Finite Group Databasebeta

← C3×#82540 #11866171

Subgroups — C3×#82540 #11866171

Order 1152.

1221 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#518C211yesmaximal, characteristic, complemented
2C2×C3×#932C211yesmaximal, characteristic, complemented
3C22×C3×#213C211yesmaximal, characteristic, complemented
4C3×#5525C211yesmaximal, characteristic, complemented
5C3×#5016C214yesmaximal, complemented
6C3×#5015C214yesmaximal, complemented
7C3×#5002C211yesmaximal, characteristic, complemented
8C3×#5001C211yesmaximal, characteristic, complemented
9C3×#4999C211yesmaximal, characteristic, complemented
10C3×#3041—31nomaximal
11#82540C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C32×C4×D4—22no—
16C32×C4×D4C2214yescomplemented
17C32×#117C2211yescharacteristic
18C32×#117C2211yescharacteristic
19C2×C32×C22⋊C4—22no—
20C2×C3×#214C2212yescomplemented
21C3×#932—24no—
22C2×C3×#213—24no—
23C2×C3×#213C2214yes—
24C2×C3×#213C2212yescomplemented
25C2×C3×#208C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C3×#895—22no—
28C3×#882—22no—
29C3×#875C2212yescomplemented
30C3×#874C2212yescomplemented
31C3×#810C2212yescomplemented
32C3×#809C2214yescomplemented
33C3×#809C2212yescomplemented
34C3×C4⋊(C3⋊C8)C2214yescomplemented
35C2×C3×AGL(1,ℤ/8)—31no—
36C22×C3×D8—31no—
37C3×#535—31no—
38C3×#518—21no—
39C3×#518D311yescharacteristic, complemented
40C3×#518C2×C311yescharacteristic, complemented
41C2×#932C2×C311yescharacteristic, complemented
42C22×#213C2×C311yescharacteristic, complemented
43#5525C2×C311yescharacteristic, complemented
44C3×#463—34no—
45C3×#459—34no—
46C3×#424—31no—
47C3×#418—31no—
48C3×#416—31no—
49#5016C2×C314yescomplemented
50#5015C2×C314yescomplemented
51#5002C2×C311yescharacteristic, complemented
52#5001C2×C311yescharacteristic, complemented
53#4999C2×C311yescharacteristic, complemented
54C24×C32—21no—
55C23×C3×D3—21no—
56C2×C32×D4—21no—
57C2×C32×D4—22no—
58C2×C32×D4D414yescomplemented
59C2×C32×D4C2312yes—
60C22×C32×C4—24no—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—21no—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12—22no—
66C2×C3×D12—22no—
67C2×C3×D12—21no—
68C2×C3×D12—24no—
69C2×C3×D12C2312yes—
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2311yescharacteristic
73C32×C4⋊C4D412yes—
74C32×C4⋊C4C2312yes—
75C32×C22⋊C4—21no—
76C32×C22⋊C4—24no—
77C32×C22⋊C4D412yes—
78C32×C42—21no—
79C32×C42C2312yes—
80C3×#214—24no—
81C3×#213—28no—
82C3×#213—24no—
83C3×(C2×D3)⋊C4—24no—
84C3×#208—22no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2312yes—
87C2×C3×C3⋊C8C2312yes—
88#3041—31noSylow
89C2×C3×#41—31no—
90C22×C3×D4—21no—
91C22×C3×D4—31no—
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C22×C311yescharacteristic
94C2×#235C22×C311yescharacteristic
95C22×D12C22×C311yescharacteristic
96C3×AGL(1,ℤ/8)—64no—
97C2×C3×SD16—32no—
98C2×C3×D8—34no—
99C2×C3×D8—32no—
100C2×C3×D8—64no—
101C2×C3×M16—31no—
102C22×C3×C8—31no—
103C3×#127—32no—
104C3×#124—32no—
105C3×#121—62no—
106C3×#120—62no—
107C3×C4×D4—24no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—24no—
111C3×C4×D4C2×D314yescomplemented
112C3×C4×D4C22×C314yescomplemented
113C3×#117—21no—
114C3×#117—21no—
115C3×#117C2×D311yescharacteristic
116C3×#117C2×D311yescharacteristic
117C3×#117C22×C311yescharacteristic
118C3×#117C22×C311yescharacteristic
119C2×C3×C22⋊C4—24no—
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—22no—
122C2×#214C22×C312yescomplemented
123#932—24no—
124C2×#213—24no—
125C2×#213C22×C314yes—
126C2×#213C22×C312yescomplemented
127C2×#208C22×C311yescharacteristic
128C22×C3⋊C8C22×C311yescharacteristic
129#895—22no—
130#882—22no—
131#875C22×C312yescomplemented
132#874C22×C312yescomplemented
133C3×C4⋊C8—34no—
134C3×#103—32no—
135C3×#102—32no—
136C3×#102—34no—
137#810C22×C312yescomplemented
138#809C22×C314yescomplemented
139#809C22×C312yescomplemented
140C4⋊(C3⋊C8)C22×C314yescomplemented
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—82no—
144C23×C32C2×D411yescharacteristic
145C22×C3×D3—22no—
146C22×C3×D3—21no—
147C22×C3×D3—42no—
148C22×C3×D3—82no—
149C32×D4—24no—
150C32×D4—22no—
151C32×D4C2×D414yes—
152C2×C32×C4—22no—
153C2×C32×C4—84no—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D414yes—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4#4112yes—
158C2×C32×C4C2411yescharacteristic
159C3×#67—42no—
160C2×C3×Dic3—21no—
161C3×D12—22no—
162C3×D12—21no—
163C3×D12—22no—
164C3×D12—21no—
165C3×D12—42no—
166C3×D12—84no—
167C3×C4×D3—42no—
168C3×Dic6—21no—
169C3×Dic6—21no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—22no—
172C2×AGL(1,ℤ/8)—31no—
173C22×D8—31no—
174#535—31no—
175#518C3×D311yescharacteristic, complemented
176#463—34no—
177#459—34no—
178#424—31no—
179#418—31no—
180#416—31no—
181C24×C3—22no—
182C24×C3—21no—
183C24×C3—21no—
184C24×C3—61no—
185C23×D3—21no—
186C3×#41—62no—
187C3×#41—62no—
188C2×C3×Q8—31no—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—32no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C2×C3×D4—62no—
201C2×C3×D4—62no—
202C2×C3×D4—61no—
203C2×C3×D4—64no—
204C2×C3×D4D1214yescomplemented
205C2×C3×D4C3×D414yescomplemented
206C2×C3×D4C22×D312yes—
207C2×C3×D4C23×C312yes—
208C22×C3×C4—28no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—61no—
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C23×C311yescharacteristic
215C2×#67—21no—
216#235—22no—
217#235—22no—
218C2×D12—22no—
219C2×D12—22no—
220C2×D12—21no—
221C2×D12—24no—
222C2×D12C23×C311yescharacteristic
223C2×D12C23×C312yes—
224C2×C4×D3—21no—
225C2×Dic6C23×C311yescharacteristic
226C3×SD16—64no—
227C3×D8—68no—
228C3×D8—64no—
229C3×M16—62no—
230C2×C3×C8—32no—
231C2×C3×C8—32no—
232C2×C3×C8—62no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4#6712yes—
236C3×C4⋊C4C3×D412yes—
237C3×C4⋊C4C22×D312yes—
238C3×C4⋊C4C23×C312yes—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—28no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4#6712yes—
248C3×C22⋊C4C3×D412yes—
249C3×C42—22no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—22no—
253C3×C42C22×D312yes—
254C3×C42C23×C312yes—
255#214—24no—
256#213—28no—
257#213—24no—
258(C2×D3)⋊C4—24no—
259#208—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8C23×C312yes—
262C2×C3⋊C8C23×C312yes—
263C22×C32—24no—
264C22×C32—22no—
265C22×C32—44no—
266C22×C32—84no—
267C22×C32#12112yes—
268C22×C32C22×D411yescharacteristic
269C2×C3×D3—41no—
270C2×C3×D3—42no—
271C2×C3×D3—42no—
272C2×C3×D3—44no—
273C2×C3×D3—84no—
274C32×C4—24no—
275C32×C4—42no—
276C32×C4#12112yes—
277C32×C4C22×D411yescharacteristic
278C32×C4C2×#4111yescharacteristic
279C3×Dic3—41no—
280C2×#41—31no—
281C22×D4—31no—
282C22×D4C2×C3×D311yescharacteristic
283AGL(1,ℤ/8)—64no—
284C2×SD16—32no—
285C2×D8—34no—
286C2×D8—32no—
287C2×D8—64no—
288C2×M16—31no—
289C22×C8—31no—
290#127—32no—
291#124—32no—
292#121—62no—
293#120—62no—
294C4×D4—22no—
295C4×D4C2×C3×D314yescomplemented
296#117C2×C3×D311yescharacteristic
297#117C2×C3×D311yescharacteristic
298C2×C22⋊C4—22no—
299C4⋊C8—34no—
300#103—32no—
301#102—32no—
302#102—34no—
303C23×C3—24no—
304C23×C3—28no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—21no—
310C23×C3—62no—
311C23×C3—61no—
312C23×C3—82no—
313C23×C3—82no—
314C23×C3—84no—
315C23×C3—122no—
316C23×C3—242no—
317C23×C3C2×#6711yescharacteristic
318C23×C3C2×C3×D411yescharacteristic
319C22×D3—22no—
320C22×D3—21no—
321C22×D3—42no—
322C22×D3—82no—
323C3×Q8—61no—
324C3×Q8—61no—
325C3×D4—28no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—24no—
332C3×D4—62no—
333C3×D4—61no—
334C3×D4—62no—
335C3×D4—61no—
336C3×D4—122no—
337C3×D4—122no—
338C3×D4—244no—
339C3×D4C2×D1214yes—
340C3×D4C2×C3×D414yes—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—61no—
350C2×C3×C4—84no—
351C2×C3×C4—84no—
352C2×C3×C4—88no—
353C2×C3×C4—122no—
354C2×C3×C4C2×D1212yes—
355C2×C3×C4#23512yes—
356C2×C3×C4C2×#6714yes—
357C2×C3×C4C2×#6711yescharacteristic
358C2×C3×C4C2×C3×D412yes—
359C2×C3×C4C2×C3×D414yes—
360C2×C3×C4C2×C3×D411yescharacteristic
361C2×C3×C4C3×#4112yes—
362C2×C3×C4C23×D311yescharacteristic
363C2×C3×C4C24×C311yescharacteristic
364#67—42no—
365C2×Dic3—21no—
366D12—22no—
367D12—21no—
368D12—22no—
369D12—21no—
370D12—42no—
371D12—84no—
372C4×D3—42no—
373Dic6—21no—
374Dic6—21no—
375C3×C8—62no—
376C3×C8—62no—
377C3⋊C8—22no—
378C3⋊C8—22no—
379C2×C32—22no—
380C2×C32—44no—
381C2×C32C2×#12111yescharacteristic
382C2×C32#57612yes—
383C3×D3—81no—
384C3×D3—82no—
385C24—21no—
386C24—61no—
387#41—62no—
388#41—62no—
389C2×Q8—31no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—32no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—61no—
398C2×D4—64no—
399C2×D4C3×D1214yescomplemented
400C2×D4C22×C3×D312yes—
401C22×C4—24no—
402C22×C4—61no—
403C22×C4C22×C3×D311yescharacteristic
404SD16—64no—
405D8—68no—
406D8—64no—
407M16—62no—
408C2×C8—32no—
409C2×C8—32no—
410C2×C8—62no—
411C4⋊C4C3×#6712yes—
412C4⋊C4C22×C3×D312yes—
413C22⋊C4—21no—
414C22⋊C4—24no—
415C22⋊C4—64no—
416C22⋊C4C3×#6712yes—
417C42—21no—
418C42C22×C3×D312yes—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—28no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—48no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—84no—
431C22×C3—84no—
432C22×C3—88no—
433C22×C3—121no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—124no—
437C22×C3—244no—
438C22×C3#93012yes—
439C22×C3C3×#12112yes—
440C22×C3C22×#6711yescharacteristic
441C22×C3C22×C3×D411yescharacteristic
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—41no—
445C2×D3—42no—
446C2×D3—84no—
447C3×C4—24no—
448C3×C4—24no—
449C3×C4—22no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—44no—
453C3×C4—44no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—121no—
457C3×C4#93012yes—
458C3×C4C3×#12112yes—
459C3×C4C22×D1211yescharacteristic
460C3×C4C2×#23511yescharacteristic
461C3×C4C22×C3×D411yescharacteristic
462C3×C4C2×C3×#4111yescharacteristic
463Dic3—41no—
464C32#304111yescharacteristic, complemented, Sylow
465C23—22no—
466C23—24no—
467C23—62no—
468C23—61no—
469C23—82no—
470C23—122no—
471C23—242no—
472C23C2×C3×#6711yescharacteristic
473Q8—61no—
474Q8—61no—
475D4—24no—
476D4—22no—
477D4—62no—
478D4—61no—
479D4—62no—
480D4—61no—
481D4—122no—
482D4—122no—
483D4—244no—
484D4C2×C3×D1214yes—
485C2×C4—22no—
486C2×C4—61no—
487C2×C4—84no—
488C2×C4—122no—
489C2×C4C2×C3×D1212yes—
490C2×C4C3×#23512yes—
491C2×C4C2×C3×#6714yes—
492C2×C4C2×C3×#6711yescharacteristic
493C2×C4C23×C3×D311yescharacteristic
494C8—62no—
495C8—62no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—42no—
501C2×C3—48no—
502C2×C3—44no—
503C2×C3—44no—
504C2×C3—242no—
505C2×C3—241no—
506C2×C3C2×#93011yescharacteristic
507C2×C3#583512yes—
508C2×C3C2×C3×#12111yescharacteristic
509C2×C3C3×#57612yes—
510D3—82no—
511D3—81no—
512C22—22no—
513C22—24no—
514C22—44no—
515C22—84no—
516C22—121no—
517C22—122no—
518C22—122no—
519C22—124no—
520C22—244no—
521C22C3×#93012yes—
522C22C22×C3×#6711yescharacteristic
523C4—24no—
524C4—42no—
525C4—121no—
526C4C3×#93012yes—
527C4C22×C3×D1211yescharacteristic
528C4C2×C3×#23511yescharacteristic
529C3—21no—
530C3#8254011yescharacteristic, direct product factor
531C3C3×#304111yescharacteristic, complemented
532C2—22no—
533C2—44no—
534C2—242no—
535C2—241no—
536C2C2×C3×#93011yescharacteristic
537C2C3×#583512yes—
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#518C2
socleC22×C32C22×D4
hypercenterC3×#518D3
hypocenterC3C3×#3041
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#518

Complements

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

NH ≅ G/Ntype
1. C32×#518C2semidirect
2. C2×C3×#932C2semidirect
3. C22×C3×#213C2semidirect
4. C3×#5525C2semidirect
5. C3×#5016C2semidirect
6. C3×#5015C2semidirect
7. C3×#5002C2semidirect
8. C3×#5001C2semidirect
9. C3×#4999C2semidirect
11. #82540C3direct (N × H)
16. C32×C4×D4C22semidirect
20. C2×C3×#214C22semidirect
24. C2×C3×#213C22semidirect
29. C3×#875C22semidirect
30. C3×#874C22semidirect
31. C3×#810C22semidirect
32. C3×#809C22semidirect
33. C3×#809C22semidirect
34. C3×C4⋊(C3⋊C8)C22semidirect
39. C3×#518D3semidirect
40. C3×#518C2×C3semidirect
41. C2×#932C2×C3semidirect
42. C22×#213C2×C3semidirect
43. #5525C2×C3semidirect
49. #5016C2×C3semidirect
50. #5015C2×C3semidirect
51. #5002C2×C3semidirect
52. #5001C2×C3semidirect
53. #4999C2×C3semidirect
58. C2×C32×D4D4semidirect
111. C3×C4×D4C2×D3semidirect
112. C3×C4×D4C22×C3semidirect
122. C2×#214C22×C3semidirect
126. C2×#213C22×C3semidirect
131. #875C22×C3semidirect
132. #874C22×C3semidirect
137. #810C22×C3semidirect
138. #809C22×C3semidirect
139. #809C22×C3semidirect
140. C4⋊(C3⋊C8)C22×C3semidirect
175. #518C3×D3semidirect
204. C2×C3×D4D12semidirect
205. C2×C3×D4C3×D4semidirect
295. C4×D4C2×C3×D3semidirect
399. C2×D4C3×D12semidirect
464. C32#3041semidirect
530. C3#82540direct (N × H)
531. C3C3×#3041semidirect

Maximal subgroups

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

subgroupindex
C32×#5182
C2×C3×#9322
C22×C3×#2132
C3×#55252
C3×#50162
C3×#50152
C3×#50022
C3×#50012
C3×#49992
C3×#30413
#825403