Finite Group Databasebeta

← #10822114

Subgroups — #10822114

Order 768.

1954 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C22×#951C211yesmaximal, characteristic, complemented
4C2×#5236C211yesmaximal, characteristic, complemented
5C2×#5237C211yesmaximal, characteristic, complemented
6C2×#5004C211yesmaximal, characteristic, complemented
7#71868C214yesmaximal, complemented
8#71857C214yesmaximal, complemented
9#71574C211yesmaximal, characteristic, complemented
10#23401—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×#41C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C3×#537—22no—
15C3×#536—21no—
16C3×#536C2212yes—
17C3×#535—21no—
18C3×#535C2212yes—
19C3×#534—22no—
20C3×#532—22no—
21C3×#525—22no—
22C3×#525—22no—
23C2×C3×#117C2211yescharacteristic
24#5825—24no—
25#5825—24no—
26C2×#951—22no—
27C2×#951—24no—
28C2×#951—24no—
29C2×#951C2214yes—
30C2×#944C2212yescomplemented
31C22×#216—21no—
32C2×#936C2212yescomplemented
33C2×#935—22no—
34C22×#214—21no—
35C2×#926C2211yescharacteristic
36C2×#924—22no—
37C22×#208C2211yescharacteristic
38C22×#208C2211yescharacteristic
39#5237—24no—
40#5236—24no—
41C2×#837C2214yes—
42C2×#836C2212yescomplemented
43C2×#833C2212yescomplemented
44#5004—22no—
45#5004C2214yescomplemented
46C2×#810—22no—
47#4532C2214yescomplemented
48C22×#137—31no—
49#3449D311yescharacteristic
50C2×#536—31no—
51C2×#421—31no—
52C2×#419—31no—
53C2×#413—31no—
54#2005—34no—
55#2004—34no—
56#1733—31no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—21no—
61C2×C3×#41—21no—
62C2×C3×#41—21no—
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C2×C3×#41—21no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—22no—
69C2×C3×#41—22no—
70C2×C3×#41—22no—
71C2×C3×#41D412yes—
72C2×C3×#41C2312yes—
73C22×C3×Q8D411yescharacteristic
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4—21no—
76C22×C3×D4D412yes—
77C22×C3×D4D411yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4D412yes—
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C22×Dic6—21no—
84C22×Dic6—21no—
85C22×Dic6—21no—
86C22×Dic6—22no—
87C22×Dic6—21no—
88C22×Dic6—22no—
89C22×Dic6—21no—
90C22×Dic6—22no—
91C22×Dic6D414yescomplemented
92C22×Dic6C2312yes—
93C3×C4⋊Q8—22no—
94C3×#127—22no—
95C3×#124—22no—
96C3×#124—22no—
97C3×#123—24no—
98C3×#123—24no—
99C3×C22⋊Q8—24no—
100C3×C22⋊Q8—24no—
101C3×#121—24no—
102C3×#121—28no—
103C3×#121—24no—
104C3×#120—24no—
105C3×#120—24no—
106C3×C4×D4—28no—
107C3×#117—21no—
108C3×#117—21no—
109C3×#117D412yes—
110C3×#117C2312yes—
111C2×C3×C4⋊C4D412yes—
112C2×C3×C22⋊C4—21no—
113C2×C3×C42—21no—
114C2×#217—24no—
115C2×#217—21no—
116#951—216no—
117#951—28no—
118#951—28no—
119#944—24no—
120#944—24no—
121C2×#216—28no—
122C2×#216—24no—
123#936—24no—
124#936—24no—
125#935—216no—
126C2×#214—28no—
127C2×#214—24no—
128#926—22no—
129#926D414yescomplemented
130C2×C4⋊Dic3—21no—
131#924—216no—
132C2×Dic3⋊C4—24no—
133C2×C4×Dic3C2312yes—
134C2×#208—22no—
135C2×#208—21no—
136C2×#208—21no—
137C2×#208—22no—
138C2×#208—22no—
139C2×#208D414yescomplemented
140C2×#208C2312yes—
141C2×#208C2312yes—
142C22×C3⋊C8—21no—
143C22×C3⋊C8—21no—
144#837—28no—
145#836—24no—
146#833—24no—
147#810—28no—
148C22×#41C2×D311yescharacteristic
149C22×#41C2×D311yescharacteristic
150C23×Q8—31no—
151C2×#137—34no—
152C2×#137—64no—
153C2×#137—64no—
154C2×#137—62no—
155C22×Q16—61no—
156C22×SD16—61no—
157C22×M16—31no—
158C22×M16—31no—
159#537—22no—
160#536—21no—
161#536—64no—
162#536—64no—
163#536C2×D312yes—
164#535—21no—
165#535C2×D312yes—
166#534—22no—
167#532—22no—
168C2×C4⋊Q8—32no—
169C2×#124—32no—
170#525—22no—
171#525—22no—
172C2×#123—62no—
173C2×C22⋊Q8—62no—
174C2×#117—31no—
175C2×#117C2×D311yescharacteristic
176#421—64no—
177C2×#104—34no—
178#419—34no—
179#419—62no—
180C2×#103—62no—
181#413—64no—
182C2×#101—32no—
183C2×#100—32no—
184#339—34no—
185C24×C3—21no—
186C24×C3C2×D412yes—
187C24×C3C2×D411yescharacteristic
188C3×#41—24no—
189C3×#41—24no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41—22no—
194C3×#41—42no—
195C3×#41—42no—
196C3×#41—42no—
197C3×#41—42no—
198C3×#41—42no—
199C3×#41—42no—
200C3×#41—42no—
201C3×#41—42no—
202C3×#41—42no—
203C3×#41—42no—
204C3×#41—42no—
205C3×#41—42no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—22no—
214C2×C3×Q8—22no—
215C2×C3×Q8C2×D412yes—
216C2×C3×D4—24no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—24no—
221C2×C3×D4—22no—
222C2×C3×D4—28no—
223C2×C3×D4—22no—
224C2×C3×D4—21no—
225C2×C3×D4—22no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—42no—
233C2×C3×D4—42no—
234C2×C3×D4C2×D412yes—
235C22×C3×C4—21no—
236C22×C3×C4—22no—
237C22×C3×C4—24no—
238C22×C3×C4—22no—
239C22×C3×C4—22no—
240C22×C3×C4—22no—
241C22×C3×C4—24no—
242C22×C3×C4—22no—
243C22×C3×C4—42no—
244C22×C3×C4—42no—
245C22×C3×C4C2×D412yes—
246C22×C3×C4C2×D412yes—
247C22×C3×C4C2×D412yes—
248C22×C3×C4C2×D412yes—
249C22×C3×C4C2×D411yescharacteristic
250C22×C3×C4C2×D411yescharacteristic
251C22×C3×C4C2×D411yescharacteristic
252C22×C3×C4C2×D411yescharacteristic
253C22×C3×C4C2×D411yescharacteristic
254C22×C3×C4#4112yes—
255C22×C3×C4C2411yescharacteristic
256C22×Dic3—24no—
257C22×Dic3—22no—
258C22×Dic3—22no—
259C22×Dic3—41no—
260C22×Dic3—41no—
261C22×Dic3—42no—
262C2×Dic6—24no—
263C2×Dic6—24no—
264C2×Dic6—24no—
265C2×Dic6—24no—
266C2×Dic6—22no—
267C2×Dic6—22no—
268C2×Dic6—22no—
269C2×Dic6—22no—
270C2×Dic6—24no—
271C2×Dic6—22no—
272C2×Dic6—41no—
273C2×Dic6—41no—
274C2×Dic6—42no—
275C2×Dic6—42no—
276C2×Dic6—42no—
277C2×Dic6—42no—
278C2×Dic6—42no—
279C2×Dic6C2×D414yes—
280C3×C4⋊C4—28no—
281C3×C4⋊C4—22no—
282C3×C4⋊C4C2×D414yes—
283C3×C22⋊C4—28no—
284C3×C22⋊C4—28no—
285C3×C22⋊C4—22no—
286C3×C22⋊C4—22no—
287C3×C42—22no—
288C3×C42—22no—
289#217—216no—
290#217—24no—
291#216—216no—
292#214—216no—
293C4⋊Dic3—24no—
294Dic3⋊C4—216no—
295C4×Dic3—22no—
296C4×Dic3C2×D414yes—
297#208—24no—
298#208—22no—
299#208—22no—
300#208—22no—
301#208—42no—
302#208C2×D414yes—
303C2×C3⋊C8—24no—
304C2×C3⋊C8—22no—
305C2×C3⋊C8—22no—
306C2×C3⋊C8—42no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—21no—
312C2×#41—21no—
313C2×#41—21no—
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41#6712yes—
322C2×#41C22×D312yes—
323C22×Q8—34no—
324C22×Q8—32no—
325C22×Q8—62no—
326C22×Q8—61no—
327C22×Q8—62no—
328C22×Q8—62no—
329C22×Q8—61no—
330C22×Q8—61no—
331C22×Q8—61no—
332C22×Q8—61no—
333C22×Q8#6711yescharacteristic
334C22×Q8C22×D311yescharacteristic
335C22×D4—21no—
336C22×D4#6712yes—
337C22×D4#6711yescharacteristic
338C22×D4C22×D311yescharacteristic
339C23×C4—21no—
340C23×C4—61no—
341C23×C4#6712yes—
342C23×C4C22×D311yescharacteristic
343#137—68no—
344#137—68no—
345#137—616no—
346C2×Q16—68no—
347C2×Q16—64no—
348C2×SD16—68no—
349C2×SD16—64no—
350C2×M16—34no—
351C2×M16—32no—
352C2×M16—32no—
353C2×M16—62no—
354C2×M16—61no—
355C2×M16—61no—
356C2×M16—62no—
357C2×M16—62no—
358C22×C8—61no—
359C22×C8—61no—
360C4⋊Q8—22no—
361C4⋊Q8—64no—
362C4⋊Q8—64no—
363#127—22no—
364#124—22no—
365#124—22no—
366#124—64no—
367#124—64no—
368#123—24no—
369#123—24no—
370#123—616no—
371C22⋊Q8—24no—
372C22⋊Q8—24no—
373C22⋊Q8—616no—
374#121—24no—
375#121—28no—
376#121—24no—
377#120—24no—
378#120—24no—
379C4×D4—28no—
380#117—21no—
381#117—21no—
382#117—34no—
383#117—62no—
384#117#6712yes—
385#117C22×D312yes—
386C2×C4⋊C4—61no—
387C2×C4⋊C4—64no—
388C2×C4⋊C4#6712yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—61no—
391C2×C22⋊C4—64no—
392C2×C42—21no—
393C2×C42—32no—
394#104—68no—
395#103—68no—
396#101—64no—
397#100—64no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—22no—
402C23×C3—42no—
403C23×C3—44no—
404C23×C3—42no—
405C23×C3—82no—
406C23×C3#12012yes—
407C23×C3#12011yescharacteristic
408C23×C3#12011yescharacteristic
409C23×C3#12112yes—
410C23×C3C22×D411yescharacteristic
411C3×Q8—22no—
412C3×Q8—22no—
413C3×Q8—41no—
414C3×Q8—41no—
415C3×Q8—41no—
416C3×Q8—41no—
417C3×Q8—42no—
418C3×D4—22no—
419C3×D4—22no—
420C3×D4—42no—
421C3×D4—42no—
422C3×D4—44no—
423C3×D4—44no—
424C3×D4—44no—
425C3×D4—41no—
426C3×D4—41no—
427C3×D4—41no—
428C3×D4—41no—
429C3×D4—42no—
430C3×D4—88no—
431C2×C3×C4—22no—
432C2×C3×C4—24no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—42no—
437C2×C3×C4—42no—
438C2×C3×C4—42no—
439C2×C3×C4—44no—
440C2×C3×C4—44no—
441C2×C3×C4—44no—
442C2×C3×C4—82no—
443C2×C3×C4#12012yes—
444C2×C3×C4#12012yes—
445C2×C3×C4#12012yes—
446C2×C3×C4#12011yescharacteristic
447C2×C3×C4#12011yescharacteristic
448C2×C3×C4#12112yes—
449C2×C3×C4#12112yes—
450C2×C3×C4#12112yes—
451C2×C3×C4#12112yes—
452C2×C3×C4#12112yes—
453C2×C3×C4#12712yes—
454C2×C3×C4#12712yes—
455C2×C3×C4C22×D411yescharacteristic
456C2×C3×C4C22×D411yescharacteristic
457C2×C3×C4C22×D411yescharacteristic
458C2×C3×C4C22×D411yescharacteristic
459C2×C3×C4C22×D411yescharacteristic
460C2×C3×C4C2×#4111yescharacteristic
461C2×Dic3—24no—
462C2×Dic3—24no—
463C2×Dic3—44no—
464C2×Dic3—42no—
465C2×Dic3—42no—
466C2×Dic3—44no—
467C2×Dic3—44no—
468Dic6—24no—
469Dic6—24no—
470Dic6—42no—
471Dic6—42no—
472Dic6—44no—
473Dic6—44no—
474C3⋊C8—24no—
475C3⋊C8—42no—
476C24—21no—
477C24C2×#6712yes—
478C24C2×#6711yescharacteristic
479#41—24no—
480#41—24no—
481#41—22no—
482#41—22no—
483#41—22no—
484#41—22no—
485#41—42no—
486#41—42no—
487#41—42no—
488#41—42no—
489#41—42no—
490#41—42no—
491#41—42no—
492#41—42no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—42no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—34no—
507C2×Q8—64no—
508C2×Q8—62no—
509C2×Q8—64no—
510C2×Q8—64no—
511C2×Q8—64no—
512C2×Q8—64no—
513C2×Q8—62no—
514C2×Q8—62no—
515C2×Q8—62no—
516C2×Q8—62no—
517C2×Q8—122no—
518C2×Q8—122no—
519C2×Q8—122no—
520C2×Q8—122no—
521C2×Q8—122no—
522C2×Q8—121no—
523C2×Q8—121no—
524C2×Q8C2×#6712yes—
525C2×D4—24no—
526C2×D4—24no—
527C2×D4—22no—
528C2×D4—22no—
529C2×D4—22no—
530C2×D4—22no—
531C2×D4—28no—
532C2×D4—22no—
533C2×D4—22no—
534C2×D4—21no—
535C2×D4—21no—
536C2×D4—21no—
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—21no—
540C2×D4—21no—
541C2×D4—42no—
542C2×D4—42no—
543C2×D4C2×#6712yes—
544C22×C4—21no—
545C22×C4—22no—
546C22×C4—24no—
547C22×C4—24no—
548C22×C4—22no—
549C22×C4—22no—
550C22×C4—22no—
551C22×C4—22no—
552C22×C4—42no—
553C22×C4—42no—
554C22×C4—64no—
555C22×C4—62no—
556C22×C4—62no—
557C22×C4—122no—
558C22×C4—121no—
559C22×C4—121no—
560C22×C4D3×D412yes—
561C22×C4D3×D412yes—
562C22×C4D3×D412yes—
563C22×C4#23712yes—
564C22×C4C2×#6712yes—
565C22×C4C2×#6711yescharacteristic
566C22×C4C2×#6711yescharacteristic
567C22×C4C2×#6711yescharacteristic
568C22×C4C2×#6711yescharacteristic
569C22×C4C2×#6711yescharacteristic
570C22×C4C23×D311yescharacteristic
571Q16—616no—
572SD16—616no—
573M16—34no—
574M16—64no—
575M16—62no—
576M16—62no—
577M16—62no—
578M16—122no—
579C2×C8—64no—
580C2×C8—62no—
581C2×C8—62no—
582C2×C8—122no—
583C4⋊C4—28no—
584C4⋊C4—22no—
585C4⋊C4—616no—
586C4⋊C4—64no—
587C4⋊C4C2×#6714yes—
588C22⋊C4—28no—
589C22⋊C4—28no—
590C22⋊C4—22no—
591C22⋊C4—22no—
592C22⋊C4—616no—
593C22⋊C4—64no—
594C42—22no—
595C42—22no—
596C42—34no—
597C42—62no—
598C22×C3—22no—
599C22×C3—22no—
600C22×C3—22no—
601C22×C3—42no—
602C22×C3—42no—
603C22×C3—44no—
604C22×C3—42no—
605C22×C3—42no—
606C22×C3—44no—
607C22×C3—44no—
608C22×C3—82no—
609C22×C3—84no—
610C22×C3#39212yes—
611C22×C3#39212yes—
612C22×C3C2×#12011yescharacteristic
613C22×C3C2×#12011yescharacteristic
614C22×C3C2×#12111yescharacteristic
615C3×C4—42no—
616C3×C4—42no—
617C3×C4—44no—
618C3×C4#39212yes—
619C3×C4#39212yes—
620C3×C4C2×#12011yescharacteristic
621C3×C4C2×#12111yescharacteristic
622C3×C4C2×#12111yescharacteristic
623C3×C4C2×#12711yescharacteristic
624Dic3—44no—
625Dic3—44no—
626C23—22no—
627C23—22no—
628C23—22no—
629C23—22no—
630C23—42no—
631C23—44no—
632C23—42no—
633C23—82no—
634C23#93712yes—
635C23#93912yes—
636C23#95311yescharacteristic
637C23#95311yescharacteristic
638C23C22×#6711yescharacteristic
639Q8—22no—
640Q8—22no—
641Q8—42no—
642Q8—41no—
643Q8—41no—
644Q8—41no—
645Q8—41no—
646Q8—64no—
647Q8—64no—
648Q8—124no—
649Q8—124no—
650Q8—122no—
651Q8—122no—
652D4—22no—
653D4—22no—
654D4—44no—
655D4—44no—
656D4—44no—
657D4—42no—
658D4—42no—
659D4—42no—
660D4—41no—
661D4—41no—
662D4—41no—
663D4—41no—
664D4—88no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—24no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—64no—
677C2×C4—64no—
678C2×C4—82no—
679C2×C4—124no—
680C2×C4—124no—
681C2×C4—124no—
682C2×C4—122no—
683C2×C4—122no—
684C2×C4#93712yes—
685C2×C4#93712yes—
686C2×C4#93712yes—
687C2×C4#93812yes—
688C2×C4#93812yes—
689C2×C4#93912yes—
690C2×C4#93912yes—
691C2×C4#93912yes—
692C2×C4#94012yes—
693C2×C4#94012yes—
694C2×C4#95311yescharacteristic
695C2×C4#95311yescharacteristic
696C2×C4C2×D3×D411yescharacteristic
697C2×C4C2×D3×D411yescharacteristic
698C2×C4C2×D3×D411yescharacteristic
699C2×C4C2×#23711yescharacteristic
700C2×C4C22×#6711yescharacteristic
701C2×C4C22×#6711yescharacteristic
702C8—64no—
703C8—122no—
704C2×C3—21no—
705C2×C3—21no—
706C2×C3—22no—
707C2×C3—22no—
708C2×C3—82no—
709C2×C3—82no—
710C2×C3C2×#39211yescharacteristic
711C2×C3#302212yes—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—42no—
716C22—42no—
717C22—44no—
718C22—42no—
719C22—42no—
720C22—44no—
721C22—44no—
722C22—82no—
723C22—84no—
724C22#522312yes—
725C22#522312yes—
726C22C2×#93711yescharacteristic
727C22C2×#93911yescharacteristic
728C22C2×#95311yescharacteristic
729C4—42no—
730C4—42no—
731C4—44no—
732C4—124no—
733C4—124no—
734C4#522312yes—
735C4#522312yes—
736C4C2×#93711yescharacteristic
737C4C2×#93811yescharacteristic
738C4C2×#93911yescharacteristic
739C4C2×#94011yescharacteristic
740C3#2340111yescharacteristic, complemented, Sylow
741C2—21no—
742C2—22no—
743C2—21no—
744C2—22no—
745C2—82no—
746C2—82no—
747C2C2×#522311yescharacteristic
748C2#8412312yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3449C2
socleC22×C3C2×#120
hypercenter#3449D3
hypocenterC3#23401
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C22×#951C2semidirect
4. C2×#5236C2semidirect
5. C2×#5237C2semidirect
6. C2×#5004C2semidirect
7. #71868C2semidirect
8. #71857C2semidirect
9. #71574C2semidirect
30. C2×#944C22semidirect
32. C2×#936C22semidirect
42. C2×#836C22semidirect
43. C2×#833C22semidirect
45. #5004C22semidirect
47. #4532C22semidirect
91. C22×Dic6D4semidirect
129. #926D4semidirect
139. C2×#208D4semidirect
740. C3#23401semidirect

Maximal subgroups

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

subgroupindex
C3×#34492
C2×#58252
C22×#9512
C2×#52362
C2×#52372
C2×#50042
#718682
#718572
#715742
#234013