Finite Group Databasebeta

← #11870520

Subgroups — #11870520

Order 1152.

956 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5728C211yesmaximal, characteristic, complemented
2C4×D3×D12C211yesmaximal, characteristic
3#10592782C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5#10591651C212yesmaximal
6Dic3×#215C212yesmaximal, complemented
7#10591549C212yesmaximal, complemented
8#10591536C212yesmaximal, complemented
9#10590300C211yesmaximal, characteristic, complemented
10#10590266C211yesmaximal, characteristic, complemented
11#10590248C211yesmaximal, characteristic, complemented
12#83351—31nomaximal
13#80408—31nomaximal
14C2×D3×D12C2211yescharacteristic
15C2×C4×D32—21no—
16C32×C4⋊Q8C2211yescharacteristic, complemented
17C2×#3755C2212yes—
18C4×C4⋊(C3⋊D3)C2211yescharacteristic
19C3×#946C2212yes—
20C3×Q8×Dic3C2212yescomplemented
21C2×C3×#215C2212yes—
22C3×#897—21no—
23C3×#892—21no—
24C3×#890—21no—
25C3×#887—21no—
26C3×C4×D12C2211yescharacteristic
27C3×C42×D3C2211yescharacteristic
28C3×#869C2211yescharacteristic, complemented
29D3×(C2×D3)⋊C4—21no—
30C4×#626—21no—
31#63894C2211yescharacteristic
32Dic3×D12C2211yescharacteristic
33D3×C4⋊Dic3C2211yescharacteristic
34#63880—21no—
35#63662C2212yescomplemented
36C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
37C3×#835C2212yescomplemented
38C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
39#63574C2212yescomplemented
40#63573C2212yescomplemented
41#63564C2212yes—
42#63562C2212yes—
43(C2×D3)⋊(C3⋊C8)C2212yes—
44Dic3×C3⋊C8C2212yescomplemented
45C3×#533—31no—
46#5728D311yescharacteristic, complemented
47C4×D3×D4—31no—
48C2×C4×D12—31no—
49C3×#461—31no—
50#5169—32no—
51Dic3×SD16—32no—
52#5084—31no—
53#5084—31no—
54#5083—31no—
55#5083—31no—
56#5083—61no—
57#5083D311yescharacteristic, complemented
58#5046—32no—
59#5043—32no—
60C4×#215—32no—
61C4×#215—32no—
62#5008—31no—
63#4925—31no—
64#4842—31no—
65#4833—31no—
66#4805—32no—
67#4788—32no—
68C22×D32—21no—
69C2×C32×Q8C2312yes—
70C2×C4⋊(C3⋊D3)C2311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3D411yescharacteristic
75C2×C3×C4×D3D411yescharacteristic
76C2×C3×C4×D3C2311yescharacteristic
77C2×C3×Dic6—21no—
78C2×#626—21no—
79C2×#624—21no—
80C2×D3×Dic3—21no—
81C2×D3×Dic3—21no—
82D3×D12—22no—
83D3×D12—22no—
84C4×D32—44no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C42C2311yescharacteristic
88#3755—24no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2312yes—
92C3×#215—24no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—22no—
97C3×C4⋊Dic3C2312yes—
98C3×C4⋊Dic3C2311yescharacteristic
99C3×Dic3⋊C4—22no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C2×C3×C3⋊C8C2312yes—
105Dic3⋊Dic3—21no—
106#3722—21no—
107(C2×D3)⋊Dic3—21no—
108#3101—91noSylow
109C2×D3×D4—31no—
110C22×D12—31no—
111C22×C4×D3—61no—
112C22×C4×D3—61no—
113C2×C3×SD16—32no—
114C3×C4⋊Q8—21no—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8C2×D311yescharacteristic, complemented
117C3×C4⋊Q8C2×D311yescharacteristic, complemented
118C3×#125—61no—
119C3×C22⋊Q8—32no—
120C3×C22⋊Q8—61no—
121C3×C4×Q8—32no—
122C3×C4×Q8—61no—
123C3×C4×D4—31no—
124C3×#117—61no—
125C2×C3×C42—31no—
126#946C2×D312yes—
127Q8×Dic3C2×D312yescomplemented
128C2×#215—32no—
129C2×#215—32no—
130C2×#215—62no—
131C2×#215C2×D312yes—
132D4×Dic3—31no—
133C4×#67—61no—
134C2×(C2×D3)⋊C4—61no—
135C2×C4⋊Dic3—31no—
136#897—21no—
137#893—31no—
138#892—21no—
139D3×C4⋊C4—31no—
140#890—21no—
141#887—21no—
142#67⋊C4—61no—
143D3×C22⋊C4—61no—
144C4×D12—31no—
145C4×D12—31no—
146C4×D12—31no—
147C4×D12—31no—
148C4×D12—61no—
149C4×D12—61no—
150C4×D12—61no—
151C4×D12C2×D311yescharacteristic
152C42×D3C2×D311yescharacteristic
153#869C2×D311yescharacteristic, complemented
154C3×C4⋊C8—31no—
155C3×#103—32no—
156#835—32no—
157#835—32no—
158#835—62no—
159#835C2×D312yescomplemented
160#832—32no—
161#830—32no—
162#821—32no—
163(C2×D3)⋊C8—32no—
164#817—32no—
165C8×Dic3—32no—
166#809—32no—
167#809—32no—
168#808—32no—
169#808—32no—
170#808—32no—
171C4⋊(C3⋊C8)—31no—
172C4⋊(C3⋊C8)—31no—
173C4⋊(C3⋊C8)—61no—
174C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
175C4×C3⋊C8—32no—
176C22×C3⋊D3—21no—
177C22×C3×D3—21no—
178C22×C3×D3C2×D411yescharacteristic
179C2×D32—24no—
180C2×D32—42no—
181C2×D32—42no—
182C32×Q8—22no—
183C32×Q8—22no—
184C2×C32×C4—22no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2411yescharacteristic
188C2×C3⋊Dic3—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)#4112yes—
191C4×C3⋊D3—42no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C3×D12—21no—
198C3×D12#4112yes—
199C3×C4×D3—21no—
200C3×C4×D3—21no—
201C3×C4×D3—41no—
202C3×C4×D3—42no—
203C3×C4×D3SD1614yescomplemented
204C3×Dic6—42no—
205#626—24no—
206#624—42no—
207D3×Dic3—42no—
208D3×Dic3—42no—
209C3⋊(C3⋊C8)—22no—
210C3×C3⋊C8—22no—
211#533—31no—
212C2×C4×D4—91no—
213#483—92no—
214#476—91no—
215#476—91no—
216#461—31no—
217#461—91no—
218#450—92no—
219C4×SD16—92no—
220C4×SD16—92no—
221#431—91no—
222C23×D3—61no—
223C23×D3—61no—
224C2×C3×Q8—22no—
225C2×C3×Q8—61no—
226C2×C3×Q8C22×D312yes—
227C2×C3×Q8C22×D312yes—
228C2×C3×D4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—61no—
233C2×#67—61no—
234C22×Dic3—61no—
235C22×Dic3—61no—
236D3×D4—62no—
237D3×D4—62no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—61no—
241C2×D12—62no—
242C2×D12—61no—
243C2×D12—62no—
244C2×D12C22×D311yescharacteristic
245C2×C4×D3—21no—
246C2×C4×D3—61no—
247C2×C4×D3—61no—
248C2×C4×D3—61no—
249C2×C4×D3—61no—
250C2×C4×D3—61no—
251C2×C4×D3—61no—
252C2×C4×D3—121no—
253C2×C4×D3—124no—
254C2×C4×D3—124no—
255C2×C4×D3#6711yescharacteristic
256C2×C4×D3#6711yescharacteristic
257C2×C4×D3C22×D311yescharacteristic
258C2×Dic6—21no—
259C3×SD16—64no—
260C2×C3×C8—32no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—32no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—41no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4—62no—
270C3×C4⋊C4—62no—
271C3×C4⋊C4—61no—
272C3×C22⋊C4—62no—
273C3×C22⋊C4—61no—
274C3×C42—21no—
275C3×C42—31no—
276C3×C42—31no—
277C3×C42—31no—
278C3×C42—62no—
279C3×C42C22×D311yescharacteristic
280C3×C42C22×D311yescharacteristic
281#217—61no—
282#215—24no—
283#215—64no—
284#215—64no—
285#215—124no—
286(C2×D3)⋊C4—22no—
287(C2×D3)⋊C4—21no—
288(C2×D3)⋊C4—61no—
289(C2×D3)⋊C4—61no—
290(C2×D3)⋊C4—61no—
291(C2×D3)⋊C4—61no—
292(C2×D3)⋊C4—61no—
293(C2×D3)⋊C4—121no—
294C4⋊Dic3—21no—
295C4⋊Dic3—22no—
296C4⋊Dic3—31no—
297C4⋊Dic3—31no—
298C4⋊Dic3—61no—
299C4⋊Dic3—61no—
300C4⋊Dic3C22×D311yescharacteristic
301C4⋊Dic3C22×D312yes—
302Dic3⋊C4—22no—
303Dic3⋊C4—61no—
304C4×Dic3—22no—
305C4×Dic3#6711yescharacteristic
306C4×Dic3#6711yescharacteristic
307C4×Dic3C22×D311yescharacteristic
308C2×C3⋊C8—32no—
309C2×C3⋊C8—32no—
310C2×C3⋊C8—62no—
311C2×C3⋊C8C22×D312yes—
312C22×C32C22×D411yescharacteristic
313C2×C3⋊D3—22no—
314C2×C3⋊D3—41no—
315C2×C3⋊D3—41no—
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—22no—
319C2×C3×D3—41no—
320C2×C3×D3—41no—
321C2×C3×D3C2×SD1612yes—
322D32—48no—
323C32×C4—21no—
324C32×C4—42no—
325C32×C4C2×SD1612yes—
326C32×C4C2×#4111yescharacteristic
327C32×C4C2×#4111yescharacteristic
328C3⋊Dic3—41no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—41no—
332C3×Dic3—42no—
333C3×Dic3C2×SD1612yes—
334C22×D4—91no—
335C23×C4—181no—
336C2×SD16—32no—
337C2×SD16—92no—
338C4⋊Q8—31no—
339C4⋊Q8D3211yescharacteristic, complemented
340#125—61no—
341C22⋊Q8—32no—
342C22⋊Q8—61no—
343C4×Q8—32no—
344C4×Q8—61no—
345C4×D4—31no—
346C4×D4—91no—
347C4×D4—91no—
348C4×D4—91no—
349C4×D4—181no—
350C4×D4—181no—
351#117—61no—
352C2×C4⋊C4—91no—
353C2×C22⋊C4—181no—
354C2×C42—31no—
355#106—92no—
356#106—92no—
357C4⋊C8—31no—
358C4⋊C8—91no—
359#103—32no—
360#103—92no—
361#102—92no—
362#102—92no—
363C22⋊C8—92no—
364C4×C8—92no—
365C23×C3—31no—
366C23×C3—61no—
367C22×D3—21no—
368C22×D3—64no—
369C22×D3—64no—
370C22×D3—61no—
371C22×D3—61no—
372C22×D3—121no—
373C22×D3—122no—
374C22×D3—122no—
375C22×D3—122no—
376C22×D3—122no—
377C22×D3C2×#6711yescharacteristic
378C3×Q8—22no—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—42no—
383C3×Q8—42no—
384C3×Q8—122no—
385C3×D4—32no—
386C3×D4—61no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—31no—
393C2×C3×C4—34no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—42no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—62no—
400C2×C3×C4—61no—
401C2×C3×C4—121no—
402C2×C3×C4—122no—
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4D3×D411yescharacteristic
405C2×C3×C4C2×#6711yescharacteristic
406C2×C3×C4C2×#6711yescharacteristic
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C23×D311yescharacteristic
409#67—64no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—121no—
415C2×Dic3—122no—
416C2×Dic3—122no—
417C2×Dic3C2×#6711yescharacteristic
418C2×Dic3C2×#6711yescharacteristic
419C2×Dic3C2×#6711yescharacteristic
420D12—21no—
421D12—32no—
422D12—32no—
423D12—62no—
424D12—64no—
425D12—61no—
426D12—61no—
427D12—121no—
428D12#23712yes—
429C4×D3—21no—
430C4×D3—21no—
431C4×D3—41no—
432C4×D3—42no—
433C4×D3—122no—
434C4×D3—122no—
435C4×D3—122no—
436C4×D3—122no—
437C4×D3—122no—
438C4×D3—122no—
439C4×D3—242no—
440C4×D3#21514yescomplemented
441Dic6—42no—
442C3×C8—62no—
443C3⋊C8—22no—
444C3⋊C8—62no—
445C3⋊C8—62no—
446C3⋊C8—122no—
447C2×C32#52511yescharacteristic
448C2×C32C22×SD1611yescharacteristic
449C2×C32#57511yescharacteristic
450C3⋊D3—42no—
451C3×D3—22no—
452C3×D3—42no—
453C24—181no—
454C2×Q8—61no—
455C2×Q8C2×D3212yes—
456C2×D4—31no—
457C2×D4—91no—
458C2×D4—181no—
459C2×D4—182no—
460C2×D4—182no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—61no—
465C22×C4—181no—
466C22×C4—181no—
467C22×C4—181no—
468C22×C4—181no—
469C22×C4—364no—
470SD16—64no—
471SD16—184no—
472C2×C8—32no—
473C2×C8—92no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—32no—
477C4⋊C4—31no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C4⋊C4—61no—
481C4⋊C4—91no—
482C4⋊C4—181no—
483C22⋊C4—62no—
484C22⋊C4—61no—
485C22⋊C4—181no—
486C22⋊C4—181no—
487C22⋊C4—181no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—62no—
492C42C2×D3211yescharacteristic
493C22×C3—21no—
494C22×C3—32no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3—62no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3C2×D3×D411yescharacteristic
501C22×C3C22×#6711yescharacteristic
502C2×D3—21no—
503C2×D3—21no—
504C2×D3—22no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—122no—
510C2×D3—128no—
511C2×D3—128no—
512C2×D3—121no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—241no—
517C2×D3—241no—
518C2×D3C2×#21512yes—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—22no—
524C3×C4—32no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—61no—
529C3×C4—61no—
530C3×C4—82no—
531C3×C4—122no—
532C3×C4—121no—
533C3×C4D3×SD1612yes—
534C3×C4C2×#21512yes—
535C3×C4C2×#23711yescharacteristic
536C3×C4D3×#4111yescharacteristic
537C3×C4D3×#4111yescharacteristic
538C3×C4D3×#4111yescharacteristic
539Dic3—21no—
540Dic3—21no—
541Dic3—41no—
542Dic3—42no—
543Dic3—121no—
544Dic3—121no—
545Dic3—241no—
546Dic3C2×#21512yes—
547C32#310111yescharacteristic, complemented, Sylow
548C23—31no—
549C23—61no—
550C23—184no—
551C23—181no—
552C23—362no—
553C23—362no—
554Q8—22no—
555Q8—22no—
556Q8—122no—
557D4—32no—
558D4—61no—
559D4—92no—
560D4—184no—
561D4—181no—
562C2×C4—22no—
563C2×C4—31no—
564C2×C4—34no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—121no—
572C2×C4—122no—
573C2×C4—181no—
574C2×C4—362no—
575C2×C4—362no—
576C2×C4—362no—
577C2×C4—362no—
578C2×C4D3×#6711yescharacteristic
579C2×C4D3×#6711yescharacteristic
580C2×C4C22×D3211yescharacteristic
581C8—62no—
582C8—182no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—62no—
587C2×C3—122no—
588C2×C3#560611yescharacteristic
589C2×C3C2×D3×SD1611yescharacteristic
590C2×C3#577711yescharacteristic
591C2×C3C22×#21511yescharacteristic
592C2×C3#582211yescharacteristic
593C2×C3#582811yescharacteristic
594D3—22no—
595D3—42no—
596D3—122no—
597D3—122no—
598D3—242no—
599C22—32no—
600C22—61no—
601C22—61no—
602C22—62no—
603C22—121no—
604C22—121no—
605C22—182no—
606C22—368no—
607C22—361no—
608C22—361no—
609C22C2×D3×#6711yescharacteristic
610C4—21no—
611C4—32no—
612C4—42no—
613C4—61no—
614C4—61no—
615C4—121no—
616C4—122no—
617C4—361no—
618C4D3×#21512yes—
619C4D3×#23711yescharacteristic
620C4#6431211yescharacteristic
621C3—21no—
622C3#8040811yescharacteristic, complemented
623C3#8335111yescharacteristic, complemented
624C2—62no—
625C2—122no—
626C2—362no—
627C2#1059530111yescharacteristic
628C2C2×D3×#21511yescharacteristic
629C2#1059531711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC2×D3×D12C22
hypercenterC4⋊Q8D32
hypocenterC32#3101
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5728C2semidirect
3. #10592782C2semidirect
4. C3×#5083C2semidirect
6. Dic3×#215C2semidirect
7. #10591549C2semidirect
8. #10591536C2semidirect
9. #10590300C2semidirect
10. #10590266C2semidirect
11. #10590248C2semidirect
16. C32×C4⋊Q8C22semidirect
20. C3×Q8×Dic3C22semidirect
28. C3×#869C22semidirect
35. #63662C22semidirect
36. C4⋊(C3⋊(C3⋊C8))C22semidirect
37. C3×#835C22semidirect
38. C3×C4⋊(C3⋊C8)C22semidirect
39. #63574C22semidirect
40. #63573C22semidirect
44. Dic3×C3⋊C8C22semidirect
46. #5728D3semidirect
57. #5083D3semidirect
116. C3×C4⋊Q8C2×D3semidirect
117. C3×C4⋊Q8C2×D3semidirect
127. Q8×Dic3C2×D3semidirect
153. #869C2×D3semidirect
159. #835C2×D3semidirect
174. C4⋊(C3⋊C8)C2×D3semidirect
203. C3×C4×D3SD16semidirect
339. C4⋊Q8D32semidirect
440. C4×D3#215semidirect
547. C32#3101semidirect
622. C3#80408semidirect
623. C3#83351semidirect

Maximal subgroups

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

subgroupindex
C3×#57282
C4×D3×D122
#105927822
C3×#50832
#105916512
Dic3×#2152
#105915492
#105915362
#105903002
#105902662
#105902482
#833513
#804083