Finite Group Databasebeta

← #10835956

Subgroups — #10835956

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82677C211yesmaximal, characteristic, complemented
2C2×#4787C211yesmaximal, characteristic, complemented
3C2×#4720C211yesmaximal, characteristic, complemented
4#73872C211yesmaximal, characteristic, complemented
5C3×#2034C211yesmaximal, characteristic, complemented
6#71684C211yesmaximal, characteristic
7#70574C211yesmaximal, characteristic, complemented
8#70563C211yesmaximal, characteristic, complemented
9#70191C211yesmaximal, characteristic, complemented
10#70186C211yesmaximal, characteristic, complemented
11#70162C211yesmaximal, characteristic, complemented
12#69962C211yesmaximal, characteristic, complemented
13#69526C211yesmaximal, characteristic, complemented
14#68892C211yesmaximal, characteristic, complemented
15#68854C211yesmaximal, characteristic, complemented
16#24667—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C22×#204C2211yescharacteristic
24C2×#893C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5222—21no—
31#5222—21no—
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4988C2211yescharacteristic
42#4975C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4957—21no—
51#4956—21no—
52#4948—21no—
53#4948—21no—
54#4787—24no—
55#4720—24no—
56#4670—21no—
57#4668—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#433—31no—
64#2620—31no—
65#2324—31no—
66#2035—31no—
67#2034D311yescharacteristic, complemented
68#2000—31no—
69#1999—31no—
70#1983—31no—
71#1972—31no—
72#1968—31no—
73#1948—31no—
74#1935—31no—
75#1934—31no—
76#1802—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—41no—
87C3×#121—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100#936—22no—
101#936—22no—
102C2×#213—22no—
103C2×#213—22no—
104C2×#213—41no—
105#930—22no—
106#930—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117#924—41no—
118#924—41no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C2×#204—22no—
129C2×#204—22no—
130C2×#204—41no—
131#893—22no—
132#893—22no—
133C3×#106—41no—
134C3×#102—22no—
135C3×#102—41no—
136C3×C22⋊C8—22no—
137C3×#95—21no—
138#832—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#830—22no—
144#821—41no—
145#816—22no—
146#815—22no—
147Dic3⋊C8—22no—
148Dic3⋊C8D414yescomplemented
149C8×Dic3—22no—
150#810—22no—
151#810—41no—
152#809—41no—
153#808—41no—
154C22×SD16—31no—
155C22×D8—31no—
156C2×#124—31no—
157C2×C22⋊Q8—31no—
158C2×#121—31no—
159C2×#121C2×D311yescharacteristic
160C2×C4×D4—31no—
161C2×#117—31no—
162#463—64no—
163#433—64no—
164C2×#106—31no—
165C2×#106C2×D311yescharacteristic
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102C2×D311yescharacteristic
175C2×#102C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8C2×D311yescharacteristic
178C2×#97—31no—
179C2×C4×C8—31no—
180#396—61no—
181#396—61no—
182#394—61no—
183#394—61no—
184#391—31no—
185#390—31no—
186#390—31no—
187#390—61no—
188#389—31no—
189#388—61no—
190#386—61no—
191#385—61no—
192#384—31no—
193#380C2×D311yescharacteristic
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C24×C3—21no—
200C24×C3C2×D411yescharacteristic
201C23×D3—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—42no—
209C2×C3×D4—81no—
210C2×C3×D4—81no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C2×#67—24no—
223C2×#67—81no—
224C2×#67—81no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12#4112yes—
238C2×C4×D3—42no—
239C2×C4×D3—81no—
240C2×C4×D3—81no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8C2×D412yes—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—81no—
261#217—22no—
262#217—22no—
263#217—42no—
264#217—42no—
265#217—42no—
266#217—81no—
267#217—81no—
268#213—44no—
269(C2×D3)⋊C4—24no—
270(C2×D3)⋊C4—42no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273C4⋊Dic3—21no—
274C4⋊Dic3—22no—
275C4⋊Dic3—22no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3#4112yes—
279Dic3⋊C4—42no—
280Dic3⋊C4—81no—
281Dic3⋊C4—81no—
282Dic3⋊C4—81no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—81no—
286C4×Dic3C2×D412yes—
287C4×Dic3#4112yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×D412yes—
294#204—44no—
295C22×Q8—31no—
296C22×D4—21no—
297C22×D4—31no—
298C22×D4—61no—
299C22×D4C22×D311yescharacteristic
300C23×C4—61no—
301C23×C4C22×D311yescharacteristic
302C2×SD16—62no—
303C2×SD16—62no—
304C2×SD16—121no—
305C2×D8—62no—
306C2×D8—62no—
307C2×D8—121no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312#124—62no—
313#124—62no—
314C22⋊Q8—121no—
315C22⋊Q8—121no—
316#121—41no—
317#121—41no—
318#121—62no—
319#121—62no—
320C4×D4—62no—
321C4×D4—62no—
322#117—62no—
323#117—62no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—61no—
351#106—41no—
352#106—121no—
353C4⋊C8—34no—
354C4⋊C8—62no—
355#103—62no—
356#103—62no—
357#103—121no—
358#102—22no—
359#102—41no—
360#102—62no—
361#102—121no—
362#102—121no—
363C22⋊C8—22no—
364C22⋊C8—62no—
365#97—62no—
366C4×C8—62no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3C22×D411yescharacteristic
387C22×D3—22no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—41no—
392C22×D3—41no—
393C22×D3—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—84no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4H(5,𝔽2)11yescharacteristic
420C2×C3×C4H(5,𝔽2)11yescharacteristic
421#67—84no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3#12112yes—
442C2×Dic3#12112yes—
443D12—22no—
444D12—22no—
445D12—41no—
446D12—41no—
447C4×D3—84no—
448Dic6—41no—
449Dic6—41no—
450Dic6—41no—
451Dic6—41no—
452C3×C8—22no—
453C3×C8—41no—
454C3⋊C8—22no—
455C3⋊C8—41no—
456C24—21no—
457C24—61no—
458C24D3×D411yescharacteristic
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—32no—
472C2×D4—42no—
473C2×D4—64no—
474C2×D4—61no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—81no—
480C2×D4—81no—
481C2×D4—241no—
482C2×D4—241no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—31no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—122no—
500C22×C4—241no—
501C22×C4—241no—
502C22×C4D3×D411yescharacteristic
503C22×C4C23×D311yescharacteristic
504SD16—124no—
505D8—124no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—32no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8D3×D412yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—32no—
525C4⋊C4—62no—
526C4⋊C4—62no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—122no—
531C4⋊C4—241no—
532C4⋊C4—241no—
533C4⋊C4—241no—
534C22⋊C4—42no—
535C22⋊C4—64no—
536C22⋊C4—62no—
537C22⋊C4—62no—
538C22⋊C4—81no—
539C22⋊C4—122no—
540C22⋊C4—122no—
541C22⋊C4—122no—
542C22⋊C4—122no—
543C22⋊C4—241no—
544C22⋊C4—241no—
545C22⋊C4—241no—
546C22⋊C4—241no—
547C42—32no—
548C42—32no—
549C42—61no—
550C42—61no—
551C42—241no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3—41no—
557C22×C3—41no—
558C22×C3—41no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3—81no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—81no—
566C22×C3C2×#12111yescharacteristic
567C22×C3#52511yescharacteristic
568C22×C3#53511yescharacteristic
569C22×C3#57611yescharacteristic
570C22×C3#57711yescharacteristic
571C22×C3#57711yescharacteristic
572C22×C3#57711yescharacteristic
573C2×D3—42no—
574C2×D3—42no—
575C2×D3—42no—
576C2×D3—81no—
577C2×D3—81no—
578C2×D3—81no—
579C2×D3—81no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4—81no—
583C3×C4—81no—
584C3×C4#54611yescharacteristic
585C3×C4#55011yescharacteristic
586C3×C4#55311yescharacteristic
587C3×C4#55511yescharacteristic
588Dic3—22no—
589Dic3—22no—
590Dic3—42no—
591Dic3—41no—
592Dic3—41no—
593Dic3—81no—
594Dic3—81no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—62no—
610C23—121no—
611C23—121no—
612C23—121no—
613C23—121no—
614C23—121no—
615C23—121no—
616C23C2×D3×D411yescharacteristic
617Q8—121no—
618Q8—121no—
619Q8—121no—
620Q8—121no—
621D4—41no—
622D4—41no—
623D4—41no—
624D4—41no—
625D4—62no—
626D4—62no—
627D4—84no—
628D4—121no—
629D4—121no—
630D4—244no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—32no—
635C2×C4—32no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—61no—
650C2×C4—61no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—62no—
654C2×C4—61no—
655C2×C4—61no—
656C2×C4—61no—
657C2×C4—61no—
658C2×C4—61no—
659C2×C4—61no—
660C2×C4—121no—
661C2×C4—121no—
662C2×C4—121no—
663C2×C4—121no—
664C2×C4—121no—
665C2×C4—121no—
666C2×C4—121no—
667C2×C4—121no—
668C2×C4—244no—
669C2×C4C2×D3×D411yescharacteristic
670C2×C4#100411yescharacteristic
671C2×C4#100411yescharacteristic
672C2×C4D3×#4111yescharacteristic
673C2×C4D3×#4111yescharacteristic
674C2×C4D3×#4111yescharacteristic
675C8—22no—
676C8—41no—
677C8—62no—
678C8—121no—
679C2×C3—81no—
680C2×C3—161no—
681C2×C3#241411yescharacteristic
682C2×C3#306011yescharacteristic
683C2×C3#306211yescharacteristic
684C2×C3#311811yescharacteristic
685C2×C3#312511yescharacteristic
686C2×C3#322111yescharacteristic
687C2×C3#322411yescharacteristic
688D3—82no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—122no—
704C22—122no—
705C22—122no—
706C22—241no—
707C22—241no—
708C22—241no—
709C22—241no—
710C22#559211yescharacteristic
711C22D3×#12111yescharacteristic
712C22#565611yescharacteristic
713C22#576211yescharacteristic
714C22#577011yescharacteristic
715C22#577411yescharacteristic
716C22#577511yescharacteristic
717C4—41no—
718C4—62no—
719C4—62no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4—121no—
724C4—122no—
725C4—121no—
726C4—241no—
727C4—241no—
728C4#559911yescharacteristic
729C4#561011yescharacteristic
730C4#566111yescharacteristic
731C4#567011yescharacteristic
732C3#2466711yescharacteristic, complemented, Sylow
733C2—81no—
734C2—161no—
735C2—242no—
736C2#7689611yescharacteristic
737C2#8012711yescharacteristic
738C2#8014311yescharacteristic
739C2#8066011yescharacteristic
740C2#8086711yescharacteristic
741C2#8131311yescharacteristic
742C2#8135211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2034C2
socleC23×C3C22×D4
prime radical#71684C2
hypercenter#2034D3
hypocenterC3#24667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
1. #82677C2semidirect
2. C2×#4787C2semidirect
3. C2×#4720C2semidirect
4. #73872C2semidirect
5. C3×#2034C2semidirect
7. #70574C2semidirect
8. #70563C2semidirect
9. #70191C2semidirect
10. #70186C2semidirect
11. #70162C2semidirect
12. #69962C2semidirect
13. #69526C2semidirect
14. #68892C2semidirect
15. #68854C2semidirect
67. #2034D3semidirect
148. Dic3⋊C8D4semidirect
732. C3#24667semidirect

Maximal subgroups

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

subgroupindex
#826772
C2×#47872
C2×#47202
#738722
C3×#20342
#716842
#705742
#705632
#701912
#701862
#701622
#699622
#695262
#688922
#688542
#246673