Finite Group Databasebeta

← C3×#87016 #11903881

Subgroups — C3×#87016 #11903881

Order 1152.

2324 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C3×#5732C213yesmaximal, complemented
3C3×#5725C213yesmaximal, complemented
4C3×#5695C216yesmaximal, complemented
5C3×#5691C216yesmaximal, complemented
6C3×#5585C213yesmaximal, complemented
7C3×#5577C211yesmaximal, characteristic, complemented
8C3×#5576C212yesmaximal, complemented
9C3×C4×#239C213yesmaximal, complemented
10C3×C4×#239C212yesmaximal, complemented
11C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
12C3×#3480—31nomaximal
13#87016C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2212yes—
15C2×C3×#239C2213yes—
16C2×C3×D3×Q8C2211yescharacteristic
17C32×C4⋊Q8C2213yes—
18C32×#125C2216yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2212yes—
21C32×C4×Q8C2213yes—
22C3×#947C2216yes—
23C3×Q8×Dic3C2211yescharacteristic
24C3×Q8×Dic3C2213yes—
25C3×#898C22112yes—
26C3×#897C2216yes—
27C3×#895C2216yes—
28C3×#895C2216yes—
29C3×#895C2216yes—
30C3×#894C22112yes—
31C3×#893C2213yes—
32C3×#893C2216yes—
33C3×#893C2216yes—
34C3×#893C2216yes—
35C3×#892C2211yescharacteristic
36C3×#892C2216yes—
37C3×#892C2216yes—
38C3×D3×C4⋊C4C2213yes—
39C3×#887C2213yes—
40C3×#875C2216yes—
41C3×#874C2213yes—
42C3×C4×D12C2216yes—
43C3×C4×D12C2213yes—
44C3×C4×D12C2216yes—
45C3×C42×D3C2213yes—
46C3×C42×D3C2216yes—
47C3×C4×Dic6C2213yes—
48C3×#559—33no—
49C3×#557—21no—
50C3×#557D311yescharacteristic, complemented
51C3×#557C2×C311yescharacteristic, complemented
52C3×#556—33no—
53C3×#553—36no—
54C3×#550—31no—
55C3×#548—32no—
56C3×#532—33no—
57C3×#529—36no—
58C3×C4×#41—33no—
59C3×C4×#41—32no—
60C2×C3×C4×Q8—31no—
61#5732C2×C313yescomplemented
62#5725C2×C313yescomplemented
63#5695C2×C316yescomplemented
64#5691C2×C316yescomplemented
65#5585C2×C313yescomplemented
66#5577C2×C311yescharacteristic, complemented
67#5576C2×C312yescomplemented
68C4×#239C2×C312yescomplemented
69C4×#239C2×C313yescomplemented
70C4×D3×Q8C2×C311yescharacteristic, complemented
71C2×C32×Q8C2313yes—
72C2×C32×Q8C2311yescharacteristic
73C3×#239—28no—
74C3×#239—212no—
75C3×D3×Q8—24no—
76C2×C3×D12C2316yes—
77C2×C3×D12C2316yes—
78C2×C3×D12C2313yes—
79C2×C3×D12C2316yes—
80C2×C3×C4×D3C2313yes—
81C2×C3×C4×D3C2316yes—
82C2×C3×C4×D3C2316yes—
83C2×C3×C4×D3C2313yes—
84C2×C3×C4×D3C2316yes—
85C2×C3×C4×D3C2311yescharacteristic
86C2×C3×Dic6C2313yes—
87C32×C4⋊C4C2313yes—
88C32×C4⋊C4C2311yescharacteristic
89C32×C4⋊C4C2316yes—
90C32×C4⋊C4C2316yes—
91C32×C4⋊C4C2316yes—
92C32×C42C2316yes—
93C32×C42C2313yes—
94C3×(C2×D3)⋊C4C2312yes—
95C3×(C2×D3)⋊C4C2316yes—
96C3×(C2×D3)⋊C4C23112yes—
97C3×(C2×D3)⋊C4C2316yes—
98C3×(C2×D3)⋊C4C23112yes—
99C3×C4⋊Dic3C2313yes—
100C3×C4⋊Dic3C2316yes—
101C3×C4⋊Dic3C2311yescharacteristic
102C3×Dic3⋊C4C2316yes—
103C3×C4×Dic3C2313yes—
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C2312yes—
106C3×C4×Dic3C2316yes—
107C3×C4×Dic3C2313yes—
108#3480—31noSylow
109C2×C3×#41—33no—
110C2×C3×#41—32no—
111C22×C3×Q8—31no—
112C2×#239C22×C313yes—
113C2×#239C22×C312yes—
114C2×D3×Q8C22×C311yescharacteristic
115C3×C4⋊Q8—23no—
116C3×C4⋊Q8C2×D313yes—
117C3×C4⋊Q8C22×C313yes—
118C3×#127—33no—
119C3×#126—312no—
120C3×#125—26no—
121C3×#125C2×D316yes—
122C3×#125C22×C316yes—
123C3×#124—36no—
124C3×#124—36no—
125C3×#123—312no—
126C3×C22⋊Q8—36no—
127C3×#121—36no—
128C3×#121—36no—
129C3×#121—36no—
130C3×C4×Q8—21no—
131C3×C4×Q8—22no—
132C3×C4×Q8—23no—
133C3×C4×Q8—31no—
134C3×C4×Q8—33no—
135C3×C4×Q8—33no—
136C3×C4×Q8—33no—
137C3×C4×Q8C2×D313yes—
138C3×C4×Q8C2×D311yescharacteristic
139C3×C4×Q8C2×D312yes—
140C3×C4×Q8C22×C311yescharacteristic
141C3×C4×Q8C22×C312yes—
142C3×C4×Q8C22×C313yes—
143C3×C4×D4—33no—
144C3×C4×D4—36no—
145C3×C4×D4—36no—
146C3×C4×D4—36no—
147C3×C4×D4—36no—
148C3×C4×D4—36no—
149C3×C4×D4—33no—
150C3×#117—31no—
151C3×#117—36no—
152C3×#117—36no—
153C2×C3×C4⋊C4—33no—
154C2×C3×C42—33no—
155C2×C3×C42—36no—
156#947C22×C316yes—
157Q8×Dic3C22×C313yes—
158Q8×Dic3C22×C311yescharacteristic
159#898C22×C3112yes—
160#897C22×C316yes—
161#895C22×C316yes—
162#895C22×C316yes—
163#895C22×C316yes—
164#894C22×C3112yes—
165#893C22×C316yes—
166#893C22×C316yes—
167#893C22×C316yes—
168#893C22×C313yes—
169#892C22×C316yes—
170#892C22×C316yes—
171#892C22×C311yescharacteristic
172D3×C4⋊C4C22×C313yes—
173#887C22×C313yes—
174#875C22×C316yes—
175#874C22×C313yes—
176C4×D12C22×C316yes—
177C4×D12C22×C316yes—
178C4×D12C22×C313yes—
179C42×D3C22×C316yes—
180C42×D3C22×C313yes—
181C4×Dic6C22×C313yes—
182C22×C3×D3C2416yes—
183C22×C3×D3C2411yescharacteristic
184C32×Q8—26no—
185C32×Q8#4114yescomplemented
186C2×C32×C4C2413yes—
187C2×C32×C4C2416yes—
188C2×C32×C4C2412yes—
189C2×C32×C4C2411yescharacteristic
190C2×C32×C4C2413yes—
191C2×C3×Dic3C2413yes—
192C2×C3×Dic3C2412yes—
193C2×C3×Dic3C2413yes—
194C2×C3×Dic3C2411yescharacteristic
195C3×D12—212no—
196C3×D12—212no—
197C3×D12—26no—
198C3×D12—46no—
199C3×C4×D3—26no—
200C3×C4×D3—212no—
201C3×C4×D3—212no—
202C3×C4×D3—46no—
203C3×C4×D3—43no—
204C3×C4×D3#4114yescomplemented
205C3×Dic6—26no—
206#559—33no—
207#557C3×D311yescharacteristic, complemented
208#556—33no—
209#553—36no—
210#550—31no—
211#548—32no—
212#532—33no—
213#529—36no—
214C4×#41—33no—
215C4×#41—32no—
216C2×C4×Q8—31no—
217C3×#41—612no—
218C3×#41—68no—
219C2×C3×Q8—21no—
220C2×C3×Q8—23no—
221C2×C3×Q8—33no—
222C2×C3×Q8—64no—
223C2×C3×Q8C22×D313yes—
224C2×C3×Q8C22×D311yescharacteristic
225C2×C3×Q8C23×C313yes—
226C2×C3×Q8C23×C311yescharacteristic
227C2×C3×D4—36no—
228C2×C3×D4—36no—
229C2×C3×D4—33no—
230C2×C3×D4—36no—
231C22×C3×C4—31no—
232C22×C3×C4—33no—
233C22×C3×C4—36no—
234C22×C3×C4—36no—
235C22×C3×C4—33no—
236C22×C3×C4—36no—
237#239—28no—
238#239—212no—
239D3×Q8—24no—
240C2×D12C23×C313yes—
241C2×D12C23×C316yes—
242C2×D12C23×C316yes—
243C2×D12C23×C316yes—
244C2×C4×D3C23×C313yes—
245C2×C4×D3C23×C316yes—
246C2×C4×D3C23×C316yes—
247C2×C4×D3C23×C316yes—
248C2×C4×D3C23×C313yes—
249C2×C4×D3C23×C311yescharacteristic
250C2×Dic6C23×C313yes—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—23no—
253C3×C4⋊C4—26no—
254C3×C4⋊C4—26no—
255C3×C4⋊C4—26no—
256C3×C4⋊C4—36no—
257C3×C4⋊C4—33no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—36no—
260C3×C4⋊C4C22×D313yes—
261C3×C4⋊C4C22×D311yescharacteristic
262C3×C4⋊C4C22×D316yes—
263C3×C4⋊C4C22×D316yes—
264C3×C4⋊C4C22×D316yes—
265C3×C4⋊C4C23×C313yes—
266C3×C4⋊C4C23×C311yescharacteristic
267C3×C4⋊C4C23×C316yes—
268C3×C4⋊C4C23×C316yes—
269C3×C4⋊C4C23×C316yes—
270C3×C22⋊C4—32no—
271C3×C22⋊C4—36no—
272C3×C22⋊C4—312no—
273C3×C22⋊C4—36no—
274C3×C22⋊C4—312no—
275C3×C42—26no—
276C3×C42—23no—
277C3×C42—32no—
278C3×C42—33no—
279C3×C42—33no—
280C3×C42—36no—
281C3×C42—33no—
282C3×C42C22×D316yes—
283C3×C42C22×D313yes—
284C3×C42C23×C313yes—
285C3×C42C23×C316yes—
286(C2×D3)⋊C4C23×C312yes—
287(C2×D3)⋊C4C23×C316yes—
288(C2×D3)⋊C4C23×C3112yes—
289(C2×D3)⋊C4C23×C316yes—
290(C2×D3)⋊C4C23×C3112yes—
291C4⋊Dic3C23×C313yes—
292C4⋊Dic3C23×C311yescharacteristic
293C4⋊Dic3C23×C316yes—
294Dic3⋊C4C23×C316yes—
295C4×Dic3C23×C313yes—
296C4×Dic3C23×C313yes—
297C4×Dic3C23×C312yes—
298C4×Dic3C23×C313yes—
299C4×Dic3C23×C316yes—
300C22×C32C2511yescharacteristic
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—26no—
304C2×C3×D3—26no—
305C2×C3×D3—26no—
306C2×C3×D3C2×#4112yes—
307C32×C4—26no—
308C32×C4—22no—
309C32×C4—23no—
310C32×C4C2×#4116yes—
311C32×C4C2×#4112yes—
312C3×Dic3—23no—
313C3×Dic3—22no—
314C3×Dic3—23no—
315C3×Dic3C2×#4112yes—
316C2×#41—33no—
317C2×#41—32no—
318C22×Q8—31no—
319C4⋊Q8C2×C3×D313yes—
320#127—33no—
321#126—312no—
322#125C2×C3×D316yes—
323#124—36no—
324#124—36no—
325#123—312no—
326C22⋊Q8—36no—
327#121—36no—
328#121—36no—
329#121—36no—
330C4×Q8—31no—
331C4×Q8—33no—
332C4×Q8—33no—
333C4×Q8—33no—
334C4×Q8C2×C3×D311yescharacteristic
335C4×Q8C2×C3×D312yes—
336C4×Q8C2×C3×D313yes—
337C4×D4—36no—
338C4×D4—36no—
339C4×D4—33no—
340C4×D4—36no—
341C4×D4—36no—
342C4×D4—33no—
343C4×D4—36no—
344#117—31no—
345#117—36no—
346#117—36no—
347C2×C4⋊C4—33no—
348C2×C42—33no—
349C2×C42—36no—
350C23×C3—31no—
351C23×C3—36no—
352C22×D3C24×C316yes—
353C22×D3C24×C311yescharacteristic
354C3×Q8—26no—
355C3×Q8—26no—
356C3×Q8—24no—
357C3×Q8—46no—
358C3×Q8—66no—
359C3×Q8#23914yescomplemented
360C3×Q8C3×#4114yescomplemented
361C3×D4—612no—
362C3×D4—612no—
363C3×D4—66no—
364C3×D4—126no—
365C2×C3×C4—23no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—23no—
369C2×C3×C4—26no—
370C2×C3×C4—31no—
371C2×C3×C4—34no—
372C2×C3×C4—32no—
373C2×C3×C4—33no—
374C2×C3×C4—33no—
375C2×C3×C4—612no—
376C2×C3×C4—66no—
377C2×C3×C4—612no—
378C2×C3×C4—126no—
379C2×C3×C4—123no—
380C2×C3×C4C23×D312yes—
381C2×C3×C4C23×D316yes—
382C2×C3×C4C23×D313yes—
383C2×C3×C4C23×D313yes—
384C2×C3×C4C23×D311yescharacteristic
385C2×C3×C4C24×C313yes—
386C2×C3×C4C24×C316yes—
387C2×C3×C4C24×C312yes—
388C2×C3×C4C24×C311yescharacteristic
389C2×C3×C4C24×C313yes—
390C2×Dic3C24×C313yes—
391C2×Dic3C24×C312yes—
392C2×Dic3C24×C313yes—
393C2×Dic3C24×C311yescharacteristic
394D12—212no—
395D12—212no—
396D12—26no—
397D12—46no—
398C4×D3—26no—
399C4×D3—212no—
400C4×D3—212no—
401C4×D3—46no—
402C4×D3—43no—
403C4×D3C3×#4114yescomplemented
404Dic6—26no—
405C2×C32C22×#4111yescharacteristic
406C2×C32C22×#4111yescharacteristic
407C2×C32#58511yescharacteristic
408C3×D3—22no—
409C3×D3—46no—
410#41—612no—
411#41—68no—
412C2×Q8—33no—
413C2×Q8—64no—
414C2×Q8C22×C3×D313yes—
415C2×Q8C22×C3×D311yescharacteristic
416C2×D4—36no—
417C2×D4—36no—
418C2×D4—33no—
419C2×D4—36no—
420C22×C4—31no—
421C22×C4—33no—
422C22×C4—36no—
423C22×C4—36no—
424C22×C4—33no—
425C22×C4—36no—
426C4⋊C4—33no—
427C4⋊C4—36no—
428C4⋊C4—31no—
429C4⋊C4—36no—
430C4⋊C4C22×C3×D313yes—
431C4⋊C4C22×C3×D311yescharacteristic
432C4⋊C4C22×C3×D316yes—
433C4⋊C4C22×C3×D316yes—
434C4⋊C4C22×C3×D316yes—
435C22⋊C4—32no—
436C22⋊C4—36no—
437C22⋊C4—312no—
438C22⋊C4—36no—
439C22⋊C4—312no—
440C42—32no—
441C42—33no—
442C42—33no—
443C42—36no—
444C42—33no—
445C42C22×C3×D316yes—
446C42C22×C3×D313yes—
447C22×C3—21no—
448C22×C3—32no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—66no—
452C22×C3—66no—
453C22×C3—66no—
454C22×C3C24×D311yescharacteristic
455C22×C3C25×C311yescharacteristic
456C2×D3—21no—
457C2×D3—21no—
458C2×D3—26no—
459C2×D3—26no—
460C2×D3—26no—
461C2×D3C2×C3×#4112yes—
462C3×C4—26no—
463C3×C4—22no—
464C3×C4—23no—
465C3×C4—26no—
466C3×C4—22no—
467C3×C4—23no—
468C3×C4—26no—
469C3×C4—22no—
470C3×C4—32no—
471C3×C4—46no—
472C3×C4—42no—
473C3×C4—43no—
474C3×C4—62no—
475C3×C4—63no—
476C3×C4—63no—
477C3×C4C2×#23916yes—
478C3×C4D3×#4112yes—
479C3×C4C2×C3×#4116yes—
480C3×C4C2×C3×#4112yes—
481Dic3—23no—
482Dic3—22no—
483Dic3—23no—
484Dic3C2×C3×#4112yes—
485C32#348011yescharacteristic, complemented, Sylow
486C23—31no—
487C23—36no—
488Q8—26no—
489Q8—66no—
490Q8C3×#23914yescomplemented
491D4—612no—
492D4—612no—
493D4—66no—
494D4—126no—
495C2×C4—31no—
496C2×C4—34no—
497C2×C4—32no—
498C2×C4—33no—
499C2×C4—33no—
500C2×C4—612no—
501C2×C4—66no—
502C2×C4—612no—
503C2×C4—126no—
504C2×C4—123no—
505C2×C4C23×C3×D312yes—
506C2×C4C23×C3×D316yes—
507C2×C4C23×C3×D313yes—
508C2×C4C23×C3×D313yes—
509C2×C4C23×C3×D311yescharacteristic
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—62no—
514C2×C3—126no—
515C2×C3C22×#23911yescharacteristic
516C2×C3C2×D3×#4111yescharacteristic
517C2×C3#596811yescharacteristic
518C2×C3C22×C3×#4111yescharacteristic
519C2×C3C22×C3×#4111yescharacteristic
520C2×C3C3×#58511yescharacteristic
521D3—22no—
522D3—46no—
523C22—32no—
524C22—61no—
525C22—61no—
526C22—66no—
527C22—66no—
528C22—66no—
529C22C24×C3×D311yescharacteristic
530C4—26no—
531C4—22no—
532C4—23no—
533C4—32no—
534C4—62no—
535C4—63no—
536C4—63no—
537C4C2×C3×#23916yes—
538C4C3×D3×#4112yes—
539C3—21no—
540C3#8701611yescharacteristic, direct product factor
541C3C3×#348011yescharacteristic, complemented
542C2—62no—
543C2—126no—
544C2C22×C3×#23911yescharacteristic
545C2C2×C3×D3×#4111yescharacteristic
546C2C3×#596811yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#557C2
socleC22×C32C25
hypercenterC3×#557D3
hypocenterC3C3×#3480
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
2. C3×#5732C2semidirect
3. C3×#5725C2semidirect
4. C3×#5695C2semidirect
5. C3×#5691C2semidirect
6. C3×#5585C2semidirect
7. C3×#5577C2semidirect
8. C3×#5576C2semidirect
9. C3×C4×#239C2semidirect
10. C3×C4×#239C2semidirect
11. C3×C4×D3×Q8C2semidirect
13. #87016C3direct (N × H)
50. C3×#557D3semidirect
51. C3×#557C2×C3semidirect
61. #5732C2×C3semidirect
62. #5725C2×C3semidirect
63. #5695C2×C3semidirect
64. #5691C2×C3semidirect
65. #5585C2×C3semidirect
66. #5577C2×C3semidirect
67. #5576C2×C3semidirect
68. C4×#239C2×C3semidirect
69. C4×#239C2×C3semidirect
70. C4×D3×Q8C2×C3semidirect
185. C32×Q8#41semidirect
204. C3×C4×D3#41semidirect
207. #557C3×D3semidirect
359. C3×Q8#239semidirect
360. C3×Q8C3×#41semidirect
403. C4×D3C3×#41semidirect
485. C32#3480semidirect
490. Q8C3×#239semidirect
540. C3#87016direct (N × H)
541. C3C3×#3480semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
C3×#57322
C3×#57252
C3×#56952
C3×#56912
C3×#55852
C3×#55772
C3×#55762
C3×C4×#2392
C3×C4×D3×Q82
C3×#34803
#870163