Finite Group Databasebeta

← C3×#71543 #11788233

Subgroups — C3×#71543 #11788233

Order 1152.

1533 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#953C211yesmaximal, characteristic
3C3×#5247C211yesmaximal, characteristic, complemented
4C2×C3×#806C212yesmaximal, complemented
5C3×#4527C212yesmaximal, complemented
6C3×#1898—31nomaximal
7#71543C311yesmaximal, characteristic, direct product factor
8C25×C32C2211yescharacteristic
9C23×C32×C4—21no—
10C22×C3×#67C412yescomplemented
11C22×C3×#67C2211yescharacteristic
12C2×C32×C22⋊C4—24no—
13C2×C32×C22⋊C4—21no—
14C2×C32×C22⋊C4C2212yes—
15C3×#953—21no—
16C3×#953—22no—
17C3×#953—21no—
18C2×C3×#217—22no—
19C2×C3×#217C412yescomplemented
20C2×C3×#217C2212yes—
21C2×C3×#217C2211yescharacteristic
22C3×#806—24no—
23C2×C3×#120—31no—
24C22×C3×C22⋊C4—21no—
25C22×C3×C22⋊C4D311yescharacteristic, complemented
26C22×C3×C22⋊C4C2×C311yescharacteristic, complemented
27C2×#953C2×C311yescharacteristic
28C2×C3×#99—32no—
29C3×#379—31no—
30#5247C2×C311yescharacteristic, complemented
31C2×#806C2×C312yescomplemented
32C3×#342—32no—
33#4527C2×C312yescomplemented
34C24×C32—22no—
35C24×C32—22no—
36C24×C32—44no—
37C24×C32C2×C412yes—
38C24×C32D411yescharacteristic
39C24×C32D412yes—
40C24×C32D411yescharacteristic
41C24×C32C2311yescharacteristic
42C23×C3×D3C2×C411yescharacteristic, complemented
43C22×C32×C4—24no—
44C22×C32×C4—22no—
45C22×C32×C4—44no—
46C2×C3×#67—22no—
47C2×C3×#67—22no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—22no—
51C2×C3×#67—22no—
52C2×C3×#67—44no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3C2×C411yescharacteristic, complemented
55C22×C3×Dic3C2×C412yescomplemented
56C32×C22⋊C4—24no—
57C32×C22⋊C4—22no—
58C32×C22⋊C4D414yescomplemented
59C3×#217—22no—
60C3×#217—22no—
61C3×#217—21no—
62C3×#217—21no—
63C3×#217—22no—
64C3×#217—22no—
65C3×#217—44no—
66#1898—31noSylow
67C25×C3—21no—
68C25×C3C2×D311yescharacteristic
69C25×C3C22×C311yescharacteristic
70C22×C3×D4—31no—
71C22×C3×D4—32no—
72C23×C3×C4—22no—
73C23×C3×C4—21no—
74C23×C3×C4—21no—
75C22×#67C3×C412yescomplemented
76C22×#67C22×C311yescharacteristic
77C3×#120—62no—
78C3×#120—61no—
79C3×#120—61no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—32no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—62no—
91C2×C3×C22⋊C4C2×D312yes—
92C2×C3×C22⋊C4C22×C312yes—
93#953—22no—
94#953—21no—
95#953—21no—
96C2×#217—22no—
97C2×#217C3×C412yescomplemented
98C2×#217C22×C312yes—
99C2×#217C22×C311yescharacteristic
100C3×#99—64no—
101#806—24no—
102C23×C32—24no—
103C23×C32—21no—
104C23×C32—21no—
105C23×C32—24no—
106C23×C32—21no—
107C23×C32—24no—
108C23×C32—21no—
109C23×C32—24no—
110C23×C32—24no—
111C23×C32—42no—
112C23×C32—42no—
113C23×C32—42no—
114C23×C32—42no—
115C23×C32—48no—
116C23×C32—48no—
117C23×C32C22⋊C412yes—
118C23×C32C22⋊C412yes—
119C23×C32C22×C411yescharacteristic
120C23×C32C2×D411yescharacteristic
121C23×C32C2×D411yescharacteristic
122C23×C32C2×D412yes—
123C23×C32#4112yes—
124C22×C3×D3—21no—
125C22×C3×D3—22no—
126C22×C3×D3—42no—
127C2×C32×C4—24no—
128C2×C32×C4—48no—
129C2×C32×C4—44no—
130C3×#67—44no—
131C3×#67—44no—
132C3×#67—41no—
133C3×#67—42no—
134C3×#67—41no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—42no—
139C2×C3×Dic3—44no—
140C2×C3×Dic3—42no—
141C2×#120—31no—
142C22×C22⋊C4C3×D311yescharacteristic, complemented
143C2×#99—32no—
144#379—31no—
145#342—32no—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3—21no—
157C24×C3—31no—
158C24×C3—44no—
159C24×C3—44no—
160C24×C3—48no—
161C24×C3C4×D312yes—
162C24×C3D1212yes—
163C24×C3#6711yescharacteristic
164C24×C3#6711yescharacteristic
165C24×C3C2×C3×C412yes—
166C24×C3C3×D412yes—
167C24×C3C3×D411yescharacteristic
168C24×C3C3×D411yescharacteristic
169C24×C3C22×D311yescharacteristic
170C24×C3C23×C311yescharacteristic
171C23×D3C2×C3×C411yescharacteristic, complemented
172C2×C3×D4—62no—
173C2×C3×D4—62no—
174C2×C3×D4—61no—
175C2×C3×D4—62no—
176C2×C3×D4—62no—
177C2×C3×D4—61no—
178C2×C3×D4—124no—
179C22×C3×C4—28no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—31no—
186C22×C3×C4—32no—
187C22×C3×C4—48no—
188C22×C3×C4—44no—
189C22×C3×C4—44no—
190C22×C3×C4—62no—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—21no—
195C2×#67—21no—
196C2×#67—22no—
197C2×#67—44no—
198C22×Dic3—22no—
199C22×Dic3C2×C3×C411yescharacteristic, complemented
200C22×Dic3C2×C3×C412yescomplemented
201C3×C22⋊C4—28no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—62no—
209C3×C22⋊C4—62no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—62no—
213C3×C22⋊C4—62no—
214C3×C22⋊C4—124no—
215C3×C22⋊C4#6714yescomplemented
216C3×C22⋊C4C3×D414yescomplemented
217#217—22no—
218#217—22no—
219#217—21no—
220#217—21no—
221#217—22no—
222#217—22no—
223#217—44no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—48no—
237C22×C32—48no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32#9914yescomplemented
241C22×C32C2×C22⋊C411yescharacteristic
242C22×C32C4×D412yes—
243C22×C32#12011yescharacteristic
244C22×C32#12112yes—
245C22×C32#12311yescharacteristic
246C2×C3×D3—41no—
247C2×C3×D3—42no—
248C2×C3×D3—82no—
249C32×C4—44no—
250C3×Dic3—41no—
251C3×Dic3—42no—
252C3×Dic3—82no—
253C25C2×C3×D311yescharacteristic
254C22×D4—31no—
255C22×D4—32no—
256C23×C4—21no—
257#120—62no—
258#120—61no—
259#120—61no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—31no—
264C2×C22⋊C4—32no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4C2×C3×D312yes—
267#99—64no—
268C23×C3—22no—
269C23×C3—28no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—22no—
278C23×C3—28no—
279C23×C3—22no—
280C23×C3—28no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—24no—
288C23×C3—21no—
289C23×C3—24no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—48no—
315C23×C3—48no—
316C23×C3—416no—
317C23×C3—48no—
318C23×C3—48no—
319C23×C3—416no—
320C23×C3—61no—
321C23×C3—62no—
322C23×C3—122no—
323C23×C3(C2×D3)⋊C412yes—
324C23×C3(C2×D3)⋊C412yes—
325C23×C3C3×C22⋊C412yes—
326C23×C3C3×C22⋊C412yes—
327C23×C3C2×C4×D311yescharacteristic
328C23×C3C2×D1211yescharacteristic
329C23×C3#23512yes—
330C23×C3C2×#6711yescharacteristic
331C23×C3C2×#6712yes—
332C23×C3C22×C3×C411yescharacteristic
333C23×C3C2×C3×D411yescharacteristic
334C23×C3C2×C3×D411yescharacteristic
335C23×C3C2×C3×D412yes—
336C23×C3C3×#4112yes—
337C22×D3—21no—
338C22×D3—22no—
339C22×D3—42no—
340C3×D4—124no—
341C3×D4—124no—
342C3×D4—121no—
343C3×D4—122no—
344C3×D4—121no—
345C2×C3×C4—28no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—416no—
349C2×C3×C4—48no—
350C2×C3×C4—48no—
351C2×C3×C4—48no—
352C2×C3×C4—44no—
353C2×C3×C4—44no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—122no—
358C2×C3×C4—124no—
359C2×C3×C4—122no—
360#67—44no—
361#67—44no—
362#67—41no—
363#67—42no—
364#67—41no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—42no—
369C2×Dic3—44no—
370C2×Dic3—42no—
371C2×C32—21no—
372C2×C32—24no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—42no—
377C2×C32#38611yescharacteristic
378C2×C32C2×#9912yes—
379C3×D3—81no—
380C24—22no—
381C24—22no—
382C24—31no—
383C24—44no—
384C24C3×C4×D312yes—
385C24C3×D1212yes—
386C24C3×#6711yescharacteristic
387C24C3×#6711yescharacteristic
388C24C22×C3×D311yescharacteristic
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—61no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—61no—
395C2×D4—124no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—31no—
399C22×C4—32no—
400C22×C4—44no—
401C22×C4—62no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—62no—
405C22⋊C4—62no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—124no—
411C22⋊C4C3×#6714yescomplemented
412C22×C3—28no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—28no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—24no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—48no—
453C22×C3—48no—
454C22×C3—48no—
455C22×C3—48no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—416no—
459C22×C3—416no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—122no—
467C22×C3—121no—
468C22×C3—242no—
469C22×C3#80614yescomplemented
470C22×C3C3×#9914yescomplemented
471C22×C3C2×(C2×D3)⋊C411yescharacteristic
472C22×C3C4×#6712yes—
473C22×C3#92911yescharacteristic
474C22×C3#93012yes—
475C22×C3#95311yescharacteristic
476C22×C3C2×C3×C22⋊C411yescharacteristic
477C22×C3C3×C4×D412yes—
478C22×C3C3×#12011yescharacteristic
479C22×C3C3×#12112yes—
480C22×C3C3×#12311yescharacteristic
481C2×D3—41no—
482C2×D3—42no—
483C2×D3—82no—
484C3×C4—48no—
485C3×C4—44no—
486C3×C4—44no—
487C3×C4—122no—
488C3×C4—121no—
489C3×C4—242no—
490Dic3—41no—
491Dic3—42no—
492Dic3—82no—
493C32#189811yescharacteristic, complemented, Sylow
494C23—21no—
495C23—21no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—24no—
500C23—21no—
501C23—24no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—48no—
508C23—48no—
509C23—61no—
510C23—62no—
511C23—122no—
512C23C3×(C2×D3)⋊C412yes—
513C23C3×(C2×D3)⋊C412yes—
514C23C2×C3×C4×D311yescharacteristic
515C23C2×C3×D1211yescharacteristic
516C23C3×#23512yes—
517C23C2×C3×#6711yescharacteristic
518C23C2×C3×#6712yes—
519D4—124no—
520D4—121no—
521D4—122no—
522D4—121no—
523D4—124no—
524C2×C4—24no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—122no—
531C2×C4—124no—
532C2×C4—122no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—28no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—24no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—21no—
546C2×C3—22no—
547C2×C3—21no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—44no—
551C2×C3—42no—
552C2×C3—42no—
553C2×C3—241no—
554C2×C3C2×#80612yes—
555C2×C3#521311yescharacteristic
556C2×C3C3×#38611yescharacteristic
557C2×C3C2×C3×#9912yes—
558D3—81no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—48no—
573C22—42no—
574C22—42no—
575C22—122no—
576C22—121no—
577C22—242no—
578C22C3×#80614yescomplemented
579C22C2×C3×(C2×D3)⋊C411yescharacteristic
580C22C3×C4×#6712yes—
581C22C3×#92911yescharacteristic
582C22C3×#93012yes—
583C22C3×#95311yescharacteristic
584C4—44no—
585C4—122no—
586C4—121no—
587C4—242no—
588C3—21no—
589C3#7154311yescharacteristic, direct product factor
590C3C3×#189811yescharacteristic, complemented
591C2—21no—
592C2—22no—
593C2—21no—
594C2—24no—
595C2—22no—
596C2—42no—
597C2—241no—
598C2C2×C3×#80612yes—
599C2C3×#521311yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC23×C3C22×C3×C4
FrattiniC24C22×C3×D3
FittingC22×C32×C22⋊C4C2
socleC22×C32C2×C22⋊C4
prime radicalC2×C3×#953C2
cosocleC24×C3C23×C3
hypercenterC22×C3×C22⋊C4D3
hypocenterC3C3×#1898
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ C22×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×C22⋊C4C2semidirect
3. C3×#5247C2semidirect
4. C2×C3×#806C2semidirect
5. C3×#4527C2semidirect
7. #71543C3direct (N × H)
10. C22×C3×#67C4semidirect
19. C2×C3×#217C4semidirect
25. C22×C3×C22⋊C4D3semidirect
26. C22×C3×C22⋊C4C2×C3semidirect
30. #5247C2×C3semidirect
31. C2×#806C2×C3semidirect
33. #4527C2×C3semidirect
42. C23×C3×D3C2×C4semidirect
54. C22×C3×Dic3C2×C4semidirect
55. C22×C3×Dic3C2×C4semidirect
58. C32×C22⋊C4D4semidirect
75. C22×#67C3×C4semidirect
97. C2×#217C3×C4semidirect
142. C22×C22⋊C4C3×D3semidirect
171. C23×D3C2×C3×C4semidirect
199. C22×Dic3C2×C3×C4semidirect
200. C22×Dic3C2×C3×C4semidirect
215. C3×C22⋊C4#67semidirect
216. C3×C22⋊C4C3×D4semidirect
240. C22×C32#99semidirect
411. C22⋊C4C3×#67semidirect
469. C22×C3#806semidirect
470. C22×C3C3×#99semidirect
493. C32#1898semidirect
578. C22C3×#806semidirect
589. C3#71543direct (N × H)
590. C3C3×#1898semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊C42
C2×C3×#9532
C3×#52472
C2×C3×#8062
C3×#45272
C3×#18983
#715433