Finite Group Databasebeta

← C2×C3×#5040 #11863599

Subgroups — C2×C3×#5040 #11863599

Order 1152.

1797 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×C32×#121C211yesmaximal, characteristic
3C22×C3×#214C212yesmaximal, complemented
4C2×C3×#830C211yesmaximal, characteristic, complemented
5C3×#5040C218yesmaximal, direct product factor
6C2×C3×#810C212yesmaximal, complemented
7C2×C3×#448—31nomaximal
8C2×#5040C311yesmaximal, characteristic, direct product factor
9C22×C32×D4—21no—
10C22×C32×D4C2211yescharacteristic
11C23×C32×C4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C22×C3×Dic6—22no—
14C22×C3×Dic6—21no—
15C22×C3×Dic6—28no—
16C22×C3×Dic6C2212yes—
17C22×C3×Dic6C2214yescomplemented
18C32×#121—22no—
19C32×#121C2214yes—
20C2×C32×C4⋊C4C2211yescharacteristic
21C2×C32×C22⋊C4—21no—
22C2×C3×#214—28no—
23C2×C3×#214C2218yescomplemented
24C22×C3×C3⋊C8C2212yes—
25C3×#830C2214yescomplemented
26C3×#810C2218yescomplemented
27C23×C3×Q8—31no—
28C23×Dic6C2×C311yescharacteristic, complemented
29C22×C3×SD16—32no—
30C2×C3×#121—21no—
31C2×C3×#121D311yescharacteristic
32C2×C3×#121C2×C311yescharacteristic
33C22×#214C2×C312yescomplemented
34C3×#448—38no—
35C2×C3×#103—32no—
36C2×C3×C22⋊C8—31no—
37C2×#830C2×C311yescharacteristic, complemented
38#5040C2×C318yesdirect product factor
39C2×#810C2×C312yescomplemented
40C24×C32—21no—
41C24×C32D411yescharacteristic
42C2×C32×D4—24no—
43C2×C32×D4—24no—
44C2×C32×D4—21no—
45C2×C32×D4—44no—
46C2×C32×D4C2312yes—
47C22×C32×C4—24no—
48C22×C32×C4—21no—
49C22×C32×C4D411yescharacteristic
50C22×C32×C4D412yes—
51C22×C32×C4C2312yes—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—28no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—24no—
56C2×C3×Dic6—28no—
57C2×C3×Dic6—216no—
58C2×C3×Dic6—22no—
59C2×C3×Dic6—24no—
60C2×C3×Dic6—24no—
61C2×C3×Dic6—216no—
62C2×C3×Dic6D418yescomplemented
63C2×C3×Dic6C2314yes—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2312yes—
66C32×C22⋊C4—24no—
67C3×#214—216no—
68C2×C3×C3⋊C8—24no—
69C2×C3×C3⋊C8C2314yes—
70C2×#448—31noSylow
71C22×C3×Q8—32no—
72C22×C3×Q8—34no—
73C22×C3×Q8—68no—
74C22×C3×Q8—62no—
75C22×C3×Q8—61no—
76C22×C3×D4—22no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C22×C311yescharacteristic
82C23×C3×C4—21no—
83C23×C3×C4—61no—
84C23×C3×C4C2×D311yescharacteristic
85C23×C3×C4C22×C311yescharacteristic
86C23×Dic3—21no—
87C22×Dic6—22no—
88C22×Dic6—21no—
89C22×Dic6—28no—
90C22×Dic6C22×C312yes—
91C22×Dic6C22×C314yescomplemented
92C2×C3×SD16—38no—
93C2×C3×SD16—68no—
94C22×C3×C8—32no—
95C3×#121—24no—
96C3×#121—22no—
97C3×#121—22no—
98C3×#121—24no—
99C3×#121C2×D314yes—
100C3×#121C22×C314yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C2×D311yescharacteristic
103C2×C3×C4⋊C4C22×C311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×#214—28no—
108C2×#214C22×C318yescomplemented
109C22×C3⋊C8C22×C312yes—
110C3×#103—38no—
111C3×C22⋊C8—34no—
112#830C22×C314yescomplemented
113#810C22×C318yescomplemented
114C23×C32—22no—
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—42no—
118C23×C32—42no—
119C23×C32—42no—
120C23×C32SD1614yescomplemented
121C23×C32C2×D412yes—
122C23×C32C2×D411yescharacteristic
123C32×D4—24no—
124C32×D4—48no—
125C32×D4—42no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—42no—
131C2×C32×C4C2×D414yes—
132C2×C32×C4C2×D412yes—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2411yescharacteristic
135C2×C3×Dic3—216no—
136C2×C3×Dic3—28no—
137C2×C3×Dic3—24no—
138C3×Dic6—216no—
139C3×Dic6—28no—
140C3×Dic6—24no—
141C3×Dic6C2×D418yescomplemented
142C3×C3⋊C8—24no—
143C23×Q8—31no—
144C22×SD16—32no—
145C2×#121C3×D311yescharacteristic
146#448—38no—
147C2×#103—32no—
148C2×C22⋊C8—31no—
149C24×C3—22no—
150C24×C3—21no—
151C24×C3—21no—
152C24×C3—21no—
153C24×C3#6711yescharacteristic
154C24×C3C3×D411yescharacteristic
155C2×C3×Q8—38no—
156C2×C3×Q8—34no—
157C2×C3×Q8—616no—
158C2×C3×Q8—68no—
159C2×C3×Q8—616no—
160C2×C3×Q8—64no—
161C2×C3×Q8—64no—
162C2×C3×Q8—62no—
163C2×C3×D4—28no—
164C2×C3×D4—22no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—44no—
173C2×C3×D4—48no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4C22×D312yes—
177C2×C3×D4C23×C312yes—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—44no—
188C22×C3×C4—68no—
189C22×C3×C4—64no—
190C22×C3×C4—62no—
191C22×C3×C4#6712yes—
192C22×C3×C4#6711yescharacteristic
193C22×C3×C4C3×D411yescharacteristic
194C22×C3×C4C3×D412yes—
195C22×C3×C4C22×D312yes—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C23×C312yes—
198C22×C3×C4C23×C311yescharacteristic
199C22×Dic3—28no—
200C22×Dic3—22no—
201C22×Dic3—24no—
202C2×Dic6—28no—
203C2×Dic6—216no—
204C2×Dic6—24no—
205C2×Dic6—22no—
206C2×Dic6—24no—
207C2×Dic6—216no—
208C2×Dic6C3×D418yescomplemented
209C2×Dic6C23×C314yes—
210C3×SD16—616no—
211C2×C3×C8—34no—
212C2×C3×C8—64no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4C22×D312yes—
218C3×C4⋊C4C23×C312yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222#214—216no—
223C2×C3⋊C8—24no—
224C2×C3⋊C8C23×C314yes—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—28no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32—42no—
231C22×C32—84no—
232C22×C32#12012yes—
233C22×C32C2×SD1612yes—
234C22×C32C2×SD1614yescomplemented
235C22×C32#13712yes—
236C22×C32C22×D411yescharacteristic
237C32×C4—22no—
238C32×C4—42no—
239C32×C4#12012yes—
240C32×C4C22×D412yes—
241C3×Dic3—28no—
242C22×Q8—32no—
243C22×Q8—34no—
244C22×Q8—68no—
245C22×Q8—61no—
246C22×Q8—62no—
247C22×D4—21no—
248C22×D4C2×C3×D311yescharacteristic
249C23×C4—61no—
250C23×C4C2×C3×D311yescharacteristic
251C2×SD16—38no—
252C2×SD16—68no—
253C22×C8—32no—
254#121—22no—
255#121C2×C3×D314yes—
256C2×C4⋊C4C2×C3×D311yescharacteristic
257C2×C22⋊C4—21no—
258#103—38no—
259C22⋊C8—34no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—44no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3#21414yescomplemented
282C23×C3C3×SD1614yescomplemented
283C23×C3C2×#6712yes—
284C23×C3C2×#6711yescharacteristic
285C23×C3C2×C3×D412yes—
286C23×C3C2×C3×D411yescharacteristic
287C3×Q8—38no—
288C3×Q8—616no—
289C3×Q8—68no—
290C3×Q8—64no—
291C3×D4—24no—
292C3×D4—24no—
293C3×D4—44no—
294C3×D4—44no—
295C3×D4—416no—
296C3×D4—48no—
297C3×D4—48no—
298C3×D4—42no—
299C3×D4—42no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—44no—
314C2×C3×C4—44no—
315C2×C3×C4—42no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—616no—
320C2×C3×C4—68no—
321C2×C3×C4—64no—
322C2×C3×C4D3×D414yes—
323C2×C3×C4C2×#6712yes—
324C2×C3×C4C2×#6711yescharacteristic
325C2×C3×C4C2×C3×D414yes—
326C2×C3×C4C2×C3×D412yes—
327C2×C3×C4C2×C3×D411yescharacteristic
328C2×C3×C4C23×D311yescharacteristic
329C2×C3×C4C24×C311yescharacteristic
330C2×Dic3—216no—
331C2×Dic3—28no—
332C2×Dic3—24no—
333Dic6—216no—
334Dic6—28no—
335Dic6—24no—
336Dic6C2×C3×D418yescomplemented
337C3×C8—64no—
338C3⋊C8—24no—
339C2×C32—24no—
340C2×C32—82no—
341C2×C32#44814yescomplemented
342C2×C32C2×#12011yescharacteristic
343C2×C32C22×SD1611yescharacteristic
344C2×C32C2×#13711yescharacteristic
345C24—21no—
346C24C3×#6711yescharacteristic
347C2×Q8—34no—
348C2×Q8—38no—
349C2×Q8—616no—
350C2×Q8—68no—
351C2×Q8—616no—
352C2×Q8—64no—
353C2×Q8—64no—
354C2×Q8—62no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4—44no—
359C2×D4C22×C3×D312yes—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—68no—
363C22×C4—64no—
364C22×C4—62no—
365C22×C4C3×#6711yescharacteristic
366C22×C4C3×#6712yes—
367C22×C4C22×C3×D312yes—
368C22×C4C22×C3×D311yescharacteristic
369SD16—616no—
370C2×C8—34no—
371C2×C8—64no—
372C4⋊C4—21no—
373C4⋊C4C22×C3×D312yes—
374C22⋊C4—24no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—28no—
378C22×C3—22no—
379C22×C3—28no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—48no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—88no—
399C22×C3—84no—
400C22×C3—84no—
401C22×C3C2×#21412yes—
402C22×C3C2×#21414yescomplemented
403C22×C3#93712yes—
404C22×C3#95112yes—
405C22×C3C3×#12012yes—
406C22×C3C2×C3×SD1612yes—
407C22×C3C2×C3×SD1614yescomplemented
408C22×C3C3×#13712yes—
409C22×C3C22×#6711yescharacteristic
410C22×C3C22×C3×D411yescharacteristic
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—44no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—68no—
420C3×C4#93712yes—
421C3×C4C3×#12012yes—
422C3×C4C2×D3×D412yes—
423C3×C4C22×C3×D412yes—
424Dic3—28no—
425C32C2×#44811yescharacteristic, complemented, Sylow
426C23—22no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C3×#21414yescomplemented
433C23C2×C3×#6712yes—
434C23C2×C3×#6711yescharacteristic
435Q8—38no—
436Q8—616no—
437Q8—68no—
438Q8—64no—
439D4—24no—
440D4—48no—
441D4—42no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—616no—
448C2×C4—68no—
449C2×C4—64no—
450C2×C4C3×D3×D414yes—
451C2×C4C2×C3×#6712yes—
452C2×C4C2×C3×#6711yescharacteristic
453C2×C4C23×C3×D311yescharacteristic
454C8—64no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—24no—
461C2×C3—44no—
462C2×C3—84no—
463C2×C3—82no—
464C2×C3—82no—
465C2×C3#504014yesdirect product factor
466C2×C3C3×#44814yescomplemented
467C2×C3C22×#21411yescharacteristic
468C2×C3C2×#93711yescharacteristic
469C2×C3C2×#95111yescharacteristic
470C2×C3C2×C3×#12011yescharacteristic
471C2×C3C22×C3×SD1611yescharacteristic
472C2×C3C2×C3×#13711yescharacteristic
473C22—22no—
474C22—22no—
475C22—28no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—84no—
480C22C2×C3×#21412yes—
481C22C2×C3×#21414yescomplemented
482C22C3×#93712yes—
483C22C3×#95112yes—
484C22C22×C3×#6711yescharacteristic
485C4—22no—
486C4—42no—
487C4—68no—
488C4C3×#93712yes—
489C4C2×C3×D3×D412yes—
490C3—21no—
491C3C2×#504011yescharacteristic, direct product factor
492C3C2×C3×#44811yescharacteristic, complemented
493C2—24no—
494C2—82no—
495C2C3×#504014yesdirect product factor
496C2C22×C3×#21411yescharacteristic
497C2C2×C3×#93711yescharacteristic
498C2C2×C3×#95111yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×#121C2
socleC23×C32C2×D4
hypercenterC2×C3×#121D3
hypocenterC3C2×C3×#448
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
1. C23×C3×Dic6C2semidirect
3. C22×C3×#214C2semidirect
4. C2×C3×#830C2semidirect
5. C3×#5040C2direct (N × H)
6. C2×C3×#810C2semidirect
8. C2×#5040C3direct (N × H)
17. C22×C3×Dic6C22semidirect
23. C2×C3×#214C22semidirect
25. C3×#830C22semidirect
26. C3×#810C22semidirect
28. C23×Dic6C2×C3semidirect
33. C22×#214C2×C3semidirect
37. C2×#830C2×C3semidirect
38. #5040C2×C3direct (N × H)
39. C2×#810C2×C3semidirect
62. C2×C3×Dic6D4semidirect
91. C22×Dic6C22×C3semidirect
108. C2×#214C22×C3semidirect
112. #830C22×C3semidirect
113. #810C22×C3semidirect
120. C23×C32SD16semidirect
141. C3×Dic6C2×D4semidirect
208. C2×Dic6C3×D4semidirect
234. C22×C32C2×SD16semidirect
281. C23×C3#214semidirect
282. C23×C3C3×SD16semidirect
336. Dic6C2×C3×D4semidirect
341. C2×C32#448semidirect
402. C22×C3C2×#214semidirect
407. C22×C3C2×C3×SD16semidirect
425. C32C2×#448semidirect
432. C23C3×#214semidirect
465. C2×C3#5040direct (N × H)
466. C2×C3C3×#448semidirect
481. C22C2×C3×#214semidirect
491. C3C2×#5040direct (N × H)
492. C3C2×C3×#448semidirect
495. C2C3×#5040direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Dic62
C2×C32×#1212
C22×C3×#2142
C2×C3×#8302
C3×#50402
C2×C3×#8102
C2×C3×#4483
C2×#50403