Finite Group Databasebeta

← #10822164

Subgroups — #10822164

Order 768.

1946 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic
3C22×#942C211yesmaximal, characteristic, complemented
4C2×#5236C211yesmaximal, characteristic, complemented
5#84062C211yesmaximal, characteristic, complemented
6C2×#5139C211yesmaximal, characteristic, complemented
7#72110C218yesmaximal, complemented
8#71987C211yesmaximal, characteristic, complemented
9#23408—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×C3×#137—22no—
14C2×C3×#137—24no—
15C2×C3×#137—24no—
16C2×C3×#137C2214yescomplemented
17C22×C3×Q16—21no—
18C22×C3×SD16—21no—
19C22×C3×M16C2211yescharacteristic
20#5833C2214yes—
21#5832—22no—
22#5830—21no—
23#5828—22no—
24#5825—21no—
25#5816—21no—
26C22×#216—21no—
27C2×#942—24no—
28C2×#942—22no—
29C2×#942—24no—
30C2×#942C2214yescomplemented
31C22×#215—21no—
32#5805—21no—
33#5804—22no—
34#5791—22no—
35C2×#926C2211yescharacteristic
36C2×#926C2211yescharacteristic
37C22×#208C2211yescharacteristic
38#5236—24no—
39C2×#837C2214yescomplemented
40#5229—22no—
41#5229C2214yescomplemented
42#5211—22no—
43#5139—24no—
44C2×#825C2214yescomplemented
45#4554C2214yescomplemented
46C22×#137—31no—
47C22×#137D311yescharacteristic, complemented
48#3449—31no—
49C2×#421—31no—
50C2×#421—31no—
51#2310—31no—
52#2007—38no—
53#1742—31no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41D412yes—
61C2×C3×#41C2312yes—
62C22×C3×Q8—21no—
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×Q8—21no—
66C22×C3×Q8—22no—
67C22×C3×Q8—21no—
68C22×C3×Q8—21no—
69C22×C3×Q8—22no—
70C22×C3×Q8D411yescharacteristic
71C22×C3×Q8D414yescomplemented
72C22×C3×Q8C2312yes—
73C22×C3×D4—21no—
74C22×C3×D4D411yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C2×#235—21no—
80C2×#235—21no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235D412yes—
86C2×#235C2312yes—
87C22×D12D411yescharacteristic
88C22×C4×D3—21no—
89C22×Dic6D411yescharacteristic
90C3×#137—216no—
91C3×#137—28no—
92C3×#137—28no—
93C2×C3×Q16—28no—
94C2×C3×Q16—24no—
95C2×C3×SD16—28no—
96C2×C3×SD16—24no—
97C2×C3×M16—22no—
98C2×C3×M16—21no—
99C2×C3×M16D414yescomplemented
100C2×C3×M16C2312yes—
101C22×C3×C8—21no—
102#953—24no—
103C2×#217—21no—
104C2×#217D412yes—
105#947—22no—
106#946—24no—
107#944—22no—
108C2×#216—28no—
109C2×#216—24no—
110#942—216no—
111#942—28no—
112#942—28no—
113C2×#215—28no—
114C2×#215—24no—
115#940—22no—
116#939—28no—
117#938—24no—
118#937—24no—
119#936—22no—
120#935—24no—
121#930—24no—
122#929—24no—
123C4×#67—28no—
124#926—21no—
125#926—21no—
126#926—21no—
127#926—21no—
128#926—22no—
129#926—22no—
130C2×C4⋊Dic3—21no—
131C2×C4⋊Dic3C2312yes—
132#924—24no—
133C2×C4×Dic3D412yes—
134C2×C4×Dic3C2312yes—
135C2×#208—21no—
136C2×#208—22no—
137C2×#208D414yescomplemented
138C2×#208C2312yes—
139C22×C3⋊C8—21no—
140#837—28no—
141C3⋊#95—28no—
142#825—28no—
143C22×#41—31no—
144C22×#41C2×D311yescharacteristic
145C23×Q8C2×D311yescharacteristic
146C2×#137—22no—
147C2×#137—24no—
148C2×#137—24no—
149C2×#137—34no—
150C2×#137—64no—
151C2×#137—62no—
152C2×#137—64no—
153C2×#137C2×D314yescomplemented
154C22×Q16—21no—
155C22×Q16—61no—
156C22×SD16—21no—
157C22×SD16—61no—
158C22×M16—31no—
159C22×M16C2×D311yescharacteristic
160#537—34no—
161#536—61no—
162#536—61no—
163#535—61no—
164#535—61no—
165#534—62no—
166#532—62no—
167#525—62no—
168#525—62no—
169C2×#117—31no—
170C2×#117—31no—
171#421—64no—
172#421—64no—
173C2×#104—34no—
174C2×#104—34no—
175#391—34no—
176#391—62no—
177#381—62no—
178#343—34no—
179C24×C3—21no—
180C24×C3C2×D411yescharacteristic
181C23×D3—21no—
182C3×#41—24no—
183C3×#41—24no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—42no—
189C3×#41—42no—
190C3×#41—42no—
191C3×#41—42no—
192C2×C3×Q8—21no—
193C2×C3×Q8—24no—
194C2×C3×Q8—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8—24no—
197C2×C3×Q8—24no—
198C2×C3×Q8—22no—
199C2×C3×Q8—24no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—41no—
206C2×C3×Q8—41no—
207C2×C3×Q8—42no—
208C2×C3×Q8—42no—
209C2×C3×Q8—42no—
210C2×C3×Q8—42no—
211C2×C3×Q8—42no—
212C2×C3×Q8C2×D414yes—
213C2×C3×Q8C2×D412yes—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—21no—
218C2×C3×D4—22no—
219C2×C3×D4—42no—
220C2×C3×D4—42no—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—21no—
223C22×C3×C4—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—41no—
228C22×C3×C4—41no—
229C22×C3×C4—42no—
230C22×C3×C4—42no—
231C22×C3×C4—42no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D412yes—
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D412yes—
240C22×C3×C4#4112yes—
241C22×C3×C4C2411yescharacteristic
242C2×#67—22no—
243C2×#67—22no—
244C2×#67—28no—
245C2×#67—42no—
246C2×#67—42no—
247C22×Dic3—21no—
248C22×Dic3—22no—
249C22×Dic3C2×D414yes—
250#235—24no—
251#235—24no—
252#235—22no—
253#235—22no—
254#235—22no—
255#235—22no—
256#235—42no—
257#235—42no—
258#235—42no—
259#235—42no—
260C2×D12—22no—
261C2×D12—22no—
262C2×D12—21no—
263C2×D12C2×D412yes—
264C2×C4×D3—22no—
265C2×C4×D3—22no—
266C2×C4×D3—42no—
267C2×C4×D3—42no—
268C2×Dic6—22no—
269C2×Dic6—22no—
270C2×Dic6—21no—
271C2×Dic6C2×D412yes—
272C3×Q16—216no—
273C3×SD16—216no—
274C3×M16—24no—
275C3×M16—22no—
276C3×M16C2×D414yes—
277C2×C3×C8—24no—
278C2×C3×C8—22no—
279#217—28no—
280#217—22no—
281#217—22no—
282#217—42no—
283#216—216no—
284#215—216no—
285(C2×D3)⋊C4—28no—
286C4⋊Dic3—22no—
287C4⋊Dic3—22no—
288C4⋊Dic3—42no—
289Dic3⋊C4—28no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292C4×Dic3—22no—
293C4×Dic3—22no—
294#208—24no—
295#208—22no—
296#208C2×D414yes—
297C2×C3⋊C8—24no—
298C2×C3⋊C8—22no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—32no—
306C2×#41—32no—
307C2×#41—61no—
308C2×#41—61no—
309C2×#41—62no—
310C2×#41—62no—
311C2×#41—62no—
312C2×#41—62no—
313C2×#41#6712yes—
314C2×#41C22×D312yes—
315C22×Q8—21no—
316C22×Q8—22no—
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—31no—
324C22×Q8#6711yescharacteristic
325C22×Q8#6714yescomplemented
326C22×Q8C22×D312yes—
327C22×D4—21no—
328C22×D4—31no—
329C22×D4—61no—
330C22×D4#6711yescharacteristic
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—61no—
334C23×C4C22×D311yescharacteristic
335#137—216no—
336#137—28no—
337#137—28no—
338#137—68no—
339#137—616no—
340#137—68no—
341C2×Q16—28no—
342C2×Q16—24no—
343C2×Q16—64no—
344C2×Q16—68no—
345C2×SD16—28no—
346C2×SD16—24no—
347C2×SD16—64no—
348C2×SD16—68no—
349C2×M16—22no—
350C2×M16—21no—
351C2×M16—34no—
352C2×M16—32no—
353C2×M16—61no—
354C2×M16—62no—
355C2×M16#6714yescomplemented
356C2×M16C22×D312yes—
357C22×C8—21no—
358C22×C8—61no—
359C4⋊Q8—62no—
360#127—62no—
361#124—62no—
362#124—62no—
363#123—64no—
364#123—64no—
365C22⋊Q8—64no—
366C22⋊Q8—64no—
367#121—64no—
368#121—68no—
369#121—64no—
370#120—64no—
371#120—64no—
372C4×D4—68no—
373#117—61no—
374#117—61no—
375#117—61no—
376#117—61no—
377#117—62no—
378#117—62no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—61no—
381C2×C22⋊C4—32no—
382C2×C22⋊C4—61no—
383C2×C42—32no—
384C2×C42—32no—
385#104—68no—
386#104—68no—
387#95—68no—
388C23×C3—22no—
389C23×C3—42no—
390C23×C3—42no—
391C23×C3—82no—
392C23×C3#12011yescharacteristic
393C23×C3#12011yescharacteristic
394C23×C3#12112yes—
395C23×C3#12712yes—
396C23×C3C22×D411yescharacteristic
397C22×D3—22no—
398C22×D3—42no—
399C22×D3—82no—
400C3×Q8—24no—
401C3×Q8—24no—
402C3×Q8—22no—
403C3×Q8—22no—
404C3×Q8—42no—
405C3×Q8—42no—
406C3×Q8—44no—
407C3×Q8—44no—
408C3×Q8—42no—
409C3×D4—22no—
410C3×D4—22no—
411C3×D4—42no—
412C3×D4—42no—
413C3×D4—44no—
414C3×D4—42no—
415C2×C3×C4—22no—
416C2×C3×C4—24no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—44no—
421C2×C3×C4—44no—
422C2×C3×C4—42no—
423C2×C3×C4—42no—
424C2×C3×C4—44no—
425C2×C3×C4#12012yes—
426C2×C3×C4#12012yes—
427C2×C3×C4#12011yescharacteristic
428C2×C3×C4#12011yescharacteristic
429C2×C3×C4#12012yes—
430C2×C3×C4#12012yes—
431C2×C3×C4#12112yes—
432C2×C3×C4#12112yes—
433C2×C3×C4#12112yes—
434C2×C3×C4#12112yes—
435C2×C3×C4#12112yes—
436C2×C3×C4#12712yes—
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C2×#4111yescharacteristic
443#67—42no—
444#67—42no—
445#67—44no—
446#67—88no—
447C2×Dic3—24no—
448C2×Dic3—22no—
449C2×Dic3—42no—
450C2×Dic3—44no—
451C2×Dic3—44no—
452C2×Dic3—42no—
453D12—22no—
454D12—22no—
455D12—42no—
456C4×D3—42no—
457C4×D3—42no—
458C4×D3—44no—
459Dic6—22no—
460Dic6—22no—
461Dic6—42no—
462C3×C8—24no—
463C3⋊C8—24no—
464C24—21no—
465C24—61no—
466C24D3×D411yescharacteristic
467#41—24no—
468#41—24no—
469#41—22no—
470#41—22no—
471#41—22no—
472#41—22no—
473#41—42no—
474#41—42no—
475#41—42no—
476#41—42no—
477#41—64no—
478#41—64no—
479#41—62no—
480#41—62no—
481#41—62no—
482#41—62no—
483#41—122no—
484#41—122no—
485#41—122no—
486#41—122no—
487C2×Q8—21no—
488C2×Q8—24no—
489C2×Q8—22no—
490C2×Q8—24no—
491C2×Q8—24no—
492C2×Q8—22no—
493C2×Q8—24no—
494C2×Q8—22no—
495C2×Q8—24no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—32no—
501C2×Q8—41no—
502C2×Q8—41no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8—61no—
509C2×Q8—62no—
510C2×Q8—62no—
511C2×Q8C2×#6712yes—
512C2×Q8C2×#6714yes—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—32no—
519C2×D4—42no—
520C2×D4—42no—
521C2×D4—62no—
522C2×D4—62no—
523C2×D4—68no—
524C2×D4—62no—
525C2×D4—61no—
526C2×D4—62no—
527C2×D4—122no—
528C2×D4—122no—
529C2×D4C2×#6712yes—
530C22×C4—21no—
531C22×C4—24no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—34no—
536C22×C4—41no—
537C22×C4—41no—
538C22×C4—42no—
539C22×C4—42no—
540C22×C4—42no—
541C22×C4—61no—
542C22×C4—62no—
543C22×C4—62no—
544C22×C4—62no—
545C22×C4—122no—
546C22×C4—122no—
547C22×C4D3×D411yescharacteristic
548C22×C4D3×D411yescharacteristic
549C22×C4D3×D411yescharacteristic
550C22×C4D3×D412yes—
551C22×C4#23712yes—
552C22×C4C2×#6712yes—
553C22×C4C2×#6711yescharacteristic
554C22×C4C2×#6711yescharacteristic
555C22×C4C2×#6712yes—
556C22×C4C23×D311yescharacteristic
557Q16—216no—
558Q16—616no—
559SD16—216no—
560SD16—616no—
561M16—24no—
562M16—22no—
563M16—34no—
564M16—64no—
565M16—62no—
566M16C2×#6714yes—
567C2×C8—24no—
568C2×C8—22no—
569C2×C8—64no—
570C2×C8—62no—
571C4⋊C4—68no—
572C4⋊C4—62no—
573C4⋊C4—62no—
574C4⋊C4—122no—
575C22⋊C4—68no—
576C22⋊C4—68no—
577C22⋊C4—62no—
578C22⋊C4—62no—
579C22⋊C4—122no—
580C42—62no—
581C42—62no—
582C42—62no—
583C42—62no—
584C22×C3—22no—
585C22×C3—22no—
586C22×C3—22no—
587C22×C3—42no—
588C22×C3—42no—
589C22×C3—42no—
590C22×C3—44no—
591C22×C3—84no—
592C22×C3#39212yes—
593C22×C3#39212yes—
594C22×C3C2×#12011yescharacteristic
595C22×C3C2×#12111yescharacteristic
596C22×C3C2×#12711yescharacteristic
597C2×D3—42no—
598C2×D3—44no—
599C2×D3—84no—
600C3×C4—42no—
601C3×C4—44no—
602C3×C4#39212yes—
603C3×C4#39212yes—
604C3×C4C2×#12011yescharacteristic
605C3×C4C2×#12011yescharacteristic
606C3×C4C2×#12111yescharacteristic
607C3×C4C2×#12111yescharacteristic
608Dic3—42no—
609Dic3—44no—
610Dic3—82no—
611C23—22no—
612C23—42no—
613C23—42no—
614C23—62no—
615C23—82no—
616C23—122no—
617C23—242no—
618C23#93711yescharacteristic
619C23#93711yescharacteristic
620C23#93912yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—24no—
624Q8—22no—
625Q8—24no—
626Q8—22no—
627Q8—44no—
628Q8—44no—
629Q8—42no—
630Q8—42no—
631Q8—42no—
632Q8—62no—
633Q8—62no—
634Q8—122no—
635D4—22no—
636D4—22no—
637D4—44no—
638D4—42no—
639D4—42no—
640D4—42no—
641D4—62no—
642D4—62no—
643D4—124no—
644D4—122no—
645D4—122no—
646D4—122no—
647D4—248no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—64no—
659C2×C4—62no—
660C2×C4—122no—
661C2×C4—124no—
662C2×C4—122no—
663C2×C4—122no—
664C2×C4—124no—
665C2×C4—124no—
666C2×C4—122no—
667C2×C4#93711yescharacteristic
668C2×C4#93711yescharacteristic
669C2×C4#93712yes—
670C2×C4#93712yes—
671C2×C4#93812yes—
672C2×C4#93812yes—
673C2×C4#93912yes—
674C2×C4#93912yes—
675C2×C4#93912yes—
676C2×C4#94012yes—
677C2×C4#95312yes—
678C2×C4#95312yes—
679C2×C4C2×D3×D411yescharacteristic
680C2×C4C2×D3×D411yescharacteristic
681C2×C4C2×#23711yescharacteristic
682C2×C4C22×#6711yescharacteristic
683C2×C4C22×#6711yescharacteristic
684C2×C4C22×#6711yescharacteristic
685C8—24no—
686C8—64no—
687C2×C3—21no—
688C2×C3—22no—
689C2×C3—22no—
690C2×C3—21no—
691C2×C3—82no—
692C2×C3C2×#39211yescharacteristic
693C2×C3#302212yes—
694D3—82no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—42no—
699C22—42no—
700C22—44no—
701C22—42no—
702C22—84no—
703C22—124no—
704C22—122no—
705C22—244no—
706C22#522312yes—
707C22#522312yes—
708C22C2×#93711yescharacteristic
709C22C2×#93911yescharacteristic
710C22C2×#94011yescharacteristic
711C4—42no—
712C4—44no—
713C4—124no—
714C4—122no—
715C4—242no—
716C4#522312yes—
717C4#522312yes—
718C4C2×#93711yescharacteristic
719C4C2×#93811yescharacteristic
720C4C2×#93911yescharacteristic
721C4C2×#95311yescharacteristic
722C3#2340811yescharacteristic, complemented, Sylow
723C2—21no—
724C2—22no—
725C2—22no—
726C2—21no—
727C2—82no—
728C2—242no—
729C2#8399312yes—
730C2C2×#522311yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC22×C3C2×#120
prime radical#87268C2
hypercenterC22×#137D3
hypocenterC3#23408
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C22×#942C2semidirect
4. C2×#5236C2semidirect
5. #84062C2semidirect
6. C2×#5139C2semidirect
7. #72110C2semidirect
8. #71987C2semidirect
16. C2×C3×#137C22semidirect
30. C2×#942C22semidirect
39. C2×#837C22semidirect
41. #5229C22semidirect
44. C2×#825C22semidirect
45. #4554C22semidirect
47. C22×#137D3semidirect
71. C22×C3×Q8D4semidirect
99. C2×C3×M16D4semidirect
137. C2×#208D4semidirect
153. C2×#137C2×D3semidirect
325. C22×Q8#67semidirect
355. C2×M16#67semidirect
722. C3#23408semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1372
#872682
C22×#9422
C2×#52362
#840622
C2×#51392
#721102
#719872
#234083