Finite Group Databasebeta

← C3⋊D3×GU(2,𝔽3) #13165977

Subgroups — C3⋊D3×GU(2,𝔽3) #13165977

Order 1728.

542 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 61 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⋊D3×#41⋊C3C211yesmaximal, characteristic
2C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
3C32×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4D3×GU(2,𝔽3)—34nomaximal
5C3⋊D3×#104—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)C411yescharacteristic, complemented
8C3⋊(C3⋊(#41⋊C3))C411yescharacteristic, complemented
9C3⋊D3×C3⋊C8—41nomaximal
10#41×C3⋊D3D311yescharacteristic
11D3×#41⋊C3—34no—
12M16×C3⋊D3—31no—
13C42×C3⋊D3—31no—
14C3⋊GU(2,𝔽3)—34no—
15C3⋊GU(2,𝔽3)—64no—
16C3×GU(2,𝔽3)D314yescomplemented
17C32×#104—31no—
18#63664—31no—
19#63652—31no—
20#63632—31no—
21C3×C4×C3⋊D3—41no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
23C3⋊(C3⋊(C3⋊C8))—41no—
24C32×C3⋊C8—41no—
25C2×GU(2,𝔽3)—91no—
26D3×#104—94no—
27C32×#41C2×D311yescharacteristic
28#3832Dic311yescharacteristic
29Q8×C3⋊D3Dic311yescharacteristic
30#3830—31no—
31D4×C3⋊D3—31no—
32#3828—31no—
33C2×C4×C3⋊D3—31no—
34C2×C4×C3⋊D3—61no—
35C3×#41⋊C3—24no—
36C3×#41⋊C3C2×D314yes—
37D3×SL(2,𝔽3)—34no—
38C3⋊(#41⋊C3)—34no—
39C32×M16—31no—
40C32×C42—31no—
41C4×C3⋊Dic3—31no—
42C4×C3⋊Dic3—61no—
43#3748—31no—
44#3743—61no—
45C8×C3⋊D3—61no—
46D3×C3⋊C8—124no—
47C2×C3×C3⋊D3—41no—
48C33×C4—41no—
49C3×C3⋊Dic3—41no—
50D3×#41—34no—
51C2×#41⋊C3—91no—
52D3×M16—94no—
53C42×D3—94no—
54GU(2,𝔽3)—91no—
55GU(2,𝔽3)—184no—
56GU(2,𝔽3)C3⋊D311yescharacteristic, direct product factor
57C3×#104—34no—
58#837—94no—
59#825—94no—
60#805—94no—
61C22×C3⋊D3—31no—
62C32×Q8C2×Dic311yescharacteristic
63C32×D4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66#638—61no—
67C2×C3⋊Dic3—31no—
68C2×C3⋊Dic3—61no—
69C4⋊(C3⋊D3)—31no—
70C4×C3⋊D3—31no—
71C4×C3⋊D3—31no—
72C4×C3⋊D3—31no—
73C4×C3⋊D3—64no—
74C4×C3⋊D3S411yescharacteristic
75#634—31no—
76C3×C4×D3—124no—
77C3×SL(2,𝔽3)—24no—
78C3×SL(2,𝔽3)C4×D314yescomplemented
79C32×C8—61no—
80C3⋊(C3⋊C8)—61no—
81C3⋊(C3⋊C8)—124no—
82C3⋊(C3⋊C8)—244no—
83C3×C3⋊C8—44no—
84C2×#104—271noSylow
85C2×C33—41no—
86C3×C3⋊D3—42no—
87C3×#41D3214yes—
88#239—34no—
89D3×Q8—34no—
90#237—94no—
91D3×D4—94no—
92#235—94no—
93C2×C4×D3—94no—
94C2×C4×D3—184no—
95#41⋊C3—24no—
96#41⋊C3—91no—
97#41⋊C3C2×C3⋊D311yescharacteristic
98C2×SL(2,𝔽3)—91no—
99C3×M16—34no—
100C3×C42—34no—
101C4×Dic3—94no—
102C4×Dic3—184no—
103#208—94no—
104C2×C3⋊C8—361no—
105ΓL(1,𝔽25)—184no—
106D3×C8—184no—
107C22×C32—31no—
108C2×C3⋊D3—31no—
109C2×C3⋊D3—62no—
110C2×C3⋊D3SL(2,ℤ/4)11yescharacteristic
111C2×C3×D3—124no—
112C32×C4—31no—
113C32×C4—44no—
114C32×C4—62no—
115C32×C4—84no—
116C32×C4C2×S411yescharacteristic
117C3⋊Dic3—31no—
118C3⋊Dic3—62no—
119C3⋊Dic3SL(2,ℤ/4)11yescharacteristic
120C3×Dic3—124no—
121C2×#41—91no—
122C2×M16—271no—
123C2×C42—271no—
124#104—31no—
125#104—271no—
126#104—271no—
127#104—271no—
128C33—41noSylow
129C22×D3—94no—
130C3×Q8D3×Dic314yes—
131C3×D4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—64no—
134C2×C3×C4—361no—
135#67—184no—
136C2×Dic3—94no—
137C2×Dic3—184no—
138D12—94no—
139C4×D3—34no—
140C4×D3—94no—
141C4×D3—94no—
142C4×D3—94no—
143C4×D3—1816no—
144Dic6—94no—
145SL(2,𝔽3)—24no—
146SL(2,𝔽3)C4×C3⋊D311yescharacteristic, complemented
147C3×C8—64no—
148C3⋊C8—41no—
149C3⋊C8—184no—
150C3⋊C8—361no—
151C3⋊C8—724no—
152C2×C32—44no—
153C2×C32—61no—
154C2×C32—84no—
155C2×C32C2×SL(2,ℤ/4)11yescharacteristic
156C3⋊D3—61no—
157C3⋊D3GU(2,𝔽3)12yesdirect product factor
158C3×D3—128no—
159#41—91no—
160#41—271no—
161#41—271no—
162#41D3×C3⋊D311yescharacteristic
163C2×Q8—91no—
164C2×D4—271no—
165C22×C4—271no—
166C22×C4—541no—
167M16—31no—
168M16—271no—
169M16—541no—
170C2×C8—541no—
171C42—31no—
172C42—271no—
173C42—541no—
174C22×C3—34no—
175C22×C3—361no—
176C2×D3—34no—
177C2×D3—94no—
178C2×D3—188no—
179C3×C4—34no—
180C3×C4—41no—
181C3×C4—68no—
182C3×C4—84no—
183C3×C4—361no—
184C3×C4D3×S414yes—
185Dic3—34no—
186Dic3—94no—
187Dic3—188no—
188C32—44no—
189C32—84no—
190C32C2×GU(2,𝔽3)11yescharacteristic, complemented
191C23—271no—
192Q8—271no—
193Q8Dic3×C3⋊D311yescharacteristic
194D4—31no—
195D4—271no—
196D4—541no—
197C2×C4—31no—
198C2×C4—61no—
199C2×C4—91no—
200C2×C4—271no—
201C2×C4—271no—
202C2×C4—271no—
203C2×C4—271no—
204C2×C4—544no—
205C2×C4—541no—
206C8—61no—
207C8—541no—
208C2×C3—41no—
209C2×C3—64no—
210C2×C3—84no—
211C2×C3—362no—
212C2×C3D3×SL(2,ℤ/4)14yes—
213D3—38no—
214D3—184no—
215C22—31no—
216C22—91no—
217C22—271no—
218C22—542no—
219C4—31no—
220C4—62no—
221C4—91no—
222C4—271no—
223C4—542no—
224C4C3⋊D3×S411yescharacteristic
225C3—41no—
226C3—84no—
227C3D3×GU(2,𝔽3)14yescomplemented
228C2—61no—
229C2—92no—
230C2—541no—
231C2C3⋊D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41C2×D3
socleC2×C32C2×SL(2,ℤ/4)
prime radicalC3⋊D3×#41⋊C3C2
cosocleC32×#41⋊C3C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
3. C32×GU(2,𝔽3)C2semidirect
7. C3⋊D3×SL(2,𝔽3)C4semidirect
8. C3⋊(C3⋊(#41⋊C3))C4semidirect
16. C3×GU(2,𝔽3)D3semidirect
22. C32×SL(2,𝔽3)C2×C4semidirect
56. GU(2,𝔽3)C3⋊D3direct (N × H)
78. C3×SL(2,𝔽3)C4×D3semidirect
146. SL(2,𝔽3)C4×C3⋊D3semidirect
157. C3⋊D3GU(2,𝔽3)direct (N × H)
190. C32C2×GU(2,𝔽3)semidirect
227. C3D3×GU(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
C3⋊(C3⋊GU(2,𝔽3))2
C32×GU(2,𝔽3)2
D3×GU(2,𝔽3)3
C3⋊D3×#1043
C3⋊D3×C3⋊C84