Finite Group Databasebeta

← #10859083

Subgroups — #10859083

Order 768.

784 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
1#82243C211yesmaximal, characteristic
2#82171C211yesmaximal, characteristic
3C3×#1736C211yesmaximal, characteristic
4#71995C211yesmaximal, characteristic
5#71989C211yesmaximal, characteristic
6#71576C211yesmaximal, characteristic
7#71559C211yesmaximal, characteristic
8#68913C211yesmaximal, characteristic, complemented
9#68910C211yesmaximal, characteristic, complemented
10#68907C211yesmaximal, characteristic, complemented
11#68904C211yesmaximal, characteristic, complemented
12#68810C211yesmaximal, characteristic, complemented
13#68807C211yesmaximal, characteristic, complemented
14#68799C211yesmaximal, characteristic, complemented
15#68789C211yesmaximal, characteristic, complemented
16#26431—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#924—21no—
21C2×#924C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#887C2211yescharacteristic
24#5212C2211yescharacteristic
25#5210—21no—
26C4×#217C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4984C2211yescharacteristic
33#4982—21no—
34#4979—21no—
35#4975C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4945—21no—
45#4944—21no—
46#4933—21no—
47C4×Dic3⋊C4—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2362—31no—
62#2099—31no—
63#2086—31no—
64#2068—31no—
65#2024—31no—
66#1987—31no—
67#1920—31no—
68#1894—31no—
69#1872—31no—
70#1865—31no—
71#1854—31no—
72#1851—31no—
73#1803—31no—
74#1736D311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C2×C3×M16—22no—
80C2×C3×M16—41no—
81C2×C3×M16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#117—22no—
85C3×#117—41no—
86C2×C3×C4⋊C4—22no—
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×C42—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98C2×#217D411yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103#924—22no—
104#924—44no—
105#924Q814yescomplemented
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#888—22no—
120#888—41no—
121#887—22no—
122#887—41no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#830—22no—
133#816—41no—
134#816—41no—
135Dic3⋊C8—41no—
136Dic3⋊C8—41no—
137#810—22no—
138#810—41no—
139#808—22no—
140#807—41no—
141C22×M16C2×D311yescharacteristic
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8—31no—
146C2×#117C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#388—61no—
164#388—61no—
165#387—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C22⋊C4—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—81no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×Q812yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C2×Dic6—81no—
216C2×Dic6C2×Q812yes—
217C3×M16—44no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4#4112yes—
232C3×C4⋊C4#4112yes—
233C3×C22⋊C4—42no—
234C3×C42—42no—
235#217—24no—
236#217—22no—
237#217—22no—
238#217—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3C2×Q812yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C22×Q8—31no—
266C22×Q8—61no—
267C23×C4—21no—
268C23×C4C22×D311yescharacteristic
269C2×M16—22no—
270C2×M16—41no—
271C2×M16—41no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C4⋊Q8—62no—
277C4⋊Q8—121no—
278C22⋊Q8—34no—
279C22⋊Q8—62no—
280C22⋊Q8—124no—
281C4×Q8—62no—
282C4×Q8—121no—
283#117—22no—
284#117—41no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
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—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313#107—121no—
314#106—62no—
315C4⋊C8—121no—
316C4⋊C8—121no—
317#103—62no—
318#103—121no—
319#103—121no—
320#103—121no—
321C22⋊C8—62no—
322#95—31no—
323#95—31no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C22×D411yescharacteristic
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—84no—
358C2×C3×C4C22⋊Q812yes—
359C2×C3×C4C22×Q811yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3AGL(1,ℤ/8)12yes—
388C2×Dic3#13712yes—
389Dic6—22no—
390Dic6—22no—
391Dic6—41no—
392Dic6—41no—
393Dic6—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24D3×D411yescharacteristic
399C2×Q8—32no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8—241no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—81no—
427C22×C4#23512yes—
428C22×C4D3×D411yescharacteristic
429C22×C4D3×Q812yes—
430C22×C4C23×D311yescharacteristic
431M16—44no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—32no—
451C4⋊C4—42no—
452C4⋊C4—64no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4#23512yes—
466C4⋊C4#23512yes—
467C22⋊C4—42no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—64no—
471C22⋊C4—122no—
472C42—42no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—122no—
478C42—122no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C22⋊Q811yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3C2×#13711yescharacteristic
491C22×C3#57711yescharacteristic
492C22×C3#57811yescharacteristic
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#53811yescharacteristic
500C3×C4#55411yescharacteristic
501C3×C4#55411yescharacteristic
502Dic3—22no—
503Dic3—22no—
504Dic3—41no—
505Dic3—42no—
506Dic3—42no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#89612yes—
517C23C2×D3×D411yescharacteristic
518Q8—62no—
519Q8—62no—
520Q8—121no—
521Q8—121no—
522Q8—244no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—84no—
556C2×C4—121no—
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#89612yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×#23511yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4#100411yescharacteristic
572C2×C4C2×D3×Q811yescharacteristic
573C8—41no—
574C8—41no—
575C8—121no—
576C8—121no—
577C2×C3—42no—
578C2×C3#245011yescharacteristic
579C2×C3#307911yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#330811yescharacteristic
582C2×C3#331211yescharacteristic
583C2×C3#332411yescharacteristic
584C2×C3#333111yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22#549011yescharacteristic
593C22#549011yescharacteristic
594C22C2×#89611yescharacteristic
595C22D3×AGL(1,ℤ/8)11yescharacteristic
596C22#577511yescharacteristic
597C22D3×#13711yescharacteristic
598C22#577911yescharacteristic
599C4—22no—
600C4—42no—
601C4—42no—
602C4—62no—
603C4—62no—
604C4—81no—
605C4—81no—
606C4—121no—
607C4—121no—
608C4—122no—
609C4—122no—
610C4—241no—
611C4—241no—
612C4#547911yescharacteristic
613C4#549411yescharacteristic
614C4#551111yescharacteristic
615C4#551111yescharacteristic
616C3#2643111yescharacteristic, complemented, Sylow
617C2—42no—
618C2#7335111yescharacteristic
619C2#7889611yescharacteristic
620C2#7890711yescharacteristic
621C2#7894011yescharacteristic
622C2#7896611yescharacteristic
623C2#7924011yescharacteristic
624C2#7924611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1736D3
hypocenterC3#26431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
8. #68913C2semidirect
9. #68910C2semidirect
10. #68907C2semidirect
11. #68904C2semidirect
12. #68810C2semidirect
13. #68807C2semidirect
14. #68799C2semidirect
15. #68789C2semidirect
105. #924Q8semidirect
616. C3#26431semidirect

Maximal subgroups

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

subgroupindex
#822432
#821712
C3×#17362
#719952
#719892
#715762
#715592
#689132
#689102
#689072
#689042
#688102
#688072
#687992
#687892
#264313