Finite Group Databasebeta

← C2×C3×C4⋊AGL(1,𝔽9) #13180926

Subgroups — C2×C3×C4⋊AGL(1,𝔽9) #13180926

Order 1728.

346 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 130 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
2C2×C3×C4×#157C211yesmaximal, characteristic
3C3×C4⋊AGL(1,𝔽9)C214yesmaximal, direct product factor
4C2×C4⋊AGL(1,𝔽9)C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C411yescharacteristic
6C22×C3×#157C2211yescharacteristic
7C2×C3×AGL(1,𝔽9)—24no—
8C2×C3×AGL(1,𝔽9)C2214yes—
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C3×C4×#157C412yes—
11C3×C4×#157C2212yes—
12C22×AGL(1,𝔽9)C2×C312yes—
13C2×C4×#157C2×C311yescharacteristic
14C4⋊AGL(1,𝔽9)C2×C314yesdirect product factor
15C22×C3×C3⋊D3C2×C411yescharacteristic
16C2×C3×#157—22no—
17C2×C3×#157C2×C412yes—
18C2×C3×#157C2×C411yescharacteristic
19C2×C3×#157D412yes—
20C2×C3×#157Q812yes—
21C2×C3×#157C2311yescharacteristic
22C3×AGL(1,𝔽9)—24no—
23C2×C33×C4C811yescharacteristic, complemented
24C2×C3×C3⋊Dic3C811yescharacteristic, complemented
25C3×C4×C3⋊D3C812yescomplemented
26C3×C4×C3⋊D3C2×C412yes—
27C2×C3×C4⋊C8—91nomaximal
28C22×#157C3×C411yescharacteristic
29C22×#157C22×C311yescharacteristic
30C2×AGL(1,𝔽9)—24no—
31C2×AGL(1,𝔽9)C22×C314yes—
32C2×C4×C3⋊D3C3×C411yescharacteristic
33C2×C3×C4×D3—121no—
34C4×#157C3×C412yes—
35C4×#157C22×C312yes—
36C22×C33C2×C811yescharacteristic
37C2×C3×C3⋊D3C4⋊C412yes—
38C2×C3×C3⋊D3C2×C811yescharacteristic
39C2×C3×C3⋊D3M1612yes—
40C2×C3×C3⋊D3C22×C411yescharacteristic
41C3×#157—22no—
42C3×#157C4⋊C412yes—
43C3×#157C2×D411yescharacteristic
44C3×#157C2×Q811yescharacteristic
45C33×C4C2×C812yescomplemented
46C3×C3⋊Dic3C2×C812yescomplemented
47C22×C3×C8—92no—
48C2×C3×C42—91no—
49C3×C4⋊C8—94no—
50C22×C3⋊D3C2×C3×C411yescharacteristic
51C22×C3×D3—121no—
52C2×#157—22no—
53C2×#157C2×C3×C412yes—
54C2×#157C2×C3×C411yescharacteristic
55C2×#157C3×D412yes—
56C2×#157C3×Q812yes—
57C2×#157C23×C311yescharacteristic
58AGL(1,𝔽9)—24no—
59C2×C32×C4—41no—
60C2×C32×C4—81no—
61C2×C32×C4C3×C811yescharacteristic, complemented
62C2×C3⋊Dic3C3×C811yescharacteristic, complemented
63C4×C3⋊D3C3×C812yescomplemented
64C4×C3⋊D3C2×C3×C412yes—
65C2×C3×Dic3—121no—
66C3×C4×D3—122no—
67C3×C4×D3—122no—
68C2×C4⋊C8—91noSylow
69C2×C33C4⋊C812yescomplemented
70C2×C33C22×C811yescharacteristic
71C3×C3⋊D3C4⋊C812yescomplemented
72C3×C3⋊D3C2×C4⋊C411yescharacteristic
73C3×C3⋊D3C2×M1611yescharacteristic
74C22×C3×C4—91no—
75C22×C3×C4—91no—
76C22×C3×C4—91no—
77C2×C4×D3—121no—
78C2×C3×C8—94no—
79C2×C3×C8—184no—
80C3×C42—92no—
81C3×C42—92no—
82C22×C32—41no—
83C22×C32—81no—
84C22×C32C2×C3×C811yescharacteristic
85C2×C3⋊D3C3×C4⋊C412yes—
86C2×C3⋊D3C2×C3×C811yescharacteristic
87C2×C3⋊D3C3×M1612yes—
88C2×C3⋊D3C22×C3×C411yescharacteristic
89C2×C3×D3—122no—
90C2×C3×D3—121no—
91C2×C3×D3—122no—
92C2×C3×D3—121no—
93#157—22no—
94#157C3×C4⋊C412yes—
95#157C2×C3×D411yescharacteristic
96#157C2×C3×Q811yescharacteristic
97C32×C4—42no—
98C32×C4—82no—
99C32×C4C2×C3×C812yescomplemented
100C3⋊Dic3C2×C3×C812yescomplemented
101C3×Dic3—122no—
102C22×C8—92no—
103C2×C42—91no—
104C4⋊C8—94no—
105C33C2×C4⋊C811yescharacteristic, complemented, Sylow
106C23×C3—91no—
107C22×D3—121no—
108C2×C3×C4—41no—
109C2×C3×C4—81no—
110C2×C3×C4—91no—
111C2×C3×C4—92no—
112C2×C3×C4—92no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—92no—
116C2×C3×C4—92no—
117C2×C3×C4—92no—
118C2×C3×C4—182no—
119C2×C3×C4AGL(1,𝔽9)11yescharacteristic, complemented
120C2×Dic3—121no—
121C4×D3—122no—
122C4×D3—122no—
123C3×C8—184no—
124C2×C32—41no—
125C2×C32—42no—
126C2×C32—82no—
127C2×C32—81no—
128C2×C32C3×C4⋊C812yescomplemented
129C2×C32C22×C3×C811yescharacteristic
130C3⋊D3C3×C4⋊C812yescomplemented
131C3⋊D3C2×C3×C4⋊C411yescharacteristic
132C3⋊D3C2×C3×M1611yescharacteristic
133C3×D3—122no—
134C3×D3—121no—
135C3×D3—121no—
136C22×C4—91no—
137C22×C4—91no—
138C22×C4—91no—
139C2×C8—94no—
140C2×C8—184no—
141C42—92no—
142C42—92no—
143C22×C3—41no—
144C22×C3—81no—
145C22×C3—91no—
146C22×C3—91no—
147C22×C3—92no—
148C22×C3—92no—
149C22×C3C2×AGL(1,𝔽9)11yescharacteristic
150C2×D3—121no—
151C2×D3—121no—
152C2×D3—122no—
153C2×D3—122no—
154C3×C4—42no—
155C3×C4—82no—
156C3×C4—91no—
157C3×C4—92no—
158C3×C4—91no—
159C3×C4—92no—
160C3×C4—182no—
161C3×C4C2×AGL(1,𝔽9)12yescomplemented
162Dic3—122no—
163C32—41no—
164C32—81no—
165C32C2×C3×C4⋊C811yescharacteristic, complemented
166C23—91no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—92no—
170C2×C4—92no—
171C2×C4—91no—
172C2×C4—92no—
173C2×C4—92no—
174C2×C4—92no—
175C2×C4—182no—
176C2×C4C3×AGL(1,𝔽9)11yescharacteristic, complemented
177C8—184no—
178C2×C3—41no—
179C2×C3—42no—
180C2×C3—81no—
181C2×C3—82no—
182C2×C3—91no—
183C2×C3—91no—
184C2×C3—92no—
185C2×C3C4⋊AGL(1,𝔽9)12yesdirect product factor
186C2×C3C22×AGL(1,𝔽9)11yescharacteristic
187D3—122no—
188D3—121no—
189D3—121no—
190C22—91no—
191C22—91no—
192C22—92no—
193C22—92no—
194C22C2×C3×AGL(1,𝔽9)11yescharacteristic
195C4—91no—
196C4—92no—
197C4—91no—
198C4—92no—
199C4—182no—
200C4C2×C3×AGL(1,𝔽9)12yescomplemented
201C3—41no—
202C3—81no—
203C3C2×C4⋊AGL(1,𝔽9)11yescharacteristic, direct product factor
204C2—91no—
205C2—91no—
206C2—92no—
207C2C3×C4⋊AGL(1,𝔽9)12yesdirect product factor
208C2C22×C3×AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC22×C3C2×AGL(1,𝔽9)
commutatorC2×C32C22×C3×C8
FrattiniC2C22×C3×AGL(1,𝔽9)
FittingC2×C33×C4C8
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×#157C23×C3
hypercenterC2×C3×C4AGL(1,𝔽9)
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊AGL(1,𝔽9)C2direct (N × H)
4. C2×C4⋊AGL(1,𝔽9)C3direct (N × H)
14. C4⋊AGL(1,𝔽9)C2×C3direct (N × H)
23. C2×C33×C4C8semidirect
24. C2×C3×C3⋊Dic3C8semidirect
25. C3×C4×C3⋊D3C8semidirect
45. C33×C4C2×C8semidirect
46. C3×C3⋊Dic3C2×C8semidirect
61. C2×C32×C4C3×C8semidirect
62. C2×C3⋊Dic3C3×C8semidirect
63. C4×C3⋊D3C3×C8semidirect
69. C2×C33C4⋊C8semidirect
71. C3×C3⋊D3C4⋊C8semidirect
99. C32×C4C2×C3×C8semidirect
100. C3⋊Dic3C2×C3×C8semidirect
105. C33C2×C4⋊C8semidirect
119. C2×C3×C4AGL(1,𝔽9)semidirect
128. C2×C32C3×C4⋊C8semidirect
130. C3⋊D3C3×C4⋊C8semidirect
161. C3×C4C2×AGL(1,𝔽9)semidirect
165. C32C2×C3×C4⋊C8semidirect
176. C2×C4C3×AGL(1,𝔽9)semidirect
185. C2×C3C4⋊AGL(1,𝔽9)direct (N × H)
200. C4C2×C3×AGL(1,𝔽9)semidirect
203. C3C2×C4⋊AGL(1,𝔽9)direct (N × H)
207. C2C3×C4⋊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
C2×C3×C4×#1572
C3×C4⋊AGL(1,𝔽9)2
C2×C4⋊AGL(1,𝔽9)3
C2×C3×C4⋊C89