Finite Group Databasebeta

← C2×C3⋊D3×GL(2,𝔽3) #13180757

Subgroups — C2×C3⋊D3×GL(2,𝔽3) #13180757

Order 1728.

1446 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3⋊D3×GL(2,𝔽3)C214yesmaximal, direct product factor
5C2×D3×GL(2,𝔽3)—34nomaximal
6C2×SD16×C3⋊D3—31nomaximal
7C22×D3×C3⋊D3—41nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3⋊D3×SL(2,𝔽3)C2212yescomplemented
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C32×GL(2,𝔽3)C2212yescomplemented
12C2×Q8×C3⋊D3D311yescharacteristic, complemented
13C2×D4×C3⋊D3—31no—
14C2×D3×SL(2,𝔽3)—34no—
15C2×C3⋊GL(2,𝔽3)—34no—
16C2×C3⋊GL(2,𝔽3)—64no—
17C2×C3×GL(2,𝔽3)D314yescomplemented
18D3×GL(2,𝔽3)—316no—
19C2×C32×SD16—31no—
20C2×#3755—31no—
21C2×#3754—31no—
22SD16×C3⋊D3—34no—
23SD16×C3⋊D3—34no—
24C2×#3744—31no—
25C2×C8×C3⋊D3—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C22×C3×C3⋊D3—41no—
28C22×C32×D3—41no—
29C2×D3×C3⋊D3—44no—
30C2×D3×C3⋊D3—48no—
31C32×SL(2,𝔽3)C2311yescharacteristic, complemented
32C22×GL(2,𝔽3)—91no—
33C2×D3×SD16—94no—
34C23×C3⋊D3—61no—
35C22×D32—124no—
36C2×C32×Q8C2×D311yescharacteristic, complemented
37C2×C32×D4—31no—
38C2×#638—61no—
39Q8×C3⋊D3—32no—
40Q8×C3⋊D3C2×D312yescomplemented
41D4×C3⋊D3—34no—
42D4×C3⋊D3—62no—
43C2×C4⋊(C3⋊D3)—31no—
44C2×C4×C3⋊D3—31no—
45C2×#634—31no—
46C2×C3×SL(2,𝔽3)—24no—
47C2×C3×SL(2,𝔽3)C2×D314yescomplemented
48D3×SL(2,𝔽3)—38no—
49C3⋊GL(2,𝔽3)—38no—
50C3⋊GL(2,𝔽3)—68no—
51C3×GL(2,𝔽3)C2×D318yescomplemented
52C32×SD16—32no—
53C32×SD16—32no—
54C2×C32×C8—31no—
55#3755—32no—
56#3755—32no—
57#3754—34no—
58C2×C3⋊(C3⋊C8)—31no—
59#3744—34no—
60C8×C3⋊D3—34no—
61C22×C33—41no—
62C2×C3⋊(C3⋊D3)—42no—
63C2×C3⋊(C3⋊D3)—44no—
64C2×C3×C3⋊D3—44no—
65C2×C3×C3⋊D3—42no—
66C2×C32×D3—42no—
67C2×C32×D3—44no—
68D3×C3⋊D3—416no—
69C2×D3×Q8—34no—
70C2×D3×D4—94no—
71C22×SL(2,𝔽3)—91no—
72C2×GL(2,𝔽3)—91no—
73C2×GL(2,𝔽3)—94no—
74C2×GL(2,𝔽3)—184no—
75C2×GL(2,𝔽3)C3⋊D311yescharacteristic, direct product factor
76C2×C3×SD16—34no—
77C2×#215—94no—
78C2×#214—94no—
79D3×SD16—916no—
80D3×SD16—916no—
81C2×#204—94no—
82C2×D3×C8—94no—
83C23×C32—61no—
84C22×C3⋊D3—61no—
85C22×C3⋊D3—64no—
86C22×C3⋊D3—124no—
87C22×C3⋊D3—124no—
88C22×C3⋊D3—244no—
89C22×C3⋊D3S411yescharacteristic
90C22×C3×D3—44no—
91C22×C3×D3—124no—
92C2×D32—1216no—
93C2×D32—1232no—
94C32×Q8—31no—
95C32×Q8C22×D311yescharacteristic, complemented
96C32×D4—32no—
97C32×D4—61no—
98C2×C32×C4—31no—
99#638—64no—
100C2×C3⋊Dic3—31no—
101C4⋊(C3⋊D3)—32no—
102C4⋊(C3⋊D3)—61no—
103C4×C3⋊D3—32no—
104C4×C3⋊D3—32no—
105#634—32no—
106#634—61no—
107C3×SL(2,𝔽3)—24no—
108C3×SL(2,𝔽3)C22×D314yescomplemented
109C32×C8—32no—
110C3⋊(C3⋊C8)—32no—
111C22×SD16—271noSylow
112C2×C33—42no—
113C2×C33—41no—
114C3⋊(C3⋊D3)—44no—
115C3×C3⋊D3—44no—
116C32×D3—44no—
117C23×D3—184no—
118C23×D3—361no—
119C2×C3×Q8D3214yescomplemented
120C2×C3×D4—34no—
121C2×#67—184no—
122D3×Q8—38no—
123D3×Q8—98no—
124D3×D4—916no—
125D3×D4—188no—
126C2×D12—94no—
127C2×C4×D3—94no—
128C2×Dic6—94no—
129C2×SL(2,𝔽3)—24no—
130C2×SL(2,𝔽3)—92no—
131C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
132GL(2,𝔽3)—92no—
133GL(2,𝔽3)—188no—
134GL(2,𝔽3)C2×C3⋊D312yesdirect product factor
135C3×SD16—38no—
136C3×SD16—38no—
137C2×C3×C8—34no—
138#215—98no—
139#215—98no—
140#214—916no—
141C2×C3⋊C8—94no—
142#204—916no—
143D3×C8—916no—
144C22×C32—44no—
145C22×C32—62no—
146C22×C32—84no—
147C22×C32—122no—
148C22×C32C2×S411yescharacteristic
149C2×C3⋊D3—62no—
150C2×C3⋊D3—128no—
151C2×C3⋊D3—122no—
152C2×C3⋊D3—128no—
153C2×C3⋊D3—1216no—
154C2×C3⋊D3—248no—
155C2×C3⋊D3—2416no—
156C2×C3⋊D3GL(2,𝔽3)14yesdirect product factor
157C2×C3⋊D3C2×S412yes—
158C2×C3×D3—48no—
159C2×C3×D3—416no—
160C2×C3×D3—1216no—
161C2×C3×D3—128no—
162D32—1264no—
163C32×C4—31no—
164C32×C4—31no—
165C3⋊Dic3—32no—
166C22×Q8—91no—
167C22×D4—271no—
168C2×SD16—31no—
169C2×SD16—271no—
170C2×SD16—274no—
171C2×SD16—274no—
172C2×SD16—271no—
173C2×SD16—271no—
174C22×C8—271no—
175C33—41noSylow
176C23×C3—64no—
177C23×C3—361no—
178C22×D3—34no—
179C22×D3—41no—
180C22×D3—1816no—
181C22×D3—184no—
182C22×D3—364no—
183C22×D3—361no—
184C22×D3—3616no—
185C22×D3—368no—
186C22×D3—724no—
187C3×Q8—34no—
188C3×Q8C2×D3214yescomplemented
189C3×D4—38no—
190C3×D4—64no—
191C2×C3×C4—34no—
192#67—1816no—
193C2×Dic3—94no—
194D12—98no—
195D12—184no—
196C4×D3—98no—
197C4×D3—98no—
198Dic6—98no—
199Dic6—184no—
200SL(2,𝔽3)—24no—
201SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
202C3×C8—38no—
203C3⋊C8—98no—
204C2×C32—44no—
205C2×C32—48no—
206C2×C32—84no—
207C2×C32—88no—
208C2×C32—122no—
209C2×C32C2×GL(2,𝔽3)12yescomplemented
210C2×C32C22×S411yescharacteristic
211C3⋊D3—122no—
212C3⋊D3—1216no—
213C3⋊D3—2416no—
214C3⋊D3C2×GL(2,𝔽3)14yesdirect product factor
215C3×D3—416no—
216C3×D3—1216no—
217C24—541no—
218C2×Q8—92no—
219C2×Q8—271no—
220C2×Q8—272no—
221C2×Q8D3×C3⋊D311yescharacteristic, complemented
222C2×D4—31no—
223C2×D4—271no—
224C2×D4—274no—
225C2×D4—542no—
226C2×D4—541no—
227C22×C4—271no—
228SD16—32no—
229SD16—32no—
230SD16—274no—
231SD16—274no—
232SD16—272no—
233SD16—272no—
234C2×C8—31no—
235C2×C8—274no—
236C2×C8—271no—
237C22×C3—41no—
238C22×C3—68no—
239C22×C3—84no—
240C22×C3—128no—
241C22×C3—364no—
242C22×C3—362no—
243C22×C3D3×S414yes—
244C2×D3—38no—
245C2×D3—316no—
246C2×D3—42no—
247C2×D3—44no—
248C2×D3—188no—
249C2×D3—362no—
250C2×D3—3632no—
251C2×D3—3616no—
252C2×D3—364no—
253C2×D3—368no—
254C2×D3—728no—
255C2×D3—7216no—
256C3×C4—34no—
257C3×C4—34no—
258Dic3—98no—
259C32—44no—
260C32—84no—
261C32C22×GL(2,𝔽3)11yescharacteristic, complemented
262C23—61no—
263C23—91no—
264C23—544no—
265C23—541no—
266C23—1084no—
267Q8—31no—
268Q8—272no—
269Q8—541no—
270Q8C2×D3×C3⋊D311yescharacteristic, complemented
271D4—32no—
272D4—61no—
273D4—272no—
274D4—541no—
275D4—544no—
276C2×C4—31no—
277C2×C4—272no—
278C2×C4—272no—
279C2×C4—271no—
280C8—32no—
281C8—272no—
282C2×C3—42no—
283C2×C3—41no—
284C2×C3—88no—
285C2×C3—84no—
286C2×C3—128no—
287C2×C3—364no—
288C2×C3D3×GL(2,𝔽3)18yescomplemented
289C2×C3C2×D3×S414yes—
290D3—316no—
291D3—44no—
292D3—364no—
293D3—368no—
294D3—7216no—
295C22—62no—
296C22—94no—
297C22—92no—
298C22—122no—
299C22—542no—
300C22—1088no—
301C22—1082no—
302C22C3⋊D3×S411yescharacteristic
303C4—31no—
304C4—31no—
305C4—272no—
306C3—41no—
307C3—84no—
308C3C2×D3×GL(2,𝔽3)14yescomplemented
309C2—94no—
310C2—122no—
311C2—1082no—
312C2C3⋊D3×GL(2,𝔽3)12yesdirect product factor
313C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×C3⋊D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C3⋊D3×GL(2,𝔽3)C2direct (N × H)
8. C2×C32×SL(2,𝔽3)C22semidirect
9. C3⋊D3×SL(2,𝔽3)C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C32×GL(2,𝔽3)C22semidirect
12. C2×Q8×C3⋊D3D3semidirect
17. C2×C3×GL(2,𝔽3)D3semidirect
31. C32×SL(2,𝔽3)C23semidirect
36. C2×C32×Q8C2×D3semidirect
40. Q8×C3⋊D3C2×D3semidirect
47. C2×C3×SL(2,𝔽3)C2×D3semidirect
51. C3×GL(2,𝔽3)C2×D3semidirect
75. C2×GL(2,𝔽3)C3⋊D3direct (N × H)
95. C32×Q8C22×D3semidirect
108. C3×SL(2,𝔽3)C22×D3semidirect
119. C2×C3×Q8D32semidirect
131. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
134. GL(2,𝔽3)C2×C3⋊D3direct (N × H)
156. C2×C3⋊D3GL(2,𝔽3)direct (N × H)
188. C3×Q8C2×D32semidirect
201. SL(2,𝔽3)C22×C3⋊D3semidirect
209. C2×C32C2×GL(2,𝔽3)semidirect
214. C3⋊D3C2×GL(2,𝔽3)direct (N × H)
221. C2×Q8D3×C3⋊D3semidirect
261. C32C22×GL(2,𝔽3)semidirect
270. Q8C2×D3×C3⋊D3semidirect
288. C2×C3D3×GL(2,𝔽3)semidirect
308. C3C2×D3×GL(2,𝔽3)semidirect
312. C2C3⋊D3×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
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C32×GL(2,𝔽3)2
C3⋊D3×GL(2,𝔽3)2
C2×D3×GL(2,𝔽3)3
C2×SD16×C3⋊D33
C22×D3×C3⋊D34