Finite Group Databasebeta

← #10731951

Subgroups — #10731951

Order 768.

1710 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#397C211yesmaximal, characteristic
2C2×#4975C211yesmaximal, characteristic, complemented
3C2×#4953C211yesmaximal, characteristic
4C2×#4952C212yesmaximal
5C2×#4942C212yesmaximal
6C2×Dic3×C22⋊C4C211yesmaximal, characteristic
7#71569C211yesmaximal, characteristic, complemented
8#69423C211yesmaximal, characteristic, complemented
9#69421C212yesmaximal, complemented
10Dic3⋊#95C211yesmaximal, characteristic, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#15150—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C3×#397—24no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4975—24no—
38#4953—24no—
39#4952—28no—
40#4942—28no—
41Dic3×C22⋊C4—24no—
42C2×#397—32no—
43C2×#397—31no—
44C2×#397D311yescharacteristic
45C2×#384—31no—
46C2×#380—32no—
47C2×C4×C22⋊C4—31no—
48#1445—32no—
49#1445—31no—
50#1445—31no—
51C4⋊#95—31no—
52C4⋊#95—32no—
53C25×C3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—22no—
59C23×Dic3—22no—
60C23×Dic3—24no—
61C23×Dic3—24no—
62C23×Dic3D414yes—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4D414yes—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217D414yes—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3D414yes—
100C3×#95—22no—
101C3×#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C24×C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4C2×D312yes—
120C22×C22⋊C4C2×D311yescharacteristic
121C22×C42—31no—
122C22×C42—31no—
123#397—24no—
124#397—68no—
125#397—64no—
126#384—64no—
127#380—68no—
128C4×C22⋊C4—64no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—31no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95C2×D312yes—
139C2×#95C2×D311yescharacteristic
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3C2×D412yes—
149C24×C3C2×D412yes—
150C24×C3C2×D412yes—
151C24×C3C2411yescharacteristic
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4#4112yes—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—28no—
187C22×Dic3—216no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—44no—
203C22×Dic3—44no—
204C22×Dic3—48no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D412yes—
211C22×Dic3#4112yes—
212C22×Dic3#4112yes—
213C22×Dic3#4114yes—
214C22×Dic3#4114yes—
215C3×C4⋊C4—44no—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—48no—
218#217—28no—
219#217—48no—
220C4⋊Dic3—44no—
221Dic3⋊C4—48no—
222Dic3⋊C4—48no—
223Dic3⋊C4—48no—
224C4×Dic3—28no—
225C4×Dic3—44no—
226C25C22×D311yescharacteristic
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—34no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4—62no—
235C23×C4—64no—
236C23×C4—64no—
237C23×C4—62no—
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—34no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4#6714yes—
269C2×C42—34no—
270C2×C42—64no—
271C2×C42—62no—
272C2×C42—64no—
273#95—22no—
274#95—24no—
275#95—62no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—64no—
280#95—64no—
281#95—62no—
282#95—62no—
283#95—64no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3#12714yes—
307C23×C3C22×D411yescharacteristic
308C23×C3C22×D411yescharacteristic
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D411yescharacteristic
311C23×C3C2×#4111yescharacteristic
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3C2×#4112yes—
316C23×C32−1+412yes—
317C23×C32−1+412yes—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12114yes—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—416no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3#12114yes—
372C2×Dic3#12114yes—
373C2×Dic3#12314yes—
374C2×Dic3#12318yes—
375C2×Dic3#12414yes—
376C2×Dic3#12618yes—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24D3×D412yes—
386C24D3×D412yes—
387C24C2×#6712yes—
388C24C23×D311yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—616no—
420C22×C4—68no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—124no—
446C22×C4—124no—
447C22×C4—128no—
448C22×C4D3×D412yes—
449C22×C4#23712yes—
450C22×C4C2×#6712yes—
451C22×C4C2×#6712yes—
452C4⋊C4—44no—
453C4⋊C4—128no—
454C4⋊C4—128no—
455C4⋊C4—128no—
456C4⋊C4—124no—
457C22⋊C4—28no—
458C22⋊C4—48no—
459C22⋊C4—68no—
460C22⋊C4—128no—
461C42—68no—
462C42—124no—
463C22×C3—24no—
464C22×C3—28no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×#12111yescharacteristic
476C22×C3C2×#12111yescharacteristic
477C22×C3C2×#12311yescharacteristic
478C22×C3C2×#12312yes—
479C22×C3C2×#12411yescharacteristic
480C22×C3C2×#12612yes—
481C22×C3C2×#12711yescharacteristic
482C22×C3#53211yescharacteristic
483C22×C3#53211yescharacteristic
484C22×C3#53211yescharacteristic
485C22×C3#53611yescharacteristic
486C22×C3#53612yes—
487C22×C3#53611yescharacteristic
488C22×C3#54112yes—
489C22×C3#54712yes—
490C22×C3#54712yes—
491C22×C3#54712yes—
492C22×C3#54712yes—
493C22×C3#55212yes—
494C22×C3#55212yes—
495C22×C3#55612yes—
496C22×C3#55612yes—
497C3×C4—24no—
498C3×C4—42no—
499C3×C4—82no—
500C3×C4—82no—
501Dic3—24no—
502Dic3—42no—
503Dic3—42no—
504Dic3—82no—
505Dic3—82no—
506Dic3#39718yescomplemented
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#94014yes—
530C23C2×D3×D411yescharacteristic
531C23C2×D3×D411yescharacteristic
532C23C2×D3×D411yescharacteristic
533C23C2×#23711yescharacteristic
534C23D3×#4112yes—
535C23D3×#4112yes—
536C23D3×#4111yescharacteristic
537C23D3×#4111yescharacteristic
538C23#101012yes—
539C23#101012yes—
540C23C22×#6711yescharacteristic
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—34no—
548C2×C4—38no—
549C2×C4—34no—
550C2×C4—38no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—1216no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4#93914yes—
600C2×C3—44no—
601C2×C3C2×#39711yescharacteristic
602C2×C3#248611yescharacteristic
603C2×C3#248611yescharacteristic
604C2×C3#249612yes—
605C2×C3#249612yes—
606C2×C3#249912yes—
607C2×C3#250112yes—
608C2×C3#254711yescharacteristic
609C2×C3#254711yescharacteristic
610C2×C3#262111yescharacteristic
611C2×C3#285611yescharacteristic
612C22—24no—
613C22—28no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22#559611yescharacteristic
624C22#559611yescharacteristic
625C22D3×#12111yescharacteristic
626C22D3×#12111yescharacteristic
627C22#564811yescharacteristic
628C22#564812yes—
629C22#565012yes—
630C22D3×#12311yescharacteristic
631C22D3×#12312yes—
632C22#565512yes—
633C22#565512yes—
634C22#567112yes—
635C22D3×#12411yescharacteristic
636C22#567511yescharacteristic
637C22#570112yes—
638C22#570212yes—
639C22D3×#12612yes—
640C22#570812yes—
641C22C2×#93911yescharacteristic
642C22C2×#94011yescharacteristic
643C22#582412yes—
644C22#582412yes—
645C22#583211yescharacteristic
646C4—24no—
647C4—38no—
648C4—42no—
649C4—64no—
650C4—82no—
651C4—82no—
652C4—122no—
653C4—122no—
654C4—242no—
655C4—242no—
656C3#1515011yescharacteristic, complemented, Sylow
657C2—44no—
658C2#7748511yescharacteristic
659C2#7749012yes—
660C2#7749112yes—
661C2#7751411yescharacteristic
662C2#7753811yescharacteristic
663C2D3×#39711yescharacteristic
664C2#8270011yescharacteristic
665C2#8285611yescharacteristic
666C2#8285912yes—
667C2#8297311yescharacteristic
668C2#8308312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#397C2
prime radicalC25×C3C23
hypercenterC2×#397D3
hypocenterC3#15150
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×#4975C2semidirect
7. #71569C2semidirect
8. #69423C2semidirect
9. #69421C2semidirect
10. Dic3⋊#95C2semidirect
11. Dic3⋊#95C2semidirect
506. Dic3#397semidirect
656. C3#15150semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3972
C2×#49752
C2×#49532
C2×#49522
C2×#49422
C2×Dic3×C22⋊C42
#715692
#694232
#694212
Dic3⋊#952
#151503