Finite Group Databasebeta

← C43⋊C3×#6910 #44535302

Subgroups — C43⋊C3×#6910 #44535302

Order 31347.

1088 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×3−1+2×C43⋊C3C312yesmaximal, complemented
2C3×3−1+2×C43⋊C3C311yesmaximal, characteristic, complemented
3C3×H(3,𝔽3)×C43⋊C3C311yesmaximal, characteristic
4#23782888C314yesmaximal, complemented
5#23782887C312yesmaximal, complemented
6#23782886C312yesmaximal, complemented
7#23782885C319yesmaximal, complemented
8#23782884C319yesmaximal, complemented
9#731×C43⋊C3C319yesmaximal, complemented
10C43×#6910C311yesmaximal, characteristic, complemented
11C33×C43⋊C3—31no—
12C33×C43⋊C3C3211yescharacteristic
13C3×#11923958C3214yescomplemented
14C3×#11923958C3212yescomplemented
15C3×#11923957—32no—
16C3×#11923957C3213yescomplemented
17C3×#11923956—32no—
18C3×#11923956C3213yescomplemented
193−1+2×C43⋊C3—32no—
203−1+2×C43⋊C3—32no—
213−1+2×C43⋊C3—32no—
223−1+2×C43⋊C3—32no—
233−1+2×C43⋊C3—32no—
243−1+2×C43⋊C3—32no—
253−1+2×C43⋊C3—32no—
263−1+2×C43⋊C3—31no—
273−1+2×C43⋊C3—32no—
283−1+2×C43⋊C3—32no—
293−1+2×C43⋊C3—32no—
303−1+2×C43⋊C3C3216yescomplemented
313−1+2×C43⋊C3C3216yescomplemented
323−1+2×C43⋊C3C3216yescomplemented
33C3×C32⋊(C43⋊C3)—32no—
34C3×C32⋊(C43⋊C3)C3212yes—
35H(3,𝔽3)×C43⋊C3—32no—
36H(3,𝔽3)×C43⋊C3—32no—
37H(3,𝔽3)×C43⋊C3—32no—
38H(3,𝔽3)×C43⋊C3C3213yes—
39H(3,𝔽3)×C43⋊C3C3213yes—
40H(3,𝔽3)×C43⋊C3C3213yes—
41C3×C9×C43⋊C3—32no—
42C3×C9×C43⋊C3C3213yescomplemented
43#16812511C32136yescomplemented
44#16812510C32118yescomplemented
45#16812509C32118yescomplemented
46C3×3−1+2×C43C3212yescomplemented
47C3×3−1+2×C43C3211yescharacteristic, complemented
48C3×H(3,𝔽3)×C43C3211yescharacteristic
49C43×#731C3219yescomplemented
50C32×C43⋊C3—32no—
51C32×C43⋊C3—33no—
52C32×C43⋊C3—33no—
53C32×C43⋊C3—33no—
54C32×C43⋊C3—93no—
55C32×C43⋊C3—99no—
56C32×C43⋊C3H(3,𝔽3)12yes—
57C32×C43⋊C3H(3,𝔽3)11yescharacteristic
58C32×C43⋊C3H(3,𝔽3)12yes—
59C32×C43⋊C3H(3,𝔽3)12yes—
60C32×C43⋊C3H(3,𝔽3)12yes—
61C32×C43⋊C3C3311yescharacteristic
62C32×C43⋊C3C3311yescharacteristic
63C32×C43⋊C3C3311yescharacteristic
64#11923958—34no—
65#11923958—34no—
66#11923958—34no—
67#11923958—32no—
68#11923958C33112yescomplemented
69#11923957—36no—
70#11923957—33no—
71#11923957—36no—
72#11923957—36no—
73#11923957—36no—
74#11923956—36no—
75#11923956—33no—
76#11923956—36no—
77#11923956—36no—
78#11923956—36no—
79C32⋊(C43⋊C3)—34no—
80C32⋊(C43⋊C3)—36no—
81C32⋊(C43⋊C3)—36no—
82C32⋊(C43⋊C3)—36no—
83C32⋊(C43⋊C3)C3316yes—
84C3×C43⋊C9—34no—
85C3×C43⋊C9C3316yescomplemented
86C9×C43⋊C3—36no—
87C9×C43⋊C3—36no—
88C9×C43⋊C3—36no—
89C9×C43⋊C3—36no—
90C9×C43⋊C3—33no—
91C33×C43—31no—
92C33×C43C3311yescharacteristic
933−1+2×C43—32no—
943−1+2×C43—32no—
953−1+2×C43—32no—
963−1+2×C43—31no—
973−1+2×C43C3316yescomplemented
98H(3,𝔽3)×C43—32no—
99H(3,𝔽3)×C43C3313yes—
100C3×C9×C43—32no—
101C3×C9×C43C3313yescomplemented
102C3×#6910—431nomaximal, Sylow
103C32×C43—33no—
104C32×C43—91no—
105C32×C43—93no—
106C32×C43C3×H(3,𝔽3)12yes—
107C32×C43C3×H(3,𝔽3)11yescharacteristic
108C32×C43C3411yescharacteristic
109C3×C43⋊C3—33no—
110C3×C43⋊C3—36no—
111C3×C43⋊C3—33no—
112C3×C43⋊C3—36no—
113C3×C43⋊C3—92no—
114C3×C43⋊C3—92no—
115C3×C43⋊C3—92no—
116C3×C43⋊C3—91no—
117C3×C43⋊C3—91no—
118C3×C43⋊C3—99no—
119C3×C43⋊C3—99no—
120C3×C43⋊C3—96no—
121C3×C43⋊C3—99no—
122C3×C43⋊C3C3×H(3,𝔽3)12yes—
123C3×C43⋊C3C3×H(3,𝔽3)12yes—
124C3×C43⋊C3C3×H(3,𝔽3)12yes—
125C3×C43⋊C3C3×H(3,𝔽3)11yescharacteristic
126C3×C43⋊C3C3×H(3,𝔽3)11yescharacteristic
127C3×C43⋊C3C3×H(3,𝔽3)11yescharacteristic
128C9×C43—36no—
129C9×C43—33no—
130C43⋊C9—312no—
131C43⋊C9—36no—
132C32×3−1+2—431no—
133C32×3−1+2—432no—
134C32×H(3,𝔽3)—431no—
135#6910—432no—
136#6910—434no—
137#6910—432no—
138#6910—439no—
139#6910—439no—
140#6910C43⋊C311yescharacteristic, direct product factor
141C3×#731—439no—
142C3×C43—31no—
143C3×C43—32no—
144C3×C43—31no—
145C3×C43—93no—
146C3×C43C32×H(3,𝔽3)11yescharacteristic
147C43⋊C3—31no—
148C43⋊C3—31no—
149C43⋊C3—32no—
150C43⋊C3—32no—
151C43⋊C3—32no—
152C43⋊C3—96no—
153C43⋊C3#691013yesdirect product factor
154C34—431no—
155C34—1291no—
156C3×3−1+2—432no—
157C3×3−1+2—433no—
158C3×3−1+2—433no—
159C3×3−1+2—434no—
160C3×3−1+2—436no—
161C3×3−1+2—436no—
162C3×3−1+2—436no—
163C3×3−1+2—1292no—
164C3×3−1+2—1292no—
165C3×3−1+2—1292no—
166C3×3−1+2—1291no—
167C3×3−1+2—1292no—
168C3×3−1+2—1292no—
169C3×3−1+2—1292no—
170C3×3−1+2—1292no—
171C3×3−1+2—1292no—
172C3×3−1+2—1292no—
173C3×3−1+2—1292no—
174C3×3−1+2—1292no—
175C3×3−1+2—1292no—
176C3×3−1+2C3×C43⋊C312yescomplemented
177C3×3−1+2C3×C43⋊C311yescharacteristic, complemented
178C3×H(3,𝔽3)—432no—
179C3×H(3,𝔽3)—433no—
180C3×H(3,𝔽3)—433no—
181C3×H(3,𝔽3)—433no—
182C3×H(3,𝔽3)—1292no—
183C3×H(3,𝔽3)—1292no—
184C3×H(3,𝔽3)—1292no—
185C3×H(3,𝔽3)—1292no—
186C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
187C32×C9—433no—
188C32×C9—1292no—
189#731—4336no—
190#731—4318no—
191#731—4318no—
192#731C3×C43⋊C319yescomplemented
193C43C3×#691011yescharacteristic, complemented, Sylow
194C33—31no—
195C33—432no—
196C33—431no—
197C33—431no—
198C33—432no—
199C33—432no—
200C33—432no—
201C33—431no—
202C33—431no—
203C33—1293no—
204C33—1293no—
205C33—1293no—
206C33—1292no—
207C33—3873no—
208C33—3879no—
209C33C32×C43⋊C311yescharacteristic
2103−1+2—32no—
2113−1+2—32no—
2123−1+2—32no—
2133−1+2—31no—
2143−1+2—4312no—
2153−1+2—1296no—
2163−1+2—1296no—
2173−1+2—1296no—
2183−1+2—1296no—
2193−1+2—1293no—
2203−1+2—1296no—
2213−1+2—1296no—
2223−1+2—1296no—
2233−1+2—1296no—
2243−1+2—1293no—
2253−1+2—1294no—
2263−1+2—1292no—
2273−1+2—1294no—
2283−1+2—1294no—
2293−1+2C32×C43⋊C316yescomplemented
230H(3,𝔽3)—32no—
231H(3,𝔽3)—436no—
232H(3,𝔽3)—1296no—
233H(3,𝔽3)—1296no—
234H(3,𝔽3)—1296no—
235H(3,𝔽3)—1294no—
236H(3,𝔽3)C32×C43⋊C313yes—
237C3×C9—32no—
238C3×C9—436no—
239C3×C9—1293no—
240C3×C9—1296no—
241C3×C9—1296no—
242C3×C9—1296no—
243C3×C9—1296no—
244C3×C9—1294no—
245C3×C9C32×C43⋊C313yescomplemented
246C32—33no—
247C32—91no—
248C32—93no—
249C32—431no—
250C32—432no—
251C32—432no—
252C32—432no—
253C32—431no—
254C32—431no—
255C32—1296no—
256C32—1293no—
257C32—1293no—
258C32—1296no—
259C32—3872no—
260C32—3872no—
261C32—3872no—
262C32—3871no—
263C32—3871no—
264C32—3876no—
265C32—3879no—
266C32—3879no—
267C32—3879no—
268C32H(3,𝔽3)×C43⋊C312yes—
269C32H(3,𝔽3)×C43⋊C311yescharacteristic
270C32C33×C43⋊C311yescharacteristic
271C9—36no—
272C9—33no—
273C9—1296no—
274C9—12912no—
275C3—31no—
276C3—32no—
277C3—31no—
278C3—93no—
279C3—433no—
280C3—1292no—
281C3—1292no—
282C3—1292no—
283C3—1291no—
284C3—1291no—
285C3—3876no—
286C3C3×H(3,𝔽3)×C43⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C43⋊C3
commutator, cosocleC32×C43C34
FrattiniC32C33×C43⋊C3
FittingC43×#6910C3
socleC3×C43C32×H(3,𝔽3)
prime radicalC3×H(3,𝔽3)×C43⋊C3C3
hypercenter#6910C43⋊C3
hypocenterC43C3×#6910
LCSG ▷ C32×C43 ▷ C3×C43 ▷ C43
UCSC1 ◁ C3 ◁ C33 ◁ #6910

Complements

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

NH ≅ G/Ntype
1. C3×3−1+2×C43⋊C3C3semidirect
2. C3×3−1+2×C43⋊C3C3semidirect
4. #23782888C3semidirect
5. #23782887C3semidirect
6. #23782886C3semidirect
7. #23782885C3semidirect
8. #23782884C3semidirect
9. #731×C43⋊C3C3semidirect
10. C43×#6910C3semidirect
13. C3×#11923958C32semidirect
14. C3×#11923958C32semidirect
16. C3×#11923957C32semidirect
18. C3×#11923956C32semidirect
30. 3−1+2×C43⋊C3C32semidirect
31. 3−1+2×C43⋊C3C32semidirect
32. 3−1+2×C43⋊C3C32semidirect
42. C3×C9×C43⋊C3C32semidirect
43. #16812511C32semidirect
44. #16812510C32semidirect
45. #16812509C32semidirect
46. C3×3−1+2×C43C32semidirect
47. C3×3−1+2×C43C32semidirect
49. C43×#731C32semidirect
68. #11923958C33semidirect
85. C3×C43⋊C9C33semidirect
97. 3−1+2×C43C33semidirect
101. C3×C9×C43C33semidirect
140. #6910C43⋊C3direct (N × H)
153. C43⋊C3#6910direct (N × H)
176. C3×3−1+2C3×C43⋊C3semidirect
177. C3×3−1+2C3×C43⋊C3semidirect
192. #731C3×C43⋊C3semidirect
193. C43C3×#6910semidirect
229. 3−1+2C32×C43⋊C3semidirect
245. C3×C9C32×C43⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C43⋊C33
C3×H(3,𝔽3)×C43⋊C33
#237828883
#237828873
#237828863
#237828853
#237828843
#731×C43⋊C33
C43×#69103
C3×#691043