Finite Group Databasebeta

← #10858828

Subgroups — #10858828

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82171C211yesmaximal, characteristic
2#82162C211yesmaximal, characteristic
3C2×C3×#340C211yesmaximal, characteristic
4#71974C211yesmaximal, characteristic
5#71972C211yesmaximal, characteristic
6#71559C211yesmaximal, characteristic
7#71555C211yesmaximal, characteristic
8#68913C211yesmaximal, characteristic, complemented
9#68910C211yesmaximal, characteristic, complemented
10#68906C211yesmaximal, characteristic, complemented
11#68904C211yesmaximal, characteristic, complemented
12#68811C211yesmaximal, characteristic, complemented
13#68808C211yesmaximal, characteristic, complemented
14#68799C211yesmaximal, characteristic, complemented
15#68790C211yesmaximal, characteristic, complemented
16#26414—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#924—21no—
21C2×#924C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#887C2211yescharacteristic
24#5212C2211yescharacteristic
25#5210—21no—
26#5210—21no—
27C4×#217—21no—
28C4×#217C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4984C2211yescharacteristic
35#4982—21no—
36#4979—21no—
37#4978—21no—
38#4976—21no—
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
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#2360—31no—
62#2100—31no—
63#2087—31no—
64#2068—31no—
65#2024—31no—
66#1987—31no—
67#1921—31no—
68#1892—31no—
69#1872—31no—
70#1863—31no—
71#1852—31no—
72#1850—31no—
73#1803—31no—
74C2×#340D311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
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#807—22no—
140#807—41no—
141C23×C8C2×D311yescharacteristic
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8—31no—
146C22×C4⋊C4C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#107—31no—
149C2×#107—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#389—61no—
164#387—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—31no—
171C4×C22⋊C4—61no—
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—42no—
196C22×C3×C4—42no—
197C22×C3×C4—81no—
198C22×C3×C4—81no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×Q812yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C2×Dic6—81no—
219C2×Dic6C2×Q812yes—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—44no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4#4112yes—
236C3×C4⋊C4#4112yes—
237#217—24no—
238#217—22no—
239#217—42no—
240#217Q1614yes—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3C2×Q812yes—
248Dic3⋊C4—24no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—42no—
260C4×Dic3—81no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C22×Q8—31no—
268C22×Q8—61no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4C22×D311yescharacteristic
272C22×C8—22no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—41no—
276C22×C8—41no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C22⋊Q8—34no—
282C22⋊Q8—62no—
283C22⋊Q8—124no—
284C4×Q8—62no—
285C4×Q8—121no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42—61no—
315#107—62no—
316#107—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#103—62no—
320#103—121no—
321#103—121no—
322#103—121no—
323C22⋊C8—62no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C2×Q1612yes—
339C23×C3AGL(1,ℤ/8)12yes—
340C23×C3C22×D411yescharacteristic
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
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—41no—
358C2×C3×C4—41no—
359C2×C3×C4—84no—
360C2×C3×C4—84no—
361C2×C3×C4C22⋊Q812yes—
362C2×C3×C4C22×Q811yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3C2×Q1612yes—
391C2×Dic3C2×Q1612yes—
392Dic6—22no—
393Dic6—22no—
394Dic6—41no—
395Dic6—41no—
396Dic6—84no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24D3×D411yescharacteristic
402C2×Q8—32no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—122no—
409C2×Q8—122no—
410C2×Q8—241no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4#23512yes—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×Q812yes—
436C22×C4C23×D311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—44no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—64no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4#23512yes—
473C4⋊C4#23512yes—
474C22⋊C4—34no—
475C22⋊C4—64no—
476C22⋊C4—62no—
477C22⋊C4—122no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42—122no—
483C42—241no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3#47714yes—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3#54611yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3C22×Q1611yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52911yescharacteristic
504C3×C4#53811yescharacteristic
505C3×C4#55411yescharacteristic
506C3×C4#55411yescharacteristic
507Dic3—22no—
508Dic3—22no—
509Dic3—42no—
510Dic3—42no—
511Dic3—41no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#89612yes—
520C23#91112yes—
521C23D3×Q1612yes—
522C23C2×D3×D411yescharacteristic
523Q8—62no—
524Q8—62no—
525Q8—121no—
526Q8—121no—
527Q8—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4#89612yes—
573C2×C4C2×#23511yescharacteristic
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×#23511yescharacteristic
576C2×C4#100411yescharacteristic
577C2×C4#100411yescharacteristic
578C2×C4C2×D3×Q811yescharacteristic
579C8—41no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3—24no—
584C2×C3#245011yescharacteristic
585C2×C3C2×#47711yescharacteristic
586C2×C3#308411yescharacteristic
587C2×C3#329911yescharacteristic
588C2×C3#329911yescharacteristic
589C2×C3#330611yescharacteristic
590C2×C3#331011yescharacteristic
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22#487514yes—
598C22#549011yescharacteristic
599C22#549011yescharacteristic
600C22C2×#89611yescharacteristic
601C22C2×#91111yescharacteristic
602C22#576211yescharacteristic
603C22#576211yescharacteristic
604C22C2×D3×Q1611yescharacteristic
605C4—22no—
606C4—42no—
607C4—42no—
608C4—62no—
609C4—62no—
610C4—81no—
611C4—81no—
612C4—122no—
613C4—122no—
614C4—121no—
615C4—121no—
616C4—241no—
617C4—241no—
618C4#547911yescharacteristic
619C4#549411yescharacteristic
620C4#551111yescharacteristic
621C4#551111yescharacteristic
622C3#2641411yescharacteristic, complemented, Sylow
623C2—24no—
624C2#7335111yescharacteristic
625C2#7876111yescharacteristic
626C2#7876211yescharacteristic
627C2#7880611yescharacteristic
628C2#7880611yescharacteristic
629C2#7913011yescharacteristic
630C2C2×#487511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#340C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#340D3
hypocenterC3#26414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
8. #68913C2semidirect
9. #68910C2semidirect
10. #68906C2semidirect
11. #68904C2semidirect
12. #68811C2semidirect
13. #68808C2semidirect
14. #68799C2semidirect
15. #68790C2semidirect
105. #924Q8semidirect
622. C3#26414semidirect

Maximal subgroups

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

subgroupindex
#821712
#821622
C2×C3×#3402
#719742
#719722
#715592
#715552
#689132
#689102
#689062
#689042
#688112
#688082
#687992
#687902
#264143