Finite Group Databasebeta

← C3×#10596412 #13180389

Subgroups — C3×#10596412 #13180389

Order 1728.

298 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 60 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×C3×#3777C211yesmaximal, characteristic, complemented
2C3×#64242C211yesmaximal, characteristic
3C3×C22⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
4C3×AGL(1,𝔽9)⋊C4C212yesmaximal, complemented
5C3×GO+(4,𝔽2)⋊C4C212yesmaximal, complemented
6#10596412C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—21no—
8C22×C3×#157C2211yescharacteristic
9C2×C3×GO+(4,𝔽2)C2211yescharacteristic
10C2×C3×AGL(1,𝔽9)C2212yes—
11C3×#3811—21no—
12C3×#3777—21no—
13C3×#3777C2212yes—
14C3×#3773C2211yescharacteristic
15C3×D32⋊C4—21no—
16C2×#3777C2×C311yescharacteristic, complemented
17#64242C2×C311yescharacteristic
18C22⋊AGL(1,𝔽9)C2×C311yescharacteristic, complemented
19AGL(1,𝔽9)⋊C4C2×C312yescomplemented
20GO+(4,𝔽2)⋊C4C2×C312yescomplemented
21C22×C3×C3⋊D3D411yescharacteristic
22C2×C3×D32—21no—
23C2×C3×#157—22no—
24C2×C3×#157—22no—
25C2×C3×#157—42no—
26C2×C3×#157D411yescharacteristic
27C2×C3×#157C2311yescharacteristic
28C3×GO+(4,𝔽2)—22no—
29C3×AGL(1,𝔽9)—22no—
30C32×#67—41no—
31C3×#626—41no—
32C3×#624—21no—
33C3×#481—91nomaximal
34C22×#157—21no—
35C22×#157C22×C311yescharacteristic
36C2×GO+(4,𝔽2)C22×C311yescharacteristic
37C2×AGL(1,𝔽9)C22×C312yes—
38C3×D3×D4—121no—
39#3811—21no—
40#3777—21no—
41#3777C22×C312yes—
42#3773C22×C311yescharacteristic
43D32⋊C4—21no—
44C22×C33SD1611yescharacteristic
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3SD1611yescharacteristic
48C2×C3×C3⋊D3C2×D411yescharacteristic
49C2×C32×D3—41no—
50C3×D32—41no—
51C3×D32—41no—
52C3×#157—21no—
53C3×#157—42no—
54C3×#157#4112yes—
55C32×Dic3—41no—
56C3×#121—91no—
57C2×C3×C4⋊C4—91no—
58C3×#106—92no—
59C3×#102—92no—
60C3×C22⋊C8—91no—
61C22×C3⋊D3C3×D411yescharacteristic
62C22×C3×D3—121no—
63C22×C3×D3—121no—
64C2×D32—21no—
65C2×#157—22no—
66C2×#157—22no—
67C2×#157—42no—
68C2×#157C3×D411yescharacteristic
69C2×#157C23×C311yescharacteristic
70GO+(4,𝔽2)—22no—
71AGL(1,𝔽9)—22no—
72C32×D4—121no—
73C3×#67—41no—
74C3×#67—41no—
75C3×#67—81no—
76C3×#67—121no—
77C3×D12—121no—
78C3×C4×D3—121no—
79#626—41no—
80#624—21no—
81#481—91noSylow
82C2×C33—21no—
83C2×C33C2×SD1611yescharacteristic
84C3×C3⋊D3—21no—
85C3×C3⋊D3#12311yescharacteristic
86C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
87C32×D3—81no—
88C2×C3×D4—91no—
89C2×C3×D4—181no—
90C22×C3×C4—91no—
91C22×C3×C4—181no—
92D3×D4—121no—
93C2×C3×C8—92no—
94C3×C4⋊C4—92no—
95C3×C4⋊C4—91no—
96C3×C4⋊C4—181no—
97C3×C22⋊C4—181no—
98C22×C32—41no—
99C22×C32—81no—
100C22×C32—121no—
101C22×C32C3×SD1611yescharacteristic
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3C3×SD1611yescharacteristic
105C2×C3⋊D3C2×C3×D411yescharacteristic
106C2×C3×D3—41no—
107C2×C3×D3—41no—
108C2×C3×D3—81no—
109C2×C3×D3—121no—
110C2×C3×D3—121no—
111C2×C3×D3—241no—
112C2×C3×D3—241no—
113C2×C3×D3—241no—
114C2×C3×D3—241no—
115D32—41no—
116D32—41no—
117#157—21no—
118#157—42no—
119#157C3×#4112yes—
120C32×C4—121no—
121C3×Dic3—41no—
122C3×Dic3—41no—
123C3×Dic3—81no—
124#121—91no—
125C2×C4⋊C4—91no—
126#106—92no—
127#102—92no—
128C22⋊C8—91no—
129C33#48111yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C23×C3—181no—
132C22×D3—121no—
133C22×D3—121no—
134C3×D4—121no—
135C3×D4—121no—
136C3×D4—182no—
137C3×D4—241no—
138C3×D4—361no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—182no—
142C2×C3×C4—181no—
143C2×C3×C4—182no—
144C2×C3×C4—362no—
145#67—41no—
146#67—121no—
147D12—121no—
148C4×D3—121no—
149C3×C8—182no—
150C2×C32—21no—
151C2×C32—41no—
152C2×C32—81no—
153C2×C32—81no—
154C2×C32—161no—
155C2×C32—241no—
156C2×C32C2×C3×SD1611yescharacteristic
157C3⋊D3—21no—
158C3⋊D3C3×#12311yescharacteristic
159C3⋊D3C3×AGL(1,ℤ/8)11yescharacteristic
160C3×D3—81no—
161C3×D3—81no—
162C3×D3—121no—
163C3×D3—121no—
164C3×D3—161no—
165C3×D3—241no—
166C2×D4—91no—
167C2×D4—181no—
168C22×C4—91no—
169C22×C4—181no—
170C2×C8—92no—
171C4⋊C4—92no—
172C4⋊C4—91no—
173C4⋊C4—181no—
174C22⋊C4—181no—
175C22×C3—41no—
176C22×C3—81no—
177C22×C3—91no—
178C22×C3—91no—
179C22×C3—121no—
180C22×C3—121no—
181C22×C3—181no—
182C22×C3—181no—
183C22×C3—241no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3AΓL(1,𝔽9)11yescharacteristic
187C2×D3—41no—
188C2×D3—121no—
189C2×D3—121no—
190C2×D3—241no—
191C2×D3—241no—
192C2×D3—241no—
193C2×D3—241no—
194C3×C4—92no—
195C3×C4—121no—
196C3×C4—121no—
197C3×C4—181no—
198C3×C4—241no—
199C3×C4—362no—
200Dic3—41no—
201C32—41no—
202C32—81no—
203C32C3×#48111yescharacteristic, complemented
204C23—91no—
205C23—181no—
206D4—121no—
207D4—182no—
208D4—361no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—182no—
212C2×C4—181no—
213C2×C4—182no—
214C2×C4—362no—
215C8—182no—
216C2×C3—21no—
217C2×C3—41no—
218C2×C3—81no—
219C2×C3—81no—
220C2×C3—91no—
221C2×C3—91no—
222C2×C3—161no—
223C2×C3—181no—
224C2×C3—241no—
225C2×C3—241no—
226C2×C3—481no—
227C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
228D3—81no—
229D3—121no—
230D3—121no—
231D3—241no—
232C22—91no—
233C22—91no—
234C22—121no—
235C22—181no—
236C22—181no—
237C22—361no—
238C22—361no—
239C22C3×AΓL(1,𝔽9)11yescharacteristic
240C4—92no—
241C4—121no—
242C4—181no—
243C4—362no—
244C3—41no—
245C3—81no—
246C3#1059641211yescharacteristic, direct product factor
247C2—21no—
248C2—91no—
249C2—91no—
250C2—181no—
251C2—241no—
252C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C2×AΓL(1,𝔽9)
commutator, cosocleC2×#157C23×C3
FrattiniC2C2×C3×AΓL(1,𝔽9)
FittingC22×C33SD16
socleC2×C33C2×SD16
prime radicalC3×#64242C2
hypercenterC22×C3AΓL(1,𝔽9)
hypocenterC32C3×#481
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3777C2semidirect
3. C3×C22⋊AGL(1,𝔽9)C2semidirect
4. C3×AGL(1,𝔽9)⋊C4C2semidirect
5. C3×GO+(4,𝔽2)⋊C4C2semidirect
6. #10596412C3direct (N × H)
16. C2×#3777C2×C3semidirect
18. C22⋊AGL(1,𝔽9)C2×C3semidirect
19. AGL(1,𝔽9)⋊C4C2×C3semidirect
20. GO+(4,𝔽2)⋊C4C2×C3semidirect
129. C33#481semidirect
203. C32C3×#481semidirect
246. C3#10596412direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37772
C3×#642422
C3×C22⋊AGL(1,𝔽9)2
C3×AGL(1,𝔽9)⋊C42
C3×GO+(4,𝔽2)⋊C42
#105964123
C3×#4819