Finite Group Databasebeta

← C2×C3×AGL(1,𝔽9)⋊C4 #13180392

Subgroups — C2×C3×AGL(1,𝔽9)⋊C4 #13180392

Order 1728.

330 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 108 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)C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×C3×#3773C211yesmaximal, characteristic
4C3×AGL(1,𝔽9)⋊C4C214yesmaximal, direct product factor
5C2×AGL(1,𝔽9)⋊C4C311yesmaximal, characteristic, direct product factor
6C22×C3×#157—21no—
7C22×C3×#157C2211yescharacteristic
8C2×C3×AGL(1,𝔽9)C414yescomplemented
9C2×C3×AGL(1,𝔽9)C2212yes—
10C2×C3×#624—21no—
11C3×#3777—21no—
12C3×#3777C2212yes—
13C3×#3773—21no—
14C3×#3773C2212yes—
15C22×AGL(1,𝔽9)C2×C311yescharacteristic
16C2×#3777C2×C311yescharacteristic
17C2×#3773C2×C311yescharacteristic
18AGL(1,𝔽9)⋊C4C2×C314yesdirect product factor
19C22×C3×C3⋊D3D411yescharacteristic
20C2×C3×#157—22no—
21C2×C3×#157—42no—
22C2×C3×#157C2×C412yes—
23C2×C3×#157D411yescharacteristic
24C2×C3×#157Q812yes—
25C2×C3×#157C2311yescharacteristic
26C3×AGL(1,𝔽9)C2×C414yescomplemented
27C2×C32×Dic3—41no—
28C3×#624—22no—
29C3×#624—41no—
30C2×C3×#106—91nomaximal
31C22×#157—21no—
32C22×#157C22×C311yescharacteristic
33C2×AGL(1,𝔽9)C3×C414yescomplemented
34C2×AGL(1,𝔽9)C22×C312yes—
35C2×C3×C4×D3—121no—
36C2×#624—21no—
37#3777—21no—
38#3777C22×C312yes—
39#3773—21no—
40#3773C22×C312yes—
41C22×C33SD1611yescharacteristic
42C2×C3×C3⋊D3C4⋊C412yes—
43C2×C3×C3⋊D3SD1611yescharacteristic
44C2×C3×C3⋊D3SD1612yes—
45C2×C3×C3⋊D3C2×D411yescharacteristic
46C3×#157—42no—
47C3×#157C4⋊C412yes—
48C3×#157C22×C411yescharacteristic
49C3×#157C2×Q811yescharacteristic
50C32×Dic3—42no—
51C22×C3×C8—91no—
52C2×C3×C4⋊C4—91no—
53C2×C3×C4⋊C4—91no—
54C3×#106—94no—
55C22×C3⋊D3C3×D411yescharacteristic
56C22×C3×D3—121no—
57C2×#157—22no—
58C2×#157—42no—
59C2×#157C2×C3×C412yes—
60C2×#157C3×D411yescharacteristic
61C2×#157C3×Q812yes—
62C2×#157C23×C311yescharacteristic
63AGL(1,𝔽9)C2×C3×C414yescomplemented
64C2×C32×C4—121no—
65C2×C3×Dic3—41no—
66C2×C3×Dic3—41no—
67C2×C3×Dic3—81no—
68C3×C4×D3—122no—
69C3×C4×D3—122no—
70#624—22no—
71#624—41no—
72C2×#106—91noSylow
73C2×C33#10612yescomplemented
74C2×C33C2×SD1611yescharacteristic
75C3×C3⋊D3#10612yescomplemented
76C3×C3⋊D3C2×C4⋊C411yescharacteristic
77C3×C3⋊D3C2×SD1611yescharacteristic
78C22×C3×C4—91no—
79C22×C3×C4—181no—
80C22×C3×C4—181no—
81C2×C4×D3—121no—
82C2×C3×C8—94no—
83C2×C3×C8—92no—
84C3×C4⋊C4—92no—
85C3×C4⋊C4—92no—
86C3×C4⋊C4—181no—
87C3×C4⋊C4—181no—
88C22×C32—41no—
89C22×C32—81no—
90C22×C32C3×SD1611yescharacteristic
91C2×C3⋊D3C3×C4⋊C412yes—
92C2×C3⋊D3C3×SD1611yescharacteristic
93C2×C3⋊D3C3×SD1612yes—
94C2×C3⋊D3C2×C3×D411yescharacteristic
95C2×C3×D3—122no—
96C2×C3×D3—121no—
97C2×C3×D3—122no—
98C2×C3×D3—121no—
99#157—42no—
100#157C3×C4⋊C412yes—
101#157C22×C3×C411yescharacteristic
102#157C2×C3×Q811yescharacteristic
103C32×C4—122no—
104C3×Dic3—42no—
105C3×Dic3—42no—
106C3×Dic3—82no—
107C22×C8—91no—
108C2×C4⋊C4—91no—
109C2×C4⋊C4—91no—
110#106—94no—
111C33C2×#10611yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—121no—
114C2×C3×C4—92no—
115C2×C3×C4—91no—
116C2×C3×C4—92no—
117C2×C3×C4—91no—
118C2×C3×C4—121no—
119C2×C3×C4—121no—
120C2×C3×C4—182no—
121C2×C3×C4—182no—
122C2×C3×C4—241no—
123C2×C3×C4—361no—
124C2×C3×C4—362no—
125C2×Dic3—41no—
126C4×D3—122no—
127C4×D3—122no—
128C3×C8—94no—
129C2×C32—41no—
130C2×C32—42no—
131C2×C32—82no—
132C2×C32—81no—
133C2×C32C3×#10612yescomplemented
134C2×C32C2×C3×SD1611yescharacteristic
135C3⋊D3C3×#10612yescomplemented
136C3⋊D3C2×C3×C4⋊C411yescharacteristic
137C3⋊D3C2×C3×SD1611yescharacteristic
138C3×D3—122no—
139C3×D3—121no—
140C3×D3—121no—
141C22×C4—91no—
142C22×C4—181no—
143C22×C4—181no—
144C2×C8—94no—
145C2×C8—92no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3—91no—
153C22×C3—91no—
154C22×C3—92no—
155C22×C3—92no—
156C22×C3AΓL(1,𝔽9)11yescharacteristic
157C2×D3—121no—
158C2×D3—121no—
159C2×D3—122no—
160C2×D3—122no—
161C3×C4—91no—
162C3×C4—92no—
163C3×C4—91no—
164C3×C4—122no—
165C3×C4—122no—
166C3×C4—242no—
167C3×C4—362no—
168Dic3—42no—
169C32—41no—
170C32—81no—
171C32C2×C3×#10611yescharacteristic, complemented
172C23—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—92no—
176C2×C4—92no—
177C2×C4—121no—
178C2×C4—182no—
179C2×C4—182no—
180C2×C4—361no—
181C2×C4—362no—
182C8—94no—
183C2×C3—42no—
184C2×C3—41no—
185C2×C3—82no—
186C2×C3—81no—
187C2×C3—91no—
188C2×C3—91no—
189C2×C3—92no—
190C2×C3AGL(1,𝔽9)⋊C412yesdirect product factor
191C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
192D3—122no—
193D3—121no—
194D3—121no—
195C22—92no—
196C22—92no—
197C22—91no—
198C22—91no—
199C22C3×AΓL(1,𝔽9)11yescharacteristic
200C4—91no—
201C4—92no—
202C4—91no—
203C4—122no—
204C4—362no—
205C3—41no—
206C3—81no—
207C3C2×AGL(1,𝔽9)⋊C411yescharacteristic, direct product factor
208C2—91no—
209C2—91no—
210C2—92no—
211C2C3×AGL(1,𝔽9)⋊C412yesdirect product factor
212C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3AΓL(1,𝔽9)
commutator#157C22×C3×C4
FrattiniC2C2×C3×AΓL(1,𝔽9)
Fitting, socleC22×C33SD16
prime radicalC22×C3×C3⋊D3D4
cosocleC2×#157C23×C3
hypocenterC32C2×C3×#106
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
4. C3×AGL(1,𝔽9)⋊C4C2direct (N × H)
5. C2×AGL(1,𝔽9)⋊C4C3direct (N × H)
8. C2×C3×AGL(1,𝔽9)C4semidirect
18. AGL(1,𝔽9)⋊C4C2×C3direct (N × H)
26. C3×AGL(1,𝔽9)C2×C4semidirect
33. C2×AGL(1,𝔽9)C3×C4semidirect
63. AGL(1,𝔽9)C2×C3×C4semidirect
73. C2×C33#106semidirect
75. C3×C3⋊D3#106semidirect
111. C33C2×#106semidirect
133. C2×C32C3×#106semidirect
135. C3⋊D3C3×#106semidirect
171. C32C2×C3×#106semidirect
190. C2×C3AGL(1,𝔽9)⋊C4direct (N × H)
207. C3C2×AGL(1,𝔽9)⋊C4direct (N × H)
211. C2C3×AGL(1,𝔽9)⋊C4direct (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×#37772
C2×C3×#37732
C3×AGL(1,𝔽9)⋊C42
C2×AGL(1,𝔽9)⋊C43
C2×C3×#1069