Finite Group Databasebeta

← #10822314

Subgroups — #10822314

Order 768.

1986 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#135C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C22×#951C211yesmaximal, characteristic, complemented
4#84087C211yesmaximal, characteristic, complemented
5C2×#5232C211yesmaximal, characteristic, complemented
6C22×#816C211yesmaximal, characteristic, complemented
7#72073C212yesmaximal, complemented
8#72058C212yesmaximal, complemented
9#72057C212yesmaximal, complemented
10#72045C212yesmaximal, complemented
11#71965C211yesmaximal, characteristic, complemented
12#23427—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C2×C3×#135—24no—
17C2×C3×#135—24no—
18C2×C3×#135—24no—
19C22×C3×Q16C2211yescharacteristic
20C22×C3×SD16C2211yescharacteristic
21C22×C3×SD16C2211yescharacteristic
22C22×C3×D8C2211yescharacteristic
23C23×C3×C8C2211yescharacteristic
24#5825—24no—
25#5825—24no—
26C2×#951—24no—
27C2×#951—24no—
28C2×#951—24no—
29C2×#944C2212yes—
30C22×#216C2212yes—
31C2×#936C2212yes—
32C2×#935—22no—
33C22×#214C2212yes—
34C2×#926C2211yescharacteristic
35C22×C4⋊Dic3C2211yescharacteristic
36C2×#924—22no—
37C22×#208C2211yescharacteristic
38#5232—24no—
39#5229C2212yes—
40C2×#835C2212yes—
41#5221C2212yes—
42#5220—22no—
43C2×#832C2212yes—
44#5212—22no—
45C2×#816—24no—
46C2×#816C2212yes—
47C2×#816C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C22×#137—31no—
51C22×#135D311yescharacteristic
52C2×#536—31no—
53C2×#417—31no—
54C22×#103—31no—
55#2311—31no—
56#2016—32no—
57#2015—32no—
58#2013—32no—
59#2012—32no—
60#1729—31no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—22no—
69C2×C3×#41—22no—
70C2×C3×#41—22no—
71C2×C3×#41D414yes—
72C2×C3×#41D414yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C23×Dic3—21no—
84C22×Dic6—22no—
85C22×Dic6—22no—
86C22×Dic6—22no—
87C22×Dic6—22no—
88C22×Dic6—21no—
89C22×Dic6—22no—
90C22×Dic6D414yes—
91C22×Dic6C2312yes—
92C3×#135—232no—
93C2×C3×Q16—22no—
94C2×C3×Q16—22no—
95C2×C3×Q16—22no—
96C2×C3×SD16—22no—
97C2×C3×SD16—22no—
98C2×C3×SD16—22no—
99C2×C3×SD16—22no—
100C2×C3×SD16—22no—
101C2×C3×SD16—22no—
102C2×C3×D8—22no—
103C2×C3×D8—22no—
104C2×C3×D8—22no—
105C22×C3×C8—22no—
106C22×C3×C8—22no—
107C22×C3×C8D414yes—
108C22×C3×C8C2311yescharacteristic
109C22×C3×C8C2311yescharacteristic
110C2×#217—24no—
111C2×#217—24no—
112C2×#217D412yes—
113#951—232no—
114#944—24no—
115#944—24no—
116C2×#216—24no—
117C2×#216—24no—
118C2×#216—24no—
119#936—24no—
120#936—24no—
121#935—216no—
122C2×#214—24no—
123C2×#214—24no—
124C2×#214—24no—
125#926—22no—
126#926—22no—
127C2×C4⋊Dic3—22no—
128C2×C4⋊Dic3—22no—
129C2×C4⋊Dic3—21no—
130C2×C4⋊Dic3D412yes—
131C2×C4⋊Dic3C2312yes—
132#924—216no—
133C2×Dic3⋊C4—24no—
134C2×C4×Dic3C2312yes—
135C2×#208—22no—
136C2×#208—22no—
137C2×#208D414yes—
138C22×C3⋊C8C2312yes—
139#835—24no—
140#832—24no—
141C3⋊#95—24no—
142#816—24no—
143#816—24no—
144C22×#41C2×D311yescharacteristic
145C22×#41C2×D311yescharacteristic
146C23×Q8—31no—
147C2×#137—64no—
148C2×#137—64no—
149C2×#137—64no—
150C2×#135—24no—
151C2×#135—24no—
152C2×#135—24no—
153C22×Q16—32no—
154C22×Q16C2×D311yescharacteristic
155C22×SD16—32no—
156C22×SD16C2×D311yescharacteristic
157C22×SD16C2×D311yescharacteristic
158C22×D8C2×D311yescharacteristic
159C22×M16—31no—
160C23×C8C2×D311yescharacteristic
161#536—64no—
162#536—64no—
163C2×C4⋊Q8—32no—
164C2×#124—32no—
165C2×#123—62no—
166C2×C22⋊Q8—62no—
167C2×#117—31no—
168C22×C4⋊C4—31no—
169#417—64no—
170C2×#103—32no—
171C2×#103—32no—
172C2×#103—32no—
173C2×#103—64no—
174C2×#102—32no—
175#391—32no—
176#390—32no—
177#386—62no—
178#380—62no—
179#336—32no—
180#336—32no—
181C24×C3—21no—
182C24×C3—21no—
183C24×C3C2×D411yescharacteristic
184C3×#41—28no—
185C3×#41—28no—
186C3×#41—28no—
187C3×#41—28no—
188C3×#41—44no—
189C3×#41—44no—
190C3×#41—44no—
191C3×#41—44no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8C2×D412yes—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—42no—
217C2×C3×D4—42no—
218C2×C3×D4—42no—
219C2×C3×D4—42no—
220C2×C3×D4C2×D412yes—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—42no—
227C22×C3×C4—42no—
228C22×C3×C4—42no—
229C22×C3×C4—42no—
230C22×C3×C4C2×D412yes—
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D412yes—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2×D412yes—
235C22×C3×C4C2×D412yes—
236C22×C3×C4#4112yes—
237C22×C3×C4C2411yescharacteristic
238C22×Dic3—24no—
239C22×Dic3—22no—
240C22×Dic3—22no—
241C22×Dic3—42no—
242C22×Dic3—42no—
243C22×Dic3—42no—
244C22×Dic3—82no—
245C22×Dic3C2×D414yes—
246C2×Dic6—28no—
247C2×Dic6—28no—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6—22no—
251C2×Dic6—22no—
252C2×Dic6—24no—
253C2×Dic6—22no—
254C2×Dic6—44no—
255C2×Dic6—44no—
256C2×Dic6—42no—
257C2×Dic6—42no—
258C2×Dic6C2×D414yes—
259C3×Q16—28no—
260C3×SD16—28no—
261C3×SD16—28no—
262C3×D8—28no—
263C2×C3×C8—28no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8C2×D412yes—
269C2×C3×C8C2×D412yes—
270#217—216no—
271#217—24no—
272#217—84no—
273#216—216no—
274#214—216no—
275C4⋊Dic3—24no—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—42no—
279Dic3⋊C4—216no—
280C4×Dic3—22no—
281C4×Dic3—22no—
282#208—28no—
283C2×C3⋊C8—22no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8C2×D414yes—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41#6714yes—
297C2×#41#6714yes—
298C22×Q8—34no—
299C22×Q8—32no—
300C22×Q8—61no—
301C22×Q8—62no—
302C22×Q8—62no—
303C22×Q8—62no—
304C22×Q8—62no—
305C22×Q8—62no—
306C22×Q8C22×D311yescharacteristic
307C22×Q8C22×D311yescharacteristic
308C22×D4—21no—
309C22×D4—21no—
310C22×D4C22×D311yescharacteristic
311C22×D4C22×D311yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—61no—
315C23×C4—61no—
316C23×C4C22×D311yescharacteristic
317#137—632no—
318#135—232no—
319C2×Q16—22no—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×Q16—64no—
323C2×Q16—64no—
324C2×Q16—64no—
325C2×SD16—22no—
326C2×SD16—22no—
327C2×SD16—22no—
328C2×SD16—22no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×SD16—64no—
332C2×SD16—64no—
333C2×SD16—64no—
334C2×D8—22no—
335C2×D8—22no—
336C2×D8—22no—
337C2×M16—34no—
338C2×M16—62no—
339C2×M16—62no—
340C22×C8—22no—
341C22×C8—22no—
342C22×C8—32no—
343C22×C8#6714yes—
344C22×C8C22×D311yescharacteristic
345C22×C8C22×D311yescharacteristic
346C4⋊Q8—64no—
347C4⋊Q8—64no—
348#124—64no—
349#124—64no—
350#123—616no—
351C22⋊Q8—616no—
352#117—62no—
353#117—62no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—64no—
357C2×C4⋊C4—62no—
358C2×C4⋊C4—62no—
359C2×C4⋊C4—61no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—64no—
362C2×C22⋊C4—64no—
363C2×C42—32no—
364#103—64no—
365#103—64no—
366#103—64no—
367#102—64no—
368#95—64no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3#12012yes—
382C23×C3#12112yes—
383C23×C3#12712yes—
384C23×C3C22×D411yescharacteristic
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—41no—
390C3×Q8—41no—
391C3×Q8—41no—
392C3×Q8—41no—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—22no—
397C3×D4—48no—
398C3×D4—48no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—41no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—48no—
410C2×C3×C4—48no—
411C2×C3×C4#12014yes—
412C2×C3×C4#12012yes—
413C2×C3×C4#12014yes—
414C2×C3×C4#12114yes—
415C2×C3×C4#12114yes—
416C2×C3×C4#12112yes—
417C2×C3×C4#12712yes—
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—42no—
429C2×Dic3—42no—
430C2×Dic3—48no—
431C2×Dic3—42no—
432C2×Dic3—44no—
433C2×Dic3—84no—
434Dic6—24no—
435Dic6—24no—
436Dic6—42no—
437Dic6—42no—
438Dic6—48no—
439C3×C8—22no—
440C3×C8—22no—
441C3⋊C8—24no—
442C24—21no—
443C24—21no—
444C24D3×D411yescharacteristic
445#41—28no—
446#41—28no—
447#41—28no—
448#41—28no—
449#41—44no—
450#41—44no—
451#41—44no—
452#41—44no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—34no—
464C2×Q8—64no—
465C2×Q8—62no—
466C2×Q8—68no—
467C2×Q8—68no—
468C2×Q8—62no—
469C2×Q8—62no—
470C2×Q8—62no—
471C2×Q8—62no—
472C2×Q8—124no—
473C2×Q8—122no—
474C2×Q8—122no—
475C2×Q8—124no—
476C2×Q8C2×#6712yes—
477C2×Q8C2×#6712yes—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4C2×#6712yes—
495C2×D4C2×#6712yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—34no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—64no—
506C22×C4—62no—
507C22×C4—62no—
508C22×C4—122no—
509C22×C4—122no—
510C22×C4—122no—
511C22×C4—242no—
512C22×C4D3×D412yes—
513C22×C4D3×D411yescharacteristic
514C22×C4D3×D412yes—
515C22×C4#23712yes—
516C22×C4C2×#6712yes—
517C22×C4C2×#6712yes—
518C22×C4C2×#6712yes—
519C22×C4C23×D311yescharacteristic
520Q16—28no—
521Q16—616no—
522SD16—28no—
523SD16—28no—
524SD16—616no—
525D8—28no—
526M16—68no—
527C2×C8—28no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—34no—
533C2×C8—62no—
534C2×C8—62no—
535C2×C8C2×#6712yes—
536C2×C8C2×#6712yes—
537C4⋊C4—616no—
538C4⋊C4—64no—
539C4⋊C4—62no—
540C4⋊C4—62no—
541C4⋊C4—122no—
542C22⋊C4—616no—
543C22⋊C4—64no—
544C22⋊C4—244no—
545C42—62no—
546C42—62no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—24no—
550C22×C3—24no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3—44no—
554C22×C3—44no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—82no—
560C22×C3—82no—
561C22×C3#39214yes—
562C22×C3C2×#12011yescharacteristic
563C22×C3C2×#12111yescharacteristic
564C22×C3C2×#12711yescharacteristic
565C22×C3#57712yes—
566C22×C3#57812yes—
567C3×C4—42no—
568C3×C4—42no—
569C3×C4#39214yes—
570C3×C4C2×#12011yescharacteristic
571C3×C4C2×#12011yescharacteristic
572C3×C4C2×#12111yescharacteristic
573C3×C4C2×#12111yescharacteristic
574Dic3—44no—
575Dic3—44no—
576Dic3—82no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23#93712yes—
590C23#93912yes—
591C23#94012yes—
592C23C2×D3×D411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599Q8—41no—
600Q8—41no—
601Q8—64no—
602Q8—64no—
603Q8—128no—
604Q8—122no—
605Q8—122no—
606D4—22no—
607D4—22no—
608D4—22no—
609D4—22no—
610D4—48no—
611D4—48no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—128no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—124no—
632C2×C4—122no—
633C2×C4—244no—
634C2×C4#93712yes—
635C2×C4#93714yes—
636C2×C4#93814yes—
637C2×C4#93912yes—
638C2×C4#93914yes—
639C2×C4#94012yes—
640C2×C4#95314yes—
641C2×C4C2×D3×D411yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4C2×#23711yescharacteristic
644C2×C4C22×#6711yescharacteristic
645C2×C4C22×#6711yescharacteristic
646C2×C4C22×#6711yescharacteristic
647C8—22no—
648C8—22no—
649C8—64no—
650C2×C3—24no—
651C2×C3—82no—
652C2×C3—82no—
653C2×C3C2×#39211yescharacteristic
654C2×C3#301312yes—
655C2×C3#304612yes—
656C2×C3#315411yescharacteristic
657C2×C3#315611yescharacteristic
658C22—24no—
659C22—22no—
660C22—22no—
661C22—24no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—82no—
672C22#522314yes—
673C22#577012yes—
674C22#577112yes—
675C22C2×#93711yescharacteristic
676C22C2×#93911yescharacteristic
677C22C2×#94011yescharacteristic
678C4—42no—
679C4—42no—
680C4—124no—
681C4—124no—
682C4—242no—
683C4#522314yes—
684C4C2×#93711yescharacteristic
685C4C2×#93811yescharacteristic
686C4C2×#93911yescharacteristic
687C4C2×#95311yescharacteristic
688C3#2342711yescharacteristic, complemented, Sylow
689C2—24no—
690C2—82no—
691C2—82no—
692C2#8390212yes—
693C2#8390511yescharacteristic
694C2#8390711yescharacteristic
695C2#8391312yes—
696C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×#135C2
socleC23×C3C22×D4
hypercenterC22×#135D3
hypocenterC3#23427
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C22×#951C2semidirect
4. #84087C2semidirect
5. C2×#5232C2semidirect
6. C22×#816C2semidirect
7. #72073C2semidirect
8. #72058C2semidirect
9. #72057C2semidirect
10. #72045C2semidirect
11. #71965C2semidirect
688. C3#23427semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1352
C2×#58252
C22×#9512
#840872
C2×#52322
C22×#8162
#720732
#720582
#720572
#720452
#719652
#234273