Finite Group Databasebeta

← #10729268

Subgroups — #10729268

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×#936C211yesmaximal, characteristic
2C2×#4951C212yesmaximal, complemented
3C2×#4655C211yesmaximal, characteristic, complemented
4C3×#1438C211yesmaximal, characteristic
5#72131C211yesmaximal, characteristic
6#71549C212yesmaximal, complemented
7#71547C212yesmaximal, complemented
8#71545C212yesmaximal, complemented
9#69424C211yesmaximal, characteristic, complemented
10#69424C211yesmaximal, characteristic, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#14932—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×#217C2212yes—
21C2×#936—28no—
22C2×#936—24no—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4951—28no—
39#4655—24no—
40C22×#124—31no—
41C2×#388—32no—
42C2×#387—31no—
43#1446—31no—
44#1446—31no—
45#1440—32no—
46#1439—32no—
47#1438—31no—
48#1438D311yescharacteristic
49#1437—32no—
50C4×#95—31no—
51C25×C3C2312yes—
52C22×C3×D4—24no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—42no—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C22×Dic6—22no—
68C22×Dic6—24no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—42no—
77C2×C3×C22⋊C4—42no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—42no—
88C2×#217—42no—
89C2×#217D418yes—
90#936—232no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3—42no—
100C2×C4×Dic3D414yes—
101C3×#95—22no—
102C3×#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8—31no—
113C23×D4C2×D311yescharacteristic
114C2×#124—64no—
115C2×#124—68no—
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—31no—
125C22×C42—32no—
126#388—68no—
127#387—64no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
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×D4—28no—
156C2×C3×D4—48no—
157C2×C3×D4—48no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4112yes—
181C22×C3×C4#4112yes—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D412yes—
213C22×Dic3C2×D414yes—
214C22×Dic3#4114yes—
215C22×Dic3#4112yes—
216C2×Dic6—28no—
217C2×Dic6—48no—
218C2×Dic6—48no—
219C2×Dic6—44no—
220C2×Dic6—44no—
221C3×C22⋊C4—216no—
222C3×C22⋊C4—48no—
223#217—216no—
224#217—216no—
225#217—48no—
226#217—48no—
227C4⋊Dic3—44no—
228Dic3⋊C4—28no—
229C4×Dic3—28no—
230C4×Dic3—48no—
231C25C22×D312yes—
232C22×Q8—64no—
233C22×Q8—62no—
234C22×Q8—62no—
235C22×Q8—62no—
236C22×Q8—122no—
237C22×D4—24no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—31no—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252#124—632no—
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—38no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
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—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C42—34no—
277C2×C42—64no—
278C2×C42—64no—
279C2×C42—64no—
280C2×C42—122no—
281#95—22no—
282#95—22no—
283#95—62no—
284#95—62no—
285#95—64no—
286#95—64no—
287#95—62no—
288#95—62no—
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×#4112yes—
332C23×C3C2×#4111yescharacteristic
333C23×C3C2×#4111yescharacteristic
334C23×C3C2×#4112yes—
335C23×C3H(5,𝔽2)12yes—
336C23×C3H(5,𝔽2)11yescharacteristic
337C23×C32−1+411yescharacteristic
338C3×D4—416no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4#12414yes—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—41no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3#12014yes—
395Dic6—416no—
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—
410C24D3×D414yes—
411C24C23×D311yescharacteristic
412C2×Q8—68no—
413C2×Q8—128no—
414C2×Q8—128no—
415C2×Q8—124no—
416C2×Q8—124no—
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—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—32no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4—32no—
443C22×C4—34no—
444C22×C4—32no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—61no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—61no—
471C22×C4—62no—
472C22×C4—61no—
473C22×C4#23512yes—
474C22×C4D3×D412yes—
475C22×C4#23714yes—
476C22×C4#23712yes—
477C22×C4#23712yes—
478C4⋊C4—68no—
479C4⋊C4—124no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—616no—
483C22⋊C4—616no—
484C22⋊C4—128no—
485C22⋊C4—128no—
486C42—68no—
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#52512yes—
505C22×C3#52511yescharacteristic
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—42no—
528C3×C4—44no—
529C3×C4—81no—
530Dic3—24no—
531Dic3—42no—
532Dic3—42no—
533Dic3—44no—
534Dic3—44no—
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×#23511yescharacteristic
572C23C2×D3×D412yes—
573C23C2×D3×D411yescharacteristic
574C23C2×#23711yescharacteristic
575C23C2×#23711yescharacteristic
576C23C2×#23712yes—
577C23#100412yes—
578C23#100411yescharacteristic
579C23D3×#4112yes—
580C23D3×#4111yescharacteristic
581C23#101011yescharacteristic
582Q8—1216no—
583D4—416no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—34no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—121no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4#88514yes—
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#549012yes—
668C22C2×#88511yescharacteristic
669C22#554211yescharacteristic
670C22#555712yes—
671C22#555911yescharacteristic
672C22#558612yes—
673C22D3×#12011yescharacteristic
674C22#559112yes—
675C22#559211yescharacteristic
676C22#559312yes—
677C22#559612yes—
678C22#560012yes—
679C22#560312yes—
680C22#560611yescharacteristic
681C22#560811yescharacteristic
682C22#561012yes—
683C22#561612yes—
684C22#565512yes—
685C22#565912yes—
686C22#566811yescharacteristic
687C22#567111yescharacteristic
688C22#571311yescharacteristic
689C22#571611yescharacteristic
690C4—24no—
691C4—42no—
692C4—44no—
693C4—64no—
694C4—81no—
695C4—124no—
696C4—124no—
697C4—122no—
698C4—122no—
699C4—241no—
700C3#1493211yescharacteristic, complemented, Sylow
701C2—84no—
702C2#7382711yescharacteristic
703C2#7472912yes—
704C2#7475912yes—
705C2#7636511yescharacteristic
706C2#7642611yescharacteristic
707C2#7666511yescharacteristic
708C2#7667912yes—
709C2#7670911yescharacteristic
710C2#7694012yes—
711C2#7696911yescharacteristic
712C2#7711611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#14932
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4951C2semidirect
3. C2×#4655C2semidirect
6. #71549C2semidirect
7. #71547C2semidirect
8. #71545C2semidirect
9. #69424C2semidirect
10. #69424C2semidirect
11. Dic3×#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×#9362
C2×#49512
C2×#46552
C3×#14382
#721312
#715492
#715472
#715452
#694242
Dic3×#952
#149323