Finite Group Databasebeta

← #44535361

Subgroups — #44535361

Order 31347.

980 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 286 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
1#23782923C312yesmaximal, complemented
2C3×3−1+2×C43⋊C3C311yesmaximal, characteristic, complemented
3#23782900C319yesmaximal, complemented
4#23782899C319yesmaximal, complemented
5#23782895C312yesmaximal, complemented
6#23782891C311yesmaximal, characteristic, complemented
7#23782888C312yesmaximal, complemented
8#23782886C311yesmaximal, characteristic, complemented
9#23782874C319yesmaximal, complemented
10C3×#16812508C312yesmaximal, complemented
11C3×#16812506C311yesmaximal, characteristic, complemented
12C43×3−1+4C311yesmaximal, characteristic, complemented
13C33×C43⋊C3C3211yescharacteristic
14#16812526—36no—
15#16812526—36no—
16#16812525C32118yescomplemented
17C3×#11923958C3212yes—
18C3×#11923958C3212yes—
19C3×#11923958C3211yescharacteristic
20C3×#11923958C3211yescharacteristic
21C3×#11923957C3213yescomplemented
22C3×#11923956C3213yescomplemented
233−1+2×C43⋊C3—33no—
243−1+2×C43⋊C3—36no—
253−1+2×C43⋊C3—36no—
26C3×C32⋊(C43⋊C3)C3212yes—
27C3×C9×C43⋊C3C3213yescomplemented
28#16812514C32118yescomplemented
29#16812512C3219yescomplemented
30#16812511C32118yescomplemented
31#16812509C3219yescomplemented
32#16812508C3216yescomplemented
33#16812508C3216yescomplemented
34#16812508C3216yescomplemented
35#16812506C3213yescomplemented
36#16812506C3213yescomplemented
37#16812506C3213yescomplemented
38C43×#737—36no—
39C43×#737C3219yescomplemented
40C3×3−1+2×C43—33no—
41C3×3−1+2×C43C3212yes—
42C3×3−1+2×C43C3211yescharacteristic
43C3×H(3,𝔽3)×C43C3211yescharacteristic
44C32×C43⋊C3—32no—
45C32×C43⋊C3—31no—
46C32×C43⋊C3—32no—
47C32×C43⋊C3—93no—
48C32×C43⋊C3C3311yescharacteristic
49C32×C43⋊C3C3311yescharacteristic
50C32×C43⋊C3C3311yescharacteristic
51#11923958—32no—
52#11923958—32no—
53#11923958—32no—
54#11923958—32no—
55#11923958—31no—
56#11923958—31no—
57#11923958—32no—
58#11923958—32no—
59#11923958C3316yes—
60#11923958C3316yes—
61#11923957—39no—
62#11923957—318no—
63#11923956—39no—
64#11923956—318no—
65C32⋊(C43⋊C3)—32no—
66C32⋊(C43⋊C3)—32no—
67C32⋊(C43⋊C3)C3316yes—
68C3×C43⋊C9—32no—
69C3×C43⋊C9—32no—
70C3×C43⋊C9C3313yes—
71C3×C43⋊C9C3313yes—
72C9×C43⋊C3—39no—
73C9×C43⋊C3—318no—
74C33×C43—31no—
75C33×C43C3311yescharacteristic
763−1+2×C43—32no—
773−1+2×C43—32no—
783−1+2×C43—318no—
793−1+2×C43H(3,𝔽3)13yes—
803−1+2×C43H(3,𝔽3)16yes—
813−1+2×C43C3316yes—
82H(3,𝔽3)×C43—32no—
83H(3,𝔽3)×C43C3313yes—
84C3×C9×C43—32no—
85C3×C9×C43—39no—
86C3×C9×C43C3313yes—
87#10624934—431nomaximal, Sylow
88C32×C43—33no—
89C32×C43—91no—
90C32×C43—271no—
91C32×C43C3×H(3,𝔽3)12yes—
92C32×C43C3×H(3,𝔽3)11yescharacteristic
93C32×C43C3411yescharacteristic
94C3×C43⋊C3—31no—
95C3×C43⋊C3—33no—
96C3×C43⋊C3—36no—
97C3×C43⋊C3—31no—
98C3×C43⋊C3—31no—
99C3×C43⋊C3—96no—
100C3×C43⋊C3—93no—
101C3×C43⋊C3—91no—
102C3×C43⋊C3—91no—
103C3×C43⋊C3—272no—
104C9×C43—36no—
105C9×C43C3×H(3,𝔽3)19yes—
106C43⋊C9—36no—
107C43⋊C9—36no—
108C43⋊C9—33no—
109C43⋊C9—33no—
1103−1+4—432no—
1113−1+4C43⋊C311yescharacteristic, complemented
112C32×3−1+2—431no—
113#6913—439no—
114#6913—439no—
115#6912—432no—
116#6911—431no—
117#6910—432no—
118#6910—431no—
119#6908—439no—
120C3×AΣL(1,𝔽27)—432no—
121C3×AΣL(1,𝔽27)—431no—
122C3×C43—31no—
123C3×C43—32no—
124C3×C43—31no—
125C3×C43—93no—
126C3×C43C32×H(3,𝔽3)11yescharacteristic
127C43⋊C3—33no—
128C43⋊C3—91no—
129C43⋊C3—91no—
130C43⋊C3—96no—
131C34—431no—
132#737—36no—
133#737—4318no—
134#737—1296no—
135#737—1296no—
136#737C3×C43⋊C319yescomplemented
137C3×3−1+2—33no—
138C3×3−1+2—431no—
139C3×3−1+2—431no—
140C3×3−1+2—433no—
141C3×3−1+2—433no—
142C3×3−1+2—432no—
143C3×3−1+2—432no—
144C3×3−1+2—1293no—
145C3×3−1+2—1296no—
146C3×3−1+2—1296no—
147C3×3−1+2C3×C43⋊C312yes—
148C3×3−1+2C3×C43⋊C311yescharacteristic
149C3×H(3,𝔽3)—432no—
150C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
151C32×C9—433no—
152#733—4318no—
153#732—439no—
154#731—4318no—
155#731—439no—
156AΣL(1,𝔽27)—436no—
157AΣL(1,𝔽27)—436no—
158AΣL(1,𝔽27)—436no—
159AΣL(1,𝔽27)—433no—
160AΣL(1,𝔽27)—433no—
161AΣL(1,𝔽27)—433no—
162C43#1062493411yescharacteristic, complemented, Sylow
163C33—31no—
164C33—431no—
165C33—431no—
166C33—431no—
167C33—1292no—
168C33—1291no—
169C33—1292no—
170C33—3873no—
171C33C32×C43⋊C311yescharacteristic
1723−1+2—32no—
1733−1+2—32no—
1743−1+2—318no—
1753−1+2—436no—
1763−1+2—436no—
1773−1+2—1292no—
1783−1+2—1292no—
1793−1+2—1291no—
1803−1+2—1291no—
1813−1+2—1292no—
1823−1+2—1292no—
1833−1+2—1292no—
1843−1+2—1292no—
1853−1+2—1299no—
1863−1+2—1299no—
1873−1+2—12918no—
1883−1+2—12918no—
1893−1+2C32⋊(C43⋊C3)13yes—
1903−1+2C32⋊(C43⋊C3)16yes—
1913−1+2C32×C43⋊C316yes—
192H(3,𝔽3)—32no—
193H(3,𝔽3)—436no—
194H(3,𝔽3)—1292no—
195H(3,𝔽3)—1292no—
196H(3,𝔽3)C32×C43⋊C313yes—
197C3×C9—32no—
198C3×C9—39no—
199C3×C9—433no—
200C3×C9—433no—
201C3×C9—1292no—
202C3×C9—1292no—
203C3×C9—1299no—
204C3×C9—12918no—
205C3×C9C32×C43⋊C313yes—
206C32—33no—
207C32—91no—
208C32—271no—
209C32—1293no—
210C32—1291no—
211C32—1296no—
212C32—1291no—
213C32—1291no—
214C32—3873no—
215C32—3876no—
216C32—3871no—
217C32—3871no—
218C32—11612no—
219C32C3×C32⋊(C43⋊C3)12yes—
220C32C3×C32⋊(C43⋊C3)11yescharacteristic
221C32C33×C43⋊C311yescharacteristic
222C9—36no—
223C9—1293no—
224C9—1293no—
225C9—1296no—
226C9—1296no—
227C9C3×C32⋊(C43⋊C3)19yes—
228C3—31no—
229C3—31no—
230C3—32no—
231C3—93no—
232C3—1293no—
233C3—3871no—
234C3—3871no—
235C3—3876no—
236C3C32×C32⋊(C43⋊C3)11yescharacteristic
subgroupG/H
centerC3C32×C32⋊(C43⋊C3)
commutator, cosocleC32×C43C34
FrattiniC32C33×C43⋊C3
FittingC43×3−1+4C3
socleC3×C43C32×H(3,𝔽3)
hypercenter3−1+4C43⋊C3
hypocenterC43#10624934
LCSG ▷ C32×C43 ▷ C3×C43 ▷ C43
UCSC1 ◁ C3 ◁ C3×3−1+2 ◁ 3−1+4

Complements

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

NH ≅ G/Ntype
1. #23782923C3semidirect
2. C3×3−1+2×C43⋊C3C3semidirect
3. #23782900C3semidirect
4. #23782899C3semidirect
5. #23782895C3semidirect
6. #23782891C3semidirect
7. #23782888C3semidirect
8. #23782886C3semidirect
9. #23782874C3semidirect
10. C3×#16812508C3semidirect
11. C3×#16812506C3semidirect
12. C43×3−1+4C3semidirect
16. #16812525C32semidirect
21. C3×#11923957C32semidirect
22. C3×#11923956C32semidirect
27. C3×C9×C43⋊C3C32semidirect
28. #16812514C32semidirect
29. #16812512C32semidirect
30. #16812511C32semidirect
31. #16812509C32semidirect
32. #16812508C32semidirect
33. #16812508C32semidirect
34. #16812508C32semidirect
35. #16812506C32semidirect
36. #16812506C32semidirect
37. #16812506C32semidirect
39. C43×#737C32semidirect
111. 3−1+4C43⋊C3semidirect
136. #737C3×C43⋊C3semidirect
162. C43#10624934semidirect

Maximal subgroups

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

subgroupindex
#237829233
C3×3−1+2×C43⋊C33
#237829003
#237828993
#237828953
#237828913
#237828883
#237828863
#237828743
C3×#168125083
C3×#168125063
C43×3−1+43
#1062493443