Finite Group Databasebeta

← C3×#10596415 #13180391

Subgroups — C3×#10596415 #13180391

Order 1728.

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

normal subgroups only

This group has 278 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
1C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic
2C2×C3×#3773C211yesmaximal, characteristic
3C3×C22⋊AGL(1,𝔽9)C211yesmaximal, characteristic
4C3×AGL(1,𝔽9)⋊C4C212yesmaximal, complemented
5C3×PSU(3,𝔽2)⋊C4C212yesmaximal, complemented
6#10596415C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2211yescharacteristic
8C2×C3×PSU(3,𝔽2)C2211yescharacteristic
9C2×C3×AGL(1,𝔽9)C2212yes—
10C2×C3×#624—21no—
11C3×C22⋊#157—21no—
12C3×#3777—21no—
13C3×#3777C2211yescharacteristic
14C3×#3773—21no—
15C3×#3773C2212yes—
16C22⋊PSU(3,𝔽2)C2×C311yescharacteristic
17C2×#3773C2×C311yescharacteristic
18C22⋊AGL(1,𝔽9)C2×C311yescharacteristic
19AGL(1,𝔽9)⋊C4C2×C312yescomplemented
20PSU(3,𝔽2)⋊C4C2×C312yescomplemented
21C22×C3×C3⋊D3D411yescharacteristic
22C2×C3×#157—22no—
23C2×C3×#157—21no—
24C2×C3×#157—21no—
25C2×C3×#157D411yescharacteristic
26C2×C3×#157C2311yescharacteristic
27C3×PSU(3,𝔽2)—22no—
28C3×AGL(1,𝔽9)—22no—
29C2×C32×Dic3—41no—
30C3×#624—22no—
31C3×#624—41no—
32C3×#483—91nomaximal
33C22×#157C22×C311yescharacteristic
34C2×PSU(3,𝔽2)C22×C311yescharacteristic
35C2×AGL(1,𝔽9)C22×C312yes—
36C2×C3×C4×D3—121no—
37C2×#624—21no—
38C22⋊#157—21no—
39#3777—21no—
40#3777C22×C311yescharacteristic
41#3773—21no—
42#3773C22×C312yes—
43C22×C33SD1611yescharacteristic
44C2×C3×C3⋊D3—21no—
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3SD1611yescharacteristic
47C2×C3×C3⋊D3C2×D411yescharacteristic
48C3×#157—21no—
49C3×#157—41no—
50C3×#157—41no—
51C3×#157#4112yes—
52C32×Dic3—42no—
53C3×C22⋊Q8—91no—
54C2×C3×C4⋊C4—91no—
55C3×#106—92no—
56C3×#103—92no—
57C3×C22⋊C8—91no—
58C22×C3⋊D3C3×D411yescharacteristic
59C22×C3×D3—121no—
60C2×#157—22no—
61C2×#157—21no—
62C2×#157—21no—
63C2×#157C3×D411yescharacteristic
64C2×#157C23×C311yescharacteristic
65PSU(3,𝔽2)—22no—
66AGL(1,𝔽9)—22no—
67C2×C32×C4—121no—
68C2×C3×Dic3—41no—
69C2×C3×Dic3—41no—
70C2×C3×Dic3—81no—
71C3×C4×D3—122no—
72C3×C4×D3—122no—
73#624—22no—
74#624—41no—
75#483—91noSylow
76C2×C33—21no—
77C2×C33C2×SD1611yescharacteristic
78C3×C3⋊D3—21no—
79C3×C3⋊D3#12311yescharacteristic
80C3×C3⋊D3#13711yescharacteristic
81C2×C3×Q8—91no—
82C22×C3×C4—91no—
83C22×C3×C4—181no—
84C2×C4×D3—121no—
85C2×C3×C8—92no—
86C3×C4⋊C4—92no—
87C3×C4⋊C4—91no—
88C3×C4⋊C4—181no—
89C3×C4⋊C4—181no—
90C3×C22⋊C4—181no—
91C22×C32—41no—
92C22×C32—81no—
93C22×C32C3×SD1611yescharacteristic
94C2×C3⋊D3—21no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3C3×SD1611yescharacteristic
97C2×C3⋊D3C2×C3×D411yescharacteristic
98C2×C3×D3—121no—
99C2×C3×D3—121no—
100C2×C3×D3—122no—
101C2×C3×D3—122no—
102#157—21no—
103#157—41no—
104#157—41no—
105#157C3×#4112yes—
106C32×C4—122no—
107C3×Dic3—42no—
108C3×Dic3—42no—
109C3×Dic3—82no—
110C22⋊Q8—91no—
111C2×C4⋊C4—91no—
112#106—92no—
113#103—92no—
114C22⋊C8—91no—
115C33#48311yescharacteristic, complemented, Sylow
116C23×C3—91no—
117C22×D3—121no—
118C3×Q8—182no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—121no—
122C2×C3×C4—121no—
123C2×C3×C4—182no—
124C2×C3×C4—182no—
125C2×C3×C4—181no—
126C2×C3×C4—181no—
127C2×C3×C4—241no—
128C2×C3×C4—361no—
129C2×Dic3—41no—
130C4×D3—122no—
131C4×D3—122no—
132C3×C8—182no—
133C2×C32—21no—
134C2×C32—42no—
135C2×C32—41no—
136C2×C32—82no—
137C2×C32—81no—
138C2×C32C2×C3×SD1611yescharacteristic
139C3⋊D3—21no—
140C3⋊D3C3×#12311yescharacteristic
141C3⋊D3C3×#13711yescharacteristic
142C3×D3—121no—
143C3×D3—121no—
144C3×D3—122no—
145C2×Q8—91no—
146C22×C4—91no—
147C22×C4—181no—
148C2×C8—92no—
149C4⋊C4—92no—
150C4⋊C4—91no—
151C4⋊C4—181no—
152C4⋊C4—181no—
153C22⋊C4—181no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3—91no—
157C22×C3—91no—
158C22×C3—181no—
159C22×C3—181no—
160C22×C3AΓL(1,𝔽9)11yescharacteristic
161C2×D3—121no—
162C2×D3—121no—
163C2×D3—122no—
164C2×D3—122no—
165C3×C4—92no—
166C3×C4—122no—
167C3×C4—122no—
168C3×C4—181no—
169C3×C4—242no—
170C3×C4—361no—
171C3×C4—361no—
172Dic3—42no—
173C32—41no—
174C32—81no—
175C32C3×#48311yescharacteristic, complemented
176C23—91no—
177Q8—182no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—121no—
181C2×C4—182no—
182C2×C4—182no—
183C2×C4—181no—
184C2×C4—181no—
185C2×C4—361no—
186C8—182no—
187C2×C3—21no—
188C2×C3—42no—
189C2×C3—41no—
190C2×C3—82no—
191C2×C3—81no—
192C2×C3—91no—
193C2×C3—91no—
194C2×C3—181no—
195C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
196D3—121no—
197D3—121no—
198D3—122no—
199C22—91no—
200C22—91no—
201C22—181no—
202C22—181no—
203C22C3×AΓL(1,𝔽9)11yescharacteristic
204C4—92no—
205C4—122no—
206C4—181no—
207C4—361no—
208C4—361no—
209C3—41no—
210C3—81no—
211C3#1059641511yescharacteristic, direct product factor
212C2—21no—
213C2—91no—
214C2—91no—
215C2—181no—
216C2C2×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 radicalC22×C3×C3⋊D3D4
hypercenterC22×C3AΓL(1,𝔽9)
hypocenterC32C3×#483
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
4. C3×AGL(1,𝔽9)⋊C4C2semidirect
5. C3×PSU(3,𝔽2)⋊C4C2semidirect
6. #10596415C3direct (N × H)
19. AGL(1,𝔽9)⋊C4C2×C3semidirect
20. PSU(3,𝔽2)⋊C4C2×C3semidirect
115. C33#483semidirect
175. C32C3×#483semidirect
211. C3#10596415direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C22⋊PSU(3,𝔽2)2
C2×C3×#37732
C3×C22⋊AGL(1,𝔽9)2
C3×AGL(1,𝔽9)⋊C42
C3×PSU(3,𝔽2)⋊C42
#105964153
C3×#4839