Finite Group Databasebeta

← C2×C3×#5753 #11918919

Subgroups — C2×C3×#5753 #11918919

Order 1152.

2949 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C212yesmaximal, complemented
2C2×C32×#131C211yesmaximal, characteristic, complemented
3C3×#5753C2116yesmaximal, direct product factor
4C2×C3×#909C216yesmaximal, complemented
5C22×C3×Dic12C213yesmaximal, complemented
6C2×C3×#904C213yesmaximal, complemented
7C2×C3×#578—31nomaximal
8C2×#5753C311yesmaximal, characteristic, direct product factor
9C2×C32×#41C2211yescharacteristic
10C3×#1010—28no—
11C3×#1010C22116yescomplemented
12C2×C3×D3×Q8—22no—
13C2×C3×#237—26no—
14C2×C3×#235C2216yes—
15C22×C3×Dic6C2216yes—
16C32×#131C2218yescomplemented
17C2×C32×M16C2213yes—
18C22×C32×C8C2213yes—
19C3×#909C22148yescomplemented
20C2×C3×Dic12C2216yes—
21C2×C3×Dic12C2213yes—
22C2×C3×Dic12C22124yescomplemented
23C3×#904C22124yescomplemented
24C2×C3×D24C2211yescharacteristic
25C2×C3×#204C2216yes—
26C2×C3×2−1+4—32no—
27C2×#1010C2×C312yescomplemented
28C3×#578—316no—
29C2×C3×#137—36no—
30C2×C3×#135—33no—
31C22×C3×Q16—33no—
32C2×C3×#131—21no—
33C2×C3×#131D311yescharacteristic, complemented
34C2×C3×#131C2×C311yescharacteristic, complemented
35#5753C2×C3116yesdirect product factor
36C2×#909C2×C316yescomplemented
37C22×Dic12C2×C313yescomplemented
38C2×#904C2×C313yescomplemented
39C32×#41D414yes—
40C32×#41C2314yes—
41C2×C32×Q8D411yescharacteristic
42C2×C32×D4D413yes—
43C22×C32×C4C2313yes—
44C2×C3×#67—26no—
45C22×C3×Dic3—26no—
46C3×D3×Q8—216no—
47C3×#237—248no—
48C3×#235—212no—
49C3×#235C23124yes—
50C2×C3×D12C2312yes—
51C2×C3×C4×D3—26no—
52C2×C3×Dic6—26no—
53C2×C3×Dic6—212no—
54C2×C3×Dic6C2316yes—
55C2×C3×Dic6C23124yes—
56C32×M16C23112yes—
57C2×C32×C8C2313yes—
58C2×C32×C8C2311yescharacteristic
59C2×C32×C8C23112yes—
60C3×Dic12C23112yes—
61C3×Dic12C23124yes—
62C3×D24C2314yes—
63C3×#204C23124yes—
64C2×#578—31noSylow
65C3×2−1+4—316no—
66C3×2−1+4—68no—
67C2×C3×#41—21no—
68C2×C3×#41—36no—
69C2×C3×#41—66no—
70C2×C3×#41C2×D311yescharacteristic
71C2×C3×#41C22×C311yescharacteristic
72C22×C3×Q8—36no—
73C22×C3×Q8—62no—
74#1010—28no—
75#1010C22×C3116yescomplemented
76C2×D3×Q8—22no—
77C2×#237—26no—
78C2×#235C22×C316yes—
79C22×Dic6C22×C316yes—
80C3×#137—348no—
81C3×#135—324no—
82C2×C3×Q16—33no—
83C2×C3×Q16—36no—
84C2×C3×Q16—324no—
85C2×C3×SD16—36no—
86C2×C3×D8—31no—
87C3×#131—28no—
88C3×#131C2×D318yescomplemented
89C3×#131C22×C318yescomplemented
90C2×C3×M16—23no—
91C2×C3×M16C2×D313yes—
92C2×C3×M16C22×C313yes—
93C22×C3×C8—23no—
94C22×C3×C8C2×D313yes—
95C22×C3×C8C22×C313yes—
96#909C22×C3148yescomplemented
97C2×Dic12C22×C313yes—
98C2×Dic12C22×C316yes—
99C2×Dic12C22×C3124yescomplemented
100#904C22×C3124yescomplemented
101C2×D24C22×C311yescharacteristic
102C2×#204C22×C316yes—
103C23×C32C2×D413yes—
104C22×C3×D3—22no—
105C32×Q8C2×D414yes—
106C32×D4C2×D4112yes—
107C2×C32×C4C2×D413yes—
108C2×C32×C4C2×D416yes—
109C2×C32×C4C2411yescharacteristic
110C2×C32×C4C2416yes—
111C3×#67—224no—
112C2×C3×Dic3—26no—
113C2×C3×Dic3—224no—
114C3×D12—22no—
115C3×D12C2414yes—
116C3×C4×D3—224no—
117C3×Dic6—26no—
118C3×Dic6—224no—
119C3×Dic6C24112yes—
120C32×C8C2416yes—
121C32×C8C2412yes—
122C2×2−1+4—32no—
123#578—316no—
124C2×#137—36no—
125C2×#135—33no—
126C22×Q16—33no—
127C2×#131C3×D311yescharacteristic, complemented
128C3×#41—24no—
129C3×#41—24no—
130C3×#41—324no—
131C3×#41—648no—
132C3×#41—612no—
133C3×#41D1214yes—
134C3×#41C3×D414yes—
135C3×#41C22×D314yes—
136C3×#41C23×C314yes—
137C2×C3×Q8—21no—
138C2×C3×Q8—36no—
139C2×C3×Q8—324no—
140C2×C3×Q8—616no—
141C2×C3×Q8—66no—
142C2×C3×Q8—612no—
143C2×C3×Q8D1211yescharacteristic
144C2×C3×Q8C3×D411yescharacteristic
145C2×C3×D4—23no—
146C2×C3×D4—32no—
147C2×C3×D4—66no—
148C2×C3×D4D1213yes—
149C2×C3×D4C3×D413yes—
150C22×C3×C4—23no—
151C22×C3×C4—66no—
152C22×C3×C4—66no—
153C22×C3×C4C22×D313yes—
154C22×C3×C4C23×C313yes—
155C2×#67—26no—
156C22×Dic3—26no—
157D3×Q8—216no—
158#237—248no—
159#235—212no—
160#235C23×C3124yes—
161C2×D12C23×C312yes—
162C2×C4×D3—26no—
163C2×Dic6—26no—
164C2×Dic6—212no—
165C2×Dic6C23×C316yes—
166C2×Dic6C23×C3124yes—
167C3×Q16—312no—
168C3×Q16—324no—
169C3×SD16—324no—
170C3×D8—34no—
171C3×M16—212no—
172C3×M16C22×D3112yes—
173C3×M16C23×C3112yes—
174C2×C3×C8—21no—
175C2×C3×C8—23no—
176C2×C3×C8—212no—
177C2×C3×C8C22×D311yescharacteristic
178C2×C3×C8C22×D313yes—
179C2×C3×C8C22×D3112yes—
180C2×C3×C8C23×C313yes—
181C2×C3×C8C23×C311yescharacteristic
182C2×C3×C8C23×C3112yes—
183Dic12C23×C3112yes—
184Dic12C23×C3124yes—
185D24C23×C314yes—
186#204C23×C3124yes—
187C22×C32—26no—
188C22×C32C22×D416yes—
189C22×C32C22×D411yescharacteristic
190C2×C3×D3—24no—
191C2×C3×D3—44no—
192C32×C4C22×D411yescharacteristic
193C32×C4C22×D416yes—
194C32×C4C2511yescharacteristic
195C3×Dic3—212no—
1962−1+4—316no—
1972−1+4—68no—
198C2×#41—36no—
199C2×#41—66no—
200C2×#41C2×C3×D311yescharacteristic
201C22×Q8—36no—
202C22×Q8—62no—
203#137—348no—
204#135—324no—
205C2×Q16—33no—
206C2×Q16—36no—
207C2×Q16—324no—
208C2×SD16—36no—
209C2×D8—31no—
210#131C2×C3×D318yescomplemented
211C2×M16C2×C3×D313yes—
212C22×C8C2×C3×D313yes—
213C23×C3—23no—
214C23×C3—62no—
215C23×C3C2×D1213yes—
216C23×C3C2×C3×D413yes—
217C22×D3—22no—
218C3×Q8—24no—
219C3×Q8—312no—
220C3×Q8—624no—
221C3×Q8—66no—
222C3×Q8C2×D1214yes—
223C3×Q8C2×C3×D414yes—
224C3×D4—212no—
225C3×D4—34no—
226C3×D4—624no—
227C3×D4—62no—
228C3×D4C2×D12112yes—
229C3×D4C2×C3×D4112yes—
230C2×C3×C4—21no—
231C2×C3×C4—26no—
232C2×C3×C4—23no—
233C2×C3×C4—26no—
234C2×C3×C4—624no—
235C2×C3×C4—66no—
236C2×C3×C4—624no—
237C2×C3×C4C2×D1216yes—
238C2×C3×C4C2×D1213yes—
239C2×C3×C4C2×C3×D413yes—
240C2×C3×C4C2×C3×D416yes—
241C2×C3×C4C23×D316yes—
242C2×C3×C4C23×D311yescharacteristic
243C2×C3×C4C24×C311yescharacteristic
244C2×C3×C4C24×C316yes—
245#67—224no—
246C2×Dic3—26no—
247C2×Dic3—224no—
248D12—22no—
249D12C24×C314yes—
250C4×D3—224no—
251Dic6—26no—
252Dic6—224no—
253Dic6C24×C3112yes—
254C3×C8—22no—
255C3×C8—26no—
256C3×C8C23×D312yes—
257C3×C8C23×D316yes—
258C3×C8C24×C316yes—
259C3×C8C24×C312yes—
260C2×C32—26no—
261C2×C32#57812yescomplemented
262C2×C32C23×D411yescharacteristic
263C3×D3—44no—
264#41—324no—
265#41—648no—
266#41—612no—
267#41C3×D1214yes—
268#41C22×C3×D314yes—
269C2×Q8—36no—
270C2×Q8—324no—
271C2×Q8—616no—
272C2×Q8—66no—
273C2×Q8—612no—
274C2×Q8C3×D1211yescharacteristic
275C2×D4—32no—
276C2×D4—66no—
277C2×D4C3×D1213yes—
278C22×C4—66no—
279C22×C4—66no—
280C22×C4C22×C3×D313yes—
281Q16—312no—
282Q16—324no—
283SD16—324no—
284D8—34no—
285M16C22×C3×D3112yes—
286C2×C8C22×C3×D311yescharacteristic
287C2×C8C22×C3×D313yes—
288C2×C8C22×C3×D3112yes—
289C22×C3—26no—
290C22×C3—26no—
291C22×C3—21no—
292C22×C3—26no—
293C22×C3—46no—
294C22×C3—64no—
295C22×C3—124no—
296C22×C3C22×D1216yes—
297C22×C3C22×D1211yescharacteristic
298C22×C3C22×C3×D416yes—
299C22×C3C22×C3×D411yescharacteristic
300C2×D3—24no—
301C2×D3—44no—
302C3×C4—21no—
303C3×C4—26no—
304C3×C4—21no—
305C3×C4—612no—
306C3×C4C22×D1216yes—
307C3×C4C22×D1211yescharacteristic
308C3×C4C22×C3×D411yescharacteristic
309C3×C4C22×C3×D416yes—
310C3×C4C24×D311yescharacteristic
311C3×C4C25×C311yescharacteristic
312Dic3—212no—
313C32C2×#57811yescharacteristic, complemented, Sylow
314C23—62no—
315C23C2×C3×D1213yes—
316Q8—312no—
317Q8—624no—
318Q8—66no—
319Q8C2×C3×D1214yes—
320D4—34no—
321D4—624no—
322D4—62no—
323D4C2×C3×D12112yes—
324C2×C4—66no—
325C2×C4—624no—
326C2×C4—624no—
327C2×C4C2×C3×D1213yes—
328C2×C4C2×C3×D1216yes—
329C2×C4C23×C3×D311yescharacteristic
330C2×C4C23×C3×D316yes—
331C8C23×C3×D312yes—
332C8C23×C3×D316yes—
333C2×C3—26no—
334C2×C3—26no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—46no—
338C2×C3—124no—
339C2×C3#575312yesdirect product factor
340C2×C3C3×#57812yescomplemented
341C2×C3C23×D1211yescharacteristic
342C2×C3C23×C3×D411yescharacteristic
343D3—44no—
344C22—26no—
345C22—64no—
346C22—124no—
347C22C22×C3×D1216yes—
348C22C22×C3×D1211yescharacteristic
349C4—612no—
350C4C22×C3×D1216yes—
351C4C22×C3×D1211yescharacteristic
352C4C24×C3×D311yescharacteristic
353C3—21no—
354C3C2×#575311yescharacteristic, direct product factor
355C3C2×C3×#57811yescharacteristic, complemented
356C2—26no—
357C2—124no—
358C2C3×#575312yesdirect product factor
359C2C23×C3×D1211yescharacteristic
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC2×C32×#131C2
socleC22×C32C22×D4
hypercenterC2×C3×#131D3
hypocenterC3C2×C3×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C2×C32×#131C2semidirect
3. C3×#5753C2direct (N × H)
4. C2×C3×#909C2semidirect
5. C22×C3×Dic12C2semidirect
6. C2×C3×#904C2semidirect
8. C2×#5753C3direct (N × H)
11. C3×#1010C22semidirect
16. C32×#131C22semidirect
19. C3×#909C22semidirect
22. C2×C3×Dic12C22semidirect
23. C3×#904C22semidirect
27. C2×#1010C2×C3semidirect
33. C2×C3×#131D3semidirect
34. C2×C3×#131C2×C3semidirect
35. #5753C2×C3direct (N × H)
36. C2×#909C2×C3semidirect
37. C22×Dic12C2×C3semidirect
38. C2×#904C2×C3semidirect
75. #1010C22×C3semidirect
88. C3×#131C2×D3semidirect
89. C3×#131C22×C3semidirect
96. #909C22×C3semidirect
99. C2×Dic12C22×C3semidirect
100. #904C22×C3semidirect
127. C2×#131C3×D3semidirect
210. #131C2×C3×D3semidirect
261. C2×C32#578semidirect
313. C32C2×#578semidirect
339. C2×C3#5753direct (N × H)
340. C2×C3C3×#578semidirect
354. C3C2×#5753direct (N × H)
355. C3C2×C3×#578semidirect
358. C2C3×#5753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C2×C32×#1312
C3×#57532
C2×C3×#9092
C22×C3×Dic122
C2×C3×#9042
C2×C3×#5783
C2×#57533