Finite Group Databasebeta

← #10690326

Subgroups — #10690326

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#447C211yesmaximal, characteristic
2#82634C211yesmaximal, characteristic, complemented
3#72131C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9041—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×D8—21no—
11C2×C3×#121C2211yescharacteristic
12C22×#217—21no—
13C22×#217—21no—
14C2×D4×Dic3—21no—
15C3×#447—22no—
16C3×#447C414yescomplemented
17C2×C3×#102—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5220—21no—
20#5220—21no—
21C2×#832—21no—
22C2×#832—21no—
23#5212C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4980—21no—
28#4951—21no—
29#4950—21no—
30#4537—22no—
31C2×#447D311yescharacteristic
32#2325—31no—
33#1786—31no—
34C2×#327—31no—
35#1438—31no—
36#1294—31no—
37#1280—31no—
38C25×C3—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—42no—
46C22×C3×D4—42no—
47C22×C3×D4C2×C412yescomplemented
48C22×C3×D4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C23×Dic3D411yescharacteristic
53C23×Dic3D411yescharacteristic
54C2×C3×D8—24no—
55C2×C3×D8—42no—
56C2×C3×D8—42no—
57C22×C3×C8—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121C2×C412yes—
61C3×#121D412yes—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#217—42no—
69C2×#217—42no—
70C2×#217—42no—
71C2×#217—42no—
72C2×#217—42no—
73C2×#217—42no—
74D4×Dic3—42no—
75D4×Dic3—42no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4×Dic3—21no—
80C22×C3⋊C8—21no—
81C3×#102—24no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8C2×C412yescomplemented
84#832—42no—
85#832—42no—
86C3⋊#95—21no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93#830—21no—
94#830—21no—
95C23×D4C2×D311yescharacteristic
96C22×D8—21no—
97C2×#121C2×D311yescharacteristic
98C2×C4×D4—61no—
99C22×C22⋊C4—61no—
100C22×C22⋊C4—61no—
101#447—22no—
102#447Dic314yescomplemented
103C2×#102—21no—
104C2×#102—61no—
105C2×#102—61no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#388—61no—
109#387—61no—
110#386—61no—
111#386—61no—
112#385—61no—
113#380—31no—
114C2×#95—31no—
115C2×#95—31no—
116#327—62no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—41no—
122C24×C3—41no—
123C24×C3—41no—
124C24×C3—41no—
125C24×C3—41no—
126C24×C3—41no—
127C24×C3—41no—
128C24×C3—41no—
129C24×C3—41no—
130C24×C3—41no—
131C24×C3—41no—
132C24×C3—41no—
133C24×C3C2×D411yescharacteristic
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C2×C3×D4—41no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—44no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4C22⋊C412yescomplemented
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4—41no—
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×C3×C4#4111yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—42no—
178C22×Dic3—41no—
179C22×Dic3—41no—
180C22×Dic3—41no—
181C22×Dic3—41no—
182C22×Dic3—41no—
183C22×Dic3—41no—
184C22×Dic3—41no—
185C22×Dic3—41no—
186C22×Dic3—41no—
187C22×Dic3—41no—
188C22×Dic3—81no—
189C22×Dic3—81no—
190C22×Dic3D812yescomplemented
191C3×D8—48no—
192C2×C3×C8—22no—
193C2×C3×C8—41no—
194C2×C3×C8—41no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C22⋊C412yescomplemented
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199#217—42no—
200#217—48no—
201#217—48no—
202#217—41no—
203#217—41no—
204#217—81no—
205#217—81no—
206C4⋊Dic3—42no—
207C4⋊Dic3—41no—
208C4⋊Dic3—41no—
209C4×Dic3—42no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212C2×C3⋊C8—41no—
213C25—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—42no—
221C22×D4—42no—
222C22×D4C2×Dic312yescomplemented
223C22×D4#6711yescharacteristic
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—61no—
227C23×C4—61no—
228C23×C4C22×D311yescharacteristic
229C2×D8—24no—
230C2×D8—42no—
231C2×D8—42no—
232C22×C8—21no—
233C22×C8—61no—
234#121—21no—
235#121—21no—
236#121C2×Dic312yes—
237#121#6712yes—
238C4×D4—122no—
239C4×D4—122no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4#6711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—122no—
250C2×C22⋊C4—122no—
251C2×C22⋊C4—122no—
252C2×C22⋊C4—122no—
253C2×C22⋊C4—122no—
254C2×C22⋊C4—122no—
255C2×C42—61no—
256#102—24no—
257#102—122no—
258#102—122no—
259C22⋊C8—21no—
260C22⋊C8—61no—
261C22⋊C8—61no—
262C22⋊C8C2×Dic312yescomplemented
263#95—61no—
264#95—61no—
265#95—61no—
266#95—61no—
267#95—61no—
268#95—61no—
269#95—61no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—24no—
275C23×C3—22no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—81no—
301C23×C3—81no—
302C23×C3—81no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3—81no—
306C23×C3—81no—
307C23×C3—81no—
308C23×C3—81no—
309C23×C3—81no—
310C23×C3—81no—
311C23×C3—81no—
312C23×C3—81no—
313C23×C3—81no—
314C23×C3—81no—
315C23×C3—81no—
316C23×C3C4×D411yescharacteristic
317C23×C3#12011yescharacteristic
318C23×C3#12111yescharacteristic
319C23×C3C2×D811yescharacteristic
320C23×C3#13511yescharacteristic
321C23×C3AGL(1,ℤ/8)11yescharacteristic
322C23×C3AGL(1,ℤ/8)11yescharacteristic
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—81no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4C2×C22⋊C411yescharacteristic
349C2×C3×C4C4×D411yescharacteristic
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12311yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—81no—
370C2×Dic3—84no—
371C2×Dic3—81no—
372C2×Dic3—81no—
373C2×Dic3—81no—
374C2×Dic3—81no—
375C2×Dic3—81no—
376C3×C8—42no—
377C3⋊C8—81no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24D3×D411yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4#21712yescomplemented
421C22×C4—21no—
422C22×C4—32no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—122no—
434C22×C4—121no—
435C22×C4—121no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—121no—
439C22×C4—121no—
440C22×C4—121no—
441C22×C4—121no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—241no—
445C22×C4—241no—
446C22×C4D3×D411yescharacteristic
447C22×C4#23711yescharacteristic
448C22×C4#23711yescharacteristic
449C22×C4C22×Dic311yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452D8—48no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—121no—
457C2×C8—121no—
458C2×C8—121no—
459C4⋊C4—21no—
460C4⋊C4—121no—
461C4⋊C4—121no—
462C4⋊C4—122no—
463C4⋊C4#21712yescomplemented
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—122no—
467C22⋊C4—128no—
468C22⋊C4—128no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C42—122no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—44no—
491C22×C3—42no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—81no—
499C22×C3—82no—
500C22×C3—81no—
501C22×C3—82no—
502C22×C3—81no—
503C22×C3—82no—
504C22×C3—81no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—82no—
508C22×C3#38611yescharacteristic
509C22×C3C4×D811yescharacteristic
510C22×C3#44211yescharacteristic
511C22×C3#44711yescharacteristic
512C22×C3#44911yescharacteristic
513C22×C3#45311yescharacteristic
514C22×C3#45411yescharacteristic
515C22×C3B(4,𝔽2)11yescharacteristic
516C22×C3B(4,𝔽2)11yescharacteristic
517C22×C3#45911yescharacteristic
518C22×C3#46311yescharacteristic
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—42no—
524Dic3—42no—
525Dic3—82no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—22no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—81no—
560C23—81no—
561C23—81no—
562C23—81no—
563C23—81no—
564C23—81no—
565C23—81no—
566C23—81no—
567C23—81no—
568C23—81no—
569C23—81no—
570C23—81no—
571C23—81no—
572C23—81no—
573C23—81no—
574C23—81no—
575C23D3×D811yescharacteristic
576C23#91111yescharacteristic
577C23#91111yescharacteristic
578C23#91211yescharacteristic
579C23D4×Dic311yescharacteristic
580C23#93711yescharacteristic
581C23#93911yescharacteristic
582D4—24no—
583D4—24no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—62no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—241no—
625C2×C4—244no—
626C2×C4—241no—
627C2×C4—241no—
628C2×C4—241no—
629C2×C4—241no—
630C2×C4—241no—
631C2×C4D4×Dic311yescharacteristic
632C2×C4#93511yescharacteristic
633C2×C4#93911yescharacteristic
634C2×C4C2×#21711yescharacteristic
635C8—42no—
636C8—241no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—82no—
642C2×C3—84no—
643C2×C3#159511yescharacteristic
644C2×C3#159711yescharacteristic
645C2×C3#159811yescharacteristic
646C2×C3#160011yescharacteristic
647C2×C3#187911yescharacteristic
648C2×C3#188311yescharacteristic
649C2×C3#188911yescharacteristic
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—81no—
675C22—82no—
676C22—82no—
677C22—81no—
678C22—82no—
679C22—81no—
680C22—82no—
681C22—81no—
682C22—82no—
683C22—82no—
684C22#503211yescharacteristic
685C22#503211yescharacteristic
686C22Dic3×D811yescharacteristic
687C22#515011yescharacteristic
688C22#515211yescharacteristic
689C22#515311yescharacteristic
690C22#515611yescharacteristic
691C22#515811yescharacteristic
692C22#519811yescharacteristic
693C22#520111yescharacteristic
694C22#522011yescharacteristic
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—42no—
700C4—122no—
701C4—242no—
702C4—241no—
703C4—241no—
704C4—241no—
705C3#904111yescharacteristic, complemented, Sylow
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2—82no—
712C2#7096811yescharacteristic
713C2#7096911yescharacteristic
714C2#7097011yescharacteristic
715C2#7097111yescharacteristic
716C2#7166911yescharacteristic
717C2#7203711yescharacteristic
718C2#7208811yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#447C2
socleC23×C3#120
cosocleC23×C3×C4C23
hypercenterC2×#447D3
hypocenterC3#9041
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
2. #82634C2semidirect
3. #72131C2semidirect
4. #72128C2semidirect
5. C2×#4537C2semidirect
6. #67675C2semidirect
7. #67674C2semidirect
16. C3×#447C4semidirect
47. C22×C3×D4C2×C4semidirect
83. C3×C22⋊C8C2×C4semidirect
102. #447Dic3semidirect
159. C2×C3×D4C22⋊C4semidirect
190. C22×Dic3D8semidirect
196. C3×C4⋊C4C22⋊C4semidirect
222. C22×D4C2×Dic3semidirect
262. C22⋊C8C2×Dic3semidirect
420. C2×D4#217semidirect
463. C4⋊C4#217semidirect
705. C3#9041semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4472
#826342
#721312
#721282
C2×#45372
#676752
#676742
#90413