Finite Group Databasebeta

← #10828526

Subgroups — #10828526

Order 768.

1104 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5045C211yesmaximal, characteristic, complemented
3C2×#5035C211yesmaximal, characteristic
4#82069C211yesmaximal, characteristic, complemented
5#71739C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#71722C211yesmaximal, characteristic, complemented
8#71722C211yesmaximal, characteristic, complemented
9#71693C211yesmaximal, characteristic, complemented
10#71678C211yesmaximal, characteristic
11#71629C211yesmaximal, characteristic, complemented
12#71628C211yesmaximal, characteristic, complemented
13#71607C211yesmaximal, characteristic, complemented
14#71606C211yesmaximal, characteristic, complemented
15#71531C211yesmaximal, characteristic, complemented
16#24068—31nomaximal, Sylow
17C3×#529—24no—
18C3×#529—24no—
19C2×C3×#126—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117—21no—
29C22×C3×C42C2211yescharacteristic
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40#5210—21no—
41#5210—21no—
42C2×#830C2211yescharacteristic
43C2×#830C2211yescharacteristic
44#5045—24no—
45#5035—24no—
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C4×Dic3⋊C4—21no—
51C4×Dic3⋊C4—21no—
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#529D311yescharacteristic
65C2×#485—31no—
66C2×#482—31no—
67#2296—31no—
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042—31no—
72#1875—31no—
73#1873—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4—21no—
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4—21no—
83C23×C3×C4D411yescharacteristic
84C23×C3×C4D411yescharacteristic
85C23×C3×C4C2311yescharacteristic
86C3×#126—28no—
87C3×#125—22no—
88C3×#125—22no—
89C3×#124—22no—
90C3×#124—22no—
91C3×#123—28no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8D414yes—
94C3×#121—22no—
95C3×#121D414yes—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×D4—28no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×#117—28no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C42—22no—
114C2×C3×C42—22no—
115C2×C3×C42—21no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×#217—21no—
121C2×#217—21no—
122#926—22no—
123#926—22no—
124#926—22no—
125#926—22no—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2311yescharacteristic
130C2×Dic3⋊C4—21no—
131C2×Dic3⋊C4—21no—
132C2×Dic3⋊C4—21no—
133C2×Dic3⋊C4—21no—
134C2×C4×Dic3—21no—
135C2×C4×Dic3—21no—
136C22×C3⋊C8C2311yescharacteristic
137C22×C3⋊C8C2311yescharacteristic
138C22×C3⋊C8C2311yescharacteristic
139C22×C3⋊C8C2311yescharacteristic
140#835—22no—
141#835—22no—
142#835—22no—
143#835—22no—
144#832—22no—
145#832—22no—
146#832—22no—
147#832—22no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152#830—22no—
153#830—22no—
154#808—22no—
155#808—22no—
156#807—22no—
157#807—22no—
158#529—24no—
159#529—24no—
160C2×#126—21no—
161C2×#125C2×D311yescharacteristic
162C2×#124C2×D311yescharacteristic
163C2×#123—21no—
164C2×C22⋊Q8C2×D311yescharacteristic
165C2×#121C2×D311yescharacteristic
166C2×C4×Q8C2×D311yescharacteristic
167C2×C4×D4—21no—
168C2×C4×D4C2×D311yescharacteristic
169C2×#117—21no—
170C2×#117—31no—
171C2×#117—31no—
172C22×C42C2×D311yescharacteristic
173#485—64no—
174#482—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189#389—31no—
190#389—31no—
191#389—31no—
192#389—31no—
193#385—61no—
194#385—61no—
195C4×C4⋊C4—61no—
196C4×C4⋊C4—61no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205C24×C3—21no—
206C24×C3C2×D411yescharacteristic
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C2×C3×D4—24no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—42no—
220C2×C3×D4—42no—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—24no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—22no—
233C22×C3×C4—41no—
234C22×C3×C4—41no—
235C22×C3×C4—41no—
236C22×C3×C4—42no—
237C22×C3×C4—42no—
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D412yes—
243C22×C3×C4C2×D412yes—
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—21no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D412yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—24no—
263C3×C42—24no—
264C3×C42—22no—
265C3×C42—22no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270#217—24no—
271#217—42no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3#4112yes—
279C4⋊Dic3#4112yes—
280Dic3⋊C4—42no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284C4×Dic3—24no—
285C4×Dic3—42no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—21no—
300C22×D4C22×D311yescharacteristic
301C23×C4—21no—
302C23×C4#6711yescharacteristic
303C23×C4#6711yescharacteristic
304C23×C4C22×D311yescharacteristic
305C22×C8—31no—
306C22×C8—31no—
307C22×C8—31no—
308C22×C8—31no—
309#126—28no—
310#125—22no—
311#125—22no—
312#124—22no—
313#124—22no—
314#123—28no—
315C22⋊Q8—22no—
316C22⋊Q8#6714yes—
317#121—22no—
318#121#6714yes—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×D4—28no—
322C4×D4—22no—
323C4×D4—22no—
324#117—28no—
325#117—62no—
326#117—62no—
327#117—62no—
328#117—62no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42#6711yescharacteristic
356C2×C42#6711yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362#106—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371C22⋊C8—62no—
372C22⋊C8—62no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—22no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—41no—
390C23×C3#12012yes—
391C23×C3C22×D411yescharacteristic
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—41no—
395C3×Q8—41no—
396C3×D4—22no—
397C3×D4—22no—
398C3×D4—48no—
399C3×D4—41no—
400C3×D4—41no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—44no—
422C2×C3×C4—48no—
423C2×C3×C4—44no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4#12012yes—
431C2×C3×C4#13512yes—
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×C3×C42−1+411yescharacteristic
437C2×C3×C42−1+411yescharacteristic
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C3⋊C8—41no—
453C3⋊C8—41no—
454C24—21no—
455C24C2×#6711yescharacteristic
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×#6712yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4C2×#6712yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C2×#6712yes—
497C22×C4C2×#6712yes—
498C22×C4C23×D311yescharacteristic
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—32no—
518C4⋊C4—32no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—122no—
526C4⋊C4—122no—
527C4⋊C4—122no—
528C4⋊C4—122no—
529C4⋊C4C2×#6712yes—
530C4⋊C4C2×#6712yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—64no—
537C22⋊C4—122no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—64no—
546C42—122no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—22no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3—42no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3C2×#12011yescharacteristic
562C22×C3#52511yescharacteristic
563C22×C3#53611yescharacteristic
564C22×C3C2×#13511yescharacteristic
565C22×C3#57511yescharacteristic
566C22×C3#57511yescharacteristic
567C22×C3#57511yescharacteristic
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—41no—
572C3×C4—42no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4—42no—
576C3×C4#54711yescharacteristic
577C3×C4#54711yescharacteristic
578C3×C4#54711yescharacteristic
579C3×C4#54711yescharacteristic
580Dic3—42no—
581Dic3—42no—
582Dic3—81no—
583Dic3—81no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23#95312yes—
598C23C22×#6711yescharacteristic
599Q8—22no—
600Q8—22no—
601Q8—41no—
602Q8—41no—
603D4—22no—
604D4—22no—
605D4—48no—
606D4—41no—
607D4—41no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4#95012yes—
650C2×C4#95312yes—
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4#101011yescharacteristic
654C2×C4#101011yescharacteristic
655C2×C4C22×#6711yescharacteristic
656C2×C4C22×#6711yescharacteristic
657C8—121no—
658C8—121no—
659C8—121no—
660C8—121no—
661C2×C3—42no—
662C2×C3—82no—
663C2×C3#240611yescharacteristic
664C2×C3#300211yescharacteristic
665C2×C3#300511yescharacteristic
666C2×C3#310211yescharacteristic
667C2×C3#311111yescharacteristic
668C2×C3#323311yescharacteristic
669C2×C3#324011yescharacteristic
670C22—22no—
671C22—22no—
672C22—22no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22C2×#95011yescharacteristic
685C22#582211yescharacteristic
686C22#582211yescharacteristic
687C22#582211yescharacteristic
688C22#582511yescharacteristic
689C22#582811yescharacteristic
690C22C2×#95311yescharacteristic
691C4—22no—
692C4—22no—
693C4—41no—
694C4—42no—
695C4—41no—
696C4—41no—
697C4—42no—
698C4—42no—
699C4—122no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4#582411yescharacteristic
704C4#582411yescharacteristic
705C4#582411yescharacteristic
706C4#582411yescharacteristic
707C3#2406811yescharacteristic, complemented, Sylow
708C2—42no—
709C2—82no—
710C2#8312211yescharacteristic
711C2#8312511yescharacteristic
712C2#8314011yescharacteristic
713C2#8314211yescharacteristic
714C2#8408011yescharacteristic
715C2#8408111yescharacteristic
716C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3#24068
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5045C2semidirect
4. #82069C2semidirect
5. #71739C2semidirect
6. #71739C2semidirect
7. #71722C2semidirect
8. #71722C2semidirect
9. #71693C2semidirect
11. #71629C2semidirect
12. #71628C2semidirect
13. #71607C2semidirect
14. #71606C2semidirect
15. #71531C2semidirect
707. C3#24068semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50452
C2×#50352
#820692
#717392
#717222
#716932
#716782
#716292
#716282
#716072
#716062
#715312
#240683