Finite Group Databasebeta

← #10733206

Subgroups — #10733206

Order 768.

2834 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#379C211yesmaximal, characteristic
2C2×#5247C212yesmaximal
3C2×#4949C211yesmaximal, characteristic, complemented
4C2×#4945C212yesmaximal, complemented
5#71549C212yesmaximal, complemented
6#71548C212yesmaximal, complemented
7#71547C212yesmaximal, complemented
8#71546C212yesmaximal, complemented
9#71545C211yesmaximal, characteristic, complemented
10#15284—31nomaximal, Sylow
11C26×C3C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×#217C2212yes—
21C22×#217C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C22×Dic3⋊C4C2212yescomplemented
24C22×C4×Dic3C2211yescharacteristic, complemented
25C3×#379—24no—
26#5247—28no—
27C2×C3⋊#95C2212yescomplemented
28C2×C3⋊#95C2212yescomplemented
29C2×C3⋊#95C2211yescharacteristic, complemented
30C2×C3⋊#95C2212yescomplemented
31C2×C3⋊#95C2211yescharacteristic, complemented
32C2×C3⋊#95C2212yescomplemented
33C2×C3⋊#95C2212yescomplemented
34#4949—24no—
35#4945—28no—
36C2×#399—31no—
37C2×#388—32no—
38C2×#379—32no—
39C2×#379D311yescharacteristic
40#1440—32no—
41#1440—32no—
42#1439—32no—
43#1439—32no—
44#1437—31no—
45C25×C3—24no—
46C25×C3—22no—
47C25×C3—24no—
48C25×C3—24no—
49C25×C3—24no—
50C25×C3—41no—
51C25×C3—42no—
52C25×C3—41no—
53C25×C3—42no—
54C25×C3C2312yes—
55C25×C3C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—41no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4—42no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4D414yes—
74C2×#217—24no—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—42no—
86C2×#217—42no—
87C2×#217—42no—
88C2×#217—42no—
89C2×#217—42no—
90C2×#217—42no—
91C2×#217D418yes—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—41no—
94C2×C4⋊Dic3—41no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—42no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3—41no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—41no—
106C3⋊#95—42no—
107C26C2×D311yescharacteristic
108C22×C4⋊C4—32no—
109C22×C4⋊C4—31no—
110C22×C22⋊C4—32no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—32no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—32no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4C2×D311yescharacteristic
117C22×C22⋊C4C2×D312yes—
118C22×C22⋊C4C2×D312yes—
119C22×C22⋊C4C2×D311yescharacteristic
120C22×C42—31no—
121#399—64no—
122#388—68no—
123#379—24no—
124#379—68no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—22no—
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—21no—
148C24×C3—21no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—216no—
152C24×C3—28no—
153C24×C3—44no—
154C24×C3—48no—
155C24×C3—48no—
156C24×C3—48no—
157C24×C3—48no—
158C24×C3—48no—
159C24×C3—48no—
160C24×C3—48no—
161C24×C3—48no—
162C24×C3—44no—
163C24×C3—44no—
164C24×C3—48no—
165C24×C3—48no—
166C24×C3—48no—
167C24×C3—48no—
168C24×C3—44no—
169C24×C3—44no—
170C24×C3—48no—
171C24×C3—44no—
172C24×C3C2×D414yes—
173C24×C3C2×D412yes—
174C24×C3#4114yes—
175C24×C3#4114yes—
176C24×C3#4114yes—
177C24×C3C2411yescharacteristic
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×Dic3—22no—
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—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3C2×D414yes—
222C22×Dic3C2×D414yes—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—48no—
225C3×C22⋊C4—48no—
226C3×C22⋊C4—44no—
227#217—216no—
228#217—48no—
229#217—48no—
230#217—48no—
231#217—48no—
232#217—48no—
233C4⋊Dic3—44no—
234Dic3⋊C4—48no—
235C4×Dic3—44no—
236C25—24no—
237C25—24no—
238C25—24no—
239C25—24no—
240C25—22no—
241C25—42no—
242C25—42no—
243C25—41no—
244C25—41no—
245C25C22×D312yes—
246C25C22×D311yescharacteristic
247C23×C4—32no—
248C23×C4—32no—
249C23×C4—32no—
250C23×C4—32no—
251C23×C4C22×D312yes—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D311yescharacteristic
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—122no—
258C2×C4⋊C4—121no—
259C2×C4⋊C4—121no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—122no—
283C2×C22⋊C4—122no—
284C2×C22⋊C4—122no—
285C2×C22⋊C4—122no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4#6714yes—
289C2×C42—64no—
290C2×C42—121no—
291#95—64no—
292#95—64no—
293#95—62no—
294#95—122no—
295#95—121no—
296#95—122no—
297#95—122no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—416no—
327C23×C3—432no—
328C23×C3—432no—
329C23×C3—432no—
330C23×C3—432no—
331C23×C3—416no—
332C23×C3—432no—
333C23×C3—416no—
334C23×C3—432no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—42no—
347C23×C3—42no—
348C23×C3—41no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—41no—
356C23×C3—42no—
357C23×C3—41no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—41no—
361C23×C3#12318yes—
362C23×C3#12414yes—
363C23×C3C22×D412yes—
364C23×C3C22×D411yescharacteristic
365C23×C3C2×#4112yes—
366C23×C3C2×#4112yes—
367C23×C3C2×#4112yes—
368C23×C3H(5,𝔽2)11yescharacteristic
369C23×C3H(5,𝔽2)11yescharacteristic
370C23×C3H(5,𝔽2)12yes—
371C23×C3H(5,𝔽2)12yes—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412C2×Dic3—42no—
413C2×Dic3—42no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416C2×Dic3—42no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—28no—
437C24—216no—
438C24—48no—
439C24—48no—
440C24—48no—
441C24—48no—
442C24—48no—
443C24—48no—
444C24—44no—
445C24—48no—
446C24—44no—
447C24—48no—
448C24—48no—
449C24—44no—
450C24—48no—
451C24—48no—
452C24—48no—
453C24—44no—
454C24—44no—
455C24—48no—
456C24—44no—
457C24#23514yes—
458C24D3×D414yes—
459C24#23714yes—
460C24#23714yes—
461C24C2×#6712yes—
462C24C23×D311yescharacteristic
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—34no—
479C22×C4—34no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—62no—
494C22×C4—62no—
495C22×C4—62no—
496C22×C4—62no—
497C22×C4—62no—
498C22×C4—62no—
499C22×C4—62no—
500C22×C4—62no—
501C22×C4—62no—
502C22×C4—62no—
503C22×C4—62no—
504C22×C4—62no—
505C22×C4—62no—
506C22×C4C2×#6712yes—
507C22×C4C2×#6712yes—
508C4⋊C4—128no—
509C4⋊C4—124no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—616no—
515C22⋊C4—128no—
516C22⋊C4—128no—
517C22⋊C4—128no—
518C22⋊C4—128no—
519C22⋊C4—128no—
520C42—124no—
521C22×C3—24no—
522C22×C3—28no—
523C22×C3—28no—
524C22×C3—28no—
525C22×C3—28no—
526C22×C3—464no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—44no—
530C22×C3—44no—
531C22×C3—44no—
532C22×C3—44no—
533C22×C3—44no—
534C22×C3—44no—
535C22×C3—44no—
536C22×C3—44no—
537C22×C3—42no—
538C22×C3—44no—
539C22×C3—44no—
540C22×C3—44no—
541C22×C3—44no—
542C22×C3—44no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—44no—
546C22×C3C2×#12312yes—
547C22×C3#52511yescharacteristic
548C22×C3#52512yes—
549C22×C3C2×#12411yescharacteristic
550C22×C3#53411yescharacteristic
551C22×C3#53512yes—
552C22×C3#53812yes—
553C22×C3#53812yes—
554C22×C3#53812yes—
555C22×C3D4212yes—
556C22×C3D4211yescharacteristic
557C22×C3#54612yes—
558C22×C3#54612yes—
559C22×C3#54612yes—
560C22×C3#55112yes—
561C22×C3#55112yes—
562C22×C3#55112yes—
563C22×C3#55111yescharacteristic
564C22×C3#55112yes—
565C22×C3#56011yescharacteristic
566C22×C3#56011yescharacteristic
567C3×C4—42no—
568C3×C4—42no—
569C3×C4—82no—
570Dic3—44no—
571Dic3—44no—
572Dic3—82no—
573Dic3—82no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—24no—
584C23—22no—
585C23—24no—
586C23—24no—
587C23—22no—
588C23—24no—
589C23—24no—
590C23—24no—
591C23—24no—
592C23—24no—
593C23—24no—
594C23—24no—
595C23—24no—
596C23—24no—
597C23—24no—
598C23—24no—
599C23—24no—
600C23—24no—
601C23—24no—
602C23—432no—
603C23—432no—
604C23—432no—
605C23—416no—
606C23—432no—
607C23—432no—
608C23—432no—
609C23—416no—
610C23—416no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—41no—
624C23—42no—
625C23—41no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23—42no—
632C23—42no—
633C23—42no—
634C23—41no—
635C23—42no—
636C23—41no—
637C23#88318yes—
638C23#93614yes—
639C23C2×#23512yes—
640C23C2×D3×D412yes—
641C23C2×#23712yes—
642C23C2×#23712yes—
643C23#100412yes—
644C23#100412yes—
645C23#100411yescharacteristic
646C23#100411yescharacteristic
647C23C22×#6711yescharacteristic
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—64no—
666C2×C4—64no—
667C2×C4—64no—
668C2×C4—64no—
669C2×C4—64no—
670C2×C4—64no—
671C2×C4—122no—
672C2×C4—122no—
673C2×C4—122no—
674C2×C4—122no—
675C2×C4—122no—
676C2×C4—122no—
677C2×C4—122no—
678C2×C4—122no—
679C2×C4—122no—
680C2×C4—122no—
681C2×C4—122no—
682C2×C4—122no—
683C2×C4—122no—
684C2×C4—122no—
685C2×C4—122no—
686C2×C4—122no—
687C2×C4—122no—
688C2×C4—122no—
689C2×C4—122no—
690C2×C4—122no—
691C2×C4—122no—
692C2×C4—122no—
693C2×C3—44no—
694C2×C3—48no—
695C2×C3#241812yes—
696C2×C3#241812yes—
697C2×C3#247212yes—
698C2×C3#248212yes—
699C2×C3#256111yescharacteristic
700C2×C3#266911yescharacteristic
701C2×C3#268512yes—
702C2×C3#269711yescharacteristic
703C2×C3#273512yes—
704C22—28no—
705C22—28no—
706C22—28no—
707C22—24no—
708C22—28no—
709C22—464no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—42no—
724C22—42no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22C2×#88312yes—
730C22#549012yes—
731C22#549412yes—
732C22#549512yes—
733C22#549512yes—
734C22#549512yes—
735C22#558612yes—
736C22#558711yescharacteristic
737C22#558712yes—
738C22#559011yescharacteristic
739C22#559112yes—
740C22#559212yes—
741C22#559312yes—
742C22#559312yes—
743C22#559511yescharacteristic
744C22#559511yescharacteristic
745C22#579111yescharacteristic
746C22C2×#93611yescharacteristic
747C22D4×#6712yes—
748C22#580411yescharacteristic
749C22#580612yes—
750C4—42no—
751C4—42no—
752C4—82no—
753C4—124no—
754C4—124no—
755C4—242no—
756C4—242no—
757C3#1528411yescharacteristic, complemented, Sylow
758C2—48no—
759C2—44no—
760C2#7382612yes—
761C2#7383212yes—
762C2#7383812yes—
763C2#7384112yes—
764C2#8211611yescharacteristic
765C2#8212812yes—
766C2#8258912yes—
767C2#8259411yescharacteristic
768C2#8260011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#379C2
prime radicalC26×C3C22
hypercenterC2×#379D3
hypocenterC3#15284
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
3. C2×#4949C2semidirect
4. C2×#4945C2semidirect
5. #71549C2semidirect
6. #71548C2semidirect
7. #71547C2semidirect
8. #71546C2semidirect
9. #71545C2semidirect
22. C22×C4⋊Dic3C22semidirect
23. C22×Dic3⋊C4C22semidirect
24. C22×C4×Dic3C22semidirect
27. C2×C3⋊#95C22semidirect
28. C2×C3⋊#95C22semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
31. C2×C3⋊#95C22semidirect
32. C2×C3⋊#95C22semidirect
33. C2×C3⋊#95C22semidirect
757. C3#15284semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3792
C2×#52472
C2×#49492
C2×#49452
#715492
#715482
#715472
#715462
#715452
#152843