Finite Group Databasebeta

← C3⋊Dic3×GL(2,𝔽3) #13165963

Subgroups — C3⋊Dic3×GL(2,𝔽3) #13165963

Order 1728.

616 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 79 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×C32×GL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)—34nomaximal
5SD16×C3⋊Dic3—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×D3×C3⋊Dic3—41nomaximal
8C32×GL(2,𝔽3)C412yescomplemented
9C2×C3×GL(2,𝔽3)D314yes—
10C2×C32×SD16—31no—
11Q8×C3⋊Dic3D311yescharacteristic, complemented
12D4×C3⋊Dic3—31no—
13Dic3×SL(2,𝔽3)—34no—
14C3⋊#859—34no—
15C3⋊#859—64no—
16#63662—31no—
17#63659—31no—
18#63644—31no—
19C8×C3⋊Dic3—31no—
20C22×C32×D3—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C2×C3×C3⋊Dic3—41no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24D3×C3⋊Dic3—44no—
25C4×GL(2,𝔽3)—91no—
26Dic3×SD16—94no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C32×D4—31no—
29C22×C3⋊Dic3—61no—
30C2×C3×SL(2,𝔽3)—24no—
31C2×C3×SL(2,𝔽3)C2×D314yes—
32C2×D3×Dic3—124no—
33C3×GL(2,𝔽3)Dic318yescomplemented
34C32×SD16—32no—
35C32×SD16—32no—
36C2×C32×C8—31no—
37#3757—61no—
38C4⋊(C3⋊Dic3)—31no—
39C4⋊(C3⋊Dic3)—31no—
40C4×C3⋊Dic3—31no—
41C2×C3⋊(C3⋊C8)—31no—
42C22×C33—41no—
43C2×C32×D3—42no—
44C2×C32×D3—42no—
45C2×C32×D3—42no—
46C3⋊(C3⋊Dic3)—42no—
47C3×C3⋊Dic3—42no—
48C2×GL(2,𝔽3)C3⋊D311yescharacteristic
49C2×C3×SD16—34no—
50Q8×Dic3—34no—
51D4×Dic3—94no—
52C4×SL(2,𝔽3)—91no—
53#859—91no—
54#859—184no—
55#835—94no—
56#832—94no—
57#817—94no—
58C8×Dic3—94no—
59C23×C32—61no—
60C22×C3×D3—44no—
61C32×Q8—31no—
62C32×Q8C4×D311yescharacteristic, complemented
63C32×D4—32no—
64C32×D4—61no—
65C2×C32×C4—31no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—124no—
69C2×C3⋊Dic3—122no—
70C2×C3⋊Dic3—244no—
71C2×C3⋊Dic3S411yescharacteristic
72C2×C3×Dic3—124no—
73C3×SL(2,𝔽3)—24no—
74C3×SL(2,𝔽3)C2×Dic314yescomplemented
75D3×Dic3—1216no—
76C32×C8—32no—
77C3⋊(C3⋊C8)—61no—
78C4×SD16—271noSylow
79C2×C33—41no—
80C2×C33—41no—
81C2×C33—41no—
82C32×D3—44no—
83C2×C3×Q8D3214yes—
84C2×C3×D4—34no—
85C22×Dic3—184no—
86C2×C4×D3—361no—
87C2×SL(2,𝔽3)—24no—
88C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
89GL(2,𝔽3)C3⋊Dic312yesdirect product factor
90C3×SD16—38no—
91C3×SD16—38no—
92C2×C3×C8—34no—
93#217—184no—
94C4⋊Dic3—94no—
95C4⋊Dic3—94no—
96C4×Dic3—94no—
97C2×C3⋊C8—94no—
98C22×C32—44no—
99C22×C32—62no—
100C22×C32—84no—
101C22×C32—121no—
102C22×C32—121no—
103C22×C32C2×S411yescharacteristic
104C2×C3×D3—48no—
105C2×C3×D3—48no—
106C2×C3×D3—48no—
107C32×C4—31no—
108C32×C4—31no—
109C3⋊Dic3—61no—
110C3⋊Dic3—121no—
111C3⋊Dic3—128no—
112C3⋊Dic3—248no—
113C3⋊Dic3GL(2,𝔽3)12yesdirect product factor
114C3×Dic3—128no—
115C2×SD16—31no—
116C4×Q8—91no—
117C4×D4—271no—
118#106—271no—
119#103—271no—
120#102—271no—
121C4×C8—271no—
122C33—41noSylow
123C23×C3—64no—
124C22×D3—41no—
125C3×Q8—34no—
126C3×Q8D3×Dic314yescomplemented
127C3×D4—38no—
128C3×D4—64no—
129C2×C3×C4—34no—
130C2×C3×C4—361no—
131C2×Dic3—34no—
132C2×Dic3—94no—
133C2×Dic3—184no—
134C2×Dic3—361no—
135C2×Dic3—368no—
136C2×Dic3—724no—
137C4×D3—364no—
138SL(2,𝔽3)—24no—
139SL(2,𝔽3)C2×C3⋊Dic311yescharacteristic, complemented
140C3×C8—38no—
141C3⋊C8—184no—
142C2×C32—44no—
143C2×C32—44no—
144C2×C32—44no—
145C2×C32—84no—
146C2×C32—84no—
147C2×C32—84no—
148C2×C32—122no—
149C2×C32C4×S411yescharacteristic
150C2×C32C2×GL(2,𝔽3)11yescharacteristic
151C2×C32#98511yescharacteristic
152C3×D3—416no—
153C2×Q8D3×C3⋊D311yescharacteristic
154C2×D4—31no—
155C22×C4—541no—
156SD16—32no—
157SD16—32no—
158C2×C8—31no—
159C2×C8—271no—
160C4⋊C4—271no—
161C4⋊C4—271no—
162C22⋊C4—541no—
163C42—271no—
164C22×C3—41no—
165C22×C3—68no—
166C22×C3—84no—
167C22×C3—124no—
168C22×C3—124no—
169C22×C3D3×S414yes—
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C3×C4—34no—
174C3×C4—34no—
175C3×C4—362no—
176Dic3—38no—
177Dic3—184no—
178Dic3—362no—
179Dic3—364no—
180Dic3—728no—
181C32—44no—
182C32—84no—
183C32C4×GL(2,𝔽3)11yescharacteristic, complemented
184C23—61no—
185Q8—31no—
186Q8D3×C3⋊Dic311yescharacteristic, complemented
187D4—32no—
188D4—61no—
189C2×C4—31no—
190C2×C4—91no—
191C2×C4—271no—
192C2×C4—541no—
193C2×C4—1082no—
194C8—32no—
195C8—541no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3—41no—
199C2×C3—84no—
200C2×C3—84no—
201C2×C3—84no—
202C2×C3—128no—
203C2×C3#6419714yes—
204C2×C3D3×GL(2,𝔽3)14yes—
205C2×C3Dic3×S414yes—
206D3—44no—
207C22—62no—
208C22—121no—
209C22—121no—
210C22C3⋊D3×S411yescharacteristic
211C4—31no—
212C4—31no—
213C4—92no—
214C4—541no—
215C4—1081no—
216C3—41no—
217C3—84no—
218C3Dic3×GL(2,𝔽3)14yescomplemented
219C2—122no—
220C2#1172417711yescharacteristic
221C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
222C2S4×C3⋊Dic311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C32×GL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
8. C32×GL(2,𝔽3)C4semidirect
11. Q8×C3⋊Dic3D3semidirect
23. C32×SL(2,𝔽3)C2×C4semidirect
33. C3×GL(2,𝔽3)Dic3semidirect
62. C32×Q8C4×D3semidirect
74. C3×SL(2,𝔽3)C2×Dic3semidirect
89. GL(2,𝔽3)C3⋊Dic3direct (N × H)
113. C3⋊Dic3GL(2,𝔽3)direct (N × H)
126. C3×Q8D3×Dic3semidirect
139. SL(2,𝔽3)C2×C3⋊Dic3semidirect
183. C32C4×GL(2,𝔽3)semidirect
186. Q8D3×C3⋊Dic3semidirect
218. C3Dic3×GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×GL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
C3⋊(C3⋊#859)2
Dic3×GL(2,𝔽3)3
SD16×C3⋊Dic33
C2×D3×C3⋊Dic34