Finite Group Databasebeta

← C22×C32×GL(2,𝔽3) #13181358

Subgroups — C22×C32×GL(2,𝔽3) #13181358

Order 1728.

1446 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 222 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×C32×GL(2,𝔽3)C216yesmaximal, direct product factor
3C22×C3×GL(2,𝔽3)C314yesmaximal, direct product factor
4C22×C32×SD16—31nomaximal
5C23×C32×D3—41nomaximal
6C2×C32×SL(2,𝔽3)C2213yescomplemented
7C32×GL(2,𝔽3)C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic, complemented
9C22×C32×D4—31no—
10C22×C3×SL(2,𝔽3)—24no—
11C22×C3×SL(2,𝔽3)C2×C314yescomplemented
12C2×C3×GL(2,𝔽3)C2×C3124yesdirect product factor
13C2×C32×SD16—36no—
14C2×C32×SD16—36no—
15C22×C32×C8—31no—
16C23×C33—41no—
17C22×C32×D3—46no—
18C22×C32×D3—48no—
19C32×SL(2,𝔽3)C2311yescharacteristic, complemented
20C22×GL(2,𝔽3)C3211yescharacteristic, direct product factor
21C22×C3×SD16—34no—
22C24×C32—61no—
23C23×C3×D3—44no—
24C2×C32×Q8—33no—
25C2×C32×Q8C2×D313yescomplemented
26C2×C32×D4—36no—
27C2×C32×D4—63no—
28C22×C32×C4—31no—
29C2×C3×SL(2,𝔽3)—212no—
30C2×C3×SL(2,𝔽3)C22×C3112yescomplemented
31C3×GL(2,𝔽3)C22×C3116yesdirect product factor
32C32×SD16—312no—
33C32×SD16—34no—
34C2×C32×C8—36no—
35C22×C33—44no—
36C22×C33—43no—
37C2×C32×D3—44no—
38C2×C32×D3—424no—
39C22×C3×Q8C3×D314yescomplemented
40C22×C3×D4—34no—
41C22×SL(2,𝔽3)—24no—
42C22×SL(2,𝔽3)C2×C3211yescharacteristic, complemented
43C2×GL(2,𝔽3)C2×C3216yesdirect product factor
44C2×C3×SD16—324no—
45C2×C3×SD16—324no—
46C22×C3×C8—34no—
47C23×C32—44no—
48C23×C32—66no—
49C23×C32—84no—
50C23×C32—124no—
51C23×C32S411yescharacteristic
52C22×C3×D3—424no—
53C22×C3×D3—432no—
54C32×Q8—33no—
55C32×Q8—61no—
56C32×Q8C22×D311yescharacteristic, complemented
57C32×D4—34no—
58C32×D4—66no—
59C2×C32×C4—33no—
60C2×C32×C4—33no—
61C3×SL(2,𝔽3)—24no—
62C3×SL(2,𝔽3)C23×C314yescomplemented
63C32×C8—34no—
64C22×SD16—31noSylow
65C2×C33—46no—
66C2×C33—41no—
67C32×D3—48no—
68C24×C3—64no—
69C23×D3—41no—
70C2×C3×Q8—312no—
71C2×C3×Q8C2×C3×D3112yescomplemented
72C2×C3×D4—324no—
73C2×C3×D4—612no—
74C22×C3×C4—34no—
75C2×SL(2,𝔽3)—212no—
76C2×SL(2,𝔽3)C22×C3213yescomplemented
77GL(2,𝔽3)C22×C3214yesdirect product factor
78C3×SD16—348no—
79C3×SD16—316no—
80C2×C3×C8—324no—
81C22×C32—412no—
82C22×C32—416no—
83C22×C32—64no—
84C22×C32—812no—
85C22×C32—816no—
86C22×C32—1212no—
87C22×C32GL(2,𝔽3)14yesdirect product factor
88C22×C32C2×S413yes—
89C2×C3×D3—416no—
90C2×C3×D3—496no—
91C32×C4—31no—
92C32×C4—33no—
93C22×Q8C32×D311yescharacteristic, complemented
94C22×D4—31no—
95C2×SD16—36no—
96C2×SD16—36no—
97C22×C8—31no—
98C33—41noSylow
99C23×C3—41no—
100C23×C3—624no—
101C23×C3—84no—
102C23×C3—1216no—
103C23×C3C3×S414yes—
104C22×D3—46no—
105C22×D3—48no—
106C3×Q8—312no—
107C3×Q8—64no—
108C3×Q8C22×C3×D314yescomplemented
109C3×D4—316no—
110C3×D4—624no—
111C2×C3×C4—312no—
112C2×C3×C4—312no—
113SL(2,𝔽3)—24no—
114SL(2,𝔽3)C23×C3211yescharacteristic, complemented
115C3×C8—316no—
116C2×C32—44no—
117C2×C32—424no—
118C2×C32—84no—
119C2×C32—824no—
120C2×C32—124no—
121C2×C32C2×GL(2,𝔽3)16yesdirect product factor
122C2×C32C22×S411yescharacteristic
123C3×D3—432no—
124C24—61no—
125C2×Q8—33no—
126C2×Q8C2×C32×D313yescomplemented
127C2×D4—36no—
128C2×D4—63no—
129C22×C4—31no—
130SD16—312no—
131SD16—34no—
132C2×C8—36no—
133C22×C3—43no—
134C22×C3—44no—
135C22×C3—616no—
136C22×C3—812no—
137C22×C3—816no—
138C22×C3—1248no—
139C22×C3C3×GL(2,𝔽3)116yesdirect product factor
140C22×C3C2×C3×S4112yes—
141C2×D3—44no—
142C2×D3—424no—
143C3×C4—34no—
144C3×C4—312no—
145C32—44no—
146C32—84no—
147C32C22×GL(2,𝔽3)11yescharacteristic, direct product factor
148C23—66no—
149C23—124no—
150C23C32×S411yescharacteristic
151Q8—33no—
152Q8—61no—
153Q8C22×C32×D311yescharacteristic, complemented
154D4—34no—
155D4—66no—
156C2×C4—33no—
157C2×C4—33no—
158C8—34no—
159C2×C3—46no—
160C2×C3—41no—
161C2×C3—824no—
162C2×C3—84no—
163C2×C3—1216no—
164C2×C3C2×C3×GL(2,𝔽3)124yesdirect product factor
165C2×C3C22×C3×S414yes—
166D3—48no—
167C22—64no—
168C22—1212no—
169C22C32×GL(2,𝔽3)14yesdirect product factor
170C22C2×C32×S413yes—
171C4—31no—
172C4—33no—
173C3—41no—
174C3—84no—
175C3C22×C3×GL(2,𝔽3)14yesdirect product factor
176C2—124no—
177C2C2×C32×GL(2,𝔽3)16yesdirect product factor
178C2C22×C32×S411yescharacteristic
subgroupG/H
center, socle, hypercenterC23×C32S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC22×C32×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C32

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C32×GL(2,𝔽3)C2direct (N × H)
3. C22×C3×GL(2,𝔽3)C3direct (N × H)
6. C2×C32×SL(2,𝔽3)C22semidirect
7. C32×GL(2,𝔽3)C22direct (N × H)
8. C22×C32×Q8D3semidirect
11. C22×C3×SL(2,𝔽3)C2×C3semidirect
12. C2×C3×GL(2,𝔽3)C2×C3direct (N × H)
19. C32×SL(2,𝔽3)C23semidirect
20. C22×GL(2,𝔽3)C32direct (N × H)
25. C2×C32×Q8C2×D3semidirect
30. C2×C3×SL(2,𝔽3)C22×C3semidirect
31. C3×GL(2,𝔽3)C22×C3direct (N × H)
39. C22×C3×Q8C3×D3semidirect
42. C22×SL(2,𝔽3)C2×C32semidirect
43. C2×GL(2,𝔽3)C2×C32direct (N × H)
56. C32×Q8C22×D3semidirect
62. C3×SL(2,𝔽3)C23×C3semidirect
71. C2×C3×Q8C2×C3×D3semidirect
76. C2×SL(2,𝔽3)C22×C32semidirect
77. GL(2,𝔽3)C22×C32direct (N × H)
87. C22×C32GL(2,𝔽3)direct (N × H)
93. C22×Q8C32×D3semidirect
108. C3×Q8C22×C3×D3semidirect
114. SL(2,𝔽3)C23×C32semidirect
121. C2×C32C2×GL(2,𝔽3)direct (N × H)
126. C2×Q8C2×C32×D3semidirect
139. C22×C3C3×GL(2,𝔽3)direct (N × H)
147. C32C22×GL(2,𝔽3)direct (N × H)
153. Q8C22×C32×D3semidirect
164. C2×C3C2×C3×GL(2,𝔽3)direct (N × H)
169. C22C32×GL(2,𝔽3)direct (N × H)
175. C3C22×C3×GL(2,𝔽3)direct (N × H)
177. C2C2×C32×GL(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×C32×GL(2,𝔽3)2
C22×C3×GL(2,𝔽3)3
C22×C32×SD163
C23×C32×D34