Finite Group Databasebeta

← C3×#10596409 #13180386

Subgroups — C3×#10596409 #13180386

Order 1728.

310 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 64 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×AΓL(1,𝔽9)C212yesmaximal
2C3×C4⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
3C3×C4×GO+(4,𝔽2)C211yesmaximal, characteristic
4C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
5C3×PSU(3,𝔽2)⋊C4C212yesmaximal, complemented
6#10596409C311yesmaximal, characteristic, direct product factor
7C2×C3×PSU(3,𝔽2)C2212yes—
8C2×C3×GO+(4,𝔽2)C2211yescharacteristic
9C2×C3×AGL(1,𝔽9)C2212yes—
10C3×AΓL(1,𝔽9)—24no—
11C3×C4×D32—21no—
12C3×C4⋊#157—21no—
13C3×C4×#157C2211yescharacteristic
14C3×#3777—21no—
15C3×#3773C2211yescharacteristic
16C3×D32⋊C4—21no—
17C2×AΓL(1,𝔽9)C2×C312yes—
18C4⋊PSU(3,𝔽2)C2×C311yescharacteristic, complemented
19C4×GO+(4,𝔽2)C2×C311yescharacteristic
20C4⋊AGL(1,𝔽9)C2×C311yescharacteristic, complemented
21PSU(3,𝔽2)⋊C4C2×C312yescomplemented
22C2×C3×D32—21no—
23C2×C3×#157—22no—
24C2×C3×#157D411yescharacteristic
25C2×C3×#157C2311yescharacteristic
26C3×PSU(3,𝔽2)—22no—
27C3×PSU(3,𝔽2)—22no—
28C3×GO+(4,𝔽2)—21no—
29C3×GO+(4,𝔽2)D412yes—
30C3×AGL(1,𝔽9)—22no—
31C3×C4×C3⋊D3D411yescharacteristic
32C32×C4×D3—41no—
33C3×#624—21no—
34C3×D3×Dic3—41no—
35C3×#461—91nomaximal
36C2×PSU(3,𝔽2)C22×C312yes—
37C2×GO+(4,𝔽2)C22×C311yescharacteristic
38C2×AGL(1,𝔽9)C22×C312yes—
39AΓL(1,𝔽9)—24no—
40C2×C3×C4×D3—121no—
41C4×D32—21no—
42C4⋊#157—21no—
43C4×#157C22×C311yescharacteristic
44#3777—21no—
45#3773C22×C311yescharacteristic
46D32⋊C4—21no—
47C2×C3×C3⋊D3C2×D411yescharacteristic
48C2×C32×D3—41no—
49C3×D32—22no—
50C3×D32—41no—
51C3×#157—21no—
52C3×#157—42no—
53C3×#157C2×D411yescharacteristic
54C3×#157#4111yescharacteristic
55C33×C4SD1611yescharacteristic, complemented
56C3×C3⋊Dic3SD1611yescharacteristic, complemented
57C32×Dic3—41no—
58C2×C3×SD16—92no—
59C3×C4⋊Q8—91no—
60C3×C4×D4—91no—
61C3×C4⋊C8—91no—
62C3×#103—92no—
63C22×C3×D3—121no—
64C2×D32—21no—
65C2×#157—22no—
66C2×#157C3×D411yescharacteristic
67C2×#157C23×C311yescharacteristic
68PSU(3,𝔽2)—22no—
69PSU(3,𝔽2)—22no—
70GO+(4,𝔽2)—21no—
71GO+(4,𝔽2)C3×D412yes—
72AGL(1,𝔽9)—22no—
73C2×C32×C4—121no—
74C4×C3⋊D3C3×D411yescharacteristic
75C2×C3×Dic3—121no—
76C3×C4×D3—41no—
77C3×C4×D3—41no—
78C3×C4×D3—81no—
79C3×C4×D3—121no—
80C3×C4×D3—121no—
81C3×C4×D3—121no—
82#624—21no—
83D3×Dic3—41no—
84#461—91noSylow
85C2×C33C2×SD1611yescharacteristic
86C3×C3⋊D3#12111yescharacteristic
87C3×C3⋊D3#13711yescharacteristic
88C32×D3—42no—
89C2×C3×Q8—92no—
90C2×C3×D4—91no—
91C22×C3×C4—181no—
92C2×C4×D3—121no—
93C3×SD16—184no—
94C2×C3×C8—92no—
95C3×C4⋊C4—91no—
96C3×C4⋊C4—181no—
97C3×C4⋊C4—181no—
98C3×C22⋊C4—181no—
99C3×C42—91no—
100C22×C32—121no—
101C2×C3⋊D3C2×C3×D411yescharacteristic
102C2×C3×D3—41no—
103C2×C3×D3—41no—
104C2×C3×D3—81no—
105C2×C3×D3—121no—
106C2×C3×D3—122no—
107C2×C3×D3—122no—
108D32—22no—
109D32—41no—
110#157—21no—
111#157—42no—
112#157C2×C3×D411yescharacteristic
113#157C3×#4111yescharacteristic
114C32×C4—41no—
115C32×C4—81no—
116C32×C4—121no—
117C32×C4C3×SD1611yescharacteristic, complemented
118C3⋊Dic3C3×SD1611yescharacteristic, complemented
119C3×Dic3—41no—
120C3×Dic3—41no—
121C3×Dic3—81no—
122C3×Dic3—121no—
123C2×SD16—92no—
124C4⋊Q8—91no—
125C4×D4—91no—
126C4⋊C8—91no—
127#103—92no—
128C33#46111yescharacteristic, complemented, Sylow
129C23×C3—181no—
130C22×D3—121no—
131C3×Q8—182no—
132C3×Q8—182no—
133C3×D4—92no—
134C3×D4—181no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—121no—
139C2×C3×C4—121no—
140C2×C3×C4—181no—
141C2×C3×C4—182no—
142C2×C3×C4—241no—
143C2×C3×C4—361no—
144C2×Dic3—121no—
145C4×D3—41no—
146C4×D3—121no—
147C4×D3—121no—
148C4×D3—121no—
149C3×C8—182no—
150C2×C32—41no—
151C2×C32—81no—
152C2×C32—122no—
153C2×C32C2×C3×SD1611yescharacteristic
154C3⋊D3C3×#12111yescharacteristic
155C3⋊D3C3×#13711yescharacteristic
156C3×D3—42no—
157C3×D3—42no—
158C3×D3—82no—
159C3×D3—121no—
160C3×D3—121no—
161C2×Q8—92no—
162C2×D4—91no—
163C22×C4—181no—
164SD16—184no—
165C2×C8—92no—
166C4⋊C4—91no—
167C4⋊C4—181no—
168C4⋊C4—181no—
169C22⋊C4—181no—
170C42—91no—
171C22×C3—91no—
172C22×C3—121no—
173C22×C3—121no—
174C22×C3—182no—
175C22×C3—241no—
176C22×C3—361no—
177C2×D3—41no—
178C2×D3—121no—
179C2×D3—122no—
180C2×D3—122no—
181C3×C4—41no—
182C3×C4—81no—
183C3×C4—91no—
184C3×C4—91no—
185C3×C4—91no—
186C3×C4—121no—
187C3×C4—121no—
188C3×C4—181no—
189C3×C4—241no—
190C3×C4—362no—
191C3×C4AΓL(1,𝔽9)11yescharacteristic, complemented
192Dic3—41no—
193Dic3—121no—
194C32—41no—
195C32—81no—
196C32C3×#46111yescharacteristic, complemented
197C23—181no—
198Q8—182no—
199Q8—182no—
200D4—92no—
201D4—181no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—121no—
206C2×C4—181no—
207C2×C4—182no—
208C2×C4—361no—
209C8—182no—
210C2×C3—41no—
211C2×C3—81no—
212C2×C3—91no—
213C2×C3—91no—
214C2×C3—122no—
215C2×C3—122no—
216C2×C3—242no—
217C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
218D3—42no—
219D3—121no—
220D3—121no—
221C22—91no—
222C22—121no—
223C22—182no—
224C22—361no—
225C4—91no—
226C4—91no—
227C4—91no—
228C4—121no—
229C4—181no—
230C4—362no—
231C4C3×AΓL(1,𝔽9)11yescharacteristic, complemented
232C3—41no—
233C3—81no—
234C3#1059640911yescharacteristic, direct product factor
235C2—91no—
236C2—91no—
237C2—122no—
238C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C2×AΓL(1,𝔽9)
commutator, cosocleC2×#157C23×C3
FrattiniC2C2×C3×AΓL(1,𝔽9)
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radicalC2×C3×GO+(4,𝔽2)C22
hypercenterC3×C4AΓL(1,𝔽9)
hypocenterC32C3×#461
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊PSU(3,𝔽2)C2semidirect
4. C3×C4⋊AGL(1,𝔽9)C2semidirect
5. C3×PSU(3,𝔽2)⋊C4C2semidirect
6. #10596409C3direct (N × H)
18. C4⋊PSU(3,𝔽2)C2×C3semidirect
20. C4⋊AGL(1,𝔽9)C2×C3semidirect
21. PSU(3,𝔽2)⋊C4C2×C3semidirect
55. C33×C4SD16semidirect
56. C3×C3⋊Dic3SD16semidirect
117. C32×C4C3×SD16semidirect
118. C3⋊Dic3C3×SD16semidirect
128. C33#461semidirect
191. C3×C4AΓL(1,𝔽9)semidirect
196. C32C3×#461semidirect
231. C4C3×AΓL(1,𝔽9)semidirect
234. C3#10596409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×AΓL(1,𝔽9)2
C3×C4⋊PSU(3,𝔽2)2
C3×C4×GO+(4,𝔽2)2
C3×C4⋊AGL(1,𝔽9)2
C3×PSU(3,𝔽2)⋊C42
#105964093
C3×#4619