Finite Group Databasebeta

← C2×C3⋊(C3⋊#859) #13173068

Subgroups — C2×C3⋊(C3⋊#859) #13173068

Order 1728.

1142 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 239 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
2C3⋊(C3⋊#859)C212yesmaximal, direct product factor
3C2×C3⋊#859—312nomaximal
4C2×#63662—31nomaximal
5C22×C3⋊(C3⋊Dic3)—41nomaximal
6C2×C32×SL(2,𝔽3)C412yescomplemented
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C22×C32×Q8D311yescharacteristic
9C22×C3×SL(2,𝔽3)D3112yes—
10C2×C4⋊(C3⋊Dic3)—31no—
11C22×C3⋊(C3⋊C8)—31no—
12C3⋊#859—324no—
13#63662—32no—
14#63662—32no—
15C23×C33—41no—
16C2×C3⋊(C3⋊Dic3)—42no—
17C2×C3⋊(C3⋊Dic3)—44no—
18C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
19C2×#859—99no—
20C2×#835—94no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8Dic312yescomplemented
24C2×C32×Q8C2×D311yescharacteristic
25C22×C32×C4—31no—
26C22×C3⋊Dic3—61no—
27C22×C3⋊Dic3—1212no—
28C2×C3×SL(2,𝔽3)Dic3124yescomplemented
29C2×C3×SL(2,𝔽3)C2×D3112yes—
30C4⋊(C3⋊Dic3)—32no—
31C4⋊(C3⋊Dic3)—61no—
32C2×C3⋊(C3⋊C8)—32no—
33C2×C3⋊(C3⋊C8)—62no—
34C22×C33—42no—
35C22×C33—42no—
36C22×C33—42no—
37C22×C33—41no—
38C3⋊(C3⋊Dic3)—44no—
39C22×C3×Q8C3⋊D314yes—
40C22×SL(2,𝔽3)C3⋊D319yes—
41C2×C4⋊Dic3—94no—
42C22×C3⋊C8—94no—
43#859—918no—
44#835—98no—
45#835—98no—
46C23×C32—412no—
47C23×C32S411yescharacteristic
48C32×Q8—32no—
49C32×Q8—31no—
50C32×Q8—61no—
51C32×Q8C2×Dic311yescharacteristic, complemented
52C2×C32×C4—32no—
53C2×C32×C4—32no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56C2×C3⋊Dic3—62no—
57C2×C3⋊Dic3—1224no—
58C2×C3⋊Dic3—1248no—
59C2×C3⋊Dic3—122no—
60C3×SL(2,𝔽3)C2×Dic3112yescomplemented
61C3⋊(C3⋊C8)—62no—
62C2×#103—271noSylow
63C2×C33—44no—
64C2×C33—41no—
65C2×C33—41no—
66C2×C33—41no—
67C2×C3×Q8—34no—
68C2×C3×Q8—38no—
69C2×C3×Q8C3⋊Dic318yescomplemented
70C2×C3×Q8C2×C3⋊D314yes—
71C22×C3×C4—34no—
72C22×Dic3—184no—
73C22×Dic3—369no—
74C2×SL(2,𝔽3)C3⋊Dic3118yescomplemented
75C2×SL(2,𝔽3)C2×C3⋊D319yes—
76C4⋊Dic3—98no—
77C4⋊Dic3—184no—
78C2×C3⋊C8—98no—
79C2×C3⋊C8—188no—
80C22×C32—412no—
81C22×C32—424no—
82C22×C32—424no—
83C22×C32—424no—
84C22×C32#22612yes—
85C22×C32GL(2,𝔽3)12yes—
86C22×C32SL(2,ℤ/4)12yes—
87C22×C32C2×S411yescharacteristic
88C32×C4—31no—
89C32×C4—31no—
90C32×C4—32no—
91C3⋊Dic3—1248no—
92C3⋊Dic3—122no—
93C22×Q8C3⋊(C3⋊D3)11yescharacteristic
94C22×C8—271no—
95C2×C4⋊C4—271no—
96#103—272no—
97#103—272no—
98C33—41noSylow
99C23×C3—49no—
100C23×C3C3⋊S414yes—
101C3×Q8—38no—
102C3×Q8—34no—
103C3×Q8—64no—
104C3×Q8C2×C3⋊Dic314yescomplemented
105C2×C3×C4—34no—
106C2×C3×C4—34no—
107C2×C3×C4—38no—
108C2×C3×C4—38no—
109C2×Dic3—188no—
110C2×Dic3—3618no—
111C2×Dic3—3636no—
112C2×Dic3—368no—
113SL(2,𝔽3)C2×C3⋊Dic319yescomplemented
114C3⋊C8—188no—
115C2×C32—412no—
116C2×C32—412no—
117C2×C32—448no—
118C2×C32—412no—
119C2×C32#85914yescomplemented
120C2×C32C2×#22611yescharacteristic
121C2×C32C2×GL(2,𝔽3)11yescharacteristic
122C2×C32C2×SL(2,ℤ/4)11yescharacteristic
123C2×Q8—31no—
124C2×Q8—32no—
125C2×Q8C3⋊(C3⋊Dic3)12yescomplemented
126C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
127C22×C4—31no—
128C22×C4—541no—
129C2×C8—272no—
130C2×C8—542no—
131C4⋊C4—272no—
132C4⋊C4—541no—
133C22×C3—49no—
134C22×C3—418no—
135C22×C3—418no—
136C22×C3—418no—
137C22×C3C3⋊#22618yes—
138C22×C3C3⋊GL(2,𝔽3)18yes—
139C22×C3C3⋊SL(2,ℤ/4)18yes—
140C22×C3C2×C3⋊S414yes—
141C3×C4—34no—
142C3×C4—34no—
143C3×C4—38no—
144Dic3—3636no—
145Dic3—368no—
146C32—412no—
147C32C2×#85911yescharacteristic, complemented
148C23C3⋊(C3⋊S4)11yescharacteristic
149Q8—32no—
150Q8—31no—
151Q8—61no—
152Q8C2×C3⋊(C3⋊Dic3)11yescharacteristic, complemented
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—32no—
156C2×C4—32no—
157C2×C4—542no—
158C2×C4—1082no—
159C8—542no—
160C2×C3—436no—
161C2×C3—49no—
162C2×C3—49no—
163C2×C3—49no—
164C2×C3C3⋊#859116yescomplemented
165C2×C3C2×C3⋊#22614yes—
166C2×C3C2×C3⋊GL(2,𝔽3)14yes—
167C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
168C22C3⋊(C3⋊#226)12yes—
169C22C3⋊(C3⋊GL(2,𝔽3))12yes—
170C22C3⋊(C3⋊SL(2,ℤ/4))12yes—
171C22C2×C3⋊(C3⋊S4)11yescharacteristic
172C4—31no—
173C4—31no—
174C4—32no—
175C4—1082no—
176C3—49no—
177C3C2×C3⋊#85914yescomplemented
178C2C3⋊(C3⋊#859)14yesdirect product factor
179C2C2×C3⋊(C3⋊#226)11yescharacteristic
180C2C2×C3⋊(C3⋊GL(2,𝔽3))11yescharacteristic
181C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, hypercenterC23C3⋊(C3⋊S4)
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FrattiniC22C2×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2direct (N × H)
6. C2×C32×SL(2,𝔽3)C4semidirect
18. C32×SL(2,𝔽3)C2×C4semidirect
23. C2×C32×Q8Dic3semidirect
28. C2×C3×SL(2,𝔽3)Dic3semidirect
51. C32×Q8C2×Dic3semidirect
60. C3×SL(2,𝔽3)C2×Dic3semidirect
69. C2×C3×Q8C3⋊Dic3semidirect
74. C2×SL(2,𝔽3)C3⋊Dic3semidirect
104. C3×Q8C2×C3⋊Dic3semidirect
113. SL(2,𝔽3)C2×C3⋊Dic3semidirect
119. C2×C32#859semidirect
125. C2×Q8C3⋊(C3⋊Dic3)semidirect
147. C32C2×#859semidirect
152. Q8C2×C3⋊(C3⋊Dic3)semidirect
164. C2×C3C3⋊#859semidirect
177. C3C2×C3⋊#859semidirect
178. C2C3⋊(C3⋊#859)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
C3⋊(C3⋊#859)2
C2×C3⋊#8593
C2×#636623
C22×C3⋊(C3⋊Dic3)4