Finite Group Databasebeta

← C22×C3⋊D3×SL(2,𝔽3) #13181463

Subgroups — C22×C3⋊D3×SL(2,𝔽3) #13181463

Order 1728.

1848 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1848 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×SL(2,𝔽3)C216yesmaximal, direct product factor
3C22×Q8×C3⋊D3C311yesmaximal, characteristic, complemented
4C22×D3×SL(2,𝔽3)—34nomaximal
5C23×C3×C3⋊D3—41nomaximal
6C2×C32×SL(2,𝔽3)C2213yescomplemented
7C3⋊D3×SL(2,𝔽3)C2214yesdirect product factor
8C22×C32×Q8C2×C311yescharacteristic, complemented
9C2×Q8×C3⋊D3—36no—
10C2×Q8×C3⋊D3C2×C316yescomplemented
11C22×C4×C3⋊D3—31no—
12C22×#634—31no—
13C22×C3×SL(2,𝔽3)—24no—
14C22×C3×SL(2,𝔽3)D314yescomplemented
15C2×D3×SL(2,𝔽3)—324no—
16C23×C33—41no—
17C22×C3×C3⋊D3—48no—
18C22×C3×C3⋊D3—46no—
19C32×SL(2,𝔽3)C2311yescharacteristic, complemented
20C22×D3×Q8—34no—
21C23×SL(2,𝔽3)—91no—
22C23×C3⋊D3A411yescharacteristic
23C23×C3×D3—124no—
24C2×C32×Q8—33no—
25C2×C32×Q8C22×C313yescomplemented
26C22×C32×C4—31no—
27C22×C3⋊Dic3—31no—
28Q8×C3⋊D3—38no—
29Q8×C3⋊D3—312no—
30Q8×C3⋊D3C22×C314yescomplemented
31C2×C4×C3⋊D3—36no—
32C2×C4×C3⋊D3—36no—
33C2×#634—36no—
34C2×#634—36no—
35C2×C3×SL(2,𝔽3)—212no—
36C2×C3×SL(2,𝔽3)C2×D3112yescomplemented
37D3×SL(2,𝔽3)—316no—
38C22×C33—44no—
39C22×C33—43no—
40C2×C3×C3⋊D3—424no—
41C2×C3×C3⋊D3—44no—
42C22×C3×Q8C3×D314yescomplemented
43C2×D3×Q8—324no—
44C2×D3×Q8—924no—
45C22×C4×D3—94no—
46C22×Dic6—94no—
47C22×SL(2,𝔽3)—24no—
48C22×SL(2,𝔽3)—96no—
49C22×SL(2,𝔽3)C3⋊D311yescharacteristic, direct product factor
50C23×C32—44no—
51C23×C32—84no—
52C23×C32C2×A411yescharacteristic
53C22×C3⋊D3SL(2,𝔽3)18yesdirect product factor
54C22×C3⋊D3C2×A416yes—
55C22×C3×D3—1224no—
56C22×C3×D3—1232no—
57C32×Q8—33no—
58C32×Q8—32no—
59C32×Q8C23×C311yescharacteristic, complemented
60C2×C32×C4—33no—
61C2×C32×C4—33no—
62C2×C3⋊Dic3—36no—
63C4×C3⋊D3—312no—
64C4×C3⋊D3—34no—
65#634—312no—
66#634—34no—
67C3×SL(2,𝔽3)—24no—
68C3×SL(2,𝔽3)C22×D314yescomplemented
69C23×Q8—91noSylow
70C2×C33—46no—
71C2×C33—41no—
72C3×C3⋊D3—48no—
73C24×C3—361no—
74C23×D3—34no—
75C2×C3×Q8—312no—
76C2×C3×Q8C2×C3×D3112yescomplemented
77C22×C3×C4—34no—
78C22×Dic3—94no—
79D3×Q8—316no—
80D3×Q8—932no—
81D3×Q8—948no—
82C2×C4×D3—924no—
83C2×C4×D3—924no—
84C2×Dic6—924no—
85C2×Dic6—924no—
86C2×SL(2,𝔽3)—212no—
87C2×SL(2,𝔽3)—94no—
88C2×SL(2,𝔽3)C2×C3⋊D313yesdirect product factor
89C22×C32—412no—
90C22×C32—416no—
91C22×C32—812no—
92C22×C32—816no—
93C22×C32C2×SL(2,𝔽3)14yescomplemented
94C22×C32C22×A413yes—
95C2×C3⋊D3C2×SL(2,𝔽3)124yesdirect product factor
96C2×C3⋊D3C22×A414yes—
97C2×C3×D3—1296no—
98C2×C3×D3—1216no—
99C32×C4—33no—
100C32×C4—31no—
101C3⋊Dic3—34no—
102C22×Q8—96no—
103C22×Q8—271no—
104C22×Q8—276no—
105C22×Q8C3×C3⋊D311yescharacteristic, complemented
106C23×C4—271no—
107C33—41noSylow
108C23×C3—41no—
109C23×C3—84no—
110C23×C3—366no—
111C23×C3—368no—
112C23×C3D3×A414yes—
113C22×D3—324no—
114C22×D3—332no—
115C3×Q8—312no—
116C3×Q8—38no—
117C3×Q8C22×C3×D314yescomplemented
118C2×C3×C4—312no—
119C2×C3×C4—312no—
120C2×Dic3—924no—
121C4×D3—916no—
122C4×D3—948no—
123Dic6—948no—
124Dic6—916no—
125SL(2,𝔽3)—24no—
126SL(2,𝔽3)C22×C3⋊D311yescharacteristic, direct product factor
127C2×C32—424no—
128C2×C32—44no—
129C2×C32—824no—
130C2×C32—84no—
131C2×C32C22×SL(2,𝔽3)16yescomplemented
132C2×C32C23×A411yescharacteristic
133C3⋊D3C22×SL(2,𝔽3)18yesdirect product factor
134C3×D3—1232no—
135C24—91no—
136C2×Q8—33no—
137C2×Q8—94no—
138C2×Q8—276no—
139C2×Q8—276no—
140C2×Q8—278no—
141C2×Q8—2712no—
142C2×Q8C2×C3×C3⋊D313yescomplemented
143C22×C4—31no—
144C22×C4—276no—
145C22×C4—276no—
146C22×C4—271no—
147C22×C3—43no—
148C22×C3—44no—
149C22×C3—812no—
150C22×C3—816no—
151C22×C3—3624no—
152C22×C3—364no—
153C22×C3D3×SL(2,𝔽3)116yescomplemented
154C22×C3C2×D3×A4112yes—
155C2×D3—396no—
156C2×D3—316no—
157C3×C4—34no—
158C3×C4—312no—
159Dic3—916no—
160C32—44no—
161C32—84no—
162C32C23×SL(2,𝔽3)11yescharacteristic, complemented
163C23—96no—
164C23—98no—
165C23A4×C3⋊D311yescharacteristic
166Q8—33no—
167Q8—32no—
168Q8—274no—
169Q8—2712no—
170Q8C22×C3×C3⋊D311yescharacteristic, complemented
171C2×C4—33no—
172C2×C4—33no—
173C2×C4—274no—
174C2×C4—2712no—
175C2×C4—276no—
176C2×C3—46no—
177C2×C3—41no—
178C2×C3—824no—
179C2×C3—84no—
180C2×C3—368no—
181C2×C3C2×D3×SL(2,𝔽3)124yescomplemented
182C2×C3C22×D3×A414yes—
183D3—332no—
184C22—924no—
185C22—94no—
186C22C3⋊D3×SL(2,𝔽3)14yesdirect product factor
187C22C2×A4×C3⋊D313yes—
188C4—31no—
189C4—33no—
190C4—274no—
191C3—41no—
192C3—84no—
193C3C22×D3×SL(2,𝔽3)14yescomplemented
194C2—98no—
195C2C2×C3⋊D3×SL(2,𝔽3)16yesdirect product factor
196C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC22×C32×Q8C2×C3
socleC23×C32C2×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3⋊D3×SL(2,𝔽3)C2direct (N × H)
3. C22×Q8×C3⋊D3C3semidirect
6. C2×C32×SL(2,𝔽3)C22semidirect
7. C3⋊D3×SL(2,𝔽3)C22direct (N × H)
8. C22×C32×Q8C2×C3semidirect
10. C2×Q8×C3⋊D3C2×C3semidirect
14. C22×C3×SL(2,𝔽3)D3semidirect
19. C32×SL(2,𝔽3)C23semidirect
25. C2×C32×Q8C22×C3semidirect
30. Q8×C3⋊D3C22×C3semidirect
36. C2×C3×SL(2,𝔽3)C2×D3semidirect
42. C22×C3×Q8C3×D3semidirect
49. C22×SL(2,𝔽3)C3⋊D3direct (N × H)
53. C22×C3⋊D3SL(2,𝔽3)direct (N × H)
59. C32×Q8C23×C3semidirect
68. C3×SL(2,𝔽3)C22×D3semidirect
76. C2×C3×Q8C2×C3×D3semidirect
88. C2×SL(2,𝔽3)C2×C3⋊D3direct (N × H)
93. C22×C32C2×SL(2,𝔽3)semidirect
95. C2×C3⋊D3C2×SL(2,𝔽3)direct (N × H)
105. C22×Q8C3×C3⋊D3semidirect
117. C3×Q8C22×C3×D3semidirect
126. SL(2,𝔽3)C22×C3⋊D3direct (N × H)
131. C2×C32C22×SL(2,𝔽3)semidirect
133. C3⋊D3C22×SL(2,𝔽3)direct (N × H)
142. C2×Q8C2×C3×C3⋊D3semidirect
153. C22×C3D3×SL(2,𝔽3)semidirect
162. C32C23×SL(2,𝔽3)semidirect
170. Q8C22×C3×C3⋊D3semidirect
181. C2×C3C2×D3×SL(2,𝔽3)semidirect
186. C22C3⋊D3×SL(2,𝔽3)direct (N × H)
193. C3C22×D3×SL(2,𝔽3)semidirect
195. C2C2×C3⋊D3×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3⋊D3×SL(2,𝔽3)2
C22×Q8×C3⋊D33
C22×D3×SL(2,𝔽3)3
C23×C3×C3⋊D34