Finite Group Databasebeta

← #10729263

Subgroups — #10729263

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#124C211yesmaximal, characteristic
2C2×#4945C212yesmaximal, complemented
3C2×#4933C211yesmaximal, characteristic, complemented
4#72134C211yesmaximal, characteristic, complemented
5#72134C211yesmaximal, characteristic, complemented
6#72131C211yesmaximal, characteristic
7#72131C211yesmaximal, characteristic
8#71549C212yesmaximal, complemented
9#71547C212yesmaximal, complemented
10#71545C212yesmaximal, complemented
11C4×C3⋊#95C211yesmaximal, characteristic, complemented
12#14932—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×#124—28no—
16C2×C3×#124—24no—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×#217C2212yes—
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×C4×Dic3C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4945—28no—
39#4933—24no—
40C22×#124D311yescharacteristic
41C2×#388—32no—
42C2×#387—31no—
43#1446—31no—
44#1446—31no—
45#1440—32no—
46#1439—32no—
47#1438—31no—
48#1438—31no—
49#1437—32no—
50C4×#95—31no—
51C25×C3C2312yes—
52C22×C3×Q8—24no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C22×C3×D4—24no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—22no—
61C22×C3×D4—42no—
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C3×#124—232no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4D418yes—
79C2×C3×C42—24no—
80C2×C3×C42D414yes—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—42no—
90C2×#217—42no—
91C2×#217—42no—
92C2×#217—42no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—24no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3—24no—
100C2×C4×Dic3—42no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8C2×D311yescharacteristic
113C23×D4C2×D311yescharacteristic
114C2×#124—28no—
115C2×#124—24no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C22⋊C4C2×D312yes—
123C22×C22⋊C4C2×D312yes—
124C22×C42—32no—
125C22×C42C2×D311yescharacteristic
126#388—68no—
127#387—64no—
128C2×#95—31no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—31no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3C2×D414yes—
154C24×C3C2411yescharacteristic
155C2×C3×Q8—28no—
156C2×C3×Q8—48no—
157C2×C3×Q8—48no—
158C2×C3×Q8—44no—
159C2×C3×Q8—44no—
160C2×C3×D4—28no—
161C2×C3×D4—48no—
162C2×C3×D4—48no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3#4114yes—
218C22×Dic3#4114yes—
219C22×Dic3#4112yes—
220C22×Dic3#4112yes—
221C3×C22⋊C4—216no—
222C3×C22⋊C4—216no—
223C3×C42—28no—
224#217—216no—
225#217—48no—
226#217—48no—
227#217—48no—
228Dic3⋊C4—28no—
229Dic3⋊C4—44no—
230C4×Dic3—48no—
231C25C22×D312yes—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—42no—
237C22×D4—24no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D312yes—
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252#124—232no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—64no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—122no—
272C2×C22⋊C4—122no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4#6718yes—
276C2×C42—24no—
277C2×C42—64no—
278C2×C42—64no—
279C2×C42—122no—
280C2×C42#6714yes—
281#95—64no—
282#95—62no—
283#95—62no—
284#95—62no—
285#95—62no—
286#95—62no—
287#95—62no—
288#95—64no—
289#95—64no—
290#95—122no—
291#95—122no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3C22×D411yescharacteristic
328C23×C3C22×D412yes—
329C23×C3C2×#4111yescharacteristic
330C23×C3C2×#4111yescharacteristic
331C23×C3C2×#4111yescharacteristic
332C23×C3C2×#4112yes—
333C23×C3C2×#4111yescharacteristic
334C23×C3C2×#4112yes—
335C23×C3H(5,𝔽2)12yes—
336C23×C3H(5,𝔽2)11yescharacteristic
337C23×C32−1+411yescharacteristic
338C3×Q8—416no—
339C3×D4—416no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4#12014yes—
361C2×C3×C4#12414yes—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—41no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24C2×#6714yes—
411C24C23×D311yescharacteristic
412C2×Q8—28no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×D4—28no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—44no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—62no—
471C22×C4#23712yes—
472C22×C4#23712yes—
473C22×C4C2×#6714yes—
474C22×C4C2×#6712yes—
475C22×C4C2×#6712yes—
476C22×C4C2×#6712yes—
477C22×C4C2×#6714yes—
478C4⋊C4—68no—
479C4⋊C4—124no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—616no—
483C22⋊C4—128no—
484C22⋊C4—128no—
485C22⋊C4—128no—
486C42—28no—
487C42—128no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3C2×#12011yescharacteristic
504C22×C3#52511yescharacteristic
505C22×C3#52512yes—
506C22×C3C2×#12411yescharacteristic
507C22×C3#52912yes—
508C22×C3#52912yes—
509C22×C3#53211yescharacteristic
510C22×C3#53211yescharacteristic
511C22×C3#53212yes—
512C22×C3#53511yescharacteristic
513C22×C3#53711yescharacteristic
514C22×C3#53812yes—
515C22×C3#54212yes—
516C22×C3#54612yes—
517C22×C3#54612yes—
518C22×C3#54612yes—
519C22×C3#54712yes—
520C22×C3#55112yes—
521C22×C3#55512yes—
522C22×C3#55611yescharacteristic
523C22×C3#55611yescharacteristic
524C22×C3#55911yescharacteristic
525C22×C3#55911yescharacteristic
526C3×C4—24no—
527C3×C4—24no—
528C3×C4—42no—
529C3×C4—44no—
530Dic3—42no—
531Dic3—42no—
532Dic3—44no—
533Dic3—44no—
534Dic3—81no—
535Dic3—81no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C2×#23711yescharacteristic
572C23C2×#23711yescharacteristic
573C23#100412yes—
574C23D3×#4112yes—
575C23D3×#4111yescharacteristic
576C23D3×#4112yes—
577C23D3×#4111yescharacteristic
578C23#100911yescharacteristic
579C23#101011yescharacteristic
580C23C22×#6711yescharacteristic
581C23C22×#6712yes—
582Q8—416no—
583D4—416no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—121no—
625C2×C4—122no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4#93614yes—
639C2×C4#95314yes—
640C2×C3—84no—
641C2×C3#238811yescharacteristic
642C2×C3#243011yescharacteristic
643C2×C3#243511yescharacteristic
644C2×C3#245912yes—
645C2×C3#247412yes—
646C2×C3#252112yes—
647C2×C3#255712yes—
648C2×C3#259211yescharacteristic
649C2×C3#261011yescharacteristic
650C2×C3#266211yescharacteristic
651C2×C3#286111yescharacteristic
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22#566912yes—
668C22#567012yes—
669C22#567111yescharacteristic
670C22#567111yescharacteristic
671C22#567812yes—
672C22#567912yes—
673C22#568012yes—
674C22#568012yes—
675C22#568111yescharacteristic
676C22#568111yescharacteristic
677C22C2×#93611yescharacteristic
678C22#580511yescharacteristic
679C22#580612yes—
680C22#580612yes—
681C22#582412yes—
682C22#582812yes—
683C22#582811yescharacteristic
684C22#582912yes—
685C22#583212yes—
686C22#583211yescharacteristic
687C22#583211yescharacteristic
688C22#583311yescharacteristic
689C22C2×#95311yescharacteristic
690C4—24no—
691C4—24no—
692C4—44no—
693C4—42no—
694C4—124no—
695C4—124no—
696C4—122no—
697C4—122no—
698C4—241no—
699C4—241no—
700C3#1493211yescharacteristic, complemented, Sylow
701C2—84no—
702C2#8296612yes—
703C2#8296812yes—
704C2#8297611yescharacteristic
705C2#8298111yescharacteristic
706C2#8299411yescharacteristic
707C2#8299712yes—
708C2#8299912yes—
709C2#8300011yescharacteristic
710C2#8405011yescharacteristic
711C2#8409111yescharacteristic
712C2#8409511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#124C2
prime radicalC23×C3×D4C22
hypercenterC22×#124D3
hypocenterC3#14932
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. C2×#4945C2semidirect
3. C2×#4933C2semidirect
4. #72134C2semidirect
5. #72134C2semidirect
8. #71549C2semidirect
9. #71547C2semidirect
10. #71545C2semidirect
11. C4×C3⋊#95C2semidirect
700. C3#14932semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×#49452
C2×#49332
#721342
#721312
#715492
#715472
#715452
C4×C3⋊#952
#149323