Finite Group Databasebeta

← C2×C3×#5062 #11864854

Subgroups — C2×C3×#5062 #11864854

Order 1152.

1621 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C211yesmaximal, characteristic, complemented
2C2×C32×#124C211yesmaximal, characteristic, complemented
3C2×C3×#932C212yesmaximal, complemented
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C3×#5062C218yesmaximal, direct product factor
6C2×C3×#805C212yesmaximal, complemented
7C2×C3×#454—31nomaximal
8C2×#5062C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2211yescharacteristic
10C22×C32×D4C2211yescharacteristic
11C3×#1007—22no—
12C3×#1007—24no—
13C3×#1007C2214yescomplemented
14C2×C3×#239—22no—
15C2×C3×D3×Q8—22no—
16C2×C3×#235—22no—
17C2×C3×#235C2212yes—
18C32×#124—22no—
19C32×#124C2214yescomplemented
20C2×C32×C22⋊C4—22no—
21C2×C32×C42C2211yescharacteristic
22C2×C3×#214—22no—
23C3×#932—24no—
24C3×#932C2218yescomplemented
25C2×C3×#213—22no—
26C2×C3×#208C2212yes—
27C3×#836C2214yescomplemented
28C3×#805C2218yescomplemented
29C2×C3×2−1+4—31no—
30C2×#1007C2×C311yescharacteristic, complemented
31C2×C3×AGL(1,ℤ/8)—32no—
32C2×C3×#124—21no—
33C2×C3×#124D311yescharacteristic, complemented
34C2×C3×#124C2×C311yescharacteristic, complemented
35C2×#932C2×C312yescomplemented
36C3×#454—38no—
37C2×C3×#104—32no—
38C2×C3×#101—31no—
39C2×#836C2×C311yescharacteristic, complemented
40#5062C2×C318yesdirect product factor
41C2×#805C2×C312yescomplemented
42C24×C32—21no—
43C2×C32×Q8—22no—
44C2×C32×Q8—22no—
45C2×C32×Q8D412yes—
46C2×C32×Q8C2312yes—
47C2×C32×D4—22no—
48C2×C32×D4—22no—
49C2×C32×D4—21no—
50C2×C32×D4C2312yes—
51C22×C32×C4—21no—
52C22×C32×C4D412yes—
53C22×C32×C4C2311yescharacteristic
54C2×C3×#67—22no—
55C3×#239—28no—
56C3×#239—28no—
57C3×D3×Q8—28no—
58C3×D3×Q8—28no—
59C3×#235—28no—
60C3×#235—28no—
61C3×#235—22no—
62C3×#235—22no—
63C3×#235D414yes—
64C3×#235C2314yes—
65C2×C3×D12—22no—
66C2×C3×D12D412yes—
67C2×C3×C4×D3—24no—
68C2×C3×C4×D3—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6D412yes—
71C32×C22⋊C4—28no—
72C32×C42—21no—
73C32×C42C2312yes—
74C3×#214—28no—
75C3×#213—28no—
76C3×#208—22no—
77C3×#208C2314yes—
78C2×C3×C3⋊C8—22no—
79C2×#454—31noSylow
80C3×2−1+4—34no—
81C3×2−1+4—64no—
82C3×2−1+4—62no—
83C2×C3×#41—32no—
84C2×C3×#41—62no—
85C2×C3×#41—62no—
86C22×C3×Q8—21no—
87C22×C3×Q8—62no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—21no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93#1007—22no—
94#1007—24no—
95#1007C22×C314yescomplemented
96C2×#239—22no—
97C2×D3×Q8—22no—
98C2×#235—22no—
99C2×#235C22×C312yes—
100C3×AGL(1,ℤ/8)—38no—
101C3×AGL(1,ℤ/8)—64no—
102C2×C3×SD16—62no—
103C2×C3×D8—62no—
104C2×C3×M16—32no—
105C3×#124—24no—
106C3×#124—22no—
107C3×#124—22no—
108C3×#124—24no—
109C3×#124C2×D314yescomplemented
110C3×#124C22×C314yescomplemented
111C2×C3×C22⋊C4—24no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C42—21no—
115C2×C3×C42C2×D311yescharacteristic
116C2×C3×C42C22×C311yescharacteristic
117C2×#214—22no—
118#932—24no—
119#932C22×C318yescomplemented
120C2×#213—22no—
121C2×#208C22×C312yes—
122C3×#104—38no—
123C3×#101—34no—
124#836C22×C314yescomplemented
125#805C22×C318yescomplemented
126C23×C32—22no—
127C23×C32—44no—
128C23×C32—41no—
129C23×C32—41no—
130C23×C32C2×D411yescharacteristic
131C22×C3×D3—22no—
132C32×Q8—24no—
133C32×Q8—24no—
134C32×D4—24no—
135C32×D4—42no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—42no—
140C2×C32×C4C2×D412yes—
141C2×C32×C4C2×D412yes—
142C2×C32×C4C2×D414yes—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2411yescharacteristic
145C3×#67—24no—
146C3×#67—24no—
147C2×C3×Dic3—22no—
148C3×D12—28no—
149C3×D12—22no—
150C3×D12C2×D414yes—
151C3×C4×D3—216no—
152C3×C4×D3—24no—
153C3×C4×D3—24no—
154C3×Dic6—28no—
155C3×Dic6—22no—
156C3×Dic6C2×D414yes—
157C3×C3⋊C8—24no—
158C2×2−1+4—31no—
159C2×AGL(1,ℤ/8)—32no—
160C2×#124C3×D311yescharacteristic, complemented
161#454—38no—
162C2×#104—32no—
163C2×#101—31no—
164C24×C3—22no—
165C24×C3—21no—
166C24×C3—21no—
167C3×#41—34no—
168C3×#41—34no—
169C3×#41—68no—
170C3×#41—68no—
171C3×#41—68no—
172C3×#41—62no—
173C3×#41—62no—
174C3×#41—68no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—32no—
182C2×C3×Q8—42no—
183C2×C3×Q8—42no—
184C2×C3×Q8—62no—
185C2×C3×Q8—68no—
186C2×C3×Q8—68no—
187C2×C3×Q8#6712yes—
188C2×C3×Q8C3×D412yes—
189C2×C3×Q8C22×D312yes—
190C2×C3×Q8C23×C312yes—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—32no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4—62no—
203C2×C3×D4—62no—
204C2×C3×D4C22×D312yes—
205C2×C3×D4C23×C312yes—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—64no—
212C22×C3×C4—62no—
213C22×C3×C4#6712yes—
214C22×C3×C4C3×D412yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C2×#67—22no—
218#239—28no—
219#239—28no—
220D3×Q8—28no—
221D3×Q8—28no—
222#235—28no—
223#235—28no—
224#235—22no—
225#235—22no—
226#235C3×D414yes—
227#235C23×C314yes—
228C2×D12—22no—
229C2×D12C3×D412yes—
230C2×C4×D3—24no—
231C2×C4×D3—22no—
232C2×Dic6—22no—
233C2×Dic6C3×D412yes—
234C3×SD16—68no—
235C3×D8—68no—
236C3×M16—34no—
237C3×M16—62no—
238C2×C3×C8—62no—
239C3×C22⋊C4—216no—
240C3×C22⋊C4—28no—
241C3×C22⋊C4—28no—
242C3×C42—22no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—22no—
246C3×C42C22×D312yes—
247C3×C42C23×C312yes—
248#214—28no—
249#213—28no—
250#208—22no—
251#208C23×C314yes—
252C2×C3⋊C8—22no—
253C22×C32—22no—
254C22×C32—44no—
255C22×C32—42no—
256C22×C32—82no—
257C22×C32—82no—
258C22×C32#12011yescharacteristic
259C22×C32#12011yescharacteristic
260C22×C32C22×D411yescharacteristic
261C2×C3×D3—24no—
262C2×C3×D3—44no—
263C32×C4—24no—
264C32×C4—42no—
265C32×C4#12012yes—
266C32×C4C22×D412yes—
267C3×Dic3—24no—
2682−1+4—34no—
2692−1+4—64no—
2702−1+4—62no—
271C2×#41—32no—
272C2×#41—62no—
273C2×#41—62no—
274C22×Q8—62no—
275C22×Q8C2×C3×D311yescharacteristic
276C22×D4C2×C3×D311yescharacteristic
277AGL(1,ℤ/8)—38no—
278AGL(1,ℤ/8)—64no—
279C2×SD16—62no—
280C2×D8—62no—
281C2×M16—32no—
282#124—22no—
283#124C2×C3×D314yescomplemented
284C2×C22⋊C4—22no—
285C2×C42C2×C3×D311yescharacteristic
286#104—38no—
287#101—34no—
288C23×C3—24no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—48no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—62no—
302C23×C3C2×#6711yescharacteristic
303C23×C3C2×C3×D411yescharacteristic
304C22×D3—22no—
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—24no—
309C3×Q8—34no—
310C3×Q8—44no—
311C3×Q8—44no—
312C3×Q8—68no—
313C3×Q8—62no—
314C3×D4—24no—
315C3×D4—24no—
316C3×D4—34no—
317C3×D4—44no—
318C3×D4—44no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—68no—
322C3×D4—64no—
323C3×D4—64no—
324C3×D4—62no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—24no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—616no—
343C2×C3×C4—62no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4D3×D412yes—
347C2×C3×C4D3×D412yes—
348C2×C3×C4C2×#6714yes—
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×C3×D412yes—
351C2×C3×C4C2×C3×D412yes—
352C2×C3×C4C2×C3×D414yes—
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C24×C311yescharacteristic
356#67—24no—
357#67—24no—
358C2×Dic3—22no—
359D12—28no—
360D12—22no—
361D12C2×C3×D414yes—
362C4×D3—216no—
363C4×D3—24no—
364C4×D3—24no—
365Dic6—28no—
366Dic6—22no—
367Dic6C2×C3×D414yes—
368C3×C8—64no—
369C3⋊C8—24no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—82no—
373C2×C32#45412yescomplemented
374C2×C32C2×#12011yescharacteristic
375C3×D3—44no—
376C24—21no—
377#41—34no—
378#41—34no—
379#41—68no—
380#41—68no—
381#41—68no—
382#41—62no—
383#41—62no—
384#41—68no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—32no—
388C2×Q8—62no—
389C2×Q8—68no—
390C2×Q8—68no—
391C2×Q8C3×#6712yes—
392C2×Q8C22×C3×D312yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—32no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4C22×C3×D312yes—
400C22×C4—21no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4C3×#6712yes—
404C22×C4C22×C3×D311yescharacteristic
405SD16—68no—
406D8—68no—
407M16—34no—
408M16—62no—
409C2×C8—62no—
410C22⋊C4—28no—
411C42—21no—
412C42C22×C3×D312yes—
413C22×C3—24no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—44no—
420C22×C3—48no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—64no—
426C22×C3—84no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—84no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—124no—
433C22×C3#93711yescharacteristic
434C22×C3#93711yescharacteristic
435C22×C3C3×#12011yescharacteristic
436C22×C3C3×#12011yescharacteristic
437C22×C3C22×#6711yescharacteristic
438C22×C3C22×C3×D411yescharacteristic
439C2×D3—24no—
440C2×D3—44no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—64no—
450C3×C4#93712yes—
451C3×C4C3×#12012yes—
452C3×C4C2×D3×D412yes—
453C3×C4C22×C3×D412yes—
454Dic3—24no—
455C32C2×#45411yescharacteristic, complemented, Sylow
456C23—22no—
457C23—44no—
458C23—41no—
459C23—41no—
460C23—62no—
461C23C2×C3×#6711yescharacteristic
462Q8—24no—
463Q8—24no—
464Q8—34no—
465Q8—68no—
466Q8—62no—
467D4—24no—
468D4—34no—
469D4—42no—
470D4—68no—
471D4—64no—
472D4—64no—
473D4—62no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—616no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4C3×D3×D412yes—
483C2×C4C3×D3×D412yes—
484C2×C4C2×C3×#6714yes—
485C2×C4C2×C3×#6711yescharacteristic
486C2×C4C23×C3×D311yescharacteristic
487C8—64no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—84no—
497C2×C3—82no—
498C2×C3—82no—
499C2×C3—124no—
500C2×C3#506212yesdirect product factor
501C2×C3C3×#45412yescomplemented
502C2×C3C2×#93711yescharacteristic
503C2×C3C2×C3×#12011yescharacteristic
504D3—44no—
505C22—22no—
506C22—44no—
507C22—42no—
508C22—64no—
509C22—82no—
510C22—82no—
511C22—124no—
512C22C3×#93711yescharacteristic
513C22C3×#93711yescharacteristic
514C22C22×C3×#6711yescharacteristic
515C4—24no—
516C4—42no—
517C4—64no—
518C4C3×#93712yes—
519C4C2×C3×D3×D412yes—
520C3—21no—
521C3C2×#506211yescharacteristic, direct product factor
522C3C2×C3×#45411yescharacteristic, complemented
523C2—21no—
524C2—21no—
525C2—82no—
526C2—124no—
527C2C3×#506212yesdirect product factor
528C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#124C2
socleC22×C32#120
hypercenterC2×C3×#124D3
hypocenterC3C2×C3×#454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C2×C32×#124C2semidirect
3. C2×C3×#932C2semidirect
4. C2×C3×#836C2semidirect
5. C3×#5062C2direct (N × H)
6. C2×C3×#805C2semidirect
8. C2×#5062C3direct (N × H)
13. C3×#1007C22semidirect
19. C32×#124C22semidirect
24. C3×#932C22semidirect
27. C3×#836C22semidirect
28. C3×#805C22semidirect
30. C2×#1007C2×C3semidirect
33. C2×C3×#124D3semidirect
34. C2×C3×#124C2×C3semidirect
35. C2×#932C2×C3semidirect
39. C2×#836C2×C3semidirect
40. #5062C2×C3direct (N × H)
41. C2×#805C2×C3semidirect
95. #1007C22×C3semidirect
109. C3×#124C2×D3semidirect
110. C3×#124C22×C3semidirect
119. #932C22×C3semidirect
124. #836C22×C3semidirect
125. #805C22×C3semidirect
160. C2×#124C3×D3semidirect
283. #124C2×C3×D3semidirect
373. C2×C32#454semidirect
455. C32C2×#454semidirect
500. C2×C3#5062direct (N × H)
501. C2×C3C3×#454semidirect
521. C3C2×#5062direct (N × H)
522. C3C2×C3×#454semidirect
527. C2C3×#5062direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C2×C32×#1242
C2×C3×#9322
C2×C3×#8362
C3×#50622
C2×C3×#8052
C2×C3×#4543
C2×#50623