Finite Group Databasebeta

← #10718261

Subgroups — #10718261

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#381C211yesmaximal, characteristic
2C2×#5211C211yesmaximal, characteristic
3C22×C3⋊#95C211yesmaximal, characteristic
4C2×#4946C211yesmaximal, characteristic
5C2×Dic3×C22⋊C4C211yesmaximal, characteristic
6#71544C211yesmaximal, characteristic
7#71544C211yesmaximal, characteristic
8#69418C212yesmaximal, complemented
9#69414C212yesmaximal, complemented
10#69405C212yesmaximal, complemented
11#69403C212yesmaximal, complemented
12#13992—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C3×#381—24no—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26#5211C418yescomplemented
27C2×C3⋊#95—24no—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4946—24no—
40Dic3×C22⋊C4—24no—
41C2×#385—31no—
42C2×#381—31no—
43C2×#381D311yescharacteristic
44C2×C4×C22⋊C4—31no—
45C22×#95—31no—
46#1443—32no—
47#1442—32no—
48#1436—32no—
49#1434—32no—
50C22⋊#95—31no—
51C22⋊#95—31no—
52C25×C3C2311yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4C2×C414yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—22no—
65C23×Dic3—22no—
66C23×Dic3—22no—
67C23×Dic3—24no—
68C23×Dic3—24no—
69C23×Dic3—22no—
70C23×Dic3—22no—
71C23×Dic3—42no—
72C23×Dic3—42no—
73C23×Dic3D414yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D414yes—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217C2×C414yes—
91C2×#217C2×C414yes—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—42no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—42no—
99C3×#95—24no—
100C3×#95—24no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C3⋊#95C2×C418yescomplemented
111C3⋊#95C2×C418yescomplemented
112C24×C4—31no—
113C24×C4—31no—
114C24×C4C2×D311yescharacteristic
115C22×C4⋊C4—32no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4C2×D311yescharacteristic
119C22×C22⋊C4C2×D311yescharacteristic
120C22×C42—31no—
121C22×C42—31no—
122#385—64no—
123#381—24no—
124#381—38no—
125C4×C22⋊C4—64no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—64no—
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3C22×C412yes—
145C24×C3C2×D412yes—
146C24×C3C2×D412yes—
147C24×C3#4112yes—
148C24×C3#4112yes—
149C24×C3#4112yes—
150C24×C3#4112yes—
151C24×C3C2411yescharacteristic
152C22×C3×C4—28no—
153C22×C3×C4—28no—
154C22×C3×C4—28no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
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×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4C22×C414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—28no—
187C22×Dic3—28no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
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—28no—
213C22×Dic3—44no—
214C22×Dic3—48no—
215C22×Dic3—48no—
216C22×Dic3—48no—
217C22×Dic3—48no—
218C22×Dic3—44no—
219C22×Dic3—44no—
220C22×Dic3C22×C412yes—
221C22×Dic3C22×C414yes—
222C22×Dic3C22×C412yes—
223C22×Dic3C2×D412yes—
224C22×Dic3C2×D412yes—
225C22×Dic3#4114yes—
226C22×Dic3#4114yes—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—44no—
229#217—28no—
230#217—28no—
231Dic3⋊C4—48no—
232C4×Dic3—44no—
233C4×Dic3—44no—
234C25C22×D311yescharacteristic
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—32no—
246C23×C4—31no—
247C23×C4—34no—
248C23×C4—64no—
249C23×C4—64no—
250C23×C4—62no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—62no—
255C23×C4—122no—
256C23×C4—122no—
257C23×C4C4×D314yes—
258C23×C4C22×D312yes—
259C23×C4C22×D312yes—
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—122no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4#6714yes—
277C2×C42—62no—
278C2×C42—62no—
279C2×C42—64no—
280C2×C42—122no—
281#95—24no—
282#95—24no—
283#95—38no—
284#95—38no—
285#95—64no—
286#95—64no—
287#95—64no—
288#95—64no—
289#95—62no—
290#95—62no—
291#95—64no—
292#95—122no—
293#95—122no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3#11714yes—
311C23×C3C4×D414yes—
312C23×C3C4×D414yes—
313C23×C3#12114yes—
314C23×C3#12314yes—
315C23×C3#12414yes—
316C23×C3#12614yes—
317C23×C3C23×C411yescharacteristic
318C23×C3C22×D411yescharacteristic
319C23×C3C22×D411yescharacteristic
320C23×C3C2×#4111yescharacteristic
321C23×C3C2×#4111yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C23×C3C2×#4111yescharacteristic
324C23×C3C2×#4112yes—
325C23×C3C2×#4112yes—
326C23×C3H(5,𝔽2)12yes—
327C23×C32−1+412yes—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—416no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—416no—
371C2×Dic3—416no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—41no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3#12314yes—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24C2×C4×D312yes—
394C24#23512yes—
395C24D3×D412yes—
396C24#23712yes—
397C24#23712yes—
398C24#23712yes—
399C24C2×#6712yes—
400C24C23×D311yescharacteristic
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—32no—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—68no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—68no—
441C22×C4—68no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
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—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—124no—
467C22×C4—128no—
468C22×C4—128no—
469C22×C4—128no—
470C22×C4—128no—
471C22×C4—124no—
472C22×C4—124no—
473C22×C4C2×C4×D314yes—
474C22×C4C2×#6712yes—
475C22×C4C2×#6712yes—
476C4⋊C4—128no—
477C22⋊C4—28no—
478C22⋊C4—44no—
479C22⋊C4—68no—
480C22⋊C4—68no—
481C42—124no—
482C42—124no—
483C22×C3—24no—
484C22×C3—28no—
485C22×C3—28no—
486C22×C3—24no—
487C22×C3—24no—
488C22×C3—24no—
489C22×C3—28no—
490C22×C3—28no—
491C22×C3—24no—
492C22×C3—24no—
493C22×C3#38818yes—
494C22×C3C2×#11711yescharacteristic
495C22×C3C2×C4×D411yescharacteristic
496C22×C3C2×C4×D411yescharacteristic
497C22×C3C4×#4112yes—
498C22×C3#52012yes—
499C22×C3C2×#12111yescharacteristic
500C22×C3C2×#12311yescharacteristic
501C22×C3C2×#12311yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3C2×#12411yescharacteristic
504C22×C3C2×#12611yescharacteristic
505C22×C3#52912yes—
506C22×C3#53411yescharacteristic
507C22×C3#53611yescharacteristic
508C22×C3#53912yes—
509C22×C3#54612yes—
510C22×C3#54712yes—
511C22×C3#55112yes—
512C22×C3#55112yes—
513C22×C3#55112yes—
514C22×C3#55212yes—
515C22×C3#55212yes—
516C22×C3#55212yes—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—44no—
520C3×C4—82no—
521Dic3—24no—
522Dic3—42no—
523Dic3—44no—
524Dic3—42no—
525Dic3—44no—
526Dic3—42no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23#87714yes—
544C23#87914yes—
545C23#67⋊C414yes—
546C23#88314yes—
547C23C4×#6714yes—
548C23#93614yes—
549C23#93914yes—
550C23C22×C4×D311yescharacteristic
551C23C2×#23511yescharacteristic
552C23C2×D3×D411yescharacteristic
553C23C2×#23711yescharacteristic
554C23C2×#23711yescharacteristic
555C23C2×#23711yescharacteristic
556C23D3×#4112yes—
557C23D3×#4112yes—
558C23#100912yes—
559C23#101012yes—
560C23C22×#6711yescharacteristic
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—34no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—1216no—
605C2×C4—1216no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—121no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C3—28no—
623C2×C3C2×#38811yescharacteristic
624C2×C3C4×#12311yescharacteristic
625C2×C3#232211yescharacteristic
626C2×C3#234312yes—
627C2×C3#235912yes—
628C2×C3#248311yescharacteristic
629C2×C3#248411yescharacteristic
630C2×C3#248611yescharacteristic
631C2×C3#248812yes—
632C2×C3#249312yes—
633C2×C3#253411yescharacteristic
634C22—28no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—28no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—24no—
644C22#494518yes—
645C22C2×#87711yescharacteristic
646C22C2×#87911yescharacteristic
647C22C2×#67⋊C411yescharacteristic
648C22C2×#88311yescharacteristic
649C22#552212yes—
650C22#552312yes—
651C22#552712yes—
652C22#553112yes—
653C22#564811yescharacteristic
654C22#564912yes—
655C22#565012yes—
656C22#565012yes—
657C22D3×#12311yescharacteristic
658C22#565311yescharacteristic
659C22#565611yescharacteristic
660C22#565812yes—
661C22#565812yes—
662C22#566512yes—
663C22C2×C4×#6711yescharacteristic
664C22C2×#93611yescharacteristic
665C22C2×#93911yescharacteristic
666C22#582412yes—
667C22#582912yes—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—64no—
672C4—82no—
673C4—124no—
674C4—122no—
675C4—122no—
676C4—122no—
677C4—124no—
678C3#1399211yescharacteristic, complemented, Sylow
679C2—28no—
680C2#7405211yescharacteristic
681C2#7405611yescharacteristic
682C2#7405912yes—
683C2#7407311yescharacteristic
684C2#7407712yes—
685C2#7413011yescharacteristic
686C2C2×#494511yescharacteristic
687C2#8224212yes—
688C2#8285611yescharacteristic
689C2#8285712yes—
690C2#8290011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#381C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#381D3
hypocenterC3#13992
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
8. #69418C2semidirect
9. #69414C2semidirect
10. #69405C2semidirect
11. #69403C2semidirect
26. #5211C4semidirect
110. C3⋊#95C2×C4semidirect
111. C3⋊#95C2×C4semidirect
678. C3#13992semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3812
C2×#52112
C22×C3⋊#952
C2×#49462
C2×Dic3×C22⋊C42
#715442
#694182
#694142
#694052
#694032
#139923