Finite Group Databasebeta

← C3×#79200 #11860365

Subgroups — C3×#79200 #11860365

Order 1152.

1027 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C3×#5522C211yesmaximal, characteristic
3C2×C3×#887C211yesmaximal, characteristic, complemented
4C32×#428C211yesmaximal, characteristic, complemented
5C3×Dic3×M16C211yesmaximal, characteristic, complemented
6C3×#5004C211yesmaximal, characteristic, complemented
7C3×#4966C211yesmaximal, characteristic
8C3×#4889C212yesmaximal, complemented
9C3×#4873C212yesmaximal, complemented
10C3×#4853C212yesmaximal, complemented
11C3×#4852C212yesmaximal, complemented
12C3×C4×#137—31nomaximal
13#79200C311yesmaximal, characteristic, direct product factor
14C2×C3×#235C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C3×Dic3⋊C4—21no—
20C2×C3×C4×Dic3—21no—
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C3×#909C418yescomplemented
24C2×C3×Dic12C2212yes—
25C2×C3×#204C2212yes—
26C3×#893C2211yescharacteristic
27C3×#893C2211yescharacteristic
28C3×#887—21no—
29C3×#887—21no—
30C3×#887C2212yes—
31C3×#887C2212yes—
32C3×#887C2211yescharacteristic
33C3×#887C2211yescharacteristic
34C3×#67⋊C4—21no—
35C3×#67⋊C4—21no—
36C3×#872—21no—
37C3×C42×D3—21no—
38C32×#107C2212yes—
39C32×#106C2212yes—
40C3×#815C2212yes—
41C3×C8×Dic3C2212yes—
42C3×#810C2212yes—
43C3×#810C2212yes—
44C3×#810C2212yes—
45C3×#809C2212yes—
46C2×C3×#137—31no—
47C3×C4×#41—31no—
48C2×C3×C4×Q8—31no—
49C2×#909C2×C311yescharacteristic
50#5522C2×C311yescharacteristic
51C2×#887C2×C311yescharacteristic, complemented
52C3×#441—32no—
53C3×#440—32no—
54C3×C4×Q16—32no—
55C3×C4×SD16—32no—
56C3×#428—21no—
57C3×#428D311yescharacteristic, complemented
58C3×#428C2×C311yescharacteristic, complemented
59C3×#419—31no—
60C3×#417—31no—
61C3×C4×M16—31no—
62Dic3×M16C2×C311yescharacteristic, complemented
63#5004C2×C311yescharacteristic, complemented
64#4966C2×C311yescharacteristic
65#4889C2×C312yescomplemented
66#4873C2×C312yescomplemented
67#4853C2×C312yescomplemented
68#4852C2×C312yescomplemented
69C22×C32×C4—21no—
70C22×C32×C4C2311yescharacteristic
71C2×C3×#67—21no—
72C22×C3×Dic3—21no—
73C22×C3×Dic3D411yescharacteristic
74C22×C3×Dic3D411yescharacteristic
75C3×#235—22no—
76C3×#235C2×C414yes—
77C2×C3×D12C2311yescharacteristic
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3—21no—
80C2×C3×C4×D3—21no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6—22no—
83C2×C3×Dic6C2×C414yescomplemented
84C2×C3×Dic6C2312yes—
85C2×C3×Dic6C2311yescharacteristic
86C32×M16C2×C414yescomplemented
87C2×C32×C8C2312yes—
88C32×C4⋊C4—21no—
89C32×C4⋊C4C2312yes—
90C32×C4⋊C4C2311yescharacteristic
91C32×C4⋊C4C2311yescharacteristic
92C32×C22⋊C4—21no—
93C32×C42—21no—
94C3×(C2×D3)⋊C4—21no—
95C3×(C2×D3)⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—22no—
99C3×Dic3⋊C4—22no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—22no—
103C3×C4×Dic3—22no—
104C3×C4×Dic3D411yescharacteristic
105C3×C4×Dic3D411yescharacteristic
106C3×C4×Dic3C2311yescharacteristic
107C3×C4×Dic3C2311yescharacteristic
108C3×#208—22no—
109C2×C3×C3⋊C8C2312yes—
110C3×Dic12C2×C418yescomplemented
111C3×#204C2×C418yescomplemented
112C4×#137—31noSylow
113C2×C3×#41—31no—
114C22×C3×Q8—31no—
115C2×#235C22×C311yescharacteristic
116C22×Dic6C22×C311yescharacteristic
117C3×#137—38no—
118C2×C3×Q16—32no—
119C2×C3×SD16—32no—
120C2×C3×M16—21no—
121C2×C3×M16—31no—
122C2×C3×M16C2×D311yescharacteristic
123C2×C3×M16C22×C311yescharacteristic
124C3×C4×Q8—32no—
125C3×C4×Q8—31no—
126C3×C4×Q8—32no—
127C3×C4×Q8—31no—
128C3×C4×Q8—61no—
129C3×C4×Q8—61no—
130C3×C4×D4—31no—
131C3×C4×D4—31no—
132C3×C4×D4—61no—
133C3×C4×D4—61no—
134C3×#117—21no—
135C3×#117—61no—
136C3×#117C2×D311yescharacteristic
137C3×#117C22×C311yescharacteristic
138C2×C3×C4⋊C4—21no—
139C2×C3×C4⋊C4—61no—
140C2×C3×C4⋊C4C2×D311yescharacteristic
141C2×C3×C4⋊C4C22×C311yescharacteristic
142C2×C3×C42—31no—
143C2×C3×C42—61no—
144C2×C3×C42—61no—
145C2×Dic3⋊C4—21no—
146C2×C4×Dic3—21no—
147C2×C4×Dic3C22×C311yescharacteristic
148C2×#208C22×C311yescharacteristic
149#909C3×C418yescomplemented
150C2×Dic12C22×C312yes—
151C2×#204C22×C312yes—
152#893C22×C311yescharacteristic
153#893C22×C311yescharacteristic
154#887—21no—
155#887—21no—
156#887C22×C311yescharacteristic
157#887C22×C312yes—
158#887C22×C312yes—
159#887C22×C311yescharacteristic
160#67⋊C4—21no—
161#67⋊C4—21no—
162#872—21no—
163C42×D3—21no—
164C3×#107—22no—
165C3×#107C2×D312yes—
166C3×#107C22×C312yes—
167C3×#106—22no—
168C3×#106C2×D312yes—
169C3×#106C22×C312yes—
170C3×#103—32no—
171C3×#103—32no—
172C3×#103—32no—
173C3×#102—32no—
174C3×#97—32no—
175C3×C4×C8—32no—
176#815C22×C312yes—
177C8×Dic3C22×C312yes—
178#810C22×C312yes—
179#810C22×C312yes—
180#810C22×C312yes—
181#809C22×C312yes—
182C23×C32C2×D411yescharacteristic
183C22×C3×D3—21no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—42no—
188C2×C32×C4C22×C412yes—
189C2×C32×C4C2×D411yescharacteristic
190C2×C32×C4#4112yes—
191C2×C32×C4C2411yescharacteristic
192C3×#67—24no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3C2×D411yescharacteristic
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C2×C3×Dic3C2×D411yescharacteristic
202C3×D12—21no—
203C3×D12C22×C412yes—
204C3×C4×D3—24no—
205C3×C4×D3—42no—
206C3×C4×D3—42no—
207C3×Dic6—24no—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6C22×C414yes—
211C3×Dic6C22×C412yes—
212C32×C8C22×C414yes—
213C3×C3⋊C8—22no—
214C2×#137—31no—
215C4×#41—31no—
216C2×C4×Q8—31no—
217#441—32no—
218#440—32no—
219C4×Q16—32no—
220C4×SD16—32no—
221#428C3×D311yescharacteristic, complemented
222#419—31no—
223#417—31no—
224C4×M16—31no—
225C3×#41—34no—
226C3×#41—62no—
227C2×C3×Q8—34no—
228C2×C3×Q8—32no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—62no—
232C2×C3×D4—31no—
233C2×C3×D4—61no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4—41no—
240C22×C3×C4—61no—
241C22×C3×C4—61no—
242C22×C3×C4—61no—
243C22×C3×C4—61no—
244C22×C3×C4C22×D311yescharacteristic
245C22×C3×C4C23×C311yescharacteristic
246C2×#67—21no—
247C22×Dic3—21no—
248C22×Dic3C3×D411yescharacteristic
249C22×Dic3C3×D411yescharacteristic
250#235—22no—
251#235C2×C3×C414yes—
252C2×D12C23×C311yescharacteristic
253C2×C4×D3—21no—
254C2×C4×D3—21no—
255C2×C4×D3—21no—
256C2×Dic6—21no—
257C2×Dic6—22no—
258C2×Dic6C2×C3×C414yescomplemented
259C2×Dic6C23×C312yes—
260C2×Dic6C23×C311yescharacteristic
261C3×Q16—38no—
262C3×SD16—38no—
263C3×M16—24no—
264C3×M16—62no—
265C3×M16C4×D314yescomplemented
266C3×M16C2×C3×C414yescomplemented
267C2×C3×C8—22no—
268C2×C3×C8—32no—
269C2×C3×C8C22×D312yes—
270C2×C3×C8C23×C312yes—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—41no—
277C3×C4⋊C4—62no—
278C3×C4⋊C4—62no—
279C3×C4⋊C4—61no—
280C3×C4⋊C4—61no—
281C3×C4⋊C4C22×D312yes—
282C3×C4⋊C4C22×D311yescharacteristic
283C3×C4⋊C4C22×D311yescharacteristic
284C3×C4⋊C4C23×C312yes—
285C3×C4⋊C4C23×C311yescharacteristic
286C3×C4⋊C4C23×C311yescharacteristic
287C3×C22⋊C4—21no—
288C3×C22⋊C4—21no—
289C3×C22⋊C4—41no—
290C3×C22⋊C4—61no—
291C3×C22⋊C4—61no—
292C3×C42—21no—
293C3×C42—21no—
294C3×C42—31no—
295C3×C42—31no—
296C3×C42—31no—
297C3×C42—31no—
298C3×C42—41no—
299C3×C42—61no—
300C3×C42—61no—
301C3×C42—62no—
302C3×C42—62no—
303(C2×D3)⋊C4—21no—
304(C2×D3)⋊C4—21no—
305Dic3⋊C4—21no—
306Dic3⋊C4—21no—
307Dic3⋊C4—22no—
308Dic3⋊C4—22no—
309C4×Dic3—21no—
310C4×Dic3—21no—
311C4×Dic3—22no—
312C4×Dic3—22no—
313C4×Dic3C3×D411yescharacteristic
314C4×Dic3C3×D411yescharacteristic
315C4×Dic3C23×C311yescharacteristic
316C4×Dic3C23×C311yescharacteristic
317#208—22no—
318C2×C3⋊C8C23×C312yes—
319Dic12C2×C3×C418yescomplemented
320#204C2×C3×C418yescomplemented
321C22×C32—21no—
322C22×C32—21no—
323C22×C32C4×D412yes—
324C22×C32C22×D411yescharacteristic
325C2×C3×D3—22no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C32×C4—41no—
329C32×C4—41no—
330C32×C4—42no—
331C32×C4C4×D412yes—
332C32×C4C23×C411yescharacteristic
333C32×C4C2×#4111yescharacteristic
334C3×Dic3—24no—
335C3×Dic3—22no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—21no—
339C3×Dic3#13712yescomplemented
340C2×#41—31no—
341C22×Q8—31no—
342#137—38no—
343C2×Q16—32no—
344C2×SD16—32no—
345C2×M16—31no—
346C2×M16C2×C3×D311yescharacteristic
347C4×Q8—32no—
348C4×Q8—32no—
349C4×Q8—31no—
350C4×Q8—31no—
351C4×Q8—61no—
352C4×Q8—61no—
353C4×D4—31no—
354C4×D4—31no—
355C4×D4—61no—
356C4×D4—61no—
357#117—61no—
358#117C2×C3×D311yescharacteristic
359C2×C4⋊C4—61no—
360C2×C4⋊C4C2×C3×D311yescharacteristic
361C2×C42—31no—
362C2×C42—61no—
363C2×C42—61no—
364#107C2×C3×D312yes—
365#106C2×C3×D312yes—
366#103—32no—
367#103—32no—
368#103—32no—
369#102—32no—
370#97—32no—
371C4×C8—32no—
372C23×C3—21no—
373C23×C3—61no—
374C23×C3D3×D411yescharacteristic
375C23×C3C2×C3×D411yescharacteristic
376C22×D3—21no—
377C3×Q8—34no—
378C3×Q8—32no—
379C3×Q8—64no—
380C3×Q8—62no—
381C3×Q8—61no—
382C3×D4—32no—
383C3×D4—64no—
384C3×D4—61no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4—42no—
404C2×C3×C4—62no—
405C2×C3×C4—64no—
406C2×C3×C4—61no—
407C2×C3×C4—62no—
408C2×C3×C4—64no—
409C2×C3×C4—82no—
410C2×C3×C4—122no—
411C2×C3×C4—122no—
412C2×C3×C4—121no—
413C2×C3×C4C2×C4×D312yes—
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4#23912yes—
416C2×C3×C4C22×C3×C412yes—
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C3×#4112yes—
419C2×C3×C4C23×D311yescharacteristic
420C2×C3×C4C24×C311yescharacteristic
421#67—24no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—41no—
427C2×Dic3C2×C3×D411yescharacteristic
428C2×Dic3C2×C3×D411yescharacteristic
429C2×Dic3C2×C3×D411yescharacteristic
430C2×Dic3C2×C3×D411yescharacteristic
431D12—21no—
432D12C22×C3×C412yes—
433C4×D3—24no—
434C4×D3—42no—
435C4×D3—42no—
436Dic6—24no—
437Dic6—22no—
438Dic6—21no—
439Dic6C22×C3×C414yes—
440Dic6C22×C3×C412yes—
441C3×C8—24no—
442C3×C8—62no—
443C3×C8C2×C4×D314yes—
444C3×C8C22×C3×C414yes—
445C3⋊C8—22no—
446C2×C32—22no—
447C2×C32C2×C4×D411yescharacteristic
448C2×C32C2×#13711yescharacteristic
449C2×C32#57511yescharacteristic
450C3×D3—42no—
451#41—34no—
452#41—62no—
453C2×Q8—32no—
454C2×Q8—31no—
455C2×Q8—34no—
456C2×Q8—61no—
457C2×Q8—62no—
458C2×D4—31no—
459C2×D4—61no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4C22×C3×D311yescharacteristic
468Q16—38no—
469SD16—38no—
470M16—62no—
471M16C3×C4×D314yescomplemented
472C2×C8—32no—
473C2×C8C22×C3×D312yes—
474C4⋊C4—21no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4C22×C3×D312yes—
480C4⋊C4C22×C3×D311yescharacteristic
481C4⋊C4C22×C3×D311yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—61no—
484C22⋊C4—61no—
485C42—21no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—31no—
490C42—61no—
491C42—61no—
492C42—62no—
493C42—62no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—21no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—62no—
503C22×C3—121no—
504C22×C3—121no—
505C22×C3#89312yes—
506C22×C3C3×C4×D412yes—
507C22×C3C2×D3×D411yescharacteristic
508C22×C3C22×C3×D411yescharacteristic
509C2×D3—22no—
510C2×D3—41no—
511C2×D3—41no—
512C3×C4—22no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—32no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—62no—
523C3×C4—64no—
524C3×C4—61no—
525C3×C4—61no—
526C3×C4—61no—
527C3×C4—82no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4#89312yes—
531C3×C4C3×C4×D412yes—
532C3×C4C22×C4×D311yescharacteristic
533C3×C4C2×#23911yescharacteristic
534C3×C4C23×C3×C411yescharacteristic
535C3×C4C2×C3×#4111yescharacteristic
536Dic3—21no—
537Dic3—21no—
538Dic3—21no—
539Dic3—24no—
540Dic3—22no—
541Dic3C3×#13712yescomplemented
542C32C4×#13711yescharacteristic, complemented, Sylow
543C23—61no—
544C23C3×D3×D411yescharacteristic
545Q8—34no—
546Q8—32no—
547Q8—64no—
548Q8—62no—
549Q8—61no—
550D4—32no—
551D4—64no—
552D4—61no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—31no—
560C2×C4—42no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4C2×C3×C4×D312yes—
570C2×C4C3×D3×D411yescharacteristic
571C2×C4C3×#23912yes—
572C2×C4C23×C3×D311yescharacteristic
573C8—62no—
574C8C2×C3×C4×D314yes—
575C2×C3—22no—
576C2×C3—22no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—42no—
581C2×C3—122no—
582C2×C3C2×#89311yescharacteristic
583C2×C3#577311yescharacteristic
584C2×C3D3×#13711yescharacteristic
585C2×C3C2×C3×C4×D411yescharacteristic
586C2×C3C2×C3×#13711yescharacteristic
587C2×C3C3×#57511yescharacteristic
588D3—42no—
589C22—21no—
590C22—21no—
591C22—62no—
592C22—121no—
593C22—121no—
594C22C3×#89312yes—
595C22C2×C3×D3×D411yescharacteristic
596C4—32no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—62no—
601C4—64no—
602C4—61no—
603C4—61no—
604C4—61no—
605C4C3×#89312yes—
606C4C22×C3×C4×D311yescharacteristic
607C4C2×C3×#23911yescharacteristic
608C3—21no—
609C3#7920011yescharacteristic, direct product factor
610C3C3×C4×#13711yescharacteristic, complemented
611C2—22no—
612C2—122no—
613C2C2×C3×#89311yescharacteristic
614C2C3×#577311yescharacteristic
615C2C3×D3×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#428C2
socleC22×C32C22×D4
prime radicalC2×C3×#235C22
cosocleC2×C3×C4C24×C3
hypercenterC3×#428D3
hypocenterC3C3×C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

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

NH ≅ G/Ntype
3. C2×C3×#887C2semidirect
4. C32×#428C2semidirect
5. C3×Dic3×M16C2semidirect
6. C3×#5004C2semidirect
8. C3×#4889C2semidirect
9. C3×#4873C2semidirect
10. C3×#4853C2semidirect
11. C3×#4852C2semidirect
13. #79200C3direct (N × H)
23. C3×#909C4semidirect
51. C2×#887C2×C3semidirect
57. C3×#428D3semidirect
58. C3×#428C2×C3semidirect
62. Dic3×M16C2×C3semidirect
63. #5004C2×C3semidirect
65. #4889C2×C3semidirect
66. #4873C2×C3semidirect
67. #4853C2×C3semidirect
68. #4852C2×C3semidirect
83. C2×C3×Dic6C2×C4semidirect
86. C32×M16C2×C4semidirect
110. C3×Dic12C2×C4semidirect
111. C3×#204C2×C4semidirect
149. #909C3×C4semidirect
221. #428C3×D3semidirect
258. C2×Dic6C2×C3×C4semidirect
265. C3×M16C4×D3semidirect
266. C3×M16C2×C3×C4semidirect
319. Dic12C2×C3×C4semidirect
320. #204C2×C3×C4semidirect
339. C3×Dic3#137semidirect
471. M16C3×C4×D3semidirect
541. Dic3C3×#137semidirect
542. C32C4×#137semidirect
609. C3#79200direct (N × H)
610. C3C3×C4×#137semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C3×#55222
C2×C3×#8872
C32×#4282
C3×Dic3×M162
C3×#50042
C3×#49662
C3×#48892
C3×#48732
C3×#48532
C3×#48522
C3×C4×#1373
#792003