Finite Group Databasebeta

← C22×C3×#983 #11923691

Subgroups — C22×C3×#983 #11923691

Order 1152.

1689 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1689 conjugacy classes of subgroups — too many to draw. Only its 220 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×#983C2112yesmaximal, direct product factor
3C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C3×#226C211yesmaximal, characteristic, complemented
5C22×#983C311yesmaximal, characteristic, direct product factor
6C22×C3×#137—31nomaximal
7C22×C3×#67—41nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2216yescomplemented
10C3×#983C22116yesdirect product factor
11C2×C3×GL(2,𝔽3)C2216yescomplemented
12C2×C3×#226C2216yescomplemented
13C22×C3×#41—31no—
14C23×C3×Q8D311yescharacteristic, complemented
15C23×SL(2,𝔽3)—21no—
16C23×SL(2,𝔽3)C2×C311yescharacteristic, complemented
17C2×#983C2×C3112yesdirect product factor
18C22×GL(2,𝔽3)C2×C311yescharacteristic, complemented
19C22×#226C2×C311yescharacteristic, complemented
20C2×C3×#137—312no—
21C2×C3×#137—312no—
22C22×C3×Q16—31no—
23C22×C3×Q16—31no—
24C22×C3×SD16—31no—
25C22×C3×SD16—31no—
26C22×C3×M16—31no—
27C24×C32—41no—
28C23×C3×D3—41no—
29C2×C3×#67—412no—
30C22×C3×Dic3—41no—
31C2×C3×SL(2,𝔽3)C2313yes—
32C2×C3×SL(2,𝔽3)C2314yescomplemented
33C3×GL(2,𝔽3)C2314yescomplemented
34C3×#226C2314yescomplemented
35C22×#137—31noSylow
36C2×C3×#41—312no—
37C2×C3×#41—66no—
38C22×C3×Q8—31no—
39C22×C3×Q8—36no—
40C22×C3×Q8—31no—
41C22×C3×Q8C2×D311yescharacteristic
42C22×C3×Q8C2×D316yescomplemented
43C22×C3×D4—31no—
44C22×C3×D4—61no—
45C23×C3×C4—31no—
46C23×C3×C4—61no—
47C22×#67—41no—
48C22×SL(2,𝔽3)—21no—
49C22×SL(2,𝔽3)—26no—
50C22×SL(2,𝔽3)C22×C311yescharacteristic
51C22×SL(2,𝔽3)C22×C316yescomplemented
52#983C22×C3116yesdirect product factor
53C2×GL(2,𝔽3)C22×C316yescomplemented
54C2×#226C22×C316yescomplemented
55C3×#137—348no—
56C3×#137—316no—
57C2×C3×Q16—36no—
58C2×C3×Q16—312no—
59C2×C3×Q16—36no—
60C2×C3×SD16—36no—
61C2×C3×SD16—312no—
62C2×C3×SD16—36no—
63C2×C3×M16—312no—
64C22×C3×C8—31no—
65C22×C3×C8—31no—
66C23×C32—46no—
67C23×C32—41no—
68C23×C32—84no—
69C22×C3×D3—46no—
70C22×C3×D3—84no—
71C3×#67—416no—
72C2×C3×Dic3—46no—
73C3×SL(2,𝔽3)C2411yescharacteristic
74C22×#41—31no—
75C23×Q8C3×D311yescharacteristic, complemented
76C2×#137—312no—
77C2×#137—312no—
78C22×Q16—31no—
79C22×Q16—31no—
80C22×SD16—31no—
81C22×SD16—31no—
82C22×M16—31no—
83C24×C3—41no—
84C24×C3—61no—
85C24×C3—81no—
86C24×C3S411yescharacteristic
87C23×D3—41no—
88C3×#41—316no—
89C3×#41—624no—
90C2×C3×Q8—33no—
91C2×C3×Q8—36no—
92C2×C3×Q8—312no—
93C2×C3×Q8—36no—
94C2×C3×Q8—63no—
95C2×C3×Q8—64no—
96C2×C3×Q8—63no—
97C2×C3×Q8C22×D313yes—
98C2×C3×Q8C22×D314yescomplemented
99C2×C3×D4—36no—
100C2×C3×D4—63no—
101C2×C3×D4—612no—
102C22×C3×C4—36no—
103C22×C3×C4—36no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—612no—
107C22×C3×C4—61no—
108C2×#67—412no—
109C22×Dic3—41no—
110C2×SL(2,𝔽3)—23no—
111C2×SL(2,𝔽3)—24no—
112C2×SL(2,𝔽3)C23×C313yes—
113C2×SL(2,𝔽3)C23×C314yescomplemented
114GL(2,𝔽3)C23×C314yescomplemented
115#226C23×C314yescomplemented
116C3×Q16—312no—
117C3×Q16—316no—
118C3×Q16—34no—
119C3×SD16—312no—
120C3×SD16—316no—
121C3×SD16—34no—
122C3×M16—316no—
123C2×C3×C8—36no—
124C2×C3×C8—36no—
125C22×C32—44no—
126C22×C32—43no—
127C22×C32—44no—
128C22×C32—812no—
129C2×C3×D3—44no—
130C2×C3×D3—812no—
131C3×Dic3—44no—
132C2×#41—312no—
133C2×#41—66no—
134C22×Q8—31no—
135C22×Q8—36no—
136C22×Q8—31no—
137C22×Q8C2×C3×D311yescharacteristic
138C22×Q8C2×C3×D316yescomplemented
139C22×D4—31no—
140C22×D4—61no—
141C23×C4—31no—
142C23×C4—61no—
143#137—348no—
144#137—316no—
145C2×Q16—36no—
146C2×Q16—312no—
147C2×Q16—36no—
148C2×SD16—36no—
149C2×SD16—312no—
150C2×SD16—36no—
151C2×M16—312no—
152C22×C8—31no—
153C22×C8—31no—
154C23×C3—24no—
155C23×C3—41no—
156C23×C3—46no—
157C23×C3—66no—
158C23×C3—84no—
159C23×C3—88no—
160C23×C3—86no—
161C23×C3—81no—
162C23×C3—124no—
163C23×C3C2×S416yes—
164C23×C3C2×S411yescharacteristic
165C22×D3—46no—
166C22×D3—84no—
167C3×Q8—33no—
168C3×Q8—34no—
169C3×Q8—34no—
170C3×Q8—66no—
171C3×Q8—61no—
172C3×Q8—66no—
173C3×Q8C23×D311yescharacteristic
174C3×D4—34no—
175C3×D4—66no—
176C3×D4—616no—
177C2×C3×C4—33no—
178C2×C3×C4—34no—
179C2×C3×C4—36no—
180C2×C3×C4—33no—
181C2×C3×C4—312no—
182C2×C3×C4—616no—
183C2×C3×C4—66no—
184#67—416no—
185C2×Dic3—46no—
186SL(2,𝔽3)—21no—
187SL(2,𝔽3)C24×C311yescharacteristic
188C3×C8—34no—
189C3×C8—34no—
190C2×C32—46no—
191C2×C32—41no—
192C2×C32—84no—
193C3×D3—84no—
194C24—61no—
195C24C3×S411yescharacteristic
196#41—316no—
197#41—624no—
198C2×Q8—33no—
199C2×Q8—36no—
200C2×Q8—312no—
201C2×Q8—36no—
202C2×Q8—63no—
203C2×Q8—64no—
204C2×Q8—63no—
205C2×Q8C22×C3×D313yes—
206C2×Q8C22×C3×D314yescomplemented
207C2×D4—36no—
208C2×D4—63no—
209C2×D4—612no—
210C22×C4—31no—
211C22×C4—36no—
212C22×C4—31no—
213C22×C4—36no—
214C22×C4—612no—
215C22×C4—61no—
216Q16—312no—
217Q16—316no—
218Q16—34no—
219SD16—312no—
220SD16—316no—
221SD16—34no—
222M16—316no—
223C2×C8—36no—
224C2×C8—36no—
225C22×C3—212no—
226C22×C3—43no—
227C22×C3—44no—
228C22×C3—44no—
229C22×C3—64no—
230C22×C3—812no—
231C22×C3—824no—
232C22×C3—84no—
233C22×C3—83no—
234C22×C3—84no—
235C22×C3—1212no—
236C22×C3#98314yesdirect product factor
237C22×C3C22×S414yes—
238C22×C3C22×S413yes—
239C2×D3—44no—
240C2×D3—812no—
241C3×C4—31no—
242C3×C4—34no—
243C3×C4—33no—
244C3×C4—64no—
245Dic3—44no—
246C32—41noSylow
247C23—24no—
248C23—66no—
249C23—124no—
250C23C2×C3×S416yes—
251C23C2×C3×S411yescharacteristic
252Q8—33no—
253Q8—34no—
254Q8—34no—
255Q8—66no—
256Q8—61no—
257Q8—66no—
258Q8C23×C3×D311yescharacteristic
259D4—34no—
260D4—66no—
261D4—616no—
262C2×C4—33no—
263C2×C4—34no—
264C2×C4—36no—
265C2×C4—33no—
266C2×C4—312no—
267C2×C4—616no—
268C2×C4—66no—
269C8—34no—
270C8—34no—
271C2×C3—24no—
272C2×C3—41no—
273C2×C3—46no—
274C2×C3—84no—
275C2×C3—88no—
276C2×C3—81no—
277C2×C3—86no—
278C2×C3—124no—
279C2×C3C2×#98316yesdirect product factor
280C2×C3C23×S411yescharacteristic
281D3—84no—
282C22—212no—
283C22—64no—
284C22—1212no—
285C22C3×#98314yesdirect product factor
286C22C22×C3×S414yes—
287C22C22×C3×S413yes—
288C4—31no—
289C4—34no—
290C4—33no—
291C4—64no—
292C3—41no—
293C3—81no—
294C3C22×#98311yescharacteristic, direct product factor
295C2—24no—
296C2—124no—
297C2C2×C3×#98316yesdirect product factor
298C2C23×C3×S411yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC23×C3×Q8D3
hypercenterC24×C3S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C2×C3×#983C2direct (N × H)
3. C22×C3×GL(2,𝔽3)C2semidirect
4. C22×C3×#226C2semidirect
5. C22×#983C3direct (N × H)
9. C22×C3×SL(2,𝔽3)C22semidirect
10. C3×#983C22direct (N × H)
11. C2×C3×GL(2,𝔽3)C22semidirect
12. C2×C3×#226C22semidirect
14. C23×C3×Q8D3semidirect
16. C23×SL(2,𝔽3)C2×C3semidirect
17. C2×#983C2×C3direct (N × H)
18. C22×GL(2,𝔽3)C2×C3semidirect
19. C22×#226C2×C3semidirect
32. C2×C3×SL(2,𝔽3)C23semidirect
33. C3×GL(2,𝔽3)C23semidirect
34. C3×#226C23semidirect
42. C22×C3×Q8C2×D3semidirect
51. C22×SL(2,𝔽3)C22×C3semidirect
52. #983C22×C3direct (N × H)
53. C2×GL(2,𝔽3)C22×C3semidirect
54. C2×#226C22×C3semidirect
75. C23×Q8C3×D3semidirect
98. C2×C3×Q8C22×D3semidirect
113. C2×SL(2,𝔽3)C23×C3semidirect
114. GL(2,𝔽3)C23×C3semidirect
115. #226C23×C3semidirect
138. C22×Q8C2×C3×D3semidirect
206. C2×Q8C22×C3×D3semidirect
236. C22×C3#983direct (N × H)
279. C2×C3C2×#983direct (N × H)
285. C22C3×#983direct (N × H)
294. C3C22×#983direct (N × H)
297. C2C2×C3×#983direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3×#9832
C22×C3×GL(2,𝔽3)2
C22×C3×#2262
C22×#9833
C22×C3×#1373
C22×C3×#674