Finite Group Databasebeta

← C23×C3×#41⋊C3 #11919868

Subgroups — C23×C3×#41⋊C3 #11919868

Order 1152.

3160 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 3160 conjugacy classes of subgroups — too many to draw. Only its 638 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×C3×#41⋊C3C2114yesmaximal, direct product factor
2C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C23×C3×#41C311yesmaximal, characteristic, complemented
4C23×#41⋊C3C313yesmaximal, direct product factor
5C23×C32×C4—41nomaximal
6C2×C3×#41⋊C3C22128yesdirect product factor
7C22×C3×SL(2,𝔽3)C2217yescomplemented
8C22×C3×#41—314no—
9C22×C3×#41C2×C3114yescomplemented
10C23×C3×Q8C2×C311yescharacteristic
11C23×C3×D4—31no—
12C24×C3×C4—31no—
13C22×#41⋊C3C2×C3142yesdirect product factor
14C23×SL(2,𝔽3)C2×C313yescomplemented
15C24×C32—41no—
16C22×C32×C4—414no—
17C3×#41⋊C3C2318yesdirect product factor
18C2×C3×SL(2,𝔽3)C2317yescomplemented
19C23×#41C3211yescharacteristic, complemented, Sylow
20C25×C3—31no—
21C2×C3×#41—356no—
22C2×C3×#41—384no—
23C2×C3×#41C22×C3128yescomplemented
24C22×C3×Q8—37no—
25C22×C3×Q8C22×C317yes—
26C22×C3×D4—314no—
27C22×C3×D4—314no—
28C23×C3×C4—31no—
29C23×C3×C4—314no—
30C23×C3×C4—314no—
31C23×C3×C4—43no—
32C23×C3×C4A411yescharacteristic
33C2×#41⋊C3C22×C3184yesdirect product factor
34C22×SL(2,𝔽3)C22×C3121yescomplemented
35C23×C32—48no—
36C23×C32—47no—
37C2×C32×C4—428no—
38C3×SL(2,𝔽3)C2411yescharacteristic, complemented
39C22×#41—314no—
40C22×#41C2×C32114yescomplemented
41C23×Q8C2×C3211yescharacteristic
42C23×D4—31no—
43C24×C4—31no—
44C24×C3—314no—
45C24×C3—43no—
46C24×C3—68no—
47C24×C3C2×A411yescharacteristic
48C3×#41—3112no—
49C3×#41—356no—
50C3×#41C23×C318yescomplemented
51C2×C3×Q8—314no—
52C2×C3×Q8—321no—
53C2×C3×Q8C23×C317yes—
54C2×C3×D4—328no—
55C2×C3×D4—384no—
56C22×C3×C4—37no—
57C22×C3×C4—37no—
58C22×C3×C4—384no—
59C22×C3×C4—328no—
60C22×C3×C4—442no—
61C22×C3×C4C2×A4114yes—
62#41⋊C3C23×C3124yesdirect product factor
63C2×SL(2,𝔽3)C23×C3121yescomplemented
64C22×C32—428no—
65C22×C32—47no—
66C32×C4—48no—
67C25—31no—
68C2×#41—356no—
69C2×#41—384no—
70C2×#41C22×C32128yescomplemented
71C22×Q8—37no—
72C22×Q8C22×C3217yes—
73C22×D4—314no—
74C22×D4—314no—
75C23×C4—31no—
76C23×C4—314no—
77C23×C4—314no—
78C23×C4C3×A411yescharacteristic
79C23×C3—328no—
80C23×C3—421no—
81C23×C3—424no—
82C23×C3—656no—
83C23×C3#41⋊C318yesdirect product factor
84C23×C3C22×A417yes—
85C3×Q8—37no—
86C3×Q8—314no—
87C3×Q8C24×C311yescharacteristic
88C3×D4—38no—
89C3×D4—356no—
90C2×C3×C4—321no—
91C2×C3×C4—37no—
92C2×C3×C4—356no—
93C2×C3×C4—38no—
94C2×C3×C4—484no—
95C2×C3×C4C22×A4128yes—
96SL(2,𝔽3)C24×C313yescomplemented
97C2×C32—414no—
98C2×C32—41no—
99C24—314no—
100C24—68no—
101C24C2×C3×A411yescharacteristic
102#41—3112no—
103#41—356no—
104#41C23×C3218yescomplemented
105C2×Q8—314no—
106C2×Q8—321no—
107C2×Q8C23×C3217yes—
108C2×D4—328no—
109C2×D4—384no—
110C22×C4—37no—
111C22×C4—37no—
112C22×C4—328no—
113C22×C4—384no—
114C22×C4C2×C3×A4114yes—
115C22×C3—38no—
116C22×C3—484no—
117C22×C3—421no—
118C22×C3—656no—
119C22×C3C2×#41⋊C3128yesdirect product factor
120C22×C3C23×A417yes—
121C3×C4—37no—
122C3×C4—31no—
123C3×C4—424no—
124C3×C4C23×A418yes—
125C32—41noSylow
126C23—328no—
127C23—656no—
128C23C3×#41⋊C318yesdirect product factor
129C23C22×C3×A417yes—
130Q8—37no—
131Q8—314no—
132Q8C24×C3211yescharacteristic
133D4—38no—
134D4—356no—
135C2×C4—37no—
136C2×C4—321no—
137C2×C4—38no—
138C2×C4—356no—
139C2×C4C22×C3×A4128yes—
140C2×C3—442no—
141C2×C3—43no—
142C2×C3—68no—
143C2×C3C22×#41⋊C3114yesdirect product factor
144C2×C3C24×A411yescharacteristic
145C22—38no—
146C22—656no—
147C22C2×C3×#41⋊C3128yesdirect product factor
148C22C23×C3×A417yes—
149C4—31no—
150C4—37no—
151C4C23×C3×A418yes—
152C3—43no—
153C3C23×#41⋊C311yescharacteristic, direct product factor
154C2—68no—
155C2C22×C3×#41⋊C3114yesdirect product factor
156C2C24×C3×A411yescharacteristic
subgroupG/H
center, hypercenterC23×C3×C4A4
commutator, cosocle, hypocenterQ8C24×C32
FrattiniC2C24×C3×A4
FittingC23×C3×#41C3
socleC24×C3C2×A4
LCSG ▷ Q8
UCSC1 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2direct (N × H)
2. C23×C3×SL(2,𝔽3)C2semidirect
3. C23×C3×#41C3semidirect
4. C23×#41⋊C3C3direct (N × H)
6. C2×C3×#41⋊C3C22direct (N × H)
7. C22×C3×SL(2,𝔽3)C22semidirect
9. C22×C3×#41C2×C3semidirect
13. C22×#41⋊C3C2×C3direct (N × H)
14. C23×SL(2,𝔽3)C2×C3semidirect
17. C3×#41⋊C3C23direct (N × H)
18. C2×C3×SL(2,𝔽3)C23semidirect
19. C23×#41C32semidirect
23. C2×C3×#41C22×C3semidirect
33. C2×#41⋊C3C22×C3direct (N × H)
34. C22×SL(2,𝔽3)C22×C3semidirect
38. C3×SL(2,𝔽3)C24semidirect
40. C22×#41C2×C32semidirect
50. C3×#41C23×C3semidirect
62. #41⋊C3C23×C3direct (N × H)
63. C2×SL(2,𝔽3)C23×C3semidirect
70. C2×#41C22×C32semidirect
83. C23×C3#41⋊C3direct (N × H)
96. SL(2,𝔽3)C24×C3semidirect
104. #41C23×C32semidirect
119. C22×C3C2×#41⋊C3direct (N × H)
128. C23C3×#41⋊C3direct (N × H)
143. C2×C3C22×#41⋊C3direct (N × H)
147. C22C2×C3×#41⋊C3direct (N × H)
153. C3C23×#41⋊C3direct (N × H)
155. C2C22×C3×#41⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C23×C3×SL(2,𝔽3)2
C23×C3×#413
C23×#41⋊C33
C23×C32×C44