Finite Group Databasebeta

← C2×C4×C3⋊AGL(1,𝔽9) #13180938

Subgroups — C2×C4×C3⋊AGL(1,𝔽9) #13180938

Order 1728.

394 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 151 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
3C4×C3⋊AGL(1,𝔽9)C214yesmaximal, direct product factor
4C2×C4×AGL(1,𝔽9)—31nomaximal
5C22×C3×#157C411yescharacteristic
6C22×C3×#157C2211yescharacteristic
7C2×C3⋊AGL(1,𝔽9)C418yesdirect product factor
8C2×C3⋊AGL(1,𝔽9)C2214yes—
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C3×C4×#157C412yes—
11C3×C4×#157C2212yes—
12C22×AGL(1,𝔽9)—32no—
13C2×C4×#157D311yescharacteristic
14C4×AGL(1,𝔽9)—34no—
15C22×C3×C3⋊D3C2×C411yescharacteristic
16C2×C3×#157C814yescomplemented
17C2×C3×#157C2×C411yescharacteristic
18C2×C3×#157C2×C414yes—
19C2×C3×#157C2×C412yes—
20C2×C3×#157C2311yescharacteristic
21C3⋊AGL(1,𝔽9)C2×C418yesdirect product factor
22C2×C33×C4C811yescharacteristic, complemented
23C2×C3×C3⋊Dic3C811yescharacteristic, complemented
24C3×C4×C3⋊D3C812yescomplemented
25C3×C4×C3⋊D3C2×C412yes—
26C2×C4×C3⋊C8—91nomaximal
27C22×#157Dic311yescharacteristic
28C22×#157C2×D311yescharacteristic
29C2×AGL(1,𝔽9)—38no—
30C2×AGL(1,𝔽9)—34no—
31C2×C4×C3⋊D3Dic311yescharacteristic
32C2×C3×C4×D3—121no—
33C4×#157Dic312yes—
34C4×#157C2×D312yes—
35C22×C33C2×C811yescharacteristic
36C2×C3×C3⋊D3C4212yes—
37C2×C3×C3⋊D3C2×C811yescharacteristic
38C2×C3×C3⋊D3C2×C812yes—
39C2×C3×C3⋊D3C22×C411yescharacteristic
40C3×#157C4212yes—
41C3×#157C2×C814yescomplemented
42C3×#157C22×C412yes—
43C33×C4C2×C812yescomplemented
44C3×C3⋊Dic3C2×C812yescomplemented
45C2×C3×C42—91no—
46C22×C3⋊C8—92no—
47C4×C3⋊C8—94no—
48C22×C3⋊D3C2×Dic311yescharacteristic
49C22×C3×D3—121no—
50C2×#157C3⋊C814yescomplemented
51C2×#157C4×D314yes—
52C2×#157C2×Dic312yes—
53C2×#157C2×Dic311yescharacteristic
54C2×#157C22×D311yescharacteristic
55AGL(1,𝔽9)—38no—
56C2×C32×C4—41no—
57C2×C32×C4—81no—
58C2×C32×C4C3⋊C811yescharacteristic, complemented
59C2×C3⋊Dic3C3⋊C811yescharacteristic, complemented
60C4×C3⋊D3C3⋊C812yescomplemented
61C4×C3⋊D3C2×Dic312yes—
62C2×C3×Dic3—121no—
63C3×C4×D3—122no—
64C3×C4×D3—122no—
65C2×C4×C8—271noSylow
66C2×C33C4×C812yescomplemented
67C2×C33C22×C811yescharacteristic
68C3×C3⋊D3C4×C812yescomplemented
69C3×C3⋊D3C2×C4211yescharacteristic
70C3×C3⋊D3C22×C811yescharacteristic
71C22×C3×C4—91no—
72C22×C3×C4—91no—
73C22×C3×C4—91no—
74C2×C4×D3—121no—
75C3×C42—92no—
76C3×C42—92no—
77C2×C3⋊C8—98no—
78C2×C3⋊C8—94no—
79C22×C32—41no—
80C22×C32—81no—
81C22×C32C2×C3⋊C811yescharacteristic
82C2×C3⋊D3C2×C3⋊C811yescharacteristic
83C2×C3⋊D3C2×C3⋊C812yes—
84C2×C3⋊D3C4×Dic312yes—
85C2×C3⋊D3C22×Dic311yescharacteristic
86C2×C3×D3—122no—
87C2×C3×D3—121no—
88C2×C3×D3—122no—
89C2×C3×D3—121no—
90#157C2×C3⋊C814yescomplemented
91#157C4×Dic312yes—
92#157C2×C4×D312yes—
93C32×C4—42no—
94C32×C4—82no—
95C32×C4C2×C3⋊C812yescomplemented
96C3⋊Dic3C2×C3⋊C812yescomplemented
97C3×Dic3—122no—
98C22×C8—272no—
99C2×C42—91no—
100C4×C8—274no—
101C33C2×C4×C811yescharacteristic, complemented, Sylow
102C23×C3—91no—
103C22×D3—121no—
104C2×C3×C4—41no—
105C2×C3×C4—81no—
106C2×C3×C4—94no—
107C2×C3×C4—94no—
108C2×C3×C4—91no—
109C2×C3×C4—92no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4—92no—
113C2×C3×C4—92no—
114C2×C3×C4AGL(1,𝔽9)11yescharacteristic, complemented
115C2×Dic3—121no—
116C4×D3—122no—
117C4×D3—122no—
118C3⋊C8—98no—
119C2×C32—41no—
120C2×C32—42no—
121C2×C32—82no—
122C2×C32—81no—
123C2×C32C4×C3⋊C812yescomplemented
124C2×C32C22×C3⋊C811yescharacteristic
125C3⋊D3C4×C3⋊C812yescomplemented
126C3⋊D3C22×C3⋊C811yescharacteristic
127C3⋊D3C2×C4×Dic311yescharacteristic
128C3×D3—122no—
129C3×D3—121no—
130C3×D3—121no—
131C22×C4—91no—
132C22×C4—91no—
133C22×C4—91no—
134C2×C8—274no—
135C2×C8—278no—
136C42—92no—
137C42—92no—
138C22×C3—41no—
139C22×C3—81no—
140C22×C3—91no—
141C22×C3—91no—
142C22×C3—92no—
143C22×C3—92no—
144C22×C3C2×AGL(1,𝔽9)11yescharacteristic
145C2×D3—121no—
146C2×D3—121no—
147C2×D3—122no—
148C2×D3—122no—
149C3×C4—42no—
150C3×C4—82no—
151C3×C4—92no—
152C3×C4—92no—
153C3×C4—94no—
154C3×C4—92no—
155C3×C4C2×AGL(1,𝔽9)12yescomplemented
156Dic3—122no—
157C32—41no—
158C32—81no—
159C32C2×C4×C3⋊C811yescharacteristic, complemented
160C23—91no—
161C2×C4—91no—
162C2×C4—92no—
163C2×C4—91no—
164C2×C4—94no—
165C2×C4—94no—
166C2×C4—91no—
167C2×C4—92no—
168C2×C4—92no—
169C2×C4C3⋊AGL(1,𝔽9)11yescharacteristic, direct product factor
170C8—278no—
171C2×C3—41no—
172C2×C3—42no—
173C2×C3—81no—
174C2×C3—82no—
175C2×C3—91no—
176C2×C3—91no—
177C2×C3—92no—
178C2×C3C4×AGL(1,𝔽9)12yescomplemented
179C2×C3C22×AGL(1,𝔽9)11yescharacteristic
180D3—122no—
181D3—121no—
182D3—121no—
183C22—91no—
184C22—91no—
185C22—92no—
186C22—92no—
187C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
188C4—92no—
189C4—94no—
190C4—92no—
191C4—92no—
192C4C2×C3⋊AGL(1,𝔽9)12yesdirect product factor
193C3—41no—
194C3—81no—
195C3C2×C4×AGL(1,𝔽9)11yescharacteristic, complemented
196C2—91no—
197C2—91no—
198C2—92no—
199C2C4×C3⋊AGL(1,𝔽9)12yesdirect product factor
200C2C22×C3⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊AGL(1,𝔽9)
commutator, hypocenterC33C2×C4×C8
FrattiniC2C22×C3⋊AGL(1,𝔽9)
FittingC2×C33×C4C8
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×#157C23
LCSG ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊AGL(1,𝔽9)C2direct (N × H)
7. C2×C3⋊AGL(1,𝔽9)C4direct (N × H)
16. C2×C3×#157C8semidirect
21. C3⋊AGL(1,𝔽9)C2×C4direct (N × H)
22. C2×C33×C4C8semidirect
23. C2×C3×C3⋊Dic3C8semidirect
24. C3×C4×C3⋊D3C8semidirect
41. C3×#157C2×C8semidirect
43. C33×C4C2×C8semidirect
44. C3×C3⋊Dic3C2×C8semidirect
50. C2×#157C3⋊C8semidirect
58. C2×C32×C4C3⋊C8semidirect
59. C2×C3⋊Dic3C3⋊C8semidirect
60. C4×C3⋊D3C3⋊C8semidirect
66. C2×C33C4×C8semidirect
68. C3×C3⋊D3C4×C8semidirect
90. #157C2×C3⋊C8semidirect
95. C32×C4C2×C3⋊C8semidirect
96. C3⋊Dic3C2×C3⋊C8semidirect
101. C33C2×C4×C8semidirect
114. C2×C3×C4AGL(1,𝔽9)semidirect
123. C2×C32C4×C3⋊C8semidirect
125. C3⋊D3C4×C3⋊C8semidirect
155. C3×C4C2×AGL(1,𝔽9)semidirect
159. C32C2×C4×C3⋊C8semidirect
169. C2×C4C3⋊AGL(1,𝔽9)direct (N × H)
178. C2×C3C4×AGL(1,𝔽9)semidirect
192. C4C2×C3⋊AGL(1,𝔽9)direct (N × H)
195. C3C2×C4×AGL(1,𝔽9)semidirect
199. C2C4×C3⋊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
C4×C3⋊AGL(1,𝔽9)2
C2×C4×AGL(1,𝔽9)3
C2×C4×C3⋊C89