Finite Group Databasebeta

← #10770773

Subgroups — #10770773

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3564C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#83123C211yesmaximal, characteristic, complemented
4#83103C211yesmaximal, characteristic, complemented
5#71898C212yesmaximal, complemented
6#71890C212yesmaximal, complemented
7#71886C212yesmaximal, complemented
8#71882C212yesmaximal
9#71837C211yesmaximal, characteristic, complemented
10#71757C211yesmaximal, characteristic, complemented
11#71745C211yesmaximal, characteristic, complemented
12#18472—31nomaximal, Sylow
13C3×#563—21no—
14C3×#563—21no—
15C3×#556—21no—
16C3×#554—21no—
17C3×#552—21no—
18C3×#552—21no—
19C3×#551—21no—
20C3×#551—21no—
21C3×#544C2211yescharacteristic
22C3×#543C2211yescharacteristic
23C3×#542—21no—
24C3×#541—21no—
25C3×#541—21no—
26C3×#539—21no—
27C3×#539C2211yescharacteristic
28C3×#538C2211yescharacteristic
29C3×#533C2211yescharacteristic
30C3×#529C2211yescharacteristic
31C2×C3×#126C2211yescharacteristic
32C2×#869C2211yescharacteristic
33#5080—21no—
34#5079—21no—
35#5064—21no—
36#5055—21no—
37#5043C2212yes—
38#5043C2212yes—
39#5043C2212yes—
40#5033C2212yes—
41#5021—21no—
42#5020—21no—
43#5020—21no—
44#5008—21no—
45#4986—21no—
46#4981C2212yes—
47#4979—21no—
48#4953C2212yes—
49#4949—21no—
50#4947—21no—
51#4937C2211yescharacteristic
52#4926C2211yescharacteristic
53#4925C2211yescharacteristic
54#4538C2212yes—
55#4538C2212yes—
56#4538C2212yes—
57#4537C2212yes—
58#4528C2212yes—
59#4525C2212yes—
60#3564D311yescharacteristic
61#3238—31no—
62#3229—31no—
63#2821—31no—
64#1612—32no—
65#1611—32no—
66#1609—32no—
67#1608—32no—
68#1521—31no—
69#1515—31no—
70#1473—31no—
71C22×Dic6—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×#126—21no—
78C3×#126—21no—
79C3×#126—21no—
80C3×#126—21no—
81C3×#126—21no—
82C3×#126—21no—
83C3×#126—21no—
84C3×#126—21no—
85C3×#126—21no—
86C3×#126—21no—
87C3×#126—21no—
88C3×#126—21no—
89C3×#126—21no—
90C3×#125—21no—
91C3×#125—21no—
92C3×#125—21no—
93C3×#125—21no—
94C3×#125—21no—
95C3×#125—21no—
96C3×#125—21no—
97C3×#125—21no—
98C3×#125—21no—
99C3×#125D411yescharacteristic
100C3×#124—21no—
101C3×#124—21no—
102C3×#124—21no—
103C3×#124D411yescharacteristic
104C3×#123—21no—
105C3×#123—21no—
106C3×#123—21no—
107C3×#123—21no—
108C3×#123—21no—
109C3×#123—21no—
110C3×#123—21no—
111C3×#123—21no—
112C3×C22⋊Q8—21no—
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8—21no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—21no—
117C3×C22⋊Q8—21no—
118C3×C22⋊Q8—21no—
119C3×C22⋊Q8—21no—
120C3×C22⋊Q8—21no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8—21no—
123C3×C22⋊Q8—21no—
124C3×C22⋊Q8—21no—
125C3×C22⋊Q8—21no—
126C3×C22⋊Q8C2311yescharacteristic
127C3×C22⋊Q8C2311yescharacteristic
128C3×C22⋊Q8C2311yescharacteristic
129C3×#121—21no—
130C3×#121C2311yescharacteristic
131C3×#120—21no—
132C3×C4×Q8—21no—
133C3×C4×Q8—21no—
134C3×C4×Q8—21no—
135C3×C4×Q8D411yescharacteristic
136C3×C4×Q8D411yescharacteristic
137C3×C4×Q8D411yescharacteristic
138C3×C4×D4—21no—
139C3×C4×D4—21no—
140C3×C4×D4—21no—
141C3×C4×D4—21no—
142C3×C4×D4—21no—
143C3×C4×D4D411yescharacteristic
144C3×#117—21no—
145C3×#117—21no—
146C3×#117—21no—
147C3×#117—21no—
148C3×#117—21no—
149C3×#117—21no—
150C2×C3×C4⋊C4—21no—
151C2×C3×C4⋊C4D411yescharacteristic
152C2×C3×C4⋊C4D411yescharacteristic
153C2×C3×C4⋊C4C2311yescharacteristic
154C2×C3×C22⋊C4—21no—
155C2×C3×C22⋊C4D411yescharacteristic
156C2×C3×C22⋊C4D411yescharacteristic
157C2×C3×C22⋊C4C2311yescharacteristic
158C2×C3×C42C2311yescharacteristic
159C2×#217—22no—
160C2×C4⋊Dic3—21no—
161C2×C4⋊Dic3C2312yes—
162C2×Dic3⋊C4—22no—
163#869—21no—
164#869—21no—
165#869—21no—
166#869—21no—
167#835—22no—
168#835—22no—
169#835—22no—
170#832—22no—
171C3⋊#95—21no—
172C3⋊#95—21no—
173C3⋊#95—22no—
174C3⋊#95—22no—
175C3⋊#95C2312yes—
176#830C2312yes—
177#830C2312yes—
178#808—22no—
179#808—22no—
180#808—22no—
181#807—22no—
182C4⋊(C3⋊C8)—21no—
183C4⋊(C3⋊C8)—21no—
184#803—21no—
185C4×C3⋊C8—21no—
186#563—21no—
187#563—21no—
188#556—21no—
189#554—21no—
190#552—21no—
191#552—21no—
192#551—21no—
193#551—21no—
194#544C2×D311yescharacteristic
195#543C2×D311yescharacteristic
196#542—21no—
197#541—21no—
198#541—21no—
199#539—21no—
200#539C2×D311yescharacteristic
201#538C2×D311yescharacteristic
202#533C2×D311yescharacteristic
203C2×C4⋊Q8—31no—
204#529C2×D311yescharacteristic
205C2×#126C2×D311yescharacteristic
206#501—61no—
207#498—61no—
208#489—61no—
209#487—61no—
210#483—32no—
211#483—32no—
212#483—32no—
213#480—32no—
214#477—61no—
215#476—61no—
216#475—61no—
217#475—61no—
218#432—31no—
219#431—31no—
220#400—61no—
221#399—61no—
222#398—32no—
223#397—32no—
224#395—61no—
225#393—61no—
226#383—31no—
227#328—32no—
228#328—32no—
229#328—32no—
230#327—32no—
231#324—32no—
232#323—32no—
233C24×C3C2×D411yescharacteristic
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8C2×D411yescharacteristic
238C2×C3×Q8C2×D411yescharacteristic
239C2×C3×Q8C2×D411yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4C2×D411yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4C2×D411yescharacteristic
250C22×C3×C4C2×D411yescharacteristic
251C22×C3×C4C2×D411yescharacteristic
252C22×C3×C4C2×D411yescharacteristic
253C22×C3×C4C2×D411yescharacteristic
254C22×C3×C4C2411yescharacteristic
255C22×Dic3—22no—
256C22×Dic3—22no—
257C2×Dic6—41no—
258C2×Dic6—41no—
259C2×Dic6—82no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—21no—
277C3×C4⋊C4—21no—
278C3×C4⋊C4—21no—
279C3×C4⋊C4—21no—
280C3×C4⋊C4—21no—
281C3×C4⋊C4—21no—
282C3×C4⋊C4—21no—
283C3×C4⋊C4—21no—
284C3×C4⋊C4—21no—
285C3×C4⋊C4—21no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4C2×D411yescharacteristic
290C3×C4⋊C4C2×D411yescharacteristic
291C3×C4⋊C4C2×D411yescharacteristic
292C3×C4⋊C4C2×D411yescharacteristic
293C3×C22⋊C4—21no—
294C3×C22⋊C4—21no—
295C3×C22⋊C4—21no—
296C3×C22⋊C4—21no—
297C3×C22⋊C4—21no—
298C3×C22⋊C4—21no—
299C3×C22⋊C4—21no—
300C3×C22⋊C4—21no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—21no—
304C3×C22⋊C4—21no—
305C3×C22⋊C4—21no—
306C3×C22⋊C4—21no—
307C3×C22⋊C4—21no—
308C3×C22⋊C4—21no—
309C3×C22⋊C4—21no—
310C3×C22⋊C4—21no—
311C3×C22⋊C4—21no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—21no—
314C3×C42—21no—
315C3×C42—21no—
316C3×C42—21no—
317C3×C42—21no—
318C3×C42—21no—
319C3×C42—21no—
320C3×C42—21no—
321C3×C42C2×D411yescharacteristic
322C3×C42C2×D411yescharacteristic
323C3×C42C2×D411yescharacteristic
324C3×C42C2×D411yescharacteristic
325#217—82no—
326C4⋊Dic3—22no—
327C4⋊Dic3—22no—
328C4⋊Dic3—41no—
329C4⋊Dic3—41no—
330Dic3⋊C4—82no—
331C2×C3⋊C8—22no—
332C2×C3⋊C8—22no—
333C22×Q8—61no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—61no—
338C4⋊Q8—61no—
339C4⋊Q8—61no—
340C4⋊Q8—61no—
341C4⋊Q8#6711yescharacteristic
342C4⋊Q8#6711yescharacteristic
343#126—21no—
344#126—21no—
345#126—21no—
346#126—21no—
347#126—21no—
348#126—21no—
349#126—21no—
350#126—21no—
351#126—21no—
352#126—21no—
353#126—21no—
354#126—21no—
355#126—21no—
356#125—21no—
357#125—21no—
358#125—21no—
359#125—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125—21no—
365#125#6711yescharacteristic
366#124—21no—
367#124—21no—
368#124—21no—
369#124#6711yescharacteristic
370#123—21no—
371#123—21no—
372#123—21no—
373#123—21no—
374#123—21no—
375#123—21no—
376#123—21no—
377#123—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8C22×D311yescharacteristic
393C22⋊Q8C22×D311yescharacteristic
394C22⋊Q8C22×D311yescharacteristic
395#121—21no—
396#121C22×D311yescharacteristic
397#120—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8#6711yescharacteristic
402C4×Q8#6711yescharacteristic
403C4×Q8#6711yescharacteristic
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4#6711yescharacteristic
410#117—21no—
411#117—21no—
412#117—21no—
413#117—21no—
414#117—21no—
415#117—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—32no—
418C2×C4⋊C4—61no—
419C2×C4⋊C4—62no—
420C2×C4⋊C4#6711yescharacteristic
421C2×C4⋊C4#6711yescharacteristic
422C2×C4⋊C4C22×D311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—62no—
425C2×C22⋊C4#6711yescharacteristic
426C2×C22⋊C4#6711yescharacteristic
427C2×C22⋊C4C22×D311yescharacteristic
428C2×C42C22×D311yescharacteristic
429#107—62no—
430#106—62no—
431#106—62no—
432#106—62no—
433C4⋊C8—61no—
434C4⋊C8—61no—
435#103—62no—
436#103—62no—
437#103—62no—
438#102—62no—
439C22⋊C8—32no—
440C22⋊C8—32no—
441#97—61no—
442C4×C8—61no—
443#95—32no—
444#95—61no—
445#95—61no—
446#95—62no—
447#95—62no—
448C23×C3—21no—
449C23×C3—21no—
450C23×C3—22no—
451C23×C3—41no—
452C23×C3—41no—
453C23×C3C22×D411yescharacteristic
454C3×Q8—41no—
455C3×Q8—41no—
456C3×Q8—41no—
457C3×Q8—81no—
458C3×Q8—81no—
459C3×Q8—81no—
460C3×D4—41no—
461C3×D4—42no—
462C3×D4—81no—
463C3×D4—81no—
464C2×C3×C4—22no—
465C2×C3×C4—21no—
466C2×C3×C4—21no—
467C2×C3×C4—21no—
468C2×C3×C4—21no—
469C2×C3×C4—21no—
470C2×C3×C4—21no—
471C2×C3×C4—21no—
472C2×C3×C4—21no—
473C2×C3×C4—21no—
474C2×C3×C4—21no—
475C2×C3×C4—22no—
476C2×C3×C4—41no—
477C2×C3×C4—41no—
478C2×C3×C4—41no—
479C2×C3×C4—41no—
480C2×C3×C4—81no—
481C2×C3×C4—81no—
482C2×C3×C4—81no—
483C2×C3×C4—81no—
484C2×C3×C4—81no—
485C2×C3×C4#12011yescharacteristic
486C2×C3×C4#12011yescharacteristic
487C2×C3×C4#12011yescharacteristic
488C2×C3×C4#12011yescharacteristic
489C2×C3×C4C22×D411yescharacteristic
490C2×C3×C4C22×D411yescharacteristic
491C2×Dic3—42no—
492C2×Dic3—42no—
493C2×Dic3—42no—
494C2×Dic3—42no—
495C2×Dic3—42no—
496C2×Dic3—42no—
497Dic6—82no—
498Dic6—82no—
499C3⋊C8—42no—
500C3⋊C8—42no—
501C24C2×#6711yescharacteristic
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—121no—
506C2×Q8—121no—
507C2×Q8—242no—
508C2×Q8C2×#6711yescharacteristic
509C2×Q8C2×#6711yescharacteristic
510C2×Q8C2×#6711yescharacteristic
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4C2×#6711yescharacteristic
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4C2×#6711yescharacteristic
523C22×C4C2×#6711yescharacteristic
524C22×C4C2×#6711yescharacteristic
525C22×C4C2×#6711yescharacteristic
526C22×C4C2×#6711yescharacteristic
527C22×C4C23×D311yescharacteristic
528C2×C8—62no—
529C2×C8—62no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—62no—
560C4⋊C4—62no—
561C4⋊C4—121no—
562C4⋊C4—121no—
563C4⋊C4—242no—
564C4⋊C4C2×#6711yescharacteristic
565C4⋊C4C2×#6711yescharacteristic
566C4⋊C4C2×#6711yescharacteristic
567C4⋊C4C2×#6711yescharacteristic
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—21no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—21no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—21no—
577C22⋊C4—21no—
578C22⋊C4—21no—
579C22⋊C4—21no—
580C22⋊C4—21no—
581C22⋊C4—21no—
582C22⋊C4—21no—
583C22⋊C4—21no—
584C22⋊C4—21no—
585C22⋊C4—21no—
586C22⋊C4—21no—
587C22⋊C4—21no—
588C22⋊C4—21no—
589C22⋊C4—242no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—21no—
595C42—21no—
596C42—21no—
597C42C2×#6711yescharacteristic
598C42C2×#6711yescharacteristic
599C42C2×#6711yescharacteristic
600C42C2×#6711yescharacteristic
601C22×C3—21no—
602C22×C3—21no—
603C22×C3—41no—
604C22×C3—41no—
605C22×C3—41no—
606C22×C3—41no—
607C22×C3—41no—
608C22×C3—41no—
609C22×C3—82no—
610C22×C3C2×#12011yescharacteristic
611C22×C3#57712yes—
612C3×C4—21no—
613C3×C4—21no—
614C3×C4—21no—
615C3×C4—21no—
616C3×C4—21no—
617C3×C4—21no—
618C3×C4—41no—
619C3×C4—41no—
620C3×C4—41no—
621C3×C4—41no—
622C3×C4—41no—
623C3×C4—41no—
624C3×C4—41no—
625C3×C4—41no—
626C3×C4—41no—
627C3×C4—41no—
628Dic3—82no—
629Dic3—82no—
630C23—21no—
631C23—21no—
632C23—22no—
633C23—41no—
634C23—41no—
635C23C22×#6711yescharacteristic
636Q8—41no—
637Q8—41no—
638Q8—41no—
639Q8—81no—
640Q8—81no—
641Q8—81no—
642Q8—242no—
643Q8—242no—
644D4—41no—
645D4—42no—
646D4—81no—
647D4—81no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—122no—
670C2×C4—122no—
671C2×C4—122no—
672C2×C4—122no—
673C2×C4—122no—
674C2×C4—122no—
675C2×C4#95311yescharacteristic
676C2×C4#95311yescharacteristic
677C2×C4#95311yescharacteristic
678C2×C4#95311yescharacteristic
679C2×C4C22×#6711yescharacteristic
680C2×C4C22×#6711yescharacteristic
681C8—122no—
682C8—122no—
683C2×C3—22no—
684C2×C3—81no—
685C2×C3—81no—
686C2×C3#301011yescharacteristic
687C2×C3#302211yescharacteristic
688C2×C3#302811yescharacteristic
689C22—21no—
690C22—21no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—82no—
698C22#583712yes—
699C22C2×#95311yescharacteristic
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—242no—
717C4—242no—
718C3#1847211yescharacteristic, complemented, Sylow
719C2—22no—
720C2—81no—
721C2—81no—
722C2#8411311yescharacteristic
723C2#8412311yescharacteristic
724C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3564C2
socleC22×C3C2×#120
prime radicalC3×#538C22
hypercenter#3564D3
hypocenterC3#18472
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3564

Complements

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

NH ≅ G/Ntype
2. #83323C2semidirect
3. #83123C2semidirect
4. #83103C2semidirect
5. #71898C2semidirect
6. #71890C2semidirect
7. #71886C2semidirect
9. #71837C2semidirect
10. #71757C2semidirect
11. #71745C2semidirect
718. C3#18472semidirect

Maximal subgroups

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

subgroupindex
C3×#35642
#833232
#831232
#831032
#718982
#718902
#718862
#718822
#718372
#717572
#717452
#184723