Finite Group Databasebeta

← #10822117

Subgroups — #10822117

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
1C2×#5816C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic, complemented
3#85735C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5C2×#5139C211yesmaximal, characteristic, complemented
6C2×#5132C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#70346C214yesmaximal, complemented
9#70344C214yesmaximal, complemented
10#23401—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5816—24no—
17#5816—24no—
18C2×#947C2212yescomplemented
19C2×#946—22no—
20C2×#944C2212yescomplemented
21C22×#216—21no—
22C2×#942—24no—
23C2×#942—22no—
24C2×#942—24no—
25C2×#942C2214yes—
26C22×#215—21no—
27C2×#929—22no—
28C2×#926C2211yescharacteristic
29C22×#208C2211yescharacteristic
30#5526—21no—
31#5526C2212yes—
32#5525—21no—
33#5525C2212yes—
34#5524—22no—
35#5524—22no—
36#5510—22no—
37#5493—22no—
38#5477—22no—
39C3×#419—22no—
40C3×#419C2214yescomplemented
41C2×C3×#103—22no—
42#5139—24no—
43C2×#825C2214yes—
44C2×#824C2212yescomplemented
45#5132—24no—
46C2×#823C2212yescomplemented
47#4532C2214yescomplemented
48C22×#137—31no—
49#3449—31no—
50C2×#536—31no—
51C2×#421—31no—
52C2×#419D311yescharacteristic, complemented
53C2×#413—31no—
54#2005—34no—
55#2004—34no—
56#1733—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C22×C3×Q8—21no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8D414yescomplemented
66C22×C3×Q8C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67D412yes—
71C2×#235—22no—
72C2×#235—21no—
73C2×#235—21no—
74C2×#235—21no—
75C2×#235—21no—
76C2×#235—21no—
77C2×#235—21no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235D412yes—
86C2×#235C2312yes—
87C22×D12D411yescharacteristic
88C22×D12C2311yescharacteristic
89C22×C4×D3—21no—
90C22×C4×D3D412yes—
91C22×Dic6D411yescharacteristic
92C22×Dic6C2311yescharacteristic
93C2×C3×M16—21no—
94C2×C3×M16—22no—
95C2×C3×M16—22no—
96C2×C3×M16C2312yes—
97C22×C3×C8—21no—
98C3×#117—21no—
99C3×#117—21no—
100C3×#117D412yes—
101C3×#117C2312yes—
102C2×C3×C4⋊C4D412yes—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—21no—
105C2×#217—21no—
106#947—24no—
107#947—24no—
108#946—216no—
109#944—24no—
110#944—24no—
111C2×#216—28no—
112C2×#216—24no—
113#942—216no—
114#942—28no—
115#942—28no—
116C2×#215—28no—
117C2×#215—24no—
118#930—28no—
119#929—216no—
120C2×(C2×D3)⋊C4—24no—
121#926—22no—
122#926D414yescomplemented
123C2×C4⋊Dic3—21no—
124C2×Dic3⋊C4—24no—
125C2×C4×Dic3C2312yes—
126C2×#208—22no—
127C2×#208—21no—
128C2×#208D414yescomplemented
129C2×#208C2312yes—
130C22×C3⋊C8—21no—
131#897—24no—
132#897—24no—
133#895—24no—
134#895—24no—
135#886—24no—
136#886—24no—
137#882—24no—
138#882—24no—
139#875—22no—
140#875—22no—
141#874—22no—
142C4×D12—28no—
143#869—22no—
144C3×#103—28no—
145#825—28no—
146#824—24no—
147#823—24no—
148C22×#41—31no—
149C22×#41—31no—
150C23×Q8C2×D311yescharacteristic
151C2×#137—34no—
152C2×#137—64no—
153C2×#137—62no—
154C2×#137—64no—
155C22×Q16—61no—
156C22×SD16—61no—
157C22×M16—31no—
158C22×M16C2×D311yescharacteristic
159#537—62no—
160#536—32no—
161#536—61no—
162#536—64no—
163#536—64no—
164#535—32no—
165#535—61no—
166#534—62no—
167#532—62no—
168C2×C4⋊Q8—32no—
169C2×#124—32no—
170#525—62no—
171#525—62no—
172C2×#123—62no—
173C2×C22⋊Q8—62no—
174C2×#117—31no—
175C2×#117C2×D311yescharacteristic
176#421—64no—
177C2×#104—34no—
178#419—22no—
179#419C2×D314yescomplemented
180C2×#103—22no—
181#413—64no—
182C2×#101—32no—
183C2×#100—32no—
184#339—34no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C23×D3C2×D412yes—
188C2×C3×Q8—24no—
189C2×C3×Q8—22no—
190C2×C3×Q8—24no—
191C2×C3×Q8—24no—
192C2×C3×Q8—24no—
193C2×C3×Q8—24no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—41no—
199C2×C3×Q8—41no—
200C2×C3×Q8—42no—
201C2×C3×Q8—42no—
202C2×C3×Q8—42no—
203C2×C3×Q8—42no—
204C2×C3×Q8—42no—
205C2×C3×Q8C2×D414yes—
206C22×C3×C4—24no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—41no—
210C22×C3×C4—41no—
211C22×C3×C4—42no—
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×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4112yes—
221C22×C3×C4C2411yescharacteristic
222C2×#67—22no—
223C2×#67—24no—
224C2×#67—22no—
225C2×#67—22no—
226C2×#67—24no—
227C2×#67—22no—
228C2×#67—42no—
229C2×#67—42no—
230C22×Dic3—21no—
231C22×Dic3—22no—
232C22×Dic3C2×D412yes—
233#235—24no—
234#235—24no—
235#235—22no—
236#235—22no—
237#235—22no—
238#235—22no—
239#235—42no—
240#235—42no—
241#235—42no—
242#235—42no—
243#235—42no—
244#235—42no—
245#235—42no—
246#235—42no—
247#235—42no—
248#235—42no—
249#235—42no—
250#235—42no—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—21no—
254C2×D12—21no—
255C2×D12—21no—
256C2×D12—21no—
257C2×D12—21no—
258C2×D12—21no—
259C2×D12—21no—
260C2×D12—28no—
261C2×D12C2×D412yes—
262C2×C4×D3—22no—
263C2×C4×D3—22no—
264C2×C4×D3—22no—
265C2×C4×D3—24no—
266C2×C4×D3—24no—
267C2×C4×D3—22no—
268C2×C4×D3—42no—
269C2×C4×D3—42no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6—21no—
273C2×Dic6—21no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—21no—
279C2×Dic6C2×D412yes—
280C3×M16—22no—
281C3×M16—22no—
282C3×M16—42no—
283C2×C3×C8—22no—
284C2×C3×C8—42no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4C2×D414yes—
287C3×C22⋊C4—22no—
288C3×C22⋊C4—22no—
289C3×C42—22no—
290C3×C42—22no—
291#217—24no—
292#216—216no—
293#215—216no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—28no—
296(C2×D3)⋊C4—216no—
297C4⋊Dic3—28no—
298C4⋊Dic3—24no—
299Dic3⋊C4—216no—
300C4×Dic3—22no—
301C4×Dic3C2×D414yes—
302#208—24no—
303#208—22no—
304#208C2×D414yes—
305C2×C3⋊C8—24no—
306C2×C3⋊C8—22no—
307C2×#41—32no—
308C2×#41—32no—
309C2×#41—62no—
310C2×#41—62no—
311C2×#41—61no—
312C2×#41—61no—
313C2×#41—61no—
314C2×#41—62no—
315C2×#41—62no—
316C2×#41—61no—
317C2×#41—61no—
318C2×#41—61no—
319C2×#41—62no—
320C2×#41—62no—
321C2×#41—62no—
322C2×#41—62no—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—21no—
330C22×Q8—21no—
331C22×Q8—31no—
332C22×Q8—31no—
333C22×Q8D1214yescomplemented
334C22×Q8C22×D312yes—
335C22×D4—32no—
336C22×D4—31no—
337C22×D4—31no—
338C22×D4—61no—
339C23×C4—21no—
340C23×C4—32no—
341C23×C4—61no—
342C23×C4C22×D311yescharacteristic
343#137—68no—
344#137—616no—
345#137—68no—
346C2×Q16—64no—
347C2×Q16—68no—
348C2×SD16—64no—
349C2×SD16—68no—
350C2×M16—21no—
351C2×M16—22no—
352C2×M16—22no—
353C2×M16—34no—
354C2×M16—32no—
355C2×M16—62no—
356C2×M16—61no—
357C2×M16C22×D312yes—
358C22×C8—21no—
359C22×C8—61no—
360C4⋊Q8—62no—
361C4⋊Q8—64no—
362C4⋊Q8—64no—
363#127—62no—
364#124—62no—
365#124—62no—
366#124—64no—
367#124—64no—
368#123—64no—
369#123—64no—
370#123—616no—
371C22⋊Q8—64no—
372C22⋊Q8—616no—
373C22⋊Q8—64no—
374#121—64no—
375#121—68no—
376#121—64no—
377#120—64no—
378#120—64no—
379C4×D4—68no—
380#117—21no—
381#117—21no—
382#117—34no—
383#117—62no—
384#117#6712yes—
385#117C22×D312yes—
386C2×C4⋊C4—64no—
387C2×C4⋊C4—61no—
388C2×C4⋊C4#6712yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—64no—
391C2×C22⋊C4—61no—
392C2×C42—21no—
393C2×C42—32no—
394#104—68no—
395#103—28no—
396#101—64no—
397#100—64no—
398C23×C3—42no—
399C23×C3#12012yes—
400C23×C3#12011yescharacteristic
401C23×C3#12011yescharacteristic
402C23×C3#12112yes—
403C23×C3C22×D411yescharacteristic
404C22×D3—22no—
405C22×D3—22no—
406C22×D3—22no—
407C22×D3—22no—
408C22×D3—44no—
409C22×D3—42no—
410C22×D3—82no—
411C3×Q8—24no—
412C3×Q8—24no—
413C3×Q8—42no—
414C3×Q8—42no—
415C3×Q8—44no—
416C3×Q8—44no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4—44no—
422C2×C3×C4—44no—
423C2×C3×C4—82no—
424C2×C3×C4#12012yes—
425C2×C3×C4#12012yes—
426C2×C3×C4#12012yes—
427C2×C3×C4#12011yescharacteristic
428C2×C3×C4#12011yescharacteristic
429C2×C3×C4#12112yes—
430C2×C3×C4#12112yes—
431C2×C3×C4#12112yes—
432C2×C3×C4#12112yes—
433C2×C3×C4#12112yes—
434C2×C3×C4#12712yes—
435C2×C3×C4#12712yes—
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C2×#4111yescharacteristic
442#67—42no—
443#67—42no—
444#67—44no—
445#67—44no—
446#67—44no—
447C2×Dic3—24no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—44no—
453C2×Dic3—42no—
454D12—22no—
455D12—22no—
456D12—41no—
457D12—41no—
458D12—41no—
459D12—41no—
460D12—42no—
461D12—88no—
462C4×D3—42no—
463C4×D3—42no—
464C4×D3—44no—
465C4×D3—44no—
466C4×D3—44no—
467Dic6—22no—
468Dic6—22no—
469Dic6—41no—
470Dic6—41no—
471Dic6—41no—
472Dic6—41no—
473Dic6—42no—
474C3×C8—42no—
475C3⋊C8—24no—
476C24—32no—
477C24—61no—
478C24D3×D411yescharacteristic
479#41—64no—
480#41—64no—
481#41—62no—
482#41—62no—
483#41—62no—
484#41—62no—
485#41—122no—
486#41—122no—
487#41—122no—
488#41—122no—
489#41—122no—
490#41—122no—
491#41—122no—
492#41—122no—
493#41—122no—
494#41—122no—
495#41—122no—
496#41—122no—
497C2×Q8—24no—
498C2×Q8—22no—
499C2×Q8—24no—
500C2×Q8—24no—
501C2×Q8—24no—
502C2×Q8—24no—
503C2×Q8—22no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—22no—
507C2×Q8—32no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—41no—
512C2×Q8—41no—
513C2×Q8—42no—
514C2×Q8—42no—
515C2×Q8—61no—
516C2×Q8—61no—
517C2×Q8—61no—
518C2×Q8—61no—
519C2×Q8—61no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8—62no—
523C2×Q8—62no—
524C2×Q8C2×D1214yes—
525C2×D4—32no—
526C2×D4—68no—
527C2×D4—64no—
528C2×D4—64no—
529C2×D4—62no—
530C2×D4—62no—
531C2×D4—62no—
532C2×D4—62no—
533C2×D4—62no—
534C2×D4—62no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—61no—
538C2×D4—61no—
539C2×D4—61no—
540C2×D4—61no—
541C2×D4—61no—
542C2×D4—122no—
543C2×D4—122no—
544C22×C4—24no—
545C22×C4—22no—
546C22×C4—22no—
547C22×C4—32no—
548C22×C4—42no—
549C22×C4—41no—
550C22×C4—41no—
551C22×C4—61no—
552C22×C4—62no—
553C22×C4—64no—
554C22×C4—64no—
555C22×C4—62no—
556C22×C4—62no—
557C22×C4—62no—
558C22×C4—62no—
559C22×C4—122no—
560C22×C4—122no—
561C22×C4C2×D1212yes—
562C22×C4#23512yes—
563C22×C4D3×D411yescharacteristic
564C22×C4D3×D411yescharacteristic
565C22×C4D3×D411yescharacteristic
566C22×C4D3×D412yes—
567C22×C4D3×D412yes—
568C22×C4C2×#6711yescharacteristic
569C22×C4C2×#6711yescharacteristic
570C22×C4C23×D311yescharacteristic
571Q16—616no—
572SD16—616no—
573M16—22no—
574M16—22no—
575M16—34no—
576M16—42no—
577M16—64no—
578M16—62no—
579C2×C8—22no—
580C2×C8—42no—
581C2×C8—64no—
582C2×C8—62no—
583C4⋊C4—22no—
584C4⋊C4—616no—
585C4⋊C4—68no—
586C4⋊C4—64no—
587C4⋊C4C2×#6714yes—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C22⋊C4—616no—
591C22⋊C4—68no—
592C22⋊C4—68no—
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#39212yes—
604C22×C3#39212yes—
605C22×C3C2×#12011yescharacteristic
606C22×C3C2×#12011yescharacteristic
607C22×C3C2×#12111yescharacteristic
608C2×D3—44no—
609C2×D3—42no—
610C2×D3—42no—
611C2×D3—44no—
612C2×D3—44no—
613C2×D3—84no—
614C2×D3—82no—
615C3×C4—44no—
616C3×C4—44no—
617C3×C4#39212yes—
618C3×C4#39212yes—
619C3×C4C2×#12011yescharacteristic
620C3×C4C2×#12111yescharacteristic
621C3×C4C2×#12111yescharacteristic
622C3×C4C2×#12711yescharacteristic
623Dic3—44no—
624Dic3—42no—
625Dic3—42no—
626C23—42no—
627C23—62no—
628C23—62no—
629C23—62no—
630C23—62no—
631C23—124no—
632C23—122no—
633C23—242no—
634C23#88212yes—
635C23#88412yes—
636C23#93711yescharacteristic
637C23#93711yescharacteristic
638C23C2×D3×D411yescharacteristic
639Q8—24no—
640Q8—24no—
641Q8—44no—
642Q8—44no—
643Q8—42no—
644Q8—42no—
645Q8—62no—
646Q8—62no—
647Q8—122no—
648Q8—121no—
649Q8—121no—
650Q8—121no—
651Q8—121no—
652D4—62no—
653D4—62no—
654D4—124no—
655D4—124no—
656D4—124no—
657D4—122no—
658D4—122no—
659D4—122no—
660D4—121no—
661D4—121no—
662D4—121no—
663D4—121no—
664D4—248no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—64no—
672C2×C4—62no—
673C2×C4—62no—
674C2×C4—62no—
675C2×C4—62no—
676C2×C4—82no—
677C2×C4—124no—
678C2×C4—122no—
679C2×C4—124no—
680C2×C4—124no—
681C2×C4—124no—
682C2×C4—122no—
683C2×C4—122no—
684C2×C4#88212yes—
685C2×C4#88212yes—
686C2×C4#88212yes—
687C2×C4#88412yes—
688C2×C4#88412yes—
689C2×C4#88412yes—
690C2×C4#93012yes—
691C2×C4#93012yes—
692C2×C4#93711yescharacteristic
693C2×C4#93711yescharacteristic
694C2×C4#94012yes—
695C2×C4#94012yes—
696C2×C4C22×D1211yescharacteristic
697C2×C4C2×#23511yescharacteristic
698C2×C4C2×D3×D411yescharacteristic
699C2×C4C2×D3×D411yescharacteristic
700C2×C4C2×D3×D411yescharacteristic
701C2×C4C22×#6711yescharacteristic
702C8—42no—
703C8—64no—
704C2×C3—21no—
705C2×C3—21no—
706C2×C3—22no—
707C2×C3—22no—
708C2×C3C2×#39211yescharacteristic
709C2×C3#302212yes—
710D3—82no—
711D3—82no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—42no—
716C22—42no—
717C22—124no—
718C22—124no—
719C22—124no—
720C22—122no—
721C22—122no—
722C22—242no—
723C22—244no—
724C22#496012yes—
725C22#496012yes—
726C22C2×#88211yescharacteristic
727C22C2×#88411yescharacteristic
728C22C2×#93711yescharacteristic
729C4—44no—
730C4—44no—
731C4—124no—
732C4—122no—
733C4—122no—
734C4#496012yes—
735C4#496012yes—
736C4C2×#88211yescharacteristic
737C4C2×#88411yescharacteristic
738C4C2×#93011yescharacteristic
739C4C2×#94011yescharacteristic
740C3#2340111yescharacteristic, complemented, Sylow
741C2—21no—
742C2—22no—
743C2—21no—
744C2—22no—
745C2—242no—
746C2—242no—
747C2C2×#496011yescharacteristic
748C2#8399312yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC22×C3C2×#120
prime radical#85735C2
hypercenterC2×#419D3
hypocenterC3#23401
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C2×#5816C2semidirect
2. C22×#942C2semidirect
4. C2×C3×#419C2semidirect
5. C2×#5139C2semidirect
6. C2×#5132C2semidirect
7. #71574C2semidirect
8. #70346C2semidirect
9. #70344C2semidirect
18. C2×#947C22semidirect
20. C2×#944C22semidirect
40. C3×#419C22semidirect
44. C2×#824C22semidirect
46. C2×#823C22semidirect
47. #4532C22semidirect
52. C2×#419D3semidirect
65. C22×C3×Q8D4semidirect
122. #926D4semidirect
128. C2×#208D4semidirect
179. #419C2×D3semidirect
333. C22×Q8D12semidirect
740. C3#23401semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C22×#9422
#857352
C2×C3×#4192
C2×#51392
C2×#51322
#715742
#703462
#703442
#234013