
73C32×D4C3×C811yescharacteristic, complemented
74#638C3×C812yescomplemented
75C4⋊(C3⋊D3)C3×C811yescharacteristic, complemented
76C4×C3⋊D3C2×C3×C411yescharacteristic
77C3×#67—122no—
78C3×D12—121no—
79C3×C4×D3—121no—
80C8×D4—91noSylow
81C2×C33—22no—
82C2×C33C22×C811yescharacteristic
83C3×C3⋊D3—22no—
84C3×C3⋊D3C4×D411yescharacteristic
85C3×C3⋊D3#13111yescharacteristic
86C2×C3×D4—91no—
87C22×C3×C4—92no—
88D3×D4—121no—
89C2×C3×C8—91no—
90C2×C3×C8—91no—
91C2×C3×C8—92no—
92C2×C3×C8—184no—
93C3×C4⋊C4—91no—
94C3×C22⋊C4—92no—
95C3×C42—91no—
96C22×C32—42no—
97C22×C32—82no—
98C22×C32C2×C3×C812yescomplemented
99C2×C3⋊D3—22no—
100C2×C3⋊D3—22no—
101C2×C3⋊D3C2×C3×C812yescomplemented
102C2×C3⋊D3C22×C3×C411yescharacteristic
103C2×C3×D3—121no—
104C2×C3×D3—122no—
105C2×C3×D3—242no—
106C2×C3×D3—242no—
107#157—22no—
108#157—21no—
109#157C2×C3×D411yescharacteristic
110#157C3×#4111yescharacteristic
111C32×C4—41no—
112C32×C4—81no—
113C32×C4C2×C3×C811yescharacteristic, complemented
114C3⋊Dic3C2×C3×C811yescharacteristic, complemented
115C3×Dic3—121no—
116C22×C8—92no—
117C4×D4—91no—
118C4⋊C8—91no—
119C22⋊C8—92no—
120C4×C8—91no—
121C33C8×D411yescharacteristic, complemented, Sylow
122C23×C3—92no—
123C22×D3—122no—
124C3×D4—41no—
125C3×D4—81no—
126C3×D4—92no—
127C3×D4—91no—
128C3×D4AGL(1,𝔽9)11yescharacteristic, direct product factor
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—92no—
133C2×C3×C4—182no—
134C2×C3×C4—182no—
135#67—122no—
136D12—121no—
137C4×D3—121no—
138C3×C8—92no—
139C3×C8—181no—
140C3×C8—182no—
141C2×C32—22no—
142C2×C32—41no—
143C2×C32—82no—
144C2×C32—81no—
145C2×C32—162no—
146C2×C32C22×C3×C811yescharacteristic
147C3⋊D3—22no—
148C3⋊D3C3×C4×D411yescharacteristic
149C3⋊D3C3×#13111yescharacteristic
150C3×D3—121no—
151C3×D3—121no—
152C3×D3—242no—
153C2×D4—91no—
154C22×C4—92no—
155C2×C8—91no—
156C2×C8—91no—
157C2×C8—92no—
158C2×C8—184no—
159C4⋊C4—91no—
160C22⋊C4—92no—
161C42—91no—
162C22×C3—42no—
163C22×C3—82no—
164C22×C3—91no—
165C22×C3—92no—
166C22×C3—182no—
167C22×C3—182no—
168C22×C3C2×AGL(1,𝔽9)12yescomplemented
169C2×D3—121no—
170C2×D3—122no—
171C2×D3—242no—
172C2×D3—242no—
173C3×C4—41no—
174C3×C4—81no—
175C3×C4—91no—
176C3×C4—91no—
177C3×C4—91no—
178C3×C4—182no—
179C3×C4—181no—
180C3×C4C2×AGL(1,𝔽9)11yescharacteristic, complemented
181Dic3—121no—
182C32—41no—
183C32—81no—
184C32C3×C8×D411yescharacteristic, complemented
185C23—92no—
186D4—92no—
187D4—91no—
188D4C3×AGL(1,𝔽9)11yescharacteristic, direct product factor
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—91no—
192C2×C4—92no—
193C2×C4—182no—
194C2×C4—182no—
195C8—92no—
196C8—182no—
197C8—181no—
198C2×C3—22no—
199C2×C3—41no—
200C2×C3—82no—
201C2×C3—81no—
202C2×C3—91no—
203C2×C3—91no—
204C2×C3—162no—
205C2×C3—182no—
206C2×C3C22×AGL(1,𝔽9)11yescharacteristic
207D3—121no—
208D3—121no—
209D3—242no—
210C22—91no—
211C22—92no—
212C22—182no—
213C22—182no—
214C22C2×C3×AGL(1,𝔽9)12yescomplemented
215C4—91no—
216C4—91no—
217C4—91no—
218C4—182no—
219C4—181no—
220C4C2×C3×AGL(1,𝔽9)11yescharacteristic, complemented
221C3—41no—
222C3—81no—
223C3D4×AGL(1,𝔽9)11yescharacteristic, direct product factor
224C2—22no—
225C2—91no—
226C2—91no—
227C2—182no—
228C2C22×C3×AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C22×AGL(1,𝔽9)
commutatorC2×C32C22×C3×C8
FrattiniC2C22×C3×AGL(1,𝔽9)
FittingC33×D4C8
socleC2×C33C22×C8
prime radicalC3×D4×C3⋊D3C4
cosocleC2×#157C23×C3
hypercenterC3×D4AGL(1,𝔽9)
hypocenterC32C3×C8×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×AGL(1,𝔽9)C2semidirect
3. C3×C22⋊AGL(1,𝔽9)C2semidirect
4. C3×C4⋊AGL(1,𝔽9)C2semidirect
5. C3×C4×AGL(1,𝔽9)C2semidirect
6. D4×AGL(1,𝔽9)C3direct (N × H)
16. C22×AGL(1,𝔽9)C2×C3semidirect
18. C22⋊AGL(1,𝔽9)C2×C3semidirect
19. C4⋊AGL(1,𝔽9)C2×C3semidirect
20. C4×AGL(1,𝔽9)C2×C3semidirect
29. C3×AGL(1,𝔽9)D4direct (N × H)
30. C33×D4C8semidirect
31. C3×#638C8semidirect
32. C3×C4⋊(C3⋊D3)C8semidirect
45. C22×C33C2×C8semidirect
48. C2×C3×C3⋊D3C2×C8semidirect
54. C33×C4C2×C8semidirect
55. C3×C3⋊Dic3C2×C8semidirect
70. AGL(1,𝔽9)C3×D4direct (N × H)
73. C32×D4C3×C8semidirect
74. #638C3×C8semidirect
75. C4⋊(C3⋊D3)C3×C8semidirect
98. C22×C32C2×C3×C8semidirect
101. C2×C3⋊D3C2×C3×C8semidirect
113. C32×C4C2×C3×C8semidirect
114. C3⋊Dic3C2×C3×C8semidirect
121. C33C8×D4semidirect
128. C3×D4AGL(1,𝔽9)direct (N × H)
168. C22×C3C2×AGL(1,𝔽9)semidirect
180. C3×C4C2×AGL(1,𝔽9)semidirect
184. C32C3×C8×D4semidirect
188. D4C3×AGL(1,𝔽9)direct (N × H)
214. C22C2×C3×AGL(1,𝔽9)semidirect
220. C4C2×C3×AGL(1,𝔽9)semidirect
223. C3D4×AGL(1,𝔽9)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,𝔽9)2
C3×D4×#1572
C3×C22⋊AGL(1,𝔽9)2
C3×C4⋊AGL(1,𝔽9)2
C3×C4×AGL(1,𝔽9)2
D4×AGL(1,𝔽9)3
C3×C8×D49