Finite Group Databasebeta

← #10858119

Subgroups — #10858119

Order 768.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2442C211yesmaximal, characteristic
2C2×#5063C211yesmaximal, characteristic, complemented
3C2×#4964C211yesmaximal, characteristic
4#73468C211yesmaximal, characteristic
5#71739C211yesmaximal, characteristic, complemented
6#71559C211yesmaximal, characteristic
7#69635C211yesmaximal, characteristic, complemented
8#69632C211yesmaximal, characteristic, complemented
9#69623C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69561C211yesmaximal, characteristic, complemented
12#69540C211yesmaximal, characteristic, complemented
13#69516C211yesmaximal, characteristic
14#69506C211yesmaximal, characteristic
15#69450C211yesmaximal, characteristic
16#26367—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#400—21no—
24C3×#398—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#385—21no—
28C3×#382C2211yescharacteristic
29C3×#380—21no—
30C3×#380C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#830C2211yescharacteristic
33#5063—24no—
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38#4964—24no—
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4945—21no—
46#4944—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4652—21no—
53#4649—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2442D311yescharacteristic
65#2362—31no—
66#2097—31no—
67#2087—31no—
68#2068—31no—
69#2059—31no—
70#1949—31no—
71#1919—31no—
72#1894—31no—
73#1872—31no—
74#1865—31no—
75#1854—31no—
76#1851—31no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C3×#125—22no—
81C3×#125—41no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—41no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100#926—22no—
101#926—41no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924Q814yes—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×#208—22no—
114C2×#208—22no—
115C2×#208—22no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133#830—22no—
134#810—22no—
135#810—22no—
136#810—22no—
137#810—41no—
138#808—22no—
139#808—41no—
140#808D414yes—
141#807—22no—
142#807—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146C22×M16—31no—
147C2×#125C2×D311yescharacteristic
148C2×C22⋊Q8—31no—
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4C2×D311yescharacteristic
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8—31no—
167#400—21no—
168#398—21no—
169#391—31no—
170#389—31no—
171#389C2×D311yescharacteristic
172#389C2×D311yescharacteristic
173#388—61no—
174#388—61no—
175#387—61no—
176#385—21no—
177#382C2×D311yescharacteristic
178#380—21no—
179#380C2×D311yescharacteristic
180C4×C4⋊C4—31no—
181C4×C4⋊C4—61no—
182C4×C22⋊C4C2×D311yescharacteristic
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336—31no—
190#336—31no—
191C24×C3C2×D411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—81no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×Q812yes—
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×Q812yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4C2×D412yes—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—42no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241#217—24no—
242#217—42no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3C2×D412yes—
248C4⋊Dic3C2×Q812yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C4×Dic3—42no—
255#208—28no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D412yes—
267C2×C3⋊C8#4112yes—
268C22×Q8—31no—
269C23×C4—21no—
270C23×C4C22×D311yescharacteristic
271C2×M16—62no—
272C2×M16—62no—
273C2×M16—62no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—31no—
278#125—22no—
279#125—41no—
280C22⋊Q8—34no—
281C22⋊Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284#117—62no—
285#117—121no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4#6711yescharacteristic
307C2×C22⋊C4#6711yescharacteristic
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#107—62no—
315#107—62no—
316#106—34no—
317#106—62no—
318#106—121no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325C22⋊C8—62no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95#6711yescharacteristic
336#95#6711yescharacteristic
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22⋊Q812yes—
344C23×C3C22×D411yescharacteristic
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—84no—
376C2×C3×C4C22⋊Q812yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×Q811yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C42−1+411yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395Dic6—22no—
396Dic6—22no—
397Dic6—41no—
398Dic6—41no—
399C3⋊C8—22no—
400C3⋊C8—22no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C24C2×#6711yescharacteristic
404C2×Q8—32no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—81no—
425C22×C4D3×Q812yes—
426C22×C4#23912yes—
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C2×#6711yescharacteristic
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C23×D311yescharacteristic
433M16—68no—
434C2×C8—32no—
435C2×C8—32no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—64no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4D3×D412yes—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—64no—
472C22⋊C4—122no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—122no—
478C42—122no—
479C42—122no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3C2×C22⋊Q811yescharacteristic
488C22×C3D4211yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57811yescharacteristic
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52511yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4D4×Q811yescharacteristic
506C3×C4#55411yescharacteristic
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#94612yes—
518C23C22×#6711yescharacteristic
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—84no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4#94612yes—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4C2×D3×Q811yescharacteristic
570C2×C4C2×#23911yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#101011yescharacteristic
573C8—62no—
574C8—62no—
575C8—121no—
576C8—121no—
577C2×C3—42no—
578C2×C3#244911yescharacteristic
579C2×C3#308011yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#326911yescharacteristic
582C2×C3#327511yescharacteristic
583C2×C3#334911yescharacteristic
584C2×C3#335111yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22D4×#6711yescharacteristic
593C22C2×#94611yescharacteristic
594C22#582211yescharacteristic
595C22#582211yescharacteristic
596C22#582411yescharacteristic
597C22#583611yescharacteristic
598C22#583811yescharacteristic
599C4—22no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—122no—
609C4—122no—
610C4—122no—
611C4—241no—
612C4#562711yescharacteristic
613C4#563211yescharacteristic
614C4#563611yescharacteristic
615C4#564511yescharacteristic
616C3#2636711yescharacteristic, complemented, Sylow
617C2—42no—
618C2#8276411yescharacteristic
619C2#8301711yescharacteristic
620C2#8348011yescharacteristic
621C2#8350811yescharacteristic
622C2#8354311yescharacteristic
623C2#8359511yescharacteristic
624C2#8361211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2442C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2442D3
hypocenterC3#26367
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. C2×#5063C2semidirect
5. #71739C2semidirect
7. #69635C2semidirect
8. #69632C2semidirect
9. #69623C2semidirect
10. #69601C2semidirect
11. #69561C2semidirect
12. #69540C2semidirect
616. C3#26367semidirect

Maximal subgroups

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

subgroupindex
C3×#24422
C2×#50632
C2×#49642
#734682
#717392
#715592
#696352
#696322
#696232
#696012
#695612
#695402
#695162
#695062
#694502
#263673