Finite Group Databasebeta

← C2×C3×#5925 #11923703

Subgroups — C2×C3×#5925 #11923703

Order 1152.

1569 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1569 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×#5925C218yesmaximal, direct product factor
3C2×C3×#986C213yesmaximal, complemented
4C2×C3×#985C213yesmaximal, complemented
5C2×#5925C311yesmaximal, characteristic, direct product factor
6C2×C3×#576—31nomaximal
7C2×C3×#239—41nomaximal
8C3×#994C2214yescomplemented
9C2×C3×#41⋊C3C2213yescomplemented
10C3×#986C22112yescomplemented
11C3×#985C22112yescomplemented
12C2×C3×GL(2,𝔽3)C2213yescomplemented
13C2×C3×#226C2211yescharacteristic, complemented
14C2×C3×H(5,𝔽2)—31no—
15C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
16C2×#994—21no—
17C2×#994C2×C311yescharacteristic, complemented
18#5925C2×C318yesdirect product factor
19C2×#986C2×C313yescomplemented
20C2×#985C2×C313yescomplemented
21C3×#576—38no—
22C3×#576—38no—
23C2×C3×AGL(1,ℤ/8)—33no—
24C2×C3×AGL(1,ℤ/8)—33no—
25C2×C3×#135—33no—
26C22×C3×D8—33no—
27C2×C3×#131—31no—
28C2×C32×Q8—41no—
29C3×#239—48no—
30C2×C3×D12—43no—
31C2×C3×C4×D3—43no—
32C3×#41⋊C3C2316yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×GL(2,𝔽3)C2316yescomplemented
35C3×#226C2312yescomplemented
36C2×#576—31noSylow
37C3×H(5,𝔽2)—34no—
38C3×H(5,𝔽2)—38no—
39C3×H(5,𝔽2)—64no—
40C3×H(5,𝔽2)C2×D314yescomplemented
41C2×C3×#41—31no—
42C2×C3×#41—33no—
43C2×C3×#41—61no—
44C2×C3×#41C2×D313yes—
45C22×C3×D4—33no—
46C22×C3×D4—33no—
47C22×C3×D4—63no—
48C2×#239—41no—
49#994—24no—
50#994C22×C314yescomplemented
51C2×#41⋊C3—23no—
52C2×#41⋊C3C22×C313yescomplemented
53#986C22×C3112yescomplemented
54#985C22×C3112yescomplemented
55C2×GL(2,𝔽3)C22×C313yescomplemented
56C2×#226C22×C311yescharacteristic, complemented
57C3×AGL(1,ℤ/8)—324no—
58C3×AGL(1,ℤ/8)—312no—
59C3×AGL(1,ℤ/8)—312no—
60C3×#135—312no—
61C3×#135—312no—
62C2×C3×Q16—31no—
63C2×C3×SD16—33no—
64C2×C3×SD16—33no—
65C2×C3×D8—33no—
66C2×C3×D8—36no—
67C2×C3×D8—324no—
68C3×#131—38no—
69C2×C3×M16—33no—
70C22×C3×C8—33no—
71C22×C3×D3—43no—
72C32×Q8—44no—
73C2×C32×C4—43no—
74C2×C3×Dic3—41no—
75C3×D12—412no—
76C3×C4×D3—412no—
77C3×SL(2,𝔽3)C2411yescharacteristic
78C2×H(5,𝔽2)—31no—
79C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
80#576—38no—
81#576—38no—
82C2×AGL(1,ℤ/8)—33no—
83C2×AGL(1,ℤ/8)—33no—
84C2×#135—33no—
85C22×D8—33no—
86C2×#131—31no—
87C24×C3—61no—
88C24×C3—63no—
89C3×#41—34no—
90C3×#41—34no—
91C3×#41—36no—
92C3×#41—312no—
93C3×#41—66no—
94C3×#41—68no—
95C3×#41C22×D316yes—
96C2×C3×Q8—31no—
97C2×C3×Q8—41no—
98C2×C3×Q8—81no—
99C2×C3×Q8S411yescharacteristic
100C2×C3×Q8C22×D311yescharacteristic
101C2×C3×D4—312no—
102C2×C3×D4—33no—
103C2×C3×D4—33no—
104C2×C3×D4—312no—
105C2×C3×D4—33no—
106C2×C3×D4—66no—
107C2×C3×D4—66no—
108C2×C3×D4—624no—
109C2×C3×D4—63no—
110C2×C3×D4—63no—
111C22×C3×C4—33no—
112C22×C3×C4—63no—
113#239—48no—
114C2×D12—43no—
115C2×C4×D3—43no—
116#41⋊C3—26no—
117#41⋊C3C23×C316yescomplemented
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C23×C311yescharacteristic
120GL(2,𝔽3)C23×C316yescomplemented
121#226C23×C312yescomplemented
122C3×Q16—32no—
123C3×Q16—32no—
124C3×SD16—312no—
125C3×SD16—36no—
126C3×SD16—36no—
127C3×D8—312no—
128C3×D8—324no—
129C3×M16—312no—
130C2×C3×C8—31no—
131C2×C3×C8—33no—
132C2×C3×C8—312no—
133C22×C32—41no—
134C2×C3×D3—46no—
135C2×C3×D3—86no—
136C32×C4—46no—
137C3×Dic3—42no—
138H(5,𝔽2)—34no—
139H(5,𝔽2)—38no—
140H(5,𝔽2)—64no—
141H(5,𝔽2)C2×C3×D314yescomplemented
142C2×#41—31no—
143C2×#41—33no—
144C2×#41—61no—
145C2×#41C2×C3×D313yes—
146C22×D4—33no—
147C22×D4—33no—
148C22×D4—63no—
149AGL(1,ℤ/8)—324no—
150AGL(1,ℤ/8)—312no—
151AGL(1,ℤ/8)—312no—
152#135—312no—
153#135—312no—
154C2×Q16—31no—
155C2×SD16—33no—
156C2×SD16—33no—
157C2×D8—33no—
158C2×D8—36no—
159C2×D8—324no—
160#131—38no—
161C2×M16—33no—
162C22×C8—33no—
163C23×C3—33no—
164C23×C3—64no—
165C23×C3—612no—
166C23×C3—63no—
167C23×C3—242no—
168C23×C3—246no—
169C22×D3—43no—
170C3×Q8—31no—
171C3×Q8—32no—
172C3×Q8—44no—
173C3×Q8—61no—
174C3×Q8—84no—
175C3×Q8C2×S414yes—
176C3×Q8C23×D311yescharacteristic
177C3×D4—36no—
178C3×D4—36no—
179C3×D4—312no—
180C3×D4—63no—
181C3×D4—63no—
182C3×D4—612no—
183C3×D4—612no—
184C2×C3×C4—31no—
185C2×C3×C4—36no—
186C2×C3×C4—36no—
187C2×C3×C4—43no—
188C2×C3×C4—61no—
189C2×C3×C4—612no—
190C2×C3×C4—83no—
191C2×C3×C4C2×S413yes—
192C2×Dic3—41no—
193D12—412no—
194C4×D3—412no—
195SL(2,𝔽3)—21no—
196SL(2,𝔽3)C24×C311yescharacteristic
197C3×C8—32no—
198C3×C8—36no—
199C2×C32—42no—
200C2×C32—41no—
201C3×D3—86no—
202C24—61no—
203C24—63no—
204#41—34no—
205#41—34no—
206#41—36no—
207#41—312no—
208#41—66no—
209#41—68no—
210#41C22×C3×D316yes—
211C2×Q8—31no—
212C2×Q8C3×S411yescharacteristic
213C2×Q8C22×C3×D311yescharacteristic
214C2×D4—312no—
215C2×D4—33no—
216C2×D4—33no—
217C2×D4—312no—
218C2×D4—33no—
219C2×D4—66no—
220C2×D4—66no—
221C2×D4—624no—
222C2×D4—63no—
223C2×D4—63no—
224C22×C4—33no—
225C22×C4—63no—
226Q16—32no—
227Q16—32no—
228SD16—312no—
229SD16—36no—
230SD16—36no—
231D8—312no—
232D8—324no—
233M16—312no—
234C2×C8—31no—
235C2×C8—33no—
236C2×C8—312no—
237C22×C3—36no—
238C22×C3—41no—
239C22×C3—66no—
240C22×C3—66no—
241C22×C3—81no—
242C22×C3—126no—
243C22×C3—244no—
244C22×C3—2412no—
245C22×C3C22×S411yescharacteristic
246C2×D3—46no—
247C2×D3—86no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—46no—
251C3×C4—62no—
252C3×C4—86no—
253C3×C4C22×S416yes—
254Dic3—42no—
255C32—41noSylow
256C23—33no—
257C23—64no—
258C23—612no—
259C23—63no—
260C23—242no—
261C23—246no—
262Q8—31no—
263Q8—32no—
264Q8—61no—
265Q8C2×C3×S414yes—
266Q8C23×C3×D311yescharacteristic
267D4—36no—
268D4—36no—
269D4—312no—
270D4—63no—
271D4—63no—
272D4—612no—
273D4—612no—
274C2×C4—31no—
275C2×C4—36no—
276C2×C4—36no—
277C2×C4—61no—
278C2×C4—612no—
279C2×C4C2×C3×S413yes—
280C8—32no—
281C8—36no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—66no—
285C2×C3—81no—
286C2×C3—82no—
287C2×C3—126no—
288C2×C3#592512yesdirect product factor
289C2×C3C23×S411yescharacteristic
290D3—86no—
291C22—36no—
292C22—66no—
293C22—66no—
294C22—126no—
295C22—244no—
296C22—2412no—
297C22C22×C3×S411yescharacteristic
298C4—31no—
299C4—31no—
300C4—62no—
301C4C22×C3×S416yes—
302C3—41no—
303C3—81no—
304C3C2×#592511yescharacteristic, direct product factor
305C2—66no—
306C2—126no—
307C2C3×#592512yesdirect product factor
308C2C23×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×#5925C2direct (N × H)
3. C2×C3×#986C2semidirect
4. C2×C3×#985C2semidirect
5. C2×#5925C3direct (N × H)
8. C3×#994C22semidirect
9. C2×C3×#41⋊C3C22semidirect
10. C3×#986C22semidirect
11. C3×#985C22semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C2×C3×H(5,𝔽2)D3semidirect
17. C2×#994C2×C3semidirect
18. #5925C2×C3direct (N × H)
19. C2×#986C2×C3semidirect
20. C2×#985C2×C3semidirect
32. C3×#41⋊C3C23semidirect
34. C3×GL(2,𝔽3)C23semidirect
35. C3×#226C23semidirect
40. C3×H(5,𝔽2)C2×D3semidirect
50. #994C22×C3semidirect
52. C2×#41⋊C3C22×C3semidirect
53. #986C22×C3semidirect
54. #985C22×C3semidirect
55. C2×GL(2,𝔽3)C22×C3semidirect
56. C2×#226C22×C3semidirect
79. C2×H(5,𝔽2)C3×D3semidirect
117. #41⋊C3C23×C3semidirect
120. GL(2,𝔽3)C23×C3semidirect
121. #226C23×C3semidirect
141. H(5,𝔽2)C2×C3×D3semidirect
288. C2×C3#5925direct (N × H)
304. C3C2×#5925direct (N × H)
307. C2C3×#5925direct (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×#59252
C2×C3×#9862
C2×C3×#9852
C2×#59253
C2×C3×#5763
C2×C3×#2394