Finite Group Databasebeta

← C3×#78571 #11860585

Subgroups — C3×#78571 #11860585

Order 1152.

1005 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C3×#5523C211yesmaximal, characteristic
3C3×#5516C211yesmaximal, characteristic, complemented
4C32×#405C211yesmaximal, characteristic, complemented
5C3×#5121C211yesmaximal, characteristic, complemented
6C3×#5113C211yesmaximal, characteristic
7C2×C3×#816C211yesmaximal, characteristic, complemented
8C3×#4716C212yesmaximal, complemented
9C3×#4710C212yesmaximal, complemented
10C3×C4×Dic12C212yesmaximal, complemented
11C3×C4×#204C212yesmaximal, complemented
12C3×#2947—31nomaximal
13#78571C311yesmaximal, characteristic, direct product factor
14C2×C3×#235C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C22×C32×C8C2211yescharacteristic
18C32×#117C2211yescharacteristic
19C3×#926C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C3×#909C418yescomplemented
22C2×C3×Dic12C2212yes—
23C2×C3×#204C2212yes—
24C3×#892—21no—
25C3×D3×C4⋊C4—21no—
26C3×#887—21no—
27C3×#887—21no—
28C3×#67⋊C4—22no—
29C3×#877—22no—
30C3×C4×D12C2212yes—
31C3×C4×Dic6C2212yes—
32C3×C4×Dic6C2212yes—
33C3×C4×Dic6C2212yes—
34C32×#97C2212yes—
35C32×C4×C8C2212yes—
36C3×#821C2211yescharacteristic
37C3×#821C2211yescharacteristic
38C3×#818C2212yes—
39C3×#817C2212yes—
40C3×#816C2212yes—
41C3×#816C2211yescharacteristic
42C3×#816C2212yes—
43C3×#816C2211yescharacteristic
44C2×C3×#137—31no—
45C3×#520—31no—
46C3×#519—31no—
47C2×#909C2×C311yescharacteristic
48#5523C2×C311yescharacteristic
49#5516C2×C311yescharacteristic, complemented
50C3×#441—32no—
51C3×#440—32no—
52C3×C4×Q16—32no—
53C3×C4×SD16—32no—
54C3×#428—31no—
55C3×#416—31no—
56C2×C3×#103—31no—
57C3×#405—21no—
58C3×#405D311yescharacteristic, complemented
59C3×#405C2×C311yescharacteristic, complemented
60#5121C2×C311yescharacteristic, complemented
61#5113C2×C311yescharacteristic
62C2×#816C2×C311yescharacteristic, complemented
63#4716C2×C312yescomplemented
64#4710C2×C312yescomplemented
65C4×Dic12C2×C312yescomplemented
66C4×#204C2×C312yescomplemented
67C22×C32×C4C2311yescharacteristic
68C2×C3×#67—21no—
69C22×C3×Dic3—21no—
70C22×C3×Dic3—21no—
71C3×#235—22no—
72C3×#235C2×C414yes—
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×C4×D3—22no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6C2×C414yescomplemented
79C2×C3×Dic6C2312yes—
80C2×C3×Dic6C2311yescharacteristic
81C32×M16C2×C414yescomplemented
82C2×C32×C8—22no—
83C2×C32×C8C2311yescharacteristic
84C2×C32×C8C2311yescharacteristic
85C2×C32×C8C2312yes—
86C32×C4⋊C4D411yescharacteristic
87C32×C4⋊C4D411yescharacteristic
88C32×C22⋊C4D412yes—
89C32×C42C2312yes—
90C3×#217—21no—
91C3×(C2×D3)⋊C4—22no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3C2312yes—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—22no—
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×C4×Dic3—22no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—22no—
102C3×C4×Dic3—21no—
103C3×Dic12C2×C418yescomplemented
104C3×#204C2×C418yescomplemented
105#2947—31noSylow
106C2×C3×#41—31no—
107C22×C3×Q8—31no—
108C2×#235C22×C311yescharacteristic
109C22×Dic6C22×C311yescharacteristic
110C3×#137—38no—
111C2×C3×Q16—32no—
112C2×C3×SD16—32no—
113C2×C3×M16—21no—
114C2×C3×M16C2×D311yescharacteristic
115C2×C3×M16C22×C311yescharacteristic
116C22×C3×C8—21no—
117C22×C3×C8C2×D311yescharacteristic
118C22×C3×C8C22×C311yescharacteristic
119C3×C4×Q8—32no—
120C3×C4×Q8—32no—
121C3×C4×Q8—32no—
122C3×C4×Q8—61no—
123C3×C4×Q8—61no—
124C3×C4×D4—32no—
125C3×C4×D4—62no—
126C3×#117—21no—
127C3×#117—31no—
128C3×#117—61no—
129C3×#117—62no—
130C3×#117C2×D311yescharacteristic
131C3×#117C22×C311yescharacteristic
132C2×C3×C4⋊C4—31no—
133C2×C3×C4⋊C4—61no—
134#926C22×C311yescharacteristic
135C2×C4⋊Dic3C22×C311yescharacteristic
136#909C3×C418yescomplemented
137C2×Dic12C22×C312yes—
138C2×#204C22×C312yes—
139#892—21no—
140D3×C4⋊C4—21no—
141#887—21no—
142#887—21no—
143#67⋊C4—22no—
144#877—22no—
145C4×D12C22×C312yes—
146C4×Dic6C22×C312yes—
147C4×Dic6C22×C312yes—
148C4×Dic6C22×C312yes—
149C3×#107—32no—
150C3×#106—32no—
151C3×#103—32no—
152C3×#103—32no—
153C3×#103—31no—
154C3×#103—31no—
155C3×#102—31no—
156C3×#102—31no—
157C3×#97—22no—
158C3×#97C2×D312yes—
159C3×#97C22×C312yes—
160C3×C4×C8—22no—
161C3×C4×C8C2×D312yes—
162C3×C4×C8C22×C312yes—
163#821C22×C311yescharacteristic
164#821C22×C311yescharacteristic
165#818C22×C312yes—
166#817C22×C312yes—
167#816C22×C312yes—
168#816C22×C311yescharacteristic
169#816C22×C312yes—
170#816C22×C311yescharacteristic
171C23×C32C2×D411yescharacteristic
172C22×C3×D3—21no—
173C2×C32×C4C22×C412yes—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4#4112yes—
178C2×C32×C4C2411yescharacteristic
179C3×#67—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—81no—
187C3×D12—21no—
188C3×D12C22×C412yes—
189C3×C4×D3—24no—
190C3×C4×D3—82no—
191C3×Dic6—24no—
192C3×Dic6—22no—
193C3×Dic6—21no—
194C3×Dic6C22×C414yes—
195C3×Dic6C22×C412yes—
196C32×C8—21no—
197C32×C8—21no—
198C32×C8C22×C414yes—
199C2×#137—31no—
200#520—31no—
201#519—31no—
202#441—32no—
203#440—32no—
204C4×Q16—32no—
205C4×SD16—32no—
206#428—31no—
207#416—31no—
208C2×#103—31no—
209#405C3×D311yescharacteristic, complemented
210C3×#41—34no—
211C3×#41—62no—
212C2×C3×Q8—34no—
213C2×C3×Q8—32no—
214C2×C3×Q8—31no—
215C2×C3×Q8—61no—
216C2×C3×Q8—62no—
217C2×C3×D4—31no—
218C2×C3×D4—61no—
219C22×C3×C4—21no—
220C22×C3×C4—61no—
221C22×C3×C4—61no—
222C22×C3×C4—61no—
223C22×C3×C4—62no—
224C22×C3×C4C22×D311yescharacteristic
225C22×C3×C4C23×C311yescharacteristic
226C2×#67—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229#235—22no—
230#235C2×C3×C414yes—
231C2×D12C23×C311yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—22no—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6C2×C3×C414yescomplemented
237C2×Dic6C23×C312yes—
238C2×Dic6C23×C311yescharacteristic
239C3×Q16—38no—
240C3×SD16—38no—
241C3×M16—24no—
242C3×M16C4×D314yescomplemented
243C3×M16C2×C3×C414yescomplemented
244C2×C3×C8—24no—
245C2×C3×C8—22no—
246C2×C3×C8—22no—
247C2×C3×C8—21no—
248C2×C3×C8—22no—
249C2×C3×C8—21no—
250C2×C3×C8C22×D311yescharacteristic
251C2×C3×C8C22×D311yescharacteristic
252C2×C3×C8C22×D312yes—
253C2×C3×C8C23×C311yescharacteristic
254C2×C3×C8C23×C311yescharacteristic
255C2×C3×C8C23×C312yes—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4—62no—
262C3×C4⋊C4—62no—
263C3×C4⋊C4—62no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4D1211yescharacteristic
266C3×C4⋊C4D1211yescharacteristic
267C3×C4⋊C4C3×D411yescharacteristic
268C3×C4⋊C4C3×D411yescharacteristic
269C3×C22⋊C4—22no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—62no—
272C3×C22⋊C4D1212yes—
273C3×C22⋊C4C3×D412yes—
274C3×C42—22no—
275C3×C42—61no—
276C3×C42—62no—
277C3×C42—62no—
278C3×C42—62no—
279C3×C42C22×D312yes—
280C3×C42C23×C312yes—
281#217—21no—
282(C2×D3)⋊C4—22no—
283C4⋊Dic3—21no—
284C4⋊Dic3C23×C312yes—
285C4⋊Dic3C23×C311yescharacteristic
286C4⋊Dic3C23×C311yescharacteristic
287Dic3⋊C4—22no—
288Dic3⋊C4—22no—
289Dic3⋊C4—22no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292C4×Dic3—22no—
293C4×Dic3—21no—
294Dic12C2×C3×C418yescomplemented
295#204C2×C3×C418yescomplemented
296C22×C32—21no—
297C22×C32—21no—
298C22×C32C4×D412yes—
299C22×C32C22×D411yescharacteristic
300C2×C3×D3—22no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4C4×D412yes—
306C32×C4C23×C411yescharacteristic
307C32×C4C2×#4111yescharacteristic
308C3×Dic3—24no—
309C3×Dic3—22no—
310C3×Dic3—42no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C2×#41—31no—
314C22×Q8—31no—
315#137—38no—
316C2×Q16—32no—
317C2×SD16—32no—
318C2×M16C2×C3×D311yescharacteristic
319C22×C8C2×C3×D311yescharacteristic
320C4×Q8—32no—
321C4×Q8—32no—
322C4×Q8—32no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×D4—32no—
326C4×D4—62no—
327#117—31no—
328#117—61no—
329#117—62no—
330#117C2×C3×D311yescharacteristic
331C2×C4⋊C4—31no—
332C2×C4⋊C4—61no—
333#107—32no—
334#106—32no—
335#103—32no—
336#103—32no—
337#103—31no—
338#103—31no—
339#102—31no—
340#102—31no—
341#97C2×C3×D312yes—
342C4×C8C2×C3×D312yes—
343C23×C3—21no—
344C23×C3—61no—
345C23×C3C2×D1211yescharacteristic
346C23×C3C2×C3×D411yescharacteristic
347C22×D3—21no—
348C3×Q8—34no—
349C3×Q8—32no—
350C3×Q8—64no—
351C3×Q8—62no—
352C3×Q8—61no—
353C3×D4—32no—
354C3×D4—64no—
355C3×D4—61no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—64no—
363C2×C3×C4—61no—
364C2×C3×C4—62no—
365C2×C3×C4—64no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—242no—
370C2×C3×C4—241no—
371C2×C3×C4C2×C4×D312yes—
372C2×C3×C4C2×D1212yes—
373C2×C3×C4C2×D1212yes—
374C2×C3×C4C2×D1211yescharacteristic
375C2×C3×C4#23512yes—
376C2×C3×C4C22×C3×C412yes—
377C2×C3×C4C2×C3×D412yes—
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C3×#4112yes—
381C2×C3×C4C23×D311yescharacteristic
382C2×C3×C4C24×C311yescharacteristic
383#67—24no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—81no—
391D12—21no—
392D12C22×C3×C412yes—
393C4×D3—24no—
394C4×D3—82no—
395Dic6—24no—
396Dic6—22no—
397Dic6—21no—
398Dic6C22×C3×C414yes—
399Dic6C22×C3×C412yes—
400C3×C8—22no—
401C3×C8—22no—
402C3×C8—21no—
403C3×C8—21no—
404C3×C8—21no—
405C3×C8—21no—
406C3×C8—24no—
407C3×C8C2×C4×D314yes—
408C3×C8C22×C3×C414yes—
409C2×C32—22no—
410C2×C32C2×C4×D411yescharacteristic
411C2×C32#57711yescharacteristic
412C2×C32#57811yescharacteristic
413C3×D3—42no—
414#41—34no—
415#41—62no—
416C2×Q8—32no—
417C2×Q8—31no—
418C2×Q8—34no—
419C2×Q8—61no—
420C2×Q8—62no—
421C2×D4—31no—
422C2×D4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4C22×C3×D311yescharacteristic
428Q16—38no—
429SD16—38no—
430M16C3×C4×D314yescomplemented
431C2×C8—22no—
432C2×C8C22×C3×D311yescharacteristic
433C2×C8C22×C3×D311yescharacteristic
434C2×C8C22×C3×D312yes—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—32no—
438C4⋊C4—61no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4C3×D1211yescharacteristic
443C4⋊C4C3×D1211yescharacteristic
444C22⋊C4—61no—
445C22⋊C4—62no—
446C22⋊C4C3×D1212yes—
447C42—61no—
448C42—62no—
449C42—62no—
450C42—62no—
451C42C22×C3×D312yes—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—62no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3C4×D1212yes—
464C22×C3C3×C4×D412yes—
465C22×C3C22×D1211yescharacteristic
466C22×C3C22×C3×D411yescharacteristic
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—62no—
480C3×C4—64no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4C4×D1212yes—
485C3×C4C3×C4×D412yes—
486C3×C4C22×C4×D311yescharacteristic
487C3×C4C2×#23511yescharacteristic
488C3×C4C23×C3×C411yescharacteristic
489C3×C4C2×C3×#4111yescharacteristic
490Dic3—24no—
491Dic3—22no—
492Dic3—42no—
493Dic3—41no—
494Dic3—41no—
495C32#294711yescharacteristic, complemented, Sylow
496C23—61no—
497C23C2×C3×D1211yescharacteristic
498Q8—34no—
499Q8—32no—
500Q8—62no—
501Q8—61no—
502Q8—64no—
503D4—32no—
504D4—64no—
505D4—61no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—62no—
513C2×C4—242no—
514C2×C4—241no—
515C2×C4C2×C3×C4×D312yes—
516C2×C4C2×C3×D1212yes—
517C2×C4C2×C3×D1212yes—
518C2×C4C2×C3×D1211yescharacteristic
519C2×C4C3×#23512yes—
520C2×C4C23×C3×D311yescharacteristic
521C8—21no—
522C8—21no—
523C8C2×C3×C4×D314yes—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—42no—
530C2×C3—122no—
531C2×C3C2×C4×D1211yescharacteristic
532C2×C3#575111yescharacteristic
533C2×C3#575311yescharacteristic
534C2×C3C2×C3×C4×D411yescharacteristic
535C2×C3C3×#57711yescharacteristic
536C2×C3C3×#57811yescharacteristic
537D3—42no—
538C22—21no—
539C22—21no—
540C22—62no—
541C22—121no—
542C22—121no—
543C22C3×C4×D1212yes—
544C22C22×C3×D1211yescharacteristic
545C4—22no—
546C4—22no—
547C4—62no—
548C4—64no—
549C4—122no—
550C4—121no—
551C4—121no—
552C4C3×C4×D1212yes—
553C4C22×C3×C4×D311yescharacteristic
554C4C2×C3×#23511yescharacteristic
555C3—21no—
556C3#7857111yescharacteristic, direct product factor
557C3C3×#294711yescharacteristic, complemented
558C2—22no—
559C2—122no—
560C2C2×C3×C4×D1211yescharacteristic
561C2C3×#575111yescharacteristic
562C2C3×#575311yescharacteristic
subgroupG/H
centerC22×C3C22×D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#405C2
socleC22×C32C22×D4
prime radicalC2×C3×#235C22
cosocleC2×C3×C4C24×C3
hypercenterC3×#405D3
hypocenterC3C3×#2947
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405

Complements

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

NH ≅ G/Ntype
3. C3×#5516C2semidirect
4. C32×#405C2semidirect
5. C3×#5121C2semidirect
7. C2×C3×#816C2semidirect
8. C3×#4716C2semidirect
9. C3×#4710C2semidirect
10. C3×C4×Dic12C2semidirect
11. C3×C4×#204C2semidirect
13. #78571C3direct (N × H)
21. C3×#909C4semidirect
49. #5516C2×C3semidirect
58. C3×#405D3semidirect
59. C3×#405C2×C3semidirect
60. #5121C2×C3semidirect
62. C2×#816C2×C3semidirect
63. #4716C2×C3semidirect
64. #4710C2×C3semidirect
65. C4×Dic12C2×C3semidirect
66. C4×#204C2×C3semidirect
78. C2×C3×Dic6C2×C4semidirect
81. C32×M16C2×C4semidirect
103. C3×Dic12C2×C4semidirect
104. C3×#204C2×C4semidirect
136. #909C3×C4semidirect
209. #405C3×D3semidirect
236. C2×Dic6C2×C3×C4semidirect
242. C3×M16C4×D3semidirect
243. C3×M16C2×C3×C4semidirect
294. Dic12C2×C3×C4semidirect
295. #204C2×C3×C4semidirect
430. M16C3×C4×D3semidirect
495. C32#2947semidirect
556. C3#78571direct (N × H)
557. C3C3×#2947semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C3×#55232
C3×#55162
C32×#4052
C3×#51212
C3×#51132
C2×C3×#8162
C3×#47162
C3×#47102
C3×C4×Dic122
C3×C4×#2042
C3×#29473
#785713