Finite Group Databasebeta

← C2×A4×AGL(1,𝔽9) #13181802

Subgroups — C2×A4×AGL(1,𝔽9) #13181802

Order 1728.

233 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C2×A4×#157C211yesmaximal, characteristic
2A4×AGL(1,𝔽9)C212yesmaximal, direct product factor
3C23×AGL(1,𝔽9)C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊D3C411yescharacteristic
5C2×C3×AGL(1,𝔽9)—41nomaximal
6A4×#157C411yescharacteristic
7A4×#157C2211yescharacteristic
8C23×#157C2×C311yescharacteristic
9C22×AGL(1,𝔽9)—31no—
10C22×AGL(1,𝔽9)—31no—
11C22×AGL(1,𝔽9)—32no—
12C22×AGL(1,𝔽9)C2×C312yescomplemented
13C2×C32×A4C811yescharacteristic, complemented
14C2×C3×#157—41no—
15A4×C3⋊D3C811yescharacteristic, complemented
16A4×C3⋊D3C2×C411yescharacteristic
17C3×AGL(1,𝔽9)—42no—
18C2×C8×A4—91nomaximal
19C23×C3⋊D3C3×C411yescharacteristic
20C22×#157—31no—
21C22×#157—31no—
22C22×#157—31no—
23C22×#157—31no—
24C22×#157C3×C411yescharacteristic
25C22×#157C22×C311yescharacteristic
26C2×D3×A4—121no—
27C2×AGL(1,𝔽9)—32no—
28C2×AGL(1,𝔽9)—31no—
29C2×AGL(1,𝔽9)—32no—
30C2×AGL(1,𝔽9)—32no—
31C2×AGL(1,𝔽9)—32no—
32C2×AGL(1,𝔽9)A411yescharacteristic, direct product factor
33C2×C3×C3⋊D3—41no—
34C32×A4C2×C811yescharacteristic, complemented
35C3×#157—41no—
36C3×#157—41no—
37C2×C4×A4—91no—
38C8×A4—92no—
39C23×C32C3×C811yescharacteristic, complemented
40C22×C3⋊D3—31no—
41C22×C3⋊D3—31no—
42C22×C3⋊D3—31no—
43C22×C3⋊D3—31no—
44C22×C3⋊D3C3×C811yescharacteristic, complemented
45C22×C3⋊D3C2×C3×C411yescharacteristic
46C2×C3×A4—41no—
47C2×C3×A4—81no—
48C2×#157—32no—
49C2×#157—31no—
50C2×#157—31no—
51C2×#157—31no—
52C2×#157—31no—
53C2×#157—31no—
54C2×#157—31no—
55C2×#157—31no—
56C2×#157C2×A411yescharacteristic
57D3×A4—121no—
58D3×A4—121no—
59AGL(1,𝔽9)—32no—
60AGL(1,𝔽9)C2×A412yesdirect product factor
61C23×C8—91noSylow
62C2×C33—41no—
63C3×C3⋊D3—41no—
64C3×C3⋊D3—41no—
65C23×D3—121no—
66C22×A4—91no—
67C4×A4—91no—
68C4×A4—91no—
69C2×C3×C8—361no—
70C22×C32—31no—
71C22×C32—31no—
72C22×C32C2×C3×C811yescharacteristic, complemented
73C2×C3⋊D3—32no—
74C2×C3⋊D3—31no—
75C2×C3⋊D3—31no—
76C2×C3⋊D3—31no—
77C2×C3⋊D3—31no—
78C2×C3⋊D3—31no—
79C2×C3⋊D3—31no—
80C2×C3⋊D3—31no—
81C2×C3⋊D3C4×A411yescharacteristic
82C2×C3×D3—481no—
83C3×A4—41no—
84C3×A4—81no—
85#157—31no—
86#157—31no—
87#157C4×A411yescharacteristic
88#157C22×A411yescharacteristic
89C23×C4—91no—
90C22×C8—92no—
91C22×C8—271no—
92C22×C8—272no—
93C22×C8—271no—
94C33—41noSylow
95C23×C3—41no—
96C22×D3—121no—
97C22×D3—121no—
98C22×D3—361no—
99C22×D3—361no—
100C22×D3—361no—
101C22×D3—361no—
102C2×A4—81no—
103C2×A4—91no—
104C2×A4—91no—
105C2×A4AGL(1,𝔽9)11yescharacteristic, direct product factor
106C2×C3×C4—361no—
107C3×C8—362no—
108C2×C32—31no—
109C2×C32—31no—
110C2×C32—161no—
111C2×C32—321no—
112C2×C32C8×A411yescharacteristic, complemented
113C3⋊D3—31no—
114C3⋊D3—31no—
115C3⋊D3C8×A411yescharacteristic, complemented
116C3⋊D3C2×C4×A411yescharacteristic
117C3×D3—481no—
118C3×D3—481no—
119C24—91no—
120C22×C4—91no—
121C22×C4—91no—
122C22×C4—271no—
123C22×C4—271no—
124C22×C4—271no—
125C22×C4—271no—
126C2×C8—91no—
127C2×C8—272no—
128C2×C8—272no—
129C2×C8—271no—
130C2×C8—272no—
131C2×C8—272no—
132C22×C3—41no—
133C22×C3—121no—
134C22×C3—121no—
135C22×C3—361no—
136C2×D3—121no—
137C2×D3—361no—
138C2×D3—361no—
139C2×D3—361no—
140C2×D3—361no—
141C2×D3—361no—
142C2×D3—362no—
143C2×D3—361no—
144C2×D3—361no—
145A4—81no—
146A4C2×AGL(1,𝔽9)11yescharacteristic, direct product factor
147C3×C4—361no—
148C3×C4—361no—
149C32—161no—
150C32—321no—
151C32C2×C8×A411yescharacteristic, complemented
152C23—91no—
153C23—91no—
154C23—271no—
155C23—271no—
156C23—271no—
157C23—271no—
158C23C3×AGL(1,𝔽9)11yescharacteristic, complemented
159C2×C4—91no—
160C2×C4—271no—
161C2×C4—271no—
162C2×C4—271no—
163C2×C4—271no—
164C2×C4—271no—
165C2×C4—272no—
166C2×C4—271no—
167C2×C4—271no—
168C8—92no—
169C8—272no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—121no—
173C2×C3—121no—
174C2×C3—321no—
175C2×C3—361no—
176C2×C3—361no—
177D3—121no—
178D3—121no—
179D3—361no—
180D3—361no—
181C22—31no—
182C22—31no—
183C22—91no—
184C22—271no—
185C22—271no—
186C22—271no—
187C22—271no—
188C22—271no—
189C22—271no—
190C22—272no—
191C22—271no—
192C22C2×C3×AGL(1,𝔽9)11yescharacteristic, complemented
193C4—91no—
194C4—91no—
195C4—271no—
196C4—271no—
197C3—41no—
198C3—41no—
199C3—321no—
200C2—31no—
201C2—31no—
202C2—91no—
203C2—91no—
204C2—271no—
205C2—271no—
206C2A4×AGL(1,𝔽9)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4×AGL(1,𝔽9)
commutator, hypocenterC22×C32C2×C3×C8
FrattiniC1G
Fitting, socleC23×C32C3×C8
prime radicalC2×A4×C3⋊D3C4
cosocleC22×#157C22×C3
LCSG ▷ C22×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. A4×AGL(1,𝔽9)C2direct (N × H)
3. C23×AGL(1,𝔽9)C3semidirect
12. C22×AGL(1,𝔽9)C2×C3semidirect
13. C2×C32×A4C8semidirect
15. A4×C3⋊D3C8semidirect
32. C2×AGL(1,𝔽9)A4direct (N × H)
34. C32×A4C2×C8semidirect
39. C23×C32C3×C8semidirect
44. C22×C3⋊D3C3×C8semidirect
60. AGL(1,𝔽9)C2×A4direct (N × H)
72. C22×C32C2×C3×C8semidirect
105. C2×A4AGL(1,𝔽9)direct (N × H)
112. C2×C32C8×A4semidirect
115. C3⋊D3C8×A4semidirect
146. A4C2×AGL(1,𝔽9)direct (N × H)
151. C32C2×C8×A4semidirect
158. C23C3×AGL(1,𝔽9)semidirect
192. C22C2×C3×AGL(1,𝔽9)semidirect
206. C2A4×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
C2×A4×#1572
A4×AGL(1,𝔽9)2
C23×AGL(1,𝔽9)3
C2×C3×AGL(1,𝔽9)4
C2×C8×A49