Finite Group Databasebeta

← #10822167

Subgroups — #10822167

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
1#87268C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C22×#909C211yesmaximal, characteristic, complemented
4C2×C3×#421C211yesmaximal, characteristic, complemented
5#82230C211yesmaximal, characteristic, complemented
6C2×#5005C211yesmaximal, characteristic, complemented
7#71987C211yesmaximal, characteristic, complemented
8#70393C218yesmaximal, complemented
9#23408—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5833C2214yes—
16#5832—22no—
17#5830—21no—
18#5828—22no—
19#5825—21no—
20C2×#951—22no—
21C2×#951—24no—
22C2×#951—24no—
23C2×#951C2214yescomplemented
24#5816—21no—
25C22×#216—21no—
26#5805—21no—
27#5804—22no—
28C22×#214—21no—
29#5791—22no—
30C2×#926C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C2×#909—22no—
33C2×#909—24no—
34C2×#909—24no—
35C2×#909C2214yescomplemented
36C22×Dic12—21no—
37C22×#204—21no—
38C3×#421—24no—
39C2×C3×#104C2214yescomplemented
40#5005—24no—
41#4973—22no—
42#4943—22no—
43#4929C2214yescomplemented
44C2×#805C2214yescomplemented
45#4554C2214yescomplemented
46C22×#137—31no—
47C22×#137—31no—
48#3449—31no—
49C2×#421—31no—
50C2×#421D311yescharacteristic, complemented
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×Q8D411yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4D411yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C23×Dic3—21no—
69C2×#235—21no—
70C2×#235—21no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235D412yes—
76C2×#235C2312yes—
77C22×D12D411yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×Dic6—21no—
81C22×Dic6—22no—
82C22×Dic6—21no—
83C22×Dic6—21no—
84C22×Dic6—22no—
85C22×Dic6—21no—
86C22×Dic6—22no—
87C22×Dic6D411yescharacteristic
88C22×Dic6D414yescomplemented
89C22×Dic6C2312yes—
90C2×C3×M16—22no—
91C2×C3×M16—21no—
92C2×C3×M16D414yescomplemented
93C2×C3×M16C2312yes—
94C22×C3×C8—21no—
95C3×#117—22no—
96C3×#117—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2312yes—
100#953—24no—
101C2×#217D412yes—
102#951—216no—
103#951—28no—
104#951—28no—
105#947—22no—
106#946—24no—
107#944—22no—
108C2×#216—28no—
109C2×#216—24no—
110#940—22no—
111#939—28no—
112#938—24no—
113#937—24no—
114#936—22no—
115#935—24no—
116C2×#214—28no—
117C2×#214—24no—
118#930—24no—
119#929—24no—
120C4×#67—28no—
121#926—21no—
122#926—21no—
123#926—21no—
124#926—21no—
125C2×C4⋊Dic3C2312yes—
126#924—24no—
127C2×C4×Dic3D412yes—
128C2×#208—21no—
129C2×#208—22no—
130C2×#208D414yescomplemented
131C2×#208C2312yes—
132C22×C3⋊C8—21no—
133#909—216no—
134#909—28no—
135#909—28no—
136C2×Dic12—28no—
137C2×Dic12—24no—
138C2×#204—28no—
139C2×#204—24no—
140C3×#104—28no—
141C3⋊#95—28no—
142#805—28no—
143C22×#41—31no—
144C22×#41C2×D311yescharacteristic
145C23×Q8—31no—
146C2×#137—34no—
147C2×#137—34no—
148C2×#137—64no—
149C2×#137—62no—
150C2×#137—64no—
151C2×#137—64no—
152C2×#137—62no—
153C2×#137—64no—
154C22×Q16—61no—
155C22×Q16—61no—
156C22×SD16—61no—
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×#117C2×D311yescharacteristic
171#421—24no—
172#421—64no—
173C2×#104—34no—
174C2×#104C2×D314yescomplemented
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—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8C2×D412yes—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—42no—
202C2×C3×D4—42no—
203C2×C3×D4C2×D412yes—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4#4112yes—
219C22×C3×C4C2411yescharacteristic
220C2×#67—22no—
221C2×#67—22no—
222C2×#67—28no—
223C2×#67—42no—
224C2×#67—42no—
225C22×Dic3—21no—
226C22×Dic3—24no—
227C22×Dic3—22no—
228C22×Dic3—41no—
229C22×Dic3—41no—
230C22×Dic3—42no—
231C22×Dic3C2×D414yes—
232#235—24no—
233#235—24no—
234#235—22no—
235#235—22no—
236#235—22no—
237#235—22no—
238#235—42no—
239#235—42no—
240#235—42no—
241#235—42no—
242C2×D12—22no—
243C2×D12—22no—
244C2×D12—21no—
245C2×D12C2×D412yes—
246C2×C4×D3—22no—
247C2×C4×D3—22no—
248C2×C4×D3—42no—
249C2×C4×D3—42no—
250C2×Dic6—24no—
251C2×Dic6—22no—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C2×Dic6—24no—
255C2×Dic6—22no—
256C2×Dic6—22no—
257C2×Dic6—22no—
258C2×Dic6—22no—
259C2×Dic6—22no—
260C2×Dic6—21no—
261C2×Dic6—24no—
262C2×Dic6—22no—
263C2×Dic6—41no—
264C2×Dic6—41no—
265C2×Dic6—42no—
266C2×Dic6—42no—
267C2×Dic6—42no—
268C2×Dic6—42no—
269C2×Dic6—42no—
270C2×Dic6C2×D412yes—
271C2×Dic6C2×D414yes—
272C3×M16—24no—
273C3×M16—22no—
274C3×M16C2×D414yes—
275C2×C3×C8—24no—
276C2×C3×C8—22no—
277C3×C4⋊C4—42no—
278C3×C22⋊C4—42no—
279C3×C42—22no—
280C3×C42—22no—
281#217—28no—
282#217—22no—
283#217—22no—
284#216—216no—
285#214—216no—
286(C2×D3)⋊C4—28no—
287C4⋊Dic3—22no—
288C4⋊Dic3—22no—
289Dic3⋊C4—28no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292#208—24no—
293#208—22no—
294#208C2×D414yes—
295C2×C3⋊C8—24no—
296C2×C3⋊C8—22no—
297Dic12—216no—
298#204—216no—
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×#41D1212yes—
314C2×#41C22×D312yes—
315C22×Q8—31no—
316C22×Q8—34no—
317C22×Q8—32no—
318C22×Q8—62no—
319C22×Q8—61no—
320C22×Q8—61no—
321C22×Q8—61no—
322C22×Q8—61no—
323C22×Q8—61no—
324C22×Q8—62no—
325C22×Q8—62no—
326C22×Q8D1211yescharacteristic
327C22×D4—21no—
328C22×D4—31no—
329C22×D4—61no—
330C22×D4D1211yescharacteristic
331C23×C4—21no—
332C23×C4—61no—
333C23×C4—61no—
334C23×C4C22×D311yescharacteristic
335#137—68no—
336#137—616no—
337#137—68no—
338#137—68no—
339#137—616no—
340#137—68no—
341C2×Q16—64no—
342C2×Q16—68no—
343C2×Q16—68no—
344C2×Q16—64no—
345C2×SD16—64no—
346C2×SD16—68no—
347C2×SD16—68no—
348C2×SD16—64no—
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—22no—
374#117—22no—
375#117—61no—
376#117—61no—
377#117—61no—
378#117—61no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—32no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—32no—
383C2×C42—32no—
384C2×C42C22×D312yes—
385#104—28no—
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—22no—
401C3×Q8—22no—
402C3×Q8—42no—
403C3×D4—22no—
404C3×D4—22no—
405C3×D4—42no—
406C3×D4—42no—
407C3×D4—44no—
408C3×D4—42no—
409C2×C3×C4—22no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—44no—
414C2×C3×C4—44no—
415C2×C3×C4—42no—
416C2×C3×C4#12012yes—
417C2×C3×C4#12012yes—
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4#12011yescharacteristic
420C2×C3×C4#12012yes—
421C2×C3×C4#12012yes—
422C2×C3×C4#12112yes—
423C2×C3×C4#12112yes—
424C2×C3×C4#12112yes—
425C2×C3×C4#12112yes—
426C2×C3×C4#12112yes—
427C2×C3×C4#12712yes—
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C2×#4111yescharacteristic
434#67—42no—
435#67—42no—
436#67—44no—
437#67—88no—
438C2×Dic3—24no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—42no—
442C2×Dic3—42no—
443C2×Dic3—44no—
444C2×Dic3—44no—
445C2×Dic3—42no—
446C2×Dic3—44no—
447D12—22no—
448D12—22no—
449D12—42no—
450C4×D3—42no—
451C4×D3—42no—
452C4×D3—44no—
453Dic6—24no—
454Dic6—22no—
455Dic6—24no—
456Dic6—22no—
457Dic6—42no—
458Dic6—42no—
459Dic6—44no—
460Dic6—44no—
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—22no—
489C2×Q8—22no—
490C2×Q8—32no—
491C2×Q8—34no—
492C2×Q8—61no—
493C2×Q8—64no—
494C2×Q8—62no—
495C2×Q8—64no—
496C2×Q8—64no—
497C2×Q8—62no—
498C2×Q8—64no—
499C2×Q8—62no—
500C2×Q8—64no—
501C2×Q8—62no—
502C2×Q8—62no—
503C2×Q8—62no—
504C2×Q8—62no—
505C2×Q8—121no—
506C2×Q8—121no—
507C2×Q8—122no—
508C2×Q8—122no—
509C2×Q8—122no—
510C2×Q8—122no—
511C2×Q8—122no—
512C2×Q8C2×D1212yes—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—21no—
517C2×D4—22no—
518C2×D4—32no—
519C2×D4—42no—
520C2×D4—42no—
521C2×D4—62no—
522C2×D4—62no—
523C2×D4—68no—
524C2×D4—61no—
525C2×D4—62no—
526C2×D4—62no—
527C2×D4—122no—
528C2×D4—122no—
529C2×D4C2×D1212yes—
530C22×C4—21no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—34no—
535C22×C4—42no—
536C22×C4—42no—
537C22×C4—61no—
538C22×C4—64no—
539C22×C4—62no—
540C22×C4—62no—
541C22×C4—62no—
542C22×C4—121no—
543C22×C4—121no—
544C22×C4—122no—
545C22×C4—122no—
546C22×C4—122no—
547C22×C4C2×D1211yescharacteristic
548C22×C4C2×D1211yescharacteristic
549C22×C4#23512yes—
550C22×C4D3×D411yescharacteristic
551C22×C4D3×D412yes—
552C22×C4D3×D412yes—
553C22×C4D3×D411yescharacteristic
554C22×C4D3×D411yescharacteristic
555C22×C4C2×#6712yes—
556C22×C4C23×D311yescharacteristic
557Q16—616no—
558Q16—616no—
559SD16—616no—
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—42no—
572C4⋊C4—68no—
573C4⋊C4—62no—
574C4⋊C4—62no—
575C22⋊C4—42no—
576C22⋊C4—68no—
577C22⋊C4—68no—
578C22⋊C4—62no—
579C22⋊C4—62no—
580C42—22no—
581C42—22no—
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—82no—
602C3×C4#39212yes—
603C3×C4#39212yes—
604C3×C4C2×#12011yescharacteristic
605C3×C4C2×#12011yescharacteristic
606C3×C4C2×#12111yescharacteristic
607C3×C4C2×#12111yescharacteristic
608Dic3—44no—
609Dic3—42no—
610Dic3—44no—
611C23—22no—
612C23—42no—
613C23—42no—
614C23—62no—
615C23—82no—
616C23—122no—
617C23—242no—
618C23#88211yescharacteristic
619C23#88211yescharacteristic
620C23#88412yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—42no—
626Q8—64no—
627Q8—62no—
628Q8—64no—
629Q8—62no—
630Q8—124no—
631Q8—124no—
632Q8—122no—
633Q8—122no—
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—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—64no—
656C2×C4—62no—
657C2×C4—64no—
658C2×C4—122no—
659C2×C4—124no—
660C2×C4—124no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4—124no—
664C2×C4—122no—
665C2×C4—122no—
666C2×C4—124no—
667C2×C4#88211yescharacteristic
668C2×C4#88211yescharacteristic
669C2×C4#88212yes—
670C2×C4#88212yes—
671C2×C4#88412yes—
672C2×C4#88412yes—
673C2×C4#88412yes—
674C2×C4#93012yes—
675C2×C4#93012yes—
676C2×C4#93712yes—
677C2×C4#93712yes—
678C2×C4#94012yes—
679C2×C4C22×D1211yescharacteristic
680C2×C4C2×#23511yescharacteristic
681C2×C4C2×D3×D411yescharacteristic
682C2×C4C2×D3×D411yescharacteristic
683C2×C4C2×D3×D411yescharacteristic
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#496012yes—
707C22#496012yes—
708C22C2×#88211yescharacteristic
709C22C2×#88411yescharacteristic
710C22C2×#94011yescharacteristic
711C4—42no—
712C4—82no—
713C4—124no—
714C4—122no—
715C4—124no—
716C4#496012yes—
717C4#496012yes—
718C4C2×#88211yescharacteristic
719C4C2×#88411yescharacteristic
720C4C2×#93011yescharacteristic
721C4C2×#93711yescharacteristic
722C3#2340811yescharacteristic, complemented, Sylow
723C2—21no—
724C2—22no—
725C2—22no—
726C2—21no—
727C2—82no—
728C2—242no—
729C2#7899512yes—
730C2C2×#496011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#421C2
socleC22×C3C2×#120
prime radical#87268C2
hypercenterC2×#421D3
hypocenterC3#23408
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C22×#909C2semidirect
4. C2×C3×#421C2semidirect
5. #82230C2semidirect
6. C2×#5005C2semidirect
7. #71987C2semidirect
8. #70393C2semidirect
23. C2×#951C22semidirect
35. C2×#909C22semidirect
39. C2×C3×#104C22semidirect
43. #4929C22semidirect
44. C2×#805C22semidirect
45. #4554C22semidirect
50. C2×#421D3semidirect
88. C22×Dic6D4semidirect
92. C2×C3×M16D4semidirect
130. C2×#208D4semidirect
174. C2×#104C2×D3semidirect
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
#872682
C22×#9512
C22×#9092
C2×C3×#4212
#822302
C2×#50052
#719872
#703932
#234083