Finite Group Databasebeta

← C2×C3×#5924 #11923702

Subgroups — C2×C3×#5924 #11923702

Order 1152.

1509 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1509 conjugacy classes of subgroups — too many to draw. Only its 204 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×#994C211yesmaximal, characteristic, complemented
2C3×#5924C218yesmaximal, direct product factor
3C2×C3×#985C213yesmaximal, complemented
4C2×C3×#984C213yesmaximal, complemented
5C2×#5924C311yesmaximal, characteristic, direct product factor
6C2×C3×#577—31nomaximal
7C2×C3×D3×Q8—41nomaximal
8C3×#994C2214yescomplemented
9C2×C3×#41⋊C3C2213yes—
10C3×#985C22112yescomplemented
11C3×#984C22112yescomplemented
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2213yescomplemented
14C2×C3×2−1+4—31no—
15C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
16C2×#994—21no—
17C2×#994C2×C311yescharacteristic, complemented
18#5924C2×C318yesdirect product factor
19C2×#985C2×C313yescomplemented
20C2×#984C2×C313yescomplemented
21C3×#577—38no—
22C3×#577—38no—
23C2×C3×#137—33no—
24C2×C3×AGL(1,ℤ/8)—33no—
25C2×C3×#135—33no—
26C22×C3×SD16—33no—
27C2×C3×#131—31no—
28C2×C32×Q8—41no—
29C3×D3×Q8—48no—
30C2×C3×C4×D3—43no—
31C2×C3×Dic6—43no—
32C3×#41⋊C3C2316yes—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×GL(2,𝔽3)C2312yescomplemented
35C3×#226C2316yescomplemented
36C2×#577—31noSylow
37C3×2−1+4—38no—
38C3×2−1+4—64no—
39C3×H(5,𝔽2)—34no—
40C3×H(5,𝔽2)C2×D314yescomplemented
41C2×C3×#41—31no—
42C2×C3×#41—33no—
43C2×C3×#41—63no—
44C2×C3×#41C2×D313yes—
45C22×C3×Q8—33no—
46C22×C3×Q8—61no—
47C22×C3×D4—33no—
48C2×D3×Q8—41no—
49#994—24no—
50#994C22×C314yescomplemented
51C2×#41⋊C3—23no—
52C2×#41⋊C3C22×C313yes—
53#985C22×C3112yescomplemented
54#984C22×C3112yescomplemented
55C2×GL(2,𝔽3)C22×C311yescharacteristic, complemented
56C2×#226C22×C313yescomplemented
57C3×#137—312no—
58C3×#137—312no—
59C3×AGL(1,ℤ/8)—324no—
60C3×#135—312no—
61C3×#135—312no—
62C2×C3×Q16—33no—
63C2×C3×SD16—33no—
64C2×C3×SD16—36no—
65C2×C3×SD16—324no—
66C2×C3×SD16—31no—
67C2×C3×D8—33no—
68C3×#131—38no—
69C2×C3×M16—33no—
70C22×C3×C8—33no—
71C22×C3×D3—41no—
72C32×Q8—44no—
73C2×C32×C4—43no—
74C2×C3×Dic3—43no—
75C3×C4×D3—412no—
76C3×Dic6—412no—
77C3×SL(2,𝔽3)C2411yescharacteristic
78C2×2−1+4—31no—
79C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
80#577—38no—
81#577—38no—
82C2×#137—33no—
83C2×AGL(1,ℤ/8)—33no—
84C2×#135—33no—
85C22×SD16—33no—
86C2×#131—31no—
87C24×C3—61no—
88C3×#41—34no—
89C3×#41—34no—
90C3×#41—36no—
91C3×#41—312no—
92C3×#41—66no—
93C3×#41—624no—
94C3×#41C22×D316yes—
95C2×C3×Q8—33no—
96C2×C3×Q8—312no—
97C2×C3×Q8—41no—
98C2×C3×Q8—66no—
99C2×C3×Q8—68no—
100C2×C3×Q8—63no—
101C2×C3×Q8—81no—
102C2×C3×Q8S411yescharacteristic
103C2×C3×Q8C22×D311yescharacteristic
104C2×C3×D4—312no—
105C2×C3×D4—33no—
106C2×C3×D4—31no—
107C2×C3×D4—33no—
108C2×C3×D4—66no—
109C2×C3×D4—63no—
110C22×C3×C4—33no—
111C22×C3×C4—63no—
112C22×C3×C4—63no—
113D3×Q8—48no—
114C2×C4×D3—43no—
115C2×Dic6—43no—
116#41⋊C3—26no—
117#41⋊C3C23×C316yes—
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C23×C311yescharacteristic
120GL(2,𝔽3)C23×C312yescomplemented
121#226C23×C316yescomplemented
122C3×Q16—36no—
123C3×Q16—36no—
124C3×SD16—312no—
125C3×SD16—324no—
126C3×SD16—32no—
127C3×SD16—32no—
128C3×D8—312no—
129C3×M16—312no—
130C2×C3×C8—31no—
131C2×C3×C8—33no—
132C2×C3×C8—312no—
133C22×C32—41no—
134C2×C3×D3—42no—
135C2×C3×D3—44no—
136C32×C4—46no—
137C3×Dic3—46no—
1382−1+4—38no—
1392−1+4—64no—
140H(5,𝔽2)—34no—
141H(5,𝔽2)C2×C3×D314yescomplemented
142C2×#41—31no—
143C2×#41—33no—
144C2×#41—63no—
145C2×#41C2×C3×D313yes—
146C22×Q8—33no—
147C22×Q8—61no—
148C22×D4—33no—
149#137—312no—
150#137—312no—
151AGL(1,ℤ/8)—324no—
152#135—312no—
153#135—312no—
154C2×Q16—33no—
155C2×SD16—33no—
156C2×SD16—36no—
157C2×SD16—324no—
158C2×SD16—31no—
159C2×D8—33no—
160#131—38no—
161C2×M16—33no—
162C22×C8—33no—
163C23×C3—33no—
164C23×C3—64no—
165C23×C3—61no—
166C23×C3—242no—
167C22×D3—41no—
168C3×Q8—31no—
169C3×Q8—36no—
170C3×Q8—44no—
171C3×Q8—63no—
172C3×Q8—612no—
173C3×Q8—84no—
174C3×Q8C2×S414yes—
175C3×Q8C23×D311yescharacteristic
176C3×D4—36no—
177C3×D4—32no—
178C3×D4—312no—
179C3×D4—63no—
180C3×D4—61no—
181C3×D4—612no—
182C2×C3×C4—31no—
183C2×C3×C4—36no—
184C2×C3×C4—36no—
185C2×C3×C4—43no—
186C2×C3×C4—612no—
187C2×C3×C4—63no—
188C2×C3×C4—612no—
189C2×C3×C4—83no—
190C2×C3×C4C2×S413yes—
191C2×Dic3—43no—
192C4×D3—412no—
193Dic6—412no—
194SL(2,𝔽3)—21no—
195SL(2,𝔽3)C24×C311yescharacteristic
196C3×C8—32no—
197C3×C8—36no—
198C2×C32—42no—
199C2×C32—41no—
200C3×D3—44no—
201C24—61no—
202#41—34no—
203#41—34no—
204#41—36no—
205#41—312no—
206#41—66no—
207#41—624no—
208#41C22×C3×D316yes—
209C2×Q8—33no—
210C2×Q8—312no—
211C2×Q8—66no—
212C2×Q8—68no—
213C2×Q8—63no—
214C2×Q8C3×S411yescharacteristic
215C2×Q8C22×C3×D311yescharacteristic
216C2×D4—312no—
217C2×D4—33no—
218C2×D4—31no—
219C2×D4—33no—
220C2×D4—66no—
221C2×D4—63no—
222C22×C4—33no—
223C22×C4—63no—
224C22×C4—63no—
225Q16—36no—
226Q16—36no—
227SD16—312no—
228SD16—324no—
229SD16—32no—
230SD16—32no—
231D8—312no—
232M16—312no—
233C2×C8—31no—
234C2×C8—33no—
235C2×C8—312no—
236C22×C3—36no—
237C22×C3—41no—
238C22×C3—66no—
239C22×C3—62no—
240C22×C3—81no—
241C22×C3—122no—
242C22×C3—244no—
243C22×C3C22×S411yescharacteristic
244C2×D3—42no—
245C2×D3—44no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—46no—
249C3×C4—66no—
250C3×C4—86no—
251C3×C4C22×S416yes—
252Dic3—46no—
253C32—41noSylow
254C23—33no—
255C23—64no—
256C23—61no—
257C23—242no—
258Q8—31no—
259Q8—36no—
260Q8—63no—
261Q8—612no—
262Q8C2×C3×S414yes—
263Q8C23×C3×D311yescharacteristic
264D4—36no—
265D4—32no—
266D4—312no—
267D4—63no—
268D4—61no—
269D4—612no—
270C2×C4—31no—
271C2×C4—36no—
272C2×C4—36no—
273C2×C4—612no—
274C2×C4—63no—
275C2×C4—612no—
276C2×C4C2×C3×S413yes—
277C8—32no—
278C8—36no—
279C2×C3—41no—
280C2×C3—42no—
281C2×C3—66no—
282C2×C3—81no—
283C2×C3—82no—
284C2×C3—122no—
285C2×C3#592412yesdirect product factor
286C2×C3C23×S411yescharacteristic
287D3—44no—
288C22—36no—
289C22—66no—
290C22—62no—
291C22—122no—
292C22—244no—
293C22C22×C3×S411yescharacteristic
294C4—31no—
295C4—31no—
296C4—66no—
297C4C22×C3×S416yes—
298C3—41no—
299C3—81no—
300C3C2×#592411yescharacteristic, direct product factor
301C2—66no—
302C2—122no—
303C2C3×#592412yesdirect product factor
304C2C23×C3×S411yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC2×C3×H(5,𝔽2)D3
hypercenterC2×C3×Q8S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C3×#5924C2direct (N × H)
3. C2×C3×#985C2semidirect
4. C2×C3×#984C2semidirect
5. C2×#5924C3direct (N × H)
8. C3×#994C22semidirect
10. C3×#985C22semidirect
11. C3×#984C22semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C2×C3×H(5,𝔽2)D3semidirect
17. C2×#994C2×C3semidirect
18. #5924C2×C3direct (N × H)
19. C2×#985C2×C3semidirect
20. C2×#984C2×C3semidirect
34. C3×GL(2,𝔽3)C23semidirect
35. C3×#226C23semidirect
40. C3×H(5,𝔽2)C2×D3semidirect
50. #994C22×C3semidirect
53. #985C22×C3semidirect
54. #984C22×C3semidirect
55. C2×GL(2,𝔽3)C22×C3semidirect
56. C2×#226C22×C3semidirect
79. C2×H(5,𝔽2)C3×D3semidirect
120. GL(2,𝔽3)C23×C3semidirect
121. #226C23×C3semidirect
141. H(5,𝔽2)C2×C3×D3semidirect
285. C2×C3#5924direct (N × H)
300. C3C2×#5924direct (N × H)
303. C2C3×#5924direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C3×#59242
C2×C3×#9852
C2×C3×#9842
C2×#59243
C2×C3×#5773
C2×C3×D3×Q84