Finite Group Databasebeta

← C2×C3×#5061 #11864808

Subgroups — C2×C3×#5061 #11864808

Order 1152.

1685 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic
2C2×C32×#124C211yesmaximal, characteristic, complemented
3C2×C3×#942C212yesmaximal, complemented
4C2×C3×#833C211yesmaximal, characteristic, complemented
5C3×#5061C218yesmaximal, direct product factor
6C2×C3×#805C212yesmaximal, complemented
7C2×C3×#455—31nomaximal
8C2×#5061C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2211yescharacteristic, complemented
10C22×C32×D4C2211yescharacteristic
11C22×C3×#67—22no—
12C3×#1004—22no—
13C3×#1004—24no—
14C3×#1004C2214yes—
15C2×C3×#237—22no—
16C2×C3×D3×D4—22no—
17C2×C3×#235C2212yes—
18C32×#124—22no—
19C32×#124C2214yescomplemented
20C2×C32×C22⋊C4—22no—
21C2×C32×C42C2211yescharacteristic, complemented
22C2×C3×#216—22no—
23C3×#942—24no—
24C3×#942C2218yescomplemented
25C2×C3×#215—22no—
26C2×C3×#208C2212yescomplemented
27C3×#833C2214yescomplemented
28C3×#805C2218yescomplemented
29C2×C3×H(5,𝔽2)—31no—
30C2×#1004C2×C311yescharacteristic
31C2×C3×#137—32no—
32C2×C3×#124—21no—
33C2×C3×#124D311yescharacteristic, complemented
34C2×C3×#124C2×C311yescharacteristic, complemented
35C2×#942C2×C312yescomplemented
36C3×#455—38no—
37C2×C3×#104—32no—
38C2×C3×#100—31no—
39C2×#833C2×C311yescharacteristic, complemented
40#5061C2×C318yesdirect product factor
41C2×#805C2×C312yescomplemented
42C24×C32D412yes—
43C23×C3×D3—22no—
44C2×C32×Q8—21no—
45C2×C32×Q8—22no—
46C2×C32×Q8—22no—
47C2×C32×Q8C2312yescomplemented
48C2×C32×D4—22no—
49C2×C32×D4—22no—
50C2×C32×D4D412yes—
51C2×C32×D4C2312yes—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C2×C3×#67—24no—
56C2×C3×#67—28no—
57C2×C3×#67—22no—
58C2×C3×#67—28no—
59C22×C3×Dic3—22no—
60C3×#237—28no—
61C3×#237—28no—
62C3×D3×D4—28no—
63C3×D3×D4—28no—
64C3×#235—22no—
65C3×#235—22no—
66C3×#235D414yes—
67C3×#235C2314yes—
68C2×C3×D12D412yes—
69C2×C3×C4×D3—22no—
70C2×C3×Dic6D412yes—
71C32×C22⋊C4—28no—
72C32×C42—21no—
73C32×C42C2312yescomplemented
74C3×#216—28no—
75C3×#215—28no—
76C3×#208—22no—
77C3×#208C2314yescomplemented
78C2×C3×C3⋊C8—22no—
79C2×#455—31noSylow
80C3×H(5,𝔽2)—34no—
81C3×H(5,𝔽2)—64no—
82C3×H(5,𝔽2)—62no—
83C2×C3×#41—32no—
84C2×C3×#41—62no—
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic, complemented
87C22×C3×Q8C22×C311yescharacteristic, complemented
88C22×C3×D4—21no—
89C22×C3×D4—62no—
90C22×C3×D4—62no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93C22×#67—22no—
94#1004—22no—
95#1004—24no—
96#1004C22×C314yes—
97C2×#237—22no—
98C2×D3×D4—22no—
99C2×#235C22×C312yes—
100C3×#137—38no—
101C3×#137—64no—
102C2×C3×Q16—62no—
103C2×C3×SD16—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, complemented
116C2×C3×C42C22×C311yescharacteristic, complemented
117C2×#216—22no—
118#942—24no—
119#942C22×C318yescomplemented
120C2×#215—22no—
121C2×#208C22×C312yescomplemented
122C3×#104—38no—
123C3×#100—34no—
124#833C22×C314yescomplemented
125#805C22×C318yescomplemented
126C23×C32—22no—
127C23×C32—22no—
128C23×C32—44no—
129C23×C32C2×D411yescharacteristic
130C23×C32C2×D414yes—
131C22×C3×D3—22no—
132C22×C3×D3—28no—
133C22×C3×D3—84no—
134C32×Q8—24no—
135C32×Q8—42no—
136C32×D4—24no—
137C32×D4—24no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—42no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4C2×D412yes—
144C2×C32×C4C2×D412yes—
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2411yescharacteristic
147C3×#67—216no—
148C3×#67—24no—
149C3×#67—24no—
150C2×C3×Dic3—28no—
151C2×C3×Dic3—22no—
152C3×D12—22no—
153C3×D12C2×D414yes—
154C3×C4×D3—24no—
155C3×C4×D3—24no—
156C3×Dic6—22no—
157C3×Dic6C2×D414yes—
158C3×C3⋊C8—24no—
159C2×H(5,𝔽2)—31no—
160C2×#137—32no—
161C2×#124C3×D311yescharacteristic, complemented
162#455—38no—
163C2×#104—32no—
164C2×#100—31no—
165C24×C3—22no—
166C24×C3—62no—
167C24×C3#6712yes—
168C24×C3C3×D412yes—
169C23×D3—22no—
170C3×#41—34no—
171C3×#41—34no—
172C3×#41—68no—
173C3×#41—68no—
174C3×#41—62no—
175C3×#41—62no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—32no—
185C2×C3×Q8—42no—
186C2×C3×Q8—42no—
187C2×C3×Q8C22×D312yescomplemented
188C2×C3×Q8C23×C312yescomplemented
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—32no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—68no—
199C2×C3×D4—68no—
200C2×C3×D4—64no—
201C2×C3×D4—68no—
202C2×C3×D4—62no—
203C2×C3×D4—68no—
204C2×C3×D4#6712yes—
205C2×C3×D4C3×D412yes—
206C2×C3×D4C22×D312yes—
207C2×C3×D4C23×C312yes—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—62no—
216C22×C3×C4—62no—
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C2×#67—24no—
220C2×#67—28no—
221C2×#67—22no—
222C2×#67—28no—
223C22×Dic3—22no—
224#237—28no—
225#237—28no—
226D3×D4—28no—
227D3×D4—28no—
228#235—22no—
229#235—22no—
230#235C3×D414yes—
231#235C23×C314yes—
232C2×D12C3×D412yes—
233C2×C4×D3—22no—
234C2×Dic6C3×D412yes—
235C3×Q16—68no—
236C3×SD16—68no—
237C3×M16—34no—
238C3×M16—62no—
239C2×C3×C8—62no—
240C3×C22⋊C4—216no—
241C3×C22⋊C4—28no—
242C3×C22⋊C4—28no—
243C3×C42—22no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—22no—
247C3×C42C22×D312yescomplemented
248C3×C42C23×C312yescomplemented
249#216—28no—
250#215—28no—
251#208—22no—
252#208C23×C314yescomplemented
253C2×C3⋊C8—22no—
254C22×C32—22no—
255C22×C32—24no—
256C22×C32—44no—
257C22×C32—44no—
258C22×C32—82no—
259C22×C32#12011yescharacteristic
260C22×C32#12011yescharacteristic
261C22×C32C22×D411yescharacteristic
262C2×C3×D3—24no—
263C2×C3×D3—44no—
264C2×C3×D3—88no—
265C32×C4—42no—
266C32×C4—42no—
267C32×C4#12012yes—
268C32×C4C22×D412yes—
269C3×Dic3—24no—
270H(5,𝔽2)—34no—
271H(5,𝔽2)—64no—
272H(5,𝔽2)—62no—
273C2×#41—32no—
274C2×#41—62no—
275C22×Q8C2×C3×D311yescharacteristic, complemented
276C22×D4—62no—
277C22×D4—62no—
278C22×D4C2×C3×D311yescharacteristic
279#137—38no—
280#137—64no—
281C2×Q16—62no—
282C2×SD16—62no—
283C2×M16—32no—
284#124—22no—
285#124C2×C3×D314yescomplemented
286C2×C22⋊C4—22no—
287C2×C42C2×C3×D311yescharacteristic, complemented
288#104—38no—
289#100—34no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—24no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—48no—
301C23×C3—68no—
302C23×C3—62no—
303C23×C3—244no—
304C23×C3C2×#6711yescharacteristic
305C23×C3C2×#6714yes—
306C23×C3C2×C3×D411yescharacteristic
307C23×C3C2×C3×D414yes—
308C22×D3—22no—
309C22×D3—28no—
310C22×D3—84no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—34no—
314C3×Q8—44no—
315C3×Q8—44no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—62no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—24no—
323C3×D4—34no—
324C3×D4—44no—
325C3×D4—44no—
326C3×D4—616no—
327C3×D4—64no—
328C3×D4—64no—
329C3×D4—62no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—68no—
350C2×C3×C4—62no—
351C2×C3×C4—64no—
352C2×C3×C4—64no—
353C2×C3×C4D3×D412yes—
354C2×C3×C4D3×D412yes—
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×C3×D412yes—
357C2×C3×C4C2×C3×D412yes—
358C2×C3×C4C2×C3×D411yescharacteristic
359C2×C3×C4C23×D311yescharacteristic
360C2×C3×C4C24×C311yescharacteristic
361#67—216no—
362#67—24no—
363#67—24no—
364C2×Dic3—28no—
365C2×Dic3—22no—
366D12—22no—
367D12C2×C3×D414yes—
368C4×D3—24no—
369C4×D3—24no—
370Dic6—22no—
371Dic6C2×C3×D414yes—
372C3×C8—64no—
373C3⋊C8—24no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—44no—
377C2×C32#45512yescomplemented
378C2×C32C2×#12011yescharacteristic
379C3×D3—44no—
380C24—62no—
381C24C3×#6712yes—
382#41—34no—
383#41—34no—
384#41—68no—
385#41—68no—
386#41—62no—
387#41—62no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—32no—
392C2×Q8C22×C3×D312yescomplemented
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—32no—
396C2×D4—68no—
397C2×D4—68no—
398C2×D4—64no—
399C2×D4—68no—
400C2×D4—62no—
401C2×D4—68no—
402C2×D4C3×#6712yes—
403C2×D4C22×C3×D312yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4C22×C3×D311yescharacteristic
409Q16—68no—
410SD16—68no—
411M16—34no—
412M16—62no—
413C2×C8—62no—
414C22⋊C4—28no—
415C42—21no—
416C42C22×C3×D312yescomplemented
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—48no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—64no—
432C22×C3—84no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—84no—
436C22×C3—124no—
437C22×C3—248no—
438C22×C3#93711yescharacteristic
439C22×C3#93711yescharacteristic
440C22×C3C3×#12011yescharacteristic
441C22×C3C3×#12011yescharacteristic
442C22×C3C22×#6711yescharacteristic
443C22×C3C22×C3×D411yescharacteristic
444C2×D3—24no—
445C2×D3—44no—
446C2×D3—88no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—44no—
450C3×C4—44no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—64no—
456C3×C4#93712yes—
457C3×C4C3×#12012yes—
458C3×C4C2×D3×D412yes—
459C3×C4C22×C3×D412yes—
460Dic3—24no—
461C32C2×#45511yescharacteristic, complemented, Sylow
462C23—22no—
463C23—22no—
464C23—44no—
465C23—68no—
466C23—62no—
467C23—244no—
468C23C2×C3×#6711yescharacteristic
469C23C2×C3×#6714yes—
470Q8—24no—
471Q8—34no—
472Q8—42no—
473Q8—62no—
474D4—24no—
475D4—24no—
476D4—34no—
477D4—616no—
478D4—64no—
479D4—64no—
480D4—62no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—68no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4C3×D3×D412yes—
491C2×C4C3×D3×D412yes—
492C2×C4C2×C3×#6711yescharacteristic
493C2×C4C23×C3×D311yescharacteristic
494C8—64no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—44no—
504C2×C3—44no—
505C2×C3—84no—
506C2×C3—124no—
507C2×C3#506112yesdirect product factor
508C2×C3C3×#45512yescomplemented
509C2×C3C2×#93711yescharacteristic
510C2×C3C2×C3×#12011yescharacteristic
511D3—44no—
512C22—22no—
513C22—24no—
514C22—44no—
515C22—44no—
516C22—64no—
517C22—82no—
518C22—124no—
519C22—248no—
520C22C3×#93711yescharacteristic
521C22C3×#93711yescharacteristic
522C22C22×C3×#6711yescharacteristic
523C4—42no—
524C4—42no—
525C4—64no—
526C4C3×#93712yes—
527C4C2×C3×D3×D412yes—
528C3—21no—
529C3C2×#506111yescharacteristic, direct product factor
530C3C2×C3×#45511yescharacteristic, complemented
531C2—21no—
532C2—21no—
533C2—44no—
534C2—124no—
535C2C3×#506112yesdirect product factor
536C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#124C2
socleC22×C32#120
prime radicalC2×C3×#1004C2
hypercenterC2×C3×#124D3
hypocenterC3C2×C3×#455
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
2. C2×C32×#124C2semidirect
3. C2×C3×#942C2semidirect
4. C2×C3×#833C2semidirect
5. C3×#5061C2direct (N × H)
6. C2×C3×#805C2semidirect
8. C2×#5061C3direct (N × H)
9. C22×C32×Q8C22semidirect
19. C32×#124C22semidirect
21. C2×C32×C42C22semidirect
24. C3×#942C22semidirect
26. C2×C3×#208C22semidirect
27. C3×#833C22semidirect
28. C3×#805C22semidirect
33. C2×C3×#124D3semidirect
34. C2×C3×#124C2×C3semidirect
35. C2×#942C2×C3semidirect
39. C2×#833C2×C3semidirect
40. #5061C2×C3direct (N × H)
41. C2×#805C2×C3semidirect
47. C2×C32×Q8C23semidirect
73. C32×C42C23semidirect
77. C3×#208C23semidirect
86. C22×C3×Q8C2×D3semidirect
87. C22×C3×Q8C22×C3semidirect
109. C3×#124C2×D3semidirect
110. C3×#124C22×C3semidirect
115. C2×C3×C42C2×D3semidirect
116. C2×C3×C42C22×C3semidirect
119. #942C22×C3semidirect
121. C2×#208C22×C3semidirect
124. #833C22×C3semidirect
125. #805C22×C3semidirect
161. C2×#124C3×D3semidirect
187. C2×C3×Q8C22×D3semidirect
188. C2×C3×Q8C23×C3semidirect
247. C3×C42C22×D3semidirect
248. C3×C42C23×C3semidirect
252. #208C23×C3semidirect
275. C22×Q8C2×C3×D3semidirect
285. #124C2×C3×D3semidirect
287. C2×C42C2×C3×D3semidirect
377. C2×C32#455semidirect
392. C2×Q8C22×C3×D3semidirect
416. C42C22×C3×D3semidirect
461. C32C2×#455semidirect
507. C2×C3#5061direct (N × H)
508. C2×C3C3×#455semidirect
529. C3C2×#5061direct (N × H)
530. C3C2×C3×#455semidirect
535. C2C3×#5061direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C2×C32×#1242
C2×C3×#9422
C2×C3×#8332
C3×#50612
C2×C3×#8052
C2×C3×#4553
C2×#50613