Finite Group Databasebeta

← #10822312

Subgroups — #10822312

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×#137C211yesmaximal, characteristic, complemented
2C22×#950C211yesmaximal, characteristic
3C2×#5816C211yesmaximal, characteristic, complemented
4#84087C211yesmaximal, characteristic, complemented
5C22×#835C211yesmaximal, characteristic, complemented
6C2×#5133C211yesmaximal, characteristic, complemented
7#72073C212yesmaximal, complemented
8#72072C212yesmaximal, complemented
9#72057C212yesmaximal, complemented
10#72055C212yesmaximal, complemented
11#71980C211yesmaximal, characteristic, complemented
12#23427—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×#137—24no—
17C2×C3×#137—24no—
18C2×C3×#137—24no—
19C22×C3×Q16C2212yes—
20C22×C3×SD16C2212yes—
21C22×C3×M16C2211yescharacteristic
22C2×#950—24no—
23C2×#950—24no—
24C2×#950—24no—
25#5816—24no—
26#5816—24no—
27C2×#947C2212yes—
28C2×#946—22no—
29C2×#944C2212yes—
30C22×#216C2211yescharacteristic
31C22×#215C2211yescharacteristic
32C22×#214C2211yescharacteristic
33C22×#213C2211yescharacteristic
34C2×#929—22no—
35C2×#926C2211yescharacteristic
36C22×C4⋊Dic3C2211yescharacteristic
37C23×C3⋊C8C2211yescharacteristic
38#5229C2212yes—
39C2×#835—24no—
40C2×#835C2212yes—
41C2×#835C2212yes—
42#5221C2212yes—
43#5220—22no—
44#5212—22no—
45#5133—24no—
46C2×#821C2212yes—
47C2×#816C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C22×#137D311yescharacteristic, complemented
51C22×#135—31no—
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×#41D414yes—
67C22×C3×Q8—21no—
68C22×C3×Q8—22no—
69C22×C3×Q8—22no—
70C22×C3×Q8—22no—
71C22×C3×Q8—22no—
72C22×C3×Q8—22no—
73C22×C3×Q8D414yes—
74C22×C3×Q8C2312yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C23×Dic3—21no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235—22no—
86C2×#235—22no—
87C2×#235—22no—
88C2×#235D414yes—
89C22×D12C2311yescharacteristic
90C22×C4×D3—21no—
91C22×Dic6C2311yescharacteristic
92C3×#137—232no—
93C2×C3×Q16—24no—
94C2×C3×Q16—24no—
95C2×C3×Q16—24no—
96C2×C3×SD16—24no—
97C2×C3×SD16—24no—
98C2×C3×SD16—24no—
99C2×C3×M16—22no—
100C2×C3×M16—22no—
101C2×C3×M16D414yes—
102C22×C3×C8C2312yes—
103C2×#217—24no—
104C2×#217D412yes—
105#950—232no—
106#947—24no—
107#947—24no—
108#946—216no—
109#944—24no—
110#944—24no—
111C2×#216—22no—
112C2×#216—22no—
113C2×#216—22no—
114C2×#215—22no—
115C2×#215—22no—
116C2×#215—22no—
117C2×#214—22no—
118C2×#214—22no—
119C2×#214—22no—
120C2×#213—22no—
121C2×#213—22no—
122C2×#213—22no—
123#929—216no—
124C2×(C2×D3)⋊C4—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—
132C2×Dic3⋊C4—24no—
133C2×C4×Dic3C2312yes—
134C22×C3⋊C8—22no—
135C22×C3⋊C8—22no—
136C22×C3⋊C8D414yes—
137C22×C3⋊C8C2311yescharacteristic
138C22×C3⋊C8C2311yescharacteristic
139#835—24no—
140#835—24no—
141C3⋊#95—24no—
142#821—24no—
143#816—24no—
144C22×#41—31no—
145C22×#41C2×D311yescharacteristic
146C23×Q8C2×D311yescharacteristic
147C2×#137—24no—
148C2×#137—24no—
149C2×#137—24no—
150C2×#135—64no—
151C2×#135—64no—
152C2×#135—64no—
153C22×Q16—31no—
154C22×Q16C2×D312yes—
155C22×SD16—31no—
156C22×SD16—31no—
157C22×SD16C2×D312yes—
158C22×D8—31no—
159C22×M16C2×D311yescharacteristic
160C23×C8—31no—
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×C3C2×D411yescharacteristic
183C23×D3—21no—
184C3×#41—28no—
185C3×#41—28no—
186C3×#41—44no—
187C3×#41—44no—
188C2×C3×Q8—24no—
189C2×C3×Q8—22no—
190C2×C3×Q8—21no—
191C2×C3×Q8—28no—
192C2×C3×Q8—28no—
193C2×C3×Q8—22no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—22no—
201C2×C3×Q8—44no—
202C2×C3×Q8—44no—
203C2×C3×Q8—42no—
204C2×C3×Q8—42no—
205C2×C3×Q8C2×D412yes—
206C2×C3×Q8C2×D414yes—
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—42no—
214C2×C3×D4—42no—
215C2×C3×D4C2×D412yes—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—22no—
219C22×C3×C4—24no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4—42no—
223C22×C3×C4—42no—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2×D412yes—
230C22×C3×C4#4112yes—
231C22×C3×C4C2411yescharacteristic
232C2×#67—24no—
233C2×#67—42no—
234C2×#67—42no—
235C22×Dic3—21no—
236C22×Dic3—22no—
237C22×Dic3—42no—
238C22×Dic3—82no—
239C22×Dic3C2×D414yes—
240#235—28no—
241#235—28no—
242#235—44no—
243#235—44no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12C2×D412yes—
250C2×C4×D3—24no—
251C2×C4×D3—42no—
252C2×C4×D3—42no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6—21no—
258C2×Dic6C2×D412yes—
259C3×Q16—216no—
260C3×SD16—216no—
261C3×M16—28no—
262C2×C3×C8—22no—
263C2×C3×C8—22no—
264C2×C3×C8C2×D414yes—
265#217—24no—
266#217—84no—
267#216—28no—
268#215—28no—
269#214—28no—
270#213—28no—
271(C2×D3)⋊C4—216no—
272C4⋊Dic3—24no—
273C4⋊Dic3—22no—
274C4⋊Dic3—22no—
275C4⋊Dic3—42no—
276Dic3⋊C4—216no—
277C4×Dic3—22no—
278C4×Dic3—22no—
279C2×C3⋊C8—28no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8C2×D412yes—
285C2×C3⋊C8C2×D412yes—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—34no—
292C2×#41—62no—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41#6714yes—
298C22×Q8—21no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—31no—
305C22×Q8#6714yes—
306C22×Q8C22×D312yes—
307C22×Q8C22×D311yescharacteristic
308C22×D4—21no—
309C22×D4—31no—
310C22×D4—61no—
311C22×D4C22×D311yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—61no—
315C23×C4—61no—
316C23×C4C22×D311yescharacteristic
317#137—232no—
318#135—632no—
319C2×Q16—24no—
320C2×Q16—24no—
321C2×Q16—24no—
322C2×Q16—62no—
323C2×Q16—62no—
324C2×Q16—62no—
325C2×SD16—24no—
326C2×SD16—24no—
327C2×SD16—24no—
328C2×SD16—62no—
329C2×SD16—62no—
330C2×SD16—62no—
331C2×SD16—62no—
332C2×SD16—62no—
333C2×SD16—62no—
334C2×D8—62no—
335C2×D8—62no—
336C2×D8—62no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16#6714yes—
340C22×C8—34no—
341C22×C8—31no—
342C22×C8—31no—
343C22×C8—62no—
344C22×C8—62no—
345C22×C8C22×D312yes—
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—41no—
373C23×C3—41no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3#12012yes—
377C23×C3#12112yes—
378C23×C3#12712yes—
379C23×C3C22×D411yescharacteristic
380C22×D3—22no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—42no—
384C22×D3—42no—
385C3×Q8—22no—
386C3×Q8—24no—
387C3×Q8—22no—
388C3×Q8—24no—
389C3×Q8—42no—
390C3×Q8—42no—
391C3×Q8—41no—
392C3×Q8—41no—
393C3×Q8—48no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—48no—
397C3×D4—41no—
398C3×D4—41no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—41no—
402C2×C3×C4—42no—
403C2×C3×C4—41no—
404C2×C3×C4—42no—
405C2×C3×C4—48no—
406C2×C3×C4—48no—
407C2×C3×C4#12014yes—
408C2×C3×C4#12012yes—
409C2×C3×C4#12014yes—
410C2×C3×C4#12114yes—
411C2×C3×C4#12114yes—
412C2×C3×C4#12112yes—
413C2×C3×C4#12712yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420#67—48no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—42no—
428C2×Dic3—44no—
429C2×Dic3—84no—
430D12—22no—
431D12—22no—
432D12—41no—
433D12—41no—
434C4×D3—48no—
435Dic6—22no—
436Dic6—22no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—24no—
440C3⋊C8—22no—
441C3⋊C8—22no—
442C24—21no—
443C24—61no—
444C24D3×D411yescharacteristic
445#41—28no—
446#41—28no—
447#41—44no—
448#41—44no—
449#41—68no—
450#41—68no—
451#41—124no—
452#41—124no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—28no—
457C2×Q8—28no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—32no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8C2×#6714yes—
477C2×Q8C2×#6712yes—
478C2×D4—24no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—32no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—64no—
488C2×D4—61no—
489C2×D4—61no—
490C2×D4—61no—
491C2×D4—61no—
492C2×D4—61no—
493C2×D4—122no—
494C2×D4—122no—
495C2×D4C2×#6712yes—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—34no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—61no—
506C22×C4—64no—
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—216no—
521Q16—68no—
522SD16—216no—
523SD16—68no—
524SD16—68no—
525D8—68no—
526M16—28no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—32no—
530C2×C8—32no—
531C2×C8—68no—
532C2×C8—61no—
533C2×C8—61no—
534C2×C8—61no—
535C2×C8—61no—
536C2×C8C2×#6714yes—
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—44no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—82no—
556C22×C3#39214yes—
557C22×C3C2×#12011yescharacteristic
558C22×C3C2×#12111yescharacteristic
559C22×C3C2×#12711yescharacteristic
560C22×C3#57712yes—
561C22×C3#57812yes—
562C2×D3—42no—
563C2×D3—44no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—82no—
567C3×C4—42no—
568C3×C4—44no—
569C3×C4#39214yes—
570C3×C4C2×#12011yescharacteristic
571C3×C4C2×#12011yescharacteristic
572C3×C4C2×#12111yescharacteristic
573C3×C4C2×#12111yescharacteristic
574Dic3—42no—
575Dic3—44no—
576Dic3—82no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—62no—
585C23—122no—
586C23—122no—
587C23—121no—
588C23—121no—
589C23#93712yes—
590C23#93912yes—
591C23#94012yes—
592C23C2×D3×D411yescharacteristic
593Q8—22no—
594Q8—24no—
595Q8—24no—
596Q8—22no—
597Q8—48no—
598Q8—42no—
599Q8—42no—
600Q8—41no—
601Q8—41no—
602Q8—62no—
603Q8—62no—
604Q8—121no—
605Q8—121no—
606D4—22no—
607D4—22no—
608D4—48no—
609D4—41no—
610D4—41no—
611D4—62no—
612D4—62no—
613D4—128no—
614D4—121no—
615D4—121no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—62no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—128no—
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—24no—
648C8—62no—
649C8—62no—
650C2×C3—24no—
651C2×C3—82no—
652C2×C3C2×#39211yescharacteristic
653C2×C3#301312yes—
654C2×C3#304612yes—
655C2×C3#315411yescharacteristic
656C2×C3#315611yescharacteristic
657D3—82no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—24no—
662C22—44no—
663C22—42no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22—124no—
668C22—122no—
669C22—242no—
670C22—241no—
671C22—241no—
672C22#522314yes—
673C22#577812yes—
674C22#577912yes—
675C22C2×#93711yescharacteristic
676C22C2×#93911yescharacteristic
677C22C2×#94011yescharacteristic
678C4—42no—
679C4—44no—
680C4—124no—
681C4—122no—
682C4—242no—
683C4#522314yes—
684C4C2×#93711yescharacteristic
685C4C2×#93811yescharacteristic
686C4C2×#93911yescharacteristic
687C4C2×#95311yescharacteristic
688C3#2342711yescharacteristic, complemented, Sylow
689C2—24no—
690C2—82no—
691C2—242no—
692C2#8397112yes—
693C2#8397411yescharacteristic
694C2#8397511yescharacteristic
695C2#8398412yes—
696C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC23×C3C22×D4
prime radicalC22×#950C2
hypercenterC22×#137D3
hypocenterC3#23427
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C2×#5816C2semidirect
4. #84087C2semidirect
5. C22×#835C2semidirect
6. C2×#5133C2semidirect
7. #72073C2semidirect
8. #72072C2semidirect
9. #72057C2semidirect
10. #72055C2semidirect
11. #71980C2semidirect
50. C22×#137D3semidirect
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×#1372
C22×#9502
C2×#58162
#840872
C22×#8352
C2×#51332
#720732
#720722
#720572
#720552
#719802
#234273