Finite Group Databasebeta

← C2×C3×#4710 #11860049

Subgroups — C2×C3×#4710 #11860049

Order 1152.

1161 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#204C211yesmaximal, characteristic
2C2×C3×C4×D12C211yesmaximal, characteristic
3C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
4C2×C32×#97C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic
6C2×C3×#818C211yesmaximal, characteristic, complemented
7C2×C3×#816C211yesmaximal, characteristic, complemented
8C3×#4710C218yesmaximal, direct product factor
9C2×C3×#440—31nomaximal
10C2×#4710C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C22×C3×Dic6C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C32×C42C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4—21no—
21C2×C3×C4×Dic3—21no—
22C2×C3×#204C418yescomplemented
23C2×C3×#204C2214yes—
24C3×C4×D12—22no—
25C3×C4×D12C2214yes—
26C3×C4×Dic6—22no—
27C3×C4×Dic6C2214yescomplemented
28C32×#97C2214yescomplemented
29C3×#821C2214yes—
30C3×#818C2214yescomplemented
31C3×#816C2214yescomplemented
32C22×C3×SD16—31no—
33C2×C3×C4×Q8—31no—
34C2×C3×C4×D4—31no—
35C22×#204C2×C311yescharacteristic
36C2×C4×D12C2×C311yescharacteristic
37C2×C4×Dic6C2×C311yescharacteristic, complemented
38C3×#440—38no—
39C2×C3×#107—31no—
40C2×C3×#103—31no—
41C2×C3×#102—31no—
42C2×C3×#97—21no—
43C2×C3×#97D311yescharacteristic, complemented
44C2×C3×#97C2×C311yescharacteristic, complemented
45C2×#821C2×C311yescharacteristic
46C2×#818C2×C311yescharacteristic, complemented
47C2×#816C2×C311yescharacteristic, complemented
48#4710C2×C318yesdirect product factor
49C23×C3×D3—21no—
50C22×C32×C4D411yescharacteristic
51C22×C32×C4D411yescharacteristic
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3—21no—
56C2×C3×D12—21no—
57C2×C3×D12—22no—
58C2×C3×D12C2×C414yes—
59C2×C3×D12C2312yes—
60C2×C3×C4×D3—24no—
61C2×C3×C4×D3—44no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—22no—
64C2×C3×Dic6C2×C414yescomplemented
65C2×C3×Dic6C2312yes—
66C2×C32×C8—22no—
67C2×C32×C8C2×C414yescomplemented
68C2×C32×C8C2312yes—
69C2×C32×C8C2312yes—
70C32×C42D412yes—
71C32×C42C2312yes—
72C3×(C2×D3)⋊C4—24no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2312yes—
76C3×C4⋊Dic3C2312yes—
77C3×Dic3⋊C4—24no—
78C3×C4×Dic3—24no—
79C3×#204C2×C4116yescomplemented
80C2×#440—31noSylow
81C22×C3×Q8—31no—
82C22×C3×D4—31no—
83C23×C3×C4—61no—
84C22×D12C22×C311yescharacteristic
85C22×C4×D3—21no—
86C22×Dic6C22×C311yescharacteristic
87C2×C3×SD16—38no—
88C2×C3×SD16—34no—
89C22×C3×C8—21no—
90C22×C3×C8—21no—
91C22×C3×C8C2×D311yescharacteristic
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C22×C311yescharacteristic
94C22×C3×C8C22×C311yescharacteristic
95C3×C4×Q8—34no—
96C3×C4×Q8—62no—
97C3×C4×D4—34no—
98C3×C4×D4—62no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—61no—
102C2×C3×C22⋊C4—61no—
103C2×C3×C42—21no—
104C2×C3×C42—61no—
105C2×C3×C42C2×D311yescharacteristic
106C2×C3×C42C22×C311yescharacteristic
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×C4⋊Dic3C22×C311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×#204C3×C418yescomplemented
113C2×#204C22×C314yes—
114C4×D12—22no—
115C4×D12C22×C314yes—
116C4×Dic6—22no—
117C4×Dic6C22×C314yescomplemented
118C3×#107—34no—
119C3×#103—34no—
120C3×#102—34no—
121C3×#97—24no—
122C3×#97C2×D314yescomplemented
123C3×#97C22×C314yescomplemented
124#821C22×C314yes—
125#818C22×C314yescomplemented
126#816C22×C314yescomplemented
127C23×C32C2×D411yescharacteristic
128C22×C3×D3—24no—
129C22×C3×D3—22no—
130C22×C3×D3—42no—
131C22×C3×D3—42no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4C22×C412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4#4112yes—
139C2×C32×C4C2411yescharacteristic
140C2×C3×Dic3—24no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—42no—
146C3×D12—24no—
147C3×D12—22no—
148C3×D12C22×C414yes—
149C3×C4×D3—48no—
150C3×Dic6—24no—
151C3×Dic6—22no—
152C3×Dic6C22×C414yescomplemented
153C32×C8—22no—
154C32×C8C22×C414yescomplemented
155C22×SD16—31no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158#440—38no—
159C2×#107—31no—
160C2×#103—31no—
161C2×#102—31no—
162C2×#97C3×D311yescharacteristic, complemented
163C24×C3—61no—
164C23×D3—21no—
165C2×C3×Q8—34no—
166C2×C3×Q8—32no—
167C2×C3×Q8—61no—
168C2×C3×Q8—62no—
169C2×C3×D4—34no—
170C2×C3×D4—32no—
171C2×C3×D4—61no—
172C2×C3×D4—62no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—61no—
179C22×C3×C4—64no—
180C22×C3×C4—124no—
181C22×C3×C4D1211yescharacteristic
182C22×C3×C4D1211yescharacteristic
183C22×C3×C4C3×D411yescharacteristic
184C22×C3×C4C3×D411yescharacteristic
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C23×C311yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C2×D12—21no—
191C2×D12—22no—
192C2×D12C2×C3×C414yes—
193C2×D12C23×C312yes—
194C2×C4×D3—24no—
195C2×C4×D3—44no—
196C2×Dic6—21no—
197C2×Dic6—22no—
198C2×Dic6C2×C3×C414yescomplemented
199C2×Dic6C23×C312yes—
200C3×SD16—316no—
201C2×C3×C8—24no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—24no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C4×D314yescomplemented
208C2×C3×C8C2×C3×C414yescomplemented
209C2×C3×C8C22×D312yes—
210C2×C3×C8C22×D312yes—
211C2×C3×C8C23×C312yes—
212C2×C3×C8C23×C312yes—
213C3×C4⋊C4—32no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—64no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C22⋊C4—64no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42—64no—
222C3×C42D1212yes—
223C3×C42C3×D412yes—
224C3×C42C22×D312yes—
225C3×C42C23×C312yes—
226(C2×D3)⋊C4—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3C23×C312yes—
230C4⋊Dic3C23×C312yes—
231Dic3⋊C4—24no—
232C4×Dic3—24no—
233#204C2×C3×C4116yescomplemented
234C22×C32C4×D412yes—
235C22×C32AGL(1,ℤ/8)12yes—
236C22×C32#13712yes—
237C22×C32C22×D411yescharacteristic
238C2×C3×D3—24no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—48no—
242C32×C4—22no—
243C32×C4—22no—
244C32×C4C4×D412yes—
245C32×C4C23×C411yescharacteristic
246C32×C4C2×#4111yescharacteristic
247C3×Dic3—24no—
248C3×Dic3—42no—
249C3×Dic3—42no—
250C22×Q8—31no—
251C22×D4—31no—
252C23×C4—61no—
253C2×SD16—38no—
254C2×SD16—34no—
255C22×C8C2×C3×D311yescharacteristic
256C22×C8C2×C3×D311yescharacteristic
257C4×Q8—34no—
258C4×Q8—62no—
259C4×D4—34no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C42—61no—
266C2×C42C2×C3×D311yescharacteristic
267#107—34no—
268#103—34no—
269#102—34no—
270#97C2×C3×D314yescomplemented
271C23×C3—21no—
272C23×C3—64no—
273C23×C3—62no—
274C23×C3—122no—
275C23×C3—122no—
276C23×C3C2×D1211yescharacteristic
277C23×C3C2×C3×D411yescharacteristic
278C22×D3—24no—
279C22×D3—22no—
280C22×D3—42no—
281C22×D3—42no—
282C3×Q8—34no—
283C3×Q8—64no—
284C3×Q8—62no—
285C3×D4—34no—
286C3×D4—64no—
287C3×D4—62no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—64no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—128no—
305C2×C3×C4—122no—
306C2×C3×C4—122no—
307C2×C3×C4C2×C4×D312yes—
308C2×C3×C4C2×D1212yes—
309C2×C3×C4C2×D1212yes—
310C2×C3×C4C2×D1211yescharacteristic
311C2×C3×C4#23512yes—
312C2×C3×C4C22×C3×C412yes—
313C2×C3×C4C2×C3×D412yes—
314C2×C3×C4C2×C3×D412yes—
315C2×C3×C4C2×C3×D411yescharacteristic
316C2×C3×C4C3×#4112yes—
317C2×C3×C4C23×D311yescharacteristic
318C2×C3×C4C24×C311yescharacteristic
319C2×Dic3—24no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325D12—24no—
326D12—22no—
327D12C22×C3×C414yes—
328C4×D3—48no—
329Dic6—24no—
330Dic6—22no—
331Dic6C22×C3×C414yescomplemented
332C3×C8—24no—
333C3×C8—22no—
334C3×C8—22no—
335C3×C8—24no—
336C3×C8C2×C4×D314yescomplemented
337C3×C8C22×C3×C414yescomplemented
338C2×C32#44014yescomplemented
339C2×C32C2×C4×D411yescharacteristic
340C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
341C2×C32C2×#13711yescharacteristic
342C3×D3—44no—
343C24—61no—
344C2×Q8—32no—
345C2×Q8—34no—
346C2×Q8—61no—
347C2×Q8—62no—
348C2×D4—32no—
349C2×D4—34no—
350C2×D4—61no—
351C2×D4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—64no—
356C22×C4—124no—
357C22×C4C3×D1211yescharacteristic
358C22×C4C3×D1211yescharacteristic
359C22×C4C22×C3×D311yescharacteristic
360SD16—316no—
361C2×C8—22no—
362C2×C8C3×C4×D314yescomplemented
363C2×C8C22×C3×D312yes—
364C2×C8C22×C3×D312yes—
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—64no—
370C22⋊C4—64no—
371C42—64no—
372C42C3×D1212yes—
373C42C22×C3×D312yes—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—64no—
379C22×C3—128no—
380C22×C3—122no—
381C22×C3—122no—
382C22×C3C4×D1212yes—
383C22×C3#90812yes—
384C22×C3#90912yes—
385C22×C3C3×C4×D412yes—
386C22×C3C3×AGL(1,ℤ/8)12yes—
387C22×C3C3×#13712yes—
388C22×C3C22×D1211yescharacteristic
389C22×C3C22×C3×D411yescharacteristic
390C2×D3—24no—
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—48no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—22no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—64no—
404C3×C4—122no—
405C3×C4—122no—
406C3×C4C4×D1212yes—
407C3×C4C3×C4×D412yes—
408C3×C4C22×C4×D311yescharacteristic
409C3×C4C2×#23511yescharacteristic
410C3×C4C23×C3×C411yescharacteristic
411C3×C4C2×C3×#4111yescharacteristic
412Dic3—24no—
413Dic3—42no—
414Dic3—42no—
415C32C2×#44011yescharacteristic, complemented, Sylow
416C23—62no—
417C23—64no—
418C23—122no—
419C23—122no—
420C23C2×C3×D1211yescharacteristic
421Q8—34no—
422Q8—62no—
423Q8—64no—
424D4—34no—
425D4—62no—
426D4—64no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—62no—
430C2×C4—64no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—128no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4C2×C3×C4×D312yes—
437C2×C4C2×C3×D1212yes—
438C2×C4C2×C3×D1212yes—
439C2×C4C2×C3×D1211yescharacteristic
440C2×C4C3×#23512yes—
441C2×C4C23×C3×D311yescharacteristic
442C8—22no—
443C8C2×C3×C4×D314yescomplemented
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—24no—
447C2×C3—21no—
448C2×C3—124no—
449C2×C3#471014yesdirect product factor
450C2×C3C3×#44014yescomplemented
451C2×C3C2×C4×D1211yescharacteristic
452C2×C3C2×#90811yescharacteristic
453C2×C3C2×#90911yescharacteristic
454C2×C3C2×C3×C4×D411yescharacteristic
455C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
456C2×C3C2×C3×#13711yescharacteristic
457D3—44no—
458C22—64no—
459C22—122no—
460C22—122no—
461C22—128no—
462C22C3×C4×D1212yes—
463C22C3×#90812yes—
464C22C3×#90912yes—
465C22C22×C3×D1211yescharacteristic
466C4—22no—
467C4—22no—
468C4—64no—
469C4—122no—
470C4—122no—
471C4C3×C4×D1212yes—
472C4C22×C3×C4×D311yescharacteristic
473C4C2×C3×#23511yescharacteristic
474C3—21no—
475C3C2×#471011yescharacteristic, direct product factor
476C3C2×C3×#44011yescharacteristic, complemented
477C2—124no—
478C2C3×#471014yesdirect product factor
479C2C2×C3×C4×D1211yescharacteristic
480C2C2×C3×#90811yescharacteristic
481C2C2×C3×#90911yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#97C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#97D3
hypocenterC3C2×C3×#440
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×#97

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×Dic6C2semidirect
4. C2×C32×#97C2semidirect
6. C2×C3×#818C2semidirect
7. C2×C3×#816C2semidirect
8. C3×#4710C2direct (N × H)
10. C2×#4710C3direct (N × H)
22. C2×C3×#204C4semidirect
27. C3×C4×Dic6C22semidirect
28. C32×#97C22semidirect
30. C3×#818C22semidirect
31. C3×#816C22semidirect
37. C2×C4×Dic6C2×C3semidirect
43. C2×C3×#97D3semidirect
44. C2×C3×#97C2×C3semidirect
46. C2×#818C2×C3semidirect
47. C2×#816C2×C3semidirect
48. #4710C2×C3direct (N × H)
64. C2×C3×Dic6C2×C4semidirect
67. C2×C32×C8C2×C4semidirect
79. C3×#204C2×C4semidirect
112. C2×#204C3×C4semidirect
117. C4×Dic6C22×C3semidirect
122. C3×#97C2×D3semidirect
123. C3×#97C22×C3semidirect
125. #818C22×C3semidirect
126. #816C22×C3semidirect
152. C3×Dic6C22×C4semidirect
154. C32×C8C22×C4semidirect
162. C2×#97C3×D3semidirect
198. C2×Dic6C2×C3×C4semidirect
207. C2×C3×C8C4×D3semidirect
208. C2×C3×C8C2×C3×C4semidirect
233. #204C2×C3×C4semidirect
270. #97C2×C3×D3semidirect
331. Dic6C22×C3×C4semidirect
336. C3×C8C2×C4×D3semidirect
337. C3×C8C22×C3×C4semidirect
338. C2×C32#440semidirect
362. C2×C8C3×C4×D3semidirect
415. C32C2×#440semidirect
443. C8C2×C3×C4×D3semidirect
449. C2×C3#4710direct (N × H)
450. C2×C3C3×#440semidirect
475. C3C2×#4710direct (N × H)
476. C3C2×C3×#440semidirect
478. C2C3×#4710direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2042
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×C32×#972
C2×C3×#8212
C2×C3×#8182
C2×C3×#8162
C3×#47102
C2×C3×#4403
C2×#47103