Finite Group Databasebeta

← C2×C3×#66330 #14517689

Subgroups — C2×C3×#66330 #14517689

Order 1944.

634 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 96 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×AΣL(1,𝔽27)C211yesmaximal, characteristic
2C3×#66330C212yesmaximal, direct product factor
3C2×C32×C3⋊Dic3—31nomaximal
4C2×C3×3−1+2⋊C4—32nomaximal
5C2×C3×(C3⋊Dic3)⋊C3—31nomaximal
6C2×#66330C311yesmaximal, characteristic, direct product factor
7C2×C3×AΣL(1,𝔽27)C412yescomplemented
8C2×C3×AΣL(1,𝔽27)C2211yescharacteristic
9C22×C34D311yescharacteristic
10C32×C3⋊Dic3—32no—
11C22×C3×3−1+2D312yes—
12C22×C3×H(3,𝔽3)D311yescharacteristic
13C3×3−1+2⋊C4—34no—
14C3×(C3⋊Dic3)⋊C3—32no—
15C22×AΣL(1,𝔽27)—21no—
16C22×AΣL(1,𝔽27)—22no—
17C22×AΣL(1,𝔽27)—21no—
18C22×AΣL(1,𝔽27)C2×C311yescharacteristic
19#66330C2×C312yesdirect product factor
20C3×AΣL(1,𝔽27)C2×C411yescharacteristic, complemented, Sylow
21C2×C3×C3⋊Dic3—31no—
22C2×C3×C3⋊Dic3—32no—
23C2×C3×C3⋊Dic3—31no—
24C2×C3×C3⋊Dic3—31no—
25C2×C32×Dic3—93no—
26C2×C32×Dic3—91no—
27C2×3−1+2⋊C4—34no—
28C2×3−1+2⋊C4—32no—
29C2×(C3⋊Dic3)⋊C3—32no—
30C2×(C3⋊Dic3)⋊C3—31no—
31C2×C3×Dic9—32no—
32C2×C34Dic312yescomplemented
33C2×C34C2×D311yescharacteristic
34C2×C3×3−1+2Dic314yescomplemented
35C2×C3×3−1+2C2×D312yes—
36C2×C3×H(3,𝔽3)Dic312yescomplemented
37C2×C3×H(3,𝔽3)C2×D311yescharacteristic
38C2×AΣL(1,𝔽27)—22no—
39C2×AΣL(1,𝔽27)—21no—
40C2×AΣL(1,𝔽27)—24no—
41C2×AΣL(1,𝔽27)—22no—
42C2×AΣL(1,𝔽27)—22no—
43C2×AΣL(1,𝔽27)—21no—
44C2×AΣL(1,𝔽27)C3×C412yescomplemented
45C2×AΣL(1,𝔽27)C22×C311yescharacteristic
46C22×C33—21no—
47C22×C33—33no—
48C22×C33—32no—
49C22×C33—31no—
50C22×C33—31no—
51C22×C33—63no—
52C22×C33C3×D311yescharacteristic
53C22×C33C3⋊D311yescharacteristic
54C3×C3⋊Dic3—32no—
55C3×C3⋊Dic3—34no—
56C3×C3⋊Dic3—32no—
57C3×C3⋊Dic3—32no—
58C32×Dic3—96no—
59C32×Dic3—92no—
60C22×3−1+2—22no—
61C22×3−1+2—34no—
62C22×3−1+2C3×D312yes—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—32no—
65C22×H(3,𝔽3)C3×D311yescharacteristic
66C22×C3×C9—32no—
673−1+2⋊C4—38no—
683−1+2⋊C4—34no—
69(C3⋊Dic3)⋊C3—34no—
70(C3⋊Dic3)⋊C3—32no—
71C3×Dic9—34no—
72C34C2×Dic311yescharacteristic, complemented
73C3×3−1+2C2×Dic312yescomplemented
74C3×H(3,𝔽3)C2×Dic311yescharacteristic, complemented
75AΣL(1,𝔽27)—21no—
76AΣL(1,𝔽27)—22no—
77AΣL(1,𝔽27)—21no—
78AΣL(1,𝔽27)C2×C3×C411yescharacteristic, complemented
79C2×C32×C4—271no—
80C2×C3⋊Dic3—31no—
81C2×C3⋊Dic3—31no—
82C2×C3×Dic3—93no—
83C2×C3×Dic3—96no—
84C2×C3×Dic3—93no—
85C2×C3×Dic3—91no—
86C2×C3×Dic3—92no—
87C2×C3×Dic3—91no—
88C2×C3×Dic3—271no—
89C2×Dic9—32no—
90C2×C33—22no—
91C2×C33—21no—
92C2×C33—33no—
93C2×C33—36no—
94C2×C33—34no—
95C2×C33—32no—
96C2×C33—32no—
97C2×C33—31no—
98C2×C33—32no—
99C2×C33—31no—
100C2×C33—66no—
101C2×C33—63no—
102C2×C33C3×Dic312yescomplemented
103C2×C33C3⋊Dic312yes—
104C2×C33C2×C3×D311yescharacteristic
105C2×C33C2×C3⋊D311yescharacteristic
106C2×3−1+2—24no—
107C2×3−1+2—22no—
108C2×3−1+2—38no—
109C2×3−1+2—34no—
110C2×3−1+2C3×Dic314yescomplemented
111C2×3−1+2C2×C3×D312yes—
112C2×H(3,𝔽3)—22no—
113C2×H(3,𝔽3)—21no—
114C2×H(3,𝔽3)—34no—
115C2×H(3,𝔽3)—32no—
116C2×H(3,𝔽3)C3×Dic312yescomplemented
117C2×H(3,𝔽3)C2×C3×D311yescharacteristic
118C2×C3×C9—34no—
119C2×C3×C9—32no—
120C22×C32—31no—
121C22×C32—31no—
122C22×C32—33no—
123C22×C32—36no—
124C22×C32—33no—
125C22×C32—31no—
126C22×C32—61no—
127C22×C32—63no—
128C22×C32—66no—
129C22×C32—63no—
130C22×C32—61no—
131C22×C32—61no—
132C22×C32—91no—
133C22×C32—181no—
134C22×C32#27211yescharacteristic
135C22×C32#27212yes—
136C22×C32C3×C3⋊D311yescharacteristic
137C32×C4—272no—
138C3⋊Dic3—32no—
139C3⋊Dic3—32no—
140C3×Dic3—96no—
141C3×Dic3—912no—
142C3×Dic3—96no—
143C3×Dic3—92no—
144C3×Dic3—94no—
145C3×Dic3—92no—
146C3×Dic3—272no—
147C22×C9—32no—
148C22×C9—62no—
149Dic9—34no—
150C33—21no—
151C33—33no—
152C33—32no—
153C33—31no—
154C33—31no—
155C33—63no—
156C33C2×C3×Dic311yescharacteristic, complemented
157C33C2×C3⋊Dic311yescharacteristic
1583−1+2—22no—
1593−1+2—34no—
1603−1+2C2×C3×Dic312yescomplemented
161H(3,𝔽3)—21no—
162H(3,𝔽3)—32no—
163H(3,𝔽3)C2×C3×Dic311yescharacteristic, complemented
164C3×C9—32no—
165C2×C3×C4—272no—
166C2×C3×C4—271no—
167C2×C3×C4—271no—
168C2×Dic3—93no—
169C2×Dic3—91no—
170C2×Dic3—271no—
171C2×C32—32no—
172C2×C32—31no—
173C2×C32—32no—
174C2×C32—31no—
175C2×C32—36no—
176C2×C32—33no—
177C2×C32—36no—
178C2×C32—312no—
179C2×C32—36no—
180C2×C32—33no—
181C2×C32—32no—
182C2×C32—31no—
183C2×C32—62no—
184C2×C32—61no—
185C2×C32—66no—
186C2×C32—63no—
187C2×C32—612no—
188C2×C32—66no—
189C2×C32—66no—
190C2×C32—63no—
191C2×C32—62no—
192C2×C32—61no—
193C2×C32—62no—
194C2×C32—61no—
195C2×C32—92no—
196C2×C32—91no—
197C2×C32—182no—
198C2×C32—181no—
199C2×C32#108412yes—
200C2×C32#108414yes—
201C2×C32C2×#27211yescharacteristic
202C2×C32C2×#27212yes—
203C2×C32C3×C3⋊Dic312yes—
204C2×C32C2×C3×C3⋊D311yescharacteristic
205C2×C9—34no—
206C2×C9—32no—
207C2×C9—64no—
208C2×C9—62no—
209C22×C3—21no—
210C22×C3—33no—
211C22×C3—32no—
212C22×C3—31no—
213C22×C3—63no—
214C22×C3—91no—
215C22×C3—181no—
216C22×C3#418311yescharacteristic
217C22×C3C3×#27211yescharacteristic
218C3×C4—274no—
219C3×C4—272no—
220C3×C4—272no—
221Dic3—96no—
222Dic3—92no—
223Dic3—272no—
224C32—33no—
225C32—31no—
226C32—31no—
227C32—36no—
228C32—33no—
229C32—31no—
230C32—61no—
231C32—63no—
232C32—66no—
233C32—63no—
234C32—61no—
235C32—61no—
236C32—91no—
237C32—181no—
238C32C2×#108411yescharacteristic
239C32C2×#108412yes—
240C32C2×C3×C3⋊Dic311yescharacteristic
241C9—32no—
242C9—62no—
243C2×C4—271noSylow
244C2×C3—22no—
245C2×C3—21no—
246C2×C3—36no—
247C2×C3—34no—
248C2×C3—32no—
249C2×C3—33no—
250C2×C3—32no—
251C2×C3—31no—
252C2×C3—66no—
253C2×C3—63no—
254C2×C3—92no—
255C2×C3—91no—
256C2×C3—182no—
257C2×C3—181no—
258C2×C3#6633012yesdirect product factor
259C2×C3C2×#418311yescharacteristic
260C2×C3C3×#108412yes—
261C2×C3C2×C3×#27211yescharacteristic
262C22C3×#418311yescharacteristic
263C4—272no—
264C3—21no—
265C3—33no—
266C3—32no—
267C3—31no—
268C3—63no—
269C3—91no—
270C3—181no—
271C3C2×#6633011yescharacteristic, direct product factor
272C3C2×C3×#108411yescharacteristic
273C2C3×#6633012yesdirect product factor
274C2C2×C3×#418311yescharacteristic
subgroupG/H
center, hypercenterC22×C3#4183
commutator, hypocenterAΣL(1,𝔽27)C2×C3×C4
FrattiniC2×C32C2×C3×C3⋊D3
Fitting, prime radicalC22×C3×AΣL(1,𝔽27)C2
socleC22×C32#272
cosocleC2×AΣL(1,𝔽27)C22×C3
LCSG ▷ AΣL(1,𝔽27)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#66330C2direct (N × H)
6. C2×#66330C3direct (N × H)
7. C2×C3×AΣL(1,𝔽27)C4semidirect
19. #66330C2×C3direct (N × H)
20. C3×AΣL(1,𝔽27)C2×C4semidirect
32. C2×C34Dic3semidirect
34. C2×C3×3−1+2Dic3semidirect
36. C2×C3×H(3,𝔽3)Dic3semidirect
44. C2×AΣL(1,𝔽27)C3×C4semidirect
72. C34C2×Dic3semidirect
73. C3×3−1+2C2×Dic3semidirect
74. C3×H(3,𝔽3)C2×Dic3semidirect
78. AΣL(1,𝔽27)C2×C3×C4semidirect
102. C2×C33C3×Dic3semidirect
110. C2×3−1+2C3×Dic3semidirect
116. C2×H(3,𝔽3)C3×Dic3semidirect
156. C33C2×C3×Dic3semidirect
160. 3−1+2C2×C3×Dic3semidirect
163. H(3,𝔽3)C2×C3×Dic3semidirect
258. C2×C3#66330direct (N × H)
271. C3C2×#66330direct (N × H)
273. C2C3×#66330direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AΣL(1,𝔽27)2
C3×#663302
C2×C32×C3⋊Dic33
C2×C3×3−1+2⋊C43
C2×C3×(C3⋊Dic3)⋊C33
C2×#663303