Finite Group Databasebeta

← C3×#78800 #11867754

Subgroups — C3×#78800 #11867754

Order 1152.

1025 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5816C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3C2×C3×#901C211yesmaximal, characteristic, complemented
4C3×#5510C211yesmaximal, characteristic
5C2×C32×#103C211yesmaximal, characteristic, complemented
6C3×#5113C211yesmaximal, characteristic, complemented
7C3×#4964C211yesmaximal, characteristic, complemented
8C3×#4816C212yesmaximal, complemented
9C3×#4813C212yesmaximal, complemented
10C3×#4812C212yesmaximal, complemented
11C3×#4807C212yesmaximal, complemented
12C3×#3063—31nomaximal
13#78800C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C2×C3×#235C2211yescharacteristic
17C22×C32×C8C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C3×#947—21no—
20C3×#946C2212yes—
21C3×#946C2212yes—
22C3×#944—21no—
23C2×C3×#216C2212yes—
24C3×#942—24no—
25C2×C3×#215C2212yes—
26C3×#930—22no—
27C3×#929—22no—
28C3×#926C2211yescharacteristic
29C2×C3×#208C2211yescharacteristic
30C3×#901—24no—
31C2×C3×ΓL(1,𝔽25)C2212yes—
32C2×C3×D3×C8C2212yes—
33C3×#897—21no—
34C3×#897C2212yes—
35C3×#895—21no—
36C3×#895C2212yes—
37C32×#103C2212yes—
38C32×#103C2212yes—
39C3×#821C2211yescharacteristic
40C3×#821C2211yescharacteristic
41C3×#816C2211yescharacteristic
42C3×#816C2211yescharacteristic
43C3×#808C2212yes—
44C3×#807C2212yes—
45C2×C3×#137—31no—
46C2×C3×#131—31no—
47C3×#537—31no—
48C3×#536—31no—
49#5816C2×C311yescharacteristic, complemented
50C2×#942C2×C311yescharacteristic, complemented
51C2×#901C2×C311yescharacteristic, complemented
52#5510C2×C311yescharacteristic
53C3×#470—32no—
54C3×#468—32no—
55C3×#467—32no—
56C3×#465—32no—
57C3×#428—31no—
58C3×#416—31no—
59C2×C3×#103—21no—
60C2×C3×#103D311yescharacteristic, complemented
61C2×C3×#103C2×C311yescharacteristic, complemented
62#5113C2×C311yescharacteristic, complemented
63#4964C2×C311yescharacteristic, complemented
64#4816C2×C312yescomplemented
65#4813C2×C312yescomplemented
66#4812C2×C312yescomplemented
67#4807C2×C312yescomplemented
68C2×C32×Q8—21no—
69C2×C32×Q8—22no—
70C2×C32×Q8—22no—
71C2×C32×Q8C2312yes—
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—21no—
76C2×C3×#67D412yes—
77C3×#235—22no—
78C3×#235—22no—
79C3×#235—22no—
80C3×#235—22no—
81C2×C3×D12—22no—
82C2×C3×D12D411yescharacteristic
83C2×C3×D12C2311yescharacteristic
84C2×C3×C4×D3—21no—
85C2×C3×C4×D3C2312yes—
86C2×C3×Dic6D411yescharacteristic
87C2×C3×Dic6C2311yescharacteristic
88C2×C32×C8—22no—
89C2×C32×C8C2311yescharacteristic
90C2×C32×C8C2311yescharacteristic
91C32×C4⋊C4—21no—
92C32×C4⋊C4C2312yes—
93C3×#217—21no—
94C3×#216—24no—
95C3×#215—24no—
96C3×(C2×D3)⋊C4—22no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—22no—
99C3×(C2×D3)⋊C4—22no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3C2311yescharacteristic
102C3×C4⋊Dic3C2311yescharacteristic
103C3×Dic3⋊C4—22no—
104C3×Dic3⋊C4—22no—
105C3×C4×Dic3—21no—
106C3×#208D414yescomplemented
107C2×C3×C3⋊C8C2312yes—
108C3×ΓL(1,𝔽25)—24no—
109C3×D3×C8—24no—
110#3063—31noSylow
111C2×C3×#41—31no—
112C2×C3×#41—31no—
113C22×C3×Q8—21no—
114C22×C3×Q8C2×D311yescharacteristic
115C22×C3×Q8C22×C311yescharacteristic
116C2×#235C22×C311yescharacteristic
117C2×#235C22×C311yescharacteristic
118C3×#137—64no—
119C2×C3×Q16—32no—
120C2×C3×SD16—32no—
121C3×#131—64no—
122C2×C3×M16—31no—
123C2×C3×M16—32no—
124C22×C3×C8—21no—
125C22×C3×C8—32no—
126C22×C3×C8C2×D311yescharacteristic
127C22×C3×C8C22×C311yescharacteristic
128C3×C4⋊Q8—61no—
129C3×#124—61no—
130C3×#123—62no—
131C3×C22⋊Q8—32no—
132C3×C22⋊Q8—32no—
133C3×C22⋊Q8—32no—
134C3×C22⋊Q8—61no—
135C3×#121—32no—
136C3×#121—61no—
137C3×#121—62no—
138C3×#117—31no—
139C2×C3×C4⋊C4—21no—
140C2×C3×C4⋊C4C2×D311yescharacteristic
141C2×C3×C4⋊C4C22×C311yescharacteristic
142#947—21no—
143#946C22×C312yes—
144#946C22×C312yes—
145#944—21no—
146C2×#216C22×C312yes—
147#942—24no—
148C2×#215C22×C312yes—
149#930—22no—
150#929—22no—
151#926C22×C311yescharacteristic
152C2×#208C22×C311yescharacteristic
153#901—24no—
154C2×ΓL(1,𝔽25)C22×C312yes—
155C2×D3×C8C22×C312yes—
156#897—21no—
157#897C22×C312yes—
158#895—21no—
159#895C22×C312yes—
160C3×#107—32no—
161C3×#106—32no—
162C3×#103—22no—
163C3×#103—22no—
164C3×#103—31no—
165C3×#103—31no—
166C3×#103C2×D312yes—
167C3×#103C2×D312yes—
168C3×#103C22×C312yes—
169C3×#103C22×C312yes—
170C3×#102—31no—
171C3×#102—31no—
172#821C22×C311yescharacteristic
173#821C22×C311yescharacteristic
174#816C22×C311yescharacteristic
175#816C22×C311yescharacteristic
176#808C22×C312yes—
177#807C22×C312yes—
178C23×C32C2×D411yescharacteristic
179C22×C3×D3—21no—
180C22×C3×D3C2×D412yes—
181C32×Q8—22no—
182C32×Q8—22no—
183C32×Q8—42no—
184C2×C32×C4—22no—
185C2×C32×C4—22no—
186C2×C32×C4—42no—
187C2×C32×C4—81no—
188C2×C32×C4C2×D411yescharacteristic
189C2×C32×C4C2×D412yes—
190C2×C32×C4#4112yes—
191C2×C32×C4C2411yescharacteristic
192C3×#67—24no—
193C3×#67—42no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3C2×D412yes—
198C3×D12—22no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—82no—
202C3×C4×D3—22no—
203C3×C4×D3—22no—
204C3×C4×D3—42no—
205C3×Dic6—22no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C32×C8—21no—
209C32×C8—21no—
210C3×C3⋊C8C2×D414yes—
211C2×#137—31no—
212C2×#131—31no—
213#537—31no—
214#536—31no—
215#470—32no—
216#468—32no—
217#467—32no—
218#465—32no—
219#428—31no—
220#416—31no—
221C2×#103C3×D311yescharacteristic, complemented
222C3×#41—62no—
223C3×#41—62no—
224C3×#41—62no—
225C3×#41—62no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—24no—
229C2×C3×Q8—22no—
230C2×C3×Q8—22no—
231C2×C3×Q8—22no—
232C2×C3×Q8—22no—
233C2×C3×Q8—22no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—41no—
237C2×C3×Q8—42no—
238C2×C3×Q8C22×D312yes—
239C2×C3×Q8C23×C312yes—
240C2×C3×D4—32no—
241C2×C3×D4—31no—
242C2×C3×D4—31no—
243C2×C3×D4—62no—
244C2×C3×D4—61no—
245C22×C3×C4—21no—
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4—32no—
251C22×C3×C4—41no—
252C22×C3×C4—41no—
253C22×C3×C4—61no—
254C22×C3×C4C22×D311yescharacteristic
255C22×C3×C4C23×C311yescharacteristic
256C2×#67—21no—
257C2×#67C3×D412yes—
258#235—22no—
259#235—22no—
260#235—22no—
261#235—22no—
262C2×D12—22no—
263C2×D12C3×D411yescharacteristic
264C2×D12C23×C311yescharacteristic
265C2×C4×D3—21no—
266C2×C4×D3C23×C312yes—
267C2×Dic6C3×D411yescharacteristic
268C2×Dic6C23×C311yescharacteristic
269C3×Q16—64no—
270C3×SD16—64no—
271C3×M16—34no—
272C3×M16—64no—
273C2×C3×C8—22no—
274C2×C3×C8—22no—
275C2×C3×C8—21no—
276C2×C3×C8—21no—
277C2×C3×C8—32no—
278C2×C3×C8—42no—
279C2×C3×C8—64no—
280C2×C3×C8C22×D311yescharacteristic
281C2×C3×C8C22×D311yescharacteristic
282C2×C3×C8C23×C311yescharacteristic
283C2×C3×C8C23×C311yescharacteristic
284C3×C4⋊C4—21no—
285C3×C4⋊C4—21no—
286C3×C4⋊C4—22no—
287C3×C4⋊C4—31no—
288C3×C4⋊C4—31no—
289C3×C4⋊C4—41no—
290C3×C4⋊C4—62no—
291C3×C4⋊C4—62no—
292C3×C4⋊C4—62no—
293C3×C4⋊C4C22×D312yes—
294C3×C4⋊C4C23×C312yes—
295C3×C22⋊C4—62no—
296C3×C22⋊C4—62no—
297C3×C22⋊C4—62no—
298C3×C22⋊C4—62no—
299C3×C22⋊C4—61no—
300C3×C42—61no—
301#217—21no—
302#216—24no—
303#215—24no—
304(C2×D3)⋊C4—22no—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—22no—
307(C2×D3)⋊C4—22no—
308C4⋊Dic3—22no—
309C4⋊Dic3C23×C311yescharacteristic
310C4⋊Dic3C23×C311yescharacteristic
311Dic3⋊C4—22no—
312Dic3⋊C4—22no—
313C4×Dic3—21no—
314#208C3×D414yescomplemented
315C2×C3⋊C8C23×C312yes—
316ΓL(1,𝔽25)—24no—
317D3×C8—24no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32#12112yes—
321C22×C32C22×D411yescharacteristic
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—22no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4#12112yes—
331C32×C4C22×D411yescharacteristic
332C32×C4C2×#4111yescharacteristic
333C3×Dic3—22no—
334C3×Dic3—41no—
335C3×Dic3—41no—
336C3×Dic3—41no—
337C2×#41—31no—
338C2×#41—31no—
339C22×Q8C2×C3×D311yescharacteristic
340#137—64no—
341C2×Q16—32no—
342C2×SD16—32no—
343#131—64no—
344C2×M16—31no—
345C2×M16—32no—
346C22×C8—32no—
347C22×C8C2×C3×D311yescharacteristic
348C4⋊Q8—61no—
349#124—61no—
350#123—62no—
351C22⋊Q8—32no—
352C22⋊Q8—32no—
353C22⋊Q8—32no—
354C22⋊Q8—61no—
355#121—32no—
356#121—61no—
357#121—62no—
358#117—31no—
359C2×C4⋊C4C2×C3×D311yescharacteristic
360#107—32no—
361#106—32no—
362#103—31no—
363#103—31no—
364#103C2×C3×D312yes—
365#103C2×C3×D312yes—
366#102—31no—
367#102—31no—
368C23×C3—21no—
369C23×C3—32no—
370C23×C3—61no—
371C23×C3D3×D411yescharacteristic
372C23×C3C2×C3×D411yescharacteristic
373C22×D3—21no—
374C22×D3C2×C3×D412yes—
375C3×Q8—24no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—22no—
380C3×Q8—42no—
381C3×Q8—44no—
382C3×Q8—42no—
383C3×Q8—42no—
384C3×Q8—62no—
385C3×Q8—61no—
386C3×Q8—61no—
387C3×D4—64no—
388C3×D4—62no—
389C3×D4—61no—
390C3×D4—61no—
391C3×D4—122no—
392C3×D4—242no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—32no—
402C2×C3×C4—42no—
403C2×C3×C4—42no—
404C2×C3×C4—44no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—81no—
413C2×C3×C4—81no—
414C2×C3×C4—82no—
415C2×C3×C4—122no—
416C2×C3×C4#23512yes—
417C2×C3×C4D3×D411yescharacteristic
418C2×C3×C4D3×D412yes—
419C2×C3×C4C2×C3×D411yescharacteristic
420C2×C3×C4C2×C3×D412yes—
421C2×C3×C4C3×#4112yes—
422C2×C3×C4C23×D311yescharacteristic
423C2×C3×C4C24×C311yescharacteristic
424#67—24no—
425#67—42no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3C2×C3×D412yes—
430D12—22no—
431D12—21no—
432D12—21no—
433D12—82no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—42no—
437Dic6—22no—
438Dic6—21no—
439Dic6—21no—
440C3×C8—21no—
441C3×C8—21no—
442C3×C8—21no—
443C3×C8—21no—
444C3×C8—34no—
445C3×C8—41no—
446C3×C8—41no—
447C3⋊C8C2×C3×D414yes—
448C2×C32—22no—
449C2×C32C2×#12111yescharacteristic
450C2×C32#57711yescharacteristic
451C2×C32#57811yescharacteristic
452C3×D3—42no—
453C3×D3—81no—
454#41—62no—
455#41—62no—
456#41—62no—
457#41—62no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—31no—
462C2×Q8—31no—
463C2×Q8C22×C3×D312yes—
464C2×D4—32no—
465C2×D4—31no—
466C2×D4—31no—
467C2×D4—62no—
468C2×D4—61no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—32no—
472C22×C4—61no—
473C22×C4C22×C3×D311yescharacteristic
474Q16—64no—
475SD16—64no—
476M16—34no—
477M16—64no—
478C2×C8—22no—
479C2×C8—32no—
480C2×C8—64no—
481C2×C8C22×C3×D311yescharacteristic
482C2×C8C22×C3×D311yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4C22×C3×D312yes—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—61no—
495C42—61no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—62no—
505C22×C3—62no—
506C22×C3—62no—
507C22×C3—121no—
508C22×C3—121no—
509C22×C3—121no—
510C22×C3#88412yes—
511C22×C3C3×#12112yes—
512C22×C3C2×D3×D411yescharacteristic
513C22×C3C22×C3×D411yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C3×C4—22no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—44no—
524C3×C4—42no—
525C3×C4—42no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—62no—
529C3×C4—82no—
530C3×C4—121no—
531C3×C4—121no—
532C3×C4—121no—
533C3×C4#88412yes—
534C3×C4C3×#12112yes—
535C3×C4C2×#23511yescharacteristic
536C3×C4C2×D3×D411yescharacteristic
537C3×C4C22×C3×D411yescharacteristic
538C3×C4C2×C3×#4111yescharacteristic
539Dic3—22no—
540Dic3—41no—
541Dic3—41no—
542Dic3—41no—
543C32#306311yescharacteristic, complemented, Sylow
544C23—32no—
545C23—61no—
546C23C3×D3×D411yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—42no—
550Q8—62no—
551Q8—61no—
552Q8—61no—
553D4—64no—
554D4—62no—
555D4—61no—
556D4—61no—
557D4—122no—
558D4—242no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—32no—
562C2×C4—42no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—81no—
569C2×C4—122no—
570C2×C4C3×#23512yes—
571C2×C4C3×D3×D411yescharacteristic
572C2×C4C3×D3×D412yes—
573C2×C4C23×C3×D311yescharacteristic
574C8—21no—
575C8—21no—
576C8—34no—
577C2×C3—24no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—122no—
584C2×C3—241no—
585C2×C3C2×#88411yescharacteristic
586C2×C3#576211yescharacteristic
587C2×C3#576611yescharacteristic
588C2×C3C2×C3×#12111yescharacteristic
589C2×C3C3×#57711yescharacteristic
590C2×C3C3×#57811yescharacteristic
591D3—42no—
592D3—81no—
593C22—21no—
594C22—21no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22—121no—
599C22—121no—
600C22—121no—
601C22C3×#88412yes—
602C22C2×C3×D3×D411yescharacteristic
603C4—42no—
604C4—42no—
605C4—62no—
606C4—121no—
607C4—121no—
608C4—121no—
609C4C3×#88412yes—
610C4C2×C3×#23511yescharacteristic
611C4C2×C3×D3×D411yescharacteristic
612C3—21no—
613C3#7880011yescharacteristic, direct product factor
614C3C3×#306311yescharacteristic, complemented
615C2—22no—
616C2—122no—
617C2—241no—
618C2C2×C3×#88411yescharacteristic
619C2C3×#576211yescharacteristic
620C2C3×#576611yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#103C2
socleC22×C32C22×D4
prime radicalC3×#5510C2
hypercenterC2×C3×#103D3
hypocenterC3C3×#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
1. C3×#5816C2semidirect
2. C2×C3×#942C2semidirect
3. C2×C3×#901C2semidirect
5. C2×C32×#103C2semidirect
6. C3×#5113C2semidirect
7. C3×#4964C2semidirect
8. C3×#4816C2semidirect
9. C3×#4813C2semidirect
10. C3×#4812C2semidirect
11. C3×#4807C2semidirect
13. #78800C3direct (N × H)
49. #5816C2×C3semidirect
50. C2×#942C2×C3semidirect
51. C2×#901C2×C3semidirect
60. C2×C3×#103D3semidirect
61. C2×C3×#103C2×C3semidirect
62. #5113C2×C3semidirect
63. #4964C2×C3semidirect
64. #4816C2×C3semidirect
65. #4813C2×C3semidirect
66. #4812C2×C3semidirect
67. #4807C2×C3semidirect
106. C3×#208D4semidirect
221. C2×#103C3×D3semidirect
314. #208C3×D4semidirect
543. C32#3063semidirect
613. C3#78800direct (N × H)
614. C3C3×#3063semidirect

Maximal subgroups

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

subgroupindex
C3×#58162
C2×C3×#9422
C2×C3×#9012
C3×#55102
C2×C32×#1032
C3×#51132
C3×#49642
C3×#48162
C3×#48132
C3×#48122
C3×#48072
C3×#30633
#788003