Finite Group Databasebeta

← #10876268

Subgroups — #10876268

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82230C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic
3C3×#1854C211yesmaximal, characteristic
4#71576C211yesmaximal, characteristic
5#70644C211yesmaximal, characteristic, complemented
6#70624C211yesmaximal, characteristic, complemented
7#70590C211yesmaximal, characteristic, complemented
8#70570C211yesmaximal, characteristic, complemented
9#69963C211yesmaximal, characteristic
10#69911C211yesmaximal, characteristic
11#69525C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68915C211yesmaximal, characteristic, complemented
14#68794C211yesmaximal, characteristic, complemented
15#68793C211yesmaximal, characteristic, complemented
16#27596—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#4977C2211yescharacteristic
31#4975C2211yescharacteristic
32Dic3×C4⋊C4C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4973C2211yescharacteristic
35#4973C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4945—21no—
45#4944—21no—
46#4943—21no—
47#4943—21no—
48#4929—21no—
49#4929—21no—
50#4654—21no—
51#4653—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2314—31no—
61#2310—31no—
62#2096—31no—
63#2095—31no—
64#2074—31no—
65#2074—31no—
66#1928—31no—
67#1928—31no—
68#1927—31no—
69#1927—31no—
70#1854—31no—
71#1854D311yescharacteristic
72#1818—31no—
73#1787—31no—
74#1787—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6—22no—
80C22×Dic6—22no—
81C22×Dic6—22no—
82C22×Dic6—41no—
83C22×Dic6—41no—
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#117—41no—
89C3×#117—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C3×#107—41no—
115C3×#106—41no—
116C3×C22⋊C8—22no—
117C3×#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#816—22no—
133#816—22no—
134#816—41no—
135#816—41no—
136#810—22no—
137#810—22no—
138#810—41no—
139#810—41no—
140#808—41no—
141#807—41no—
142C23×Q8—31no—
143C2×#117—31no—
144C2×#117C2×D311yescharacteristic
145C2×#107—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×#106C2×D311yescharacteristic
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#103—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#391—31no—
160#391—31no—
161#391—31no—
162#391—31no—
163#391—61no—
164#391—61no—
165#388—61no—
166#388—61no—
167#387—61no—
168#387—61no—
169#384—31no—
170#382—31no—
171#381—61no—
172#381—61no—
173#380—31no—
174#380C2×D311yescharacteristic
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
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—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4112yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—24no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—24no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—44no—
221C2×Dic6—44no—
222C2×Dic6—44no—
223C2×Dic6—44no—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C2×Dic6#4112yes—
227C2×Dic6#4112yes—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C42—42no—
245#217—22no—
246#217—22no—
247#217—81no—
248C4⋊Dic3—22no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—81no—
261C4×Dic3—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C22×Q8—31no—
269C22×Q8—31no—
270C22×Q8—62no—
271C22×Q8—62no—
272C22×Q8—61no—
273C22×Q8—62no—
274C22×Q8—62no—
275C22×Q8—121no—
276C22×Q8—121no—
277C23×C4—61no—
278C23×C4C22×D311yescharacteristic
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#117—41no—
284#117—41no—
285#117—62no—
286#117—62no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—61no—
302C2×C42—21no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—61no—
306C2×C42—61no—
307#107—41no—
308#107—121no—
309#106—41no—
310#106—121no—
311#103—62no—
312#103—62no—
313#103—62no—
314#103—62no—
315#103—121no—
316#103—121no—
317#103—121no—
318#103—121no—
319C22⋊C8—22no—
320C22⋊C8—62no—
321#95—21no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3#12312yes—
342C23×C3C22×D411yescharacteristic
343C2×C3×C4—21no—
344C2×C3×C4—21no—
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—41no—
360C2×C3×C4—41no—
361C2×C3×C4#12312yes—
362C2×C3×C4C2×#4111yescharacteristic
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—21no—
381C2×Dic3—21no—
382C2×Dic3—48no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393Dic6—22no—
394Dic6—22no—
395Dic6—22no—
396Dic6—22no—
397Dic6—48no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C24D3×D411yescharacteristic
407C2×Q8—32no—
408C2×Q8—32no—
409C2×Q8—64no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—124no—
421C2×Q8—124no—
422C2×Q8—124no—
423C2×Q8—124no—
424C2×Q8—122no—
425C2×Q8—122no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—64no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4#23512yes—
446C22×C4D3×D411yescharacteristic
447C22×C4#23912yes—
448C22×C4C23×D311yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—241no—
476C4⋊C4—241no—
477C22⋊C4—42no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—81no—
481C22⋊C4—241no—
482C42—42no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—241no—
488C42—241no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3C2×#12311yescharacteristic
497C22×C3#52511yescharacteristic
498C22×C3#53511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C22×C3#57811yescharacteristic
502C22×C3#57811yescharacteristic
503C3×C4—22no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4#55111yescharacteristic
509C3×C4#55111yescharacteristic
510C3×C4#55111yescharacteristic
511C3×C4#55111yescharacteristic
512Dic3—41no—
513Dic3—41no—
514Dic3—41no—
515Dic3—41no—
516Dic3—42no—
517Dic3—42no—
518Dic3—81no—
519Dic3—81no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#89412yes—
527C23C2×D3×D411yescharacteristic
528Q8—62no—
529Q8—62no—
530Q8—62no—
531Q8—62no—
532Q8—128no—
533Q8—121no—
534Q8—121no—
535Q8—121no—
536Q8—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
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—41no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—128no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4#89412yes—
581C2×C4C2×#23511yescharacteristic
582C2×C4#100411yescharacteristic
583C2×C4#100411yescharacteristic
584C2×C4C2×#23911yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3—42no—
592C2×C3#248211yescharacteristic
593C2×C3#309711yescharacteristic
594C2×C3#309711yescharacteristic
595C2×C3#312111yescharacteristic
596C2×C3#312111yescharacteristic
597C2×C3#322111yescharacteristic
598C2×C3#322311yescharacteristic
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22C2×#89411yescharacteristic
607C22#559211yescharacteristic
608C22#565611yescharacteristic
609C22#577011yescharacteristic
610C22#577111yescharacteristic
611C22#577511yescharacteristic
612C22#577911yescharacteristic
613C4—22no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—121no—
620C4—121no—
621C4—121no—
622C4—122no—
623C4—122no—
624C4—241no—
625C4—241no—
626C4#555611yescharacteristic
627C4#555611yescharacteristic
628C4#563711yescharacteristic
629C4#563711yescharacteristic
630C3#2759611yescharacteristic, complemented, Sylow
631C2—42no—
632C2#7461411yescharacteristic
633C2#7917511yescharacteristic
634C2#7922011yescharacteristic
635C2#8012711yescharacteristic
636C2#8027111yescharacteristic
637C2#8135311yescharacteristic
638C2#8149711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1854D3
hypocenterC3#27596
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
5. #70644C2semidirect
6. #70624C2semidirect
7. #70590C2semidirect
8. #70570C2semidirect
12. #68915C2semidirect
13. #68915C2semidirect
14. #68794C2semidirect
15. #68793C2semidirect
630. C3#27596semidirect

Maximal subgroups

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

subgroupindex
#822302
#738682
C3×#18542
#715762
#706442
#706242
#705902
#705702
#699632
#699112
#695252
#689152
#687942
#687932
#275963