Finite Group Databasebeta

← #10822115

Subgroups — #10822115

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
1#87268C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C2×#5526C211yesmaximal, characteristic, complemented
4C2×C3×#421C211yesmaximal, characteristic, complemented
5C2×#5120C211yesmaximal, characteristic, complemented
6C2×#5132C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#70389C214yesmaximal, complemented
9#70386C214yesmaximal, complemented
10#23401—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5833—22no—
17#5832—22no—
18#5830C2212yes—
19#5828—22no—
20#5825C2212yes—
21C2×#951—22no—
22C2×#951—24no—
23C2×#951—24no—
24C2×#951C2214yes—
25#5816—21no—
26C22×#216—21no—
27#5805—21no—
28#5804—22no—
29C22×#214—21no—
30#5791—22no—
31C2×#926C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5526—24no—
34#5526—24no—
35C2×#897—22no—
36C2×#886—22no—
37C2×#875C2212yescomplemented
38C2×#869C2212yescomplemented
39C3×#421—24no—
40C2×C3×#104C2214yes—
41C2×#824C2212yescomplemented
42#5132—24no—
43C2×#823C2212yescomplemented
44#5120—22no—
45#5120C2214yescomplemented
46C2×#816—22no—
47#4523C2214yescomplemented
48C22×#137—31no—
49#3449—31no—
50C2×#536—31no—
51C2×#421D311yescharacteristic, complemented
52C2×#419—31no—
53C2×#413—31no—
54#2005—34no—
55#2004—34no—
56#1733—31no—
57C2×C3×#41—21no—
58C2×C3×#41—21no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41D412yes—
64C2×C3×#41C2312yes—
65C22×C3×Q8D411yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4D411yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67D412yes—
71C23×Dic3—21no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—21no—
76C2×#235—21no—
77C2×#235—21no—
78C2×#235—21no—
79C2×#235—22no—
80C22×D12C2311yescharacteristic
81C22×C4×D3D412yes—
82C22×Dic6—21no—
83C22×Dic6—21no—
84C22×Dic6—21no—
85C22×Dic6—22no—
86C22×Dic6—21no—
87C22×Dic6—22no—
88C22×Dic6—21no—
89C22×Dic6—22no—
90C22×Dic6D414yescomplemented
91C22×Dic6C2311yescharacteristic
92C22×Dic6C2312yes—
93C2×C3×M16—21no—
94C2×C3×M16—22no—
95C2×C3×M16—22no—
96C2×C3×M16C2312yes—
97C22×C3×C8—21no—
98C3×#117—22no—
99C3×#117D414yescomplemented
100C2×C3×C4⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C42C2312yes—
103#953—24no—
104C2×#217—21no—
105#951—216no—
106#951—28no—
107#951—28no—
108#947—22no—
109#946—24no—
110#944—22no—
111C2×#216—28no—
112C2×#216—24no—
113#940—22no—
114#939—28no—
115#938—24no—
116#937—24no—
117#936—22no—
118#935—24no—
119C2×#214—28no—
120C2×#214—24no—
121#930—24no—
122#929—24no—
123C4×#67—28no—
124C2×(C2×D3)⋊C4—24no—
125#926—21no—
126#926—21no—
127#926D412yes—
128#926C2312yes—
129C2×C4⋊Dic3—24no—
130C2×C4⋊Dic3D412yes—
131#924—24no—
132C2×C4×Dic3—21no—
133C2×#208—21no—
134C2×#208—22no—
135C2×#208D414yescomplemented
136C2×#208C2312yes—
137C22×C3⋊C8—21no—
138#897—216no—
139#886—216no—
140#875—24no—
141#875—24no—
142#869—24no—
143#869—24no—
144C3×#104—28no—
145#824—24no—
146#823—24no—
147#816—28no—
148C22×#41—31no—
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×M16C2×D311yescharacteristic
159#537—62no—
160#536—32no—
161#536—64no—
162#536—61no—
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—24no—
177C2×#104C2×D314yes—
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×D411yescharacteristic
187C23×D3C2×D412yes—
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—
198C2×C3×Q8—21no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8C2×D412yes—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4C2×D412yes—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D412yes—
224C22×C3×C4#4112yes—
225C22×C3×C4C2411yescharacteristic
226C2×#67—24no—
227C2×#67—22no—
228C2×#67—22no—
229C2×#67—24no—
230C2×#67—28no—
231C22×Dic3—24no—
232C22×Dic3—22no—
233C22×Dic3—22no—
234C22×Dic3—41no—
235C22×Dic3—41no—
236C22×Dic3—42no—
237C22×Dic3C2×D412yes—
238#235—42no—
239#235—42no—
240#235—42no—
241#235—42no—
242#235—42no—
243#235—42no—
244#235—42no—
245#235—42no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×C4×D3—24no—
253C2×C4×D3—22no—
254C2×C4×D3—22no—
255C2×C4×D3—24no—
256C2×Dic6—24no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C2×Dic6—22no—
261C2×Dic6—22no—
262C2×Dic6—22no—
263C2×Dic6—22no—
264C2×Dic6—24no—
265C2×Dic6—21no—
266C2×Dic6—21no—
267C2×Dic6—21no—
268C2×Dic6—22no—
269C2×Dic6—21no—
270C2×Dic6—21no—
271C2×Dic6—21no—
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×M16—22no—
281C3×M16—22no—
282C3×M16—42no—
283C2×C3×C8—22no—
284C2×C3×C8—42no—
285C3×C4⋊C4—24no—
286C3×C22⋊C4—24no—
287C3×C42—22no—
288C3×C42C2×D414yes—
289#217—28no—
290#217—22no—
291#217—22no—
292#216—216no—
293#214—216no—
294(C2×D3)⋊C4—216no—
295(C2×D3)⋊C4—28no—
296C4⋊Dic3—216no—
297C4⋊Dic3—22no—
298C4⋊Dic3C2×D414yes—
299Dic3⋊C4—28no—
300C4×Dic3—22no—
301C4×Dic3—22no—
302#208—24no—
303#208—22no—
304#208C2×D414yes—
305C2×C3⋊C8—24no—
306C2×C3⋊C8—22no—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—61no—
314C2×#41—61no—
315C2×#41—62no—
316C2×#41—61no—
317C2×#41—62no—
318C2×#41—61no—
319C2×#41—62no—
320C2×#41—62no—
321C2×#41D1212yes—
322C2×#41C22×D312yes—
323C22×Q8—31no—
324C22×Q8—34no—
325C22×Q8—32no—
326C22×Q8—61no—
327C22×Q8—62no—
328C22×Q8—61no—
329C22×Q8—61no—
330C22×Q8—62no—
331C22×Q8—62no—
332C22×Q8—61no—
333C22×Q8—61no—
334C22×Q8D1211yescharacteristic
335C22×D4—21no—
336C22×D4—32no—
337C22×D4—31no—
338C22×D4D1211yescharacteristic
339C23×C4—21no—
340C23×C4—32no—
341C23×C4—61no—
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—21no—
351C2×M16—22no—
352C2×M16—22no—
353C2×M16—34no—
354C2×M16—32no—
355C2×M16—61no—
356C2×M16—62no—
357C2×M16C22×D312yes—
358C22×C8—21no—
359C22×C8—61no—
360C4⋊Q8—64no—
361C4⋊Q8—62no—
362C4⋊Q8—64no—
363#127—62no—
364#124—64no—
365#124—62no—
366#124—62no—
367#124—64no—
368#123—616no—
369#123—64no—
370#123—64no—
371C22⋊Q8—616no—
372C22⋊Q8—64no—
373C22⋊Q8—64no—
374#121—64no—
375#121—64no—
376#121—68no—
377#120—64no—
378#120—64no—
379C4×D4—68no—
380#117—22no—
381#117—32no—
382#117—32no—
383#117—61no—
384#117—61no—
385#117#6714yescomplemented
386C2×C4⋊C4—21no—
387C2×C4⋊C4—32no—
388C2×C4⋊C4—64no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—64no—
391C2×C22⋊C4—61no—
392C2×C42—61no—
393C2×C42C22×D312yes—
394#104—28no—
395#103—68no—
396#101—64no—
397#100—64no—
398C23×C3—22no—
399C23×C3—42no—
400C23×C3—42no—
401C23×C3—82no—
402C23×C3#12011yescharacteristic
403C23×C3#12011yescharacteristic
404C23×C3#12012yes—
405C23×C3#12112yes—
406C23×C3C22×D411yescharacteristic
407C22×D3—22no—
408C22×D3—22no—
409C22×D3—22no—
410C22×D3—44no—
411C3×Q8—22no—
412C3×Q8—22no—
413C3×Q8—42no—
414C3×D4—22no—
415C3×D4—22no—
416C3×D4—42no—
417C3×D4—42no—
418C3×D4—44no—
419C3×D4—42no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—44no—
423C2×C3×C4—42no—
424C2×C3×C4—42no—
425C2×C3×C4—42no—
426C2×C3×C4—44no—
427C2×C3×C4#12012yes—
428C2×C3×C4#12012yes—
429C2×C3×C4#12011yescharacteristic
430C2×C3×C4#12011yescharacteristic
431C2×C3×C4#12012yes—
432C2×C3×C4#12112yes—
433C2×C3×C4#12112yes—
434C2×C3×C4#12112yes—
435C2×C3×C4#12112yes—
436C2×C3×C4#12112yes—
437C2×C3×C4#12712yes—
438C2×C3×C4#12712yes—
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C22×D411yescharacteristic
444C2×C3×C4C2×#4111yescharacteristic
445#67—44no—
446#67—44no—
447#67—88no—
448C2×Dic3—24no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—24no—
452C2×Dic3—22no—
453C2×Dic3—42no—
454C2×Dic3—42no—
455C2×Dic3—44no—
456C2×Dic3—44no—
457C2×Dic3—82no—
458D12—41no—
459D12—41no—
460D12—41no—
461D12—41no—
462C4×D3—44no—
463C4×D3—44no—
464Dic6—24no—
465Dic6—24no—
466Dic6—41no—
467Dic6—42no—
468Dic6—41no—
469Dic6—41no—
470Dic6—42no—
471Dic6—41no—
472Dic6—44no—
473Dic6—44no—
474C3×C8—42no—
475C3⋊C8—24no—
476C24—21no—
477C24—32no—
478C24D3×D411yescharacteristic
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—122no—
490#41—122no—
491#41—122no—
492#41—122no—
493#41—122no—
494#41—122no—
495#41—122no—
496#41—122no—
497C2×Q8—21no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—34no—
501C2×Q8—64no—
502C2×Q8—61no—
503C2×Q8—61no—
504C2×Q8—61no—
505C2×Q8—61no—
506C2×Q8—61no—
507C2×Q8—62no—
508C2×Q8—61no—
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—121no—
518C2×Q8—121no—
519C2×Q8—122no—
520C2×Q8—122no—
521C2×Q8—122no—
522C2×Q8—122no—
523C2×Q8—122no—
524C2×Q8C2×D1212yes—
525C2×D4—22no—
526C2×D4—22no—
527C2×D4—22no—
528C2×D4—21no—
529C2×D4—22no—
530C2×D4—42no—
531C2×D4—42no—
532C2×D4—62no—
533C2×D4—62no—
534C2×D4—64no—
535C2×D4—64no—
536C2×D4—68no—
537C2×D4—61no—
538C2×D4—61no—
539C2×D4—61no—
540C2×D4—61no—
541C2×D4—61no—
542C2×D4—61no—
543C2×D4C2×D1212yes—
544C22×C4—21no—
545C22×C4—22no—
546C22×C4—22no—
547C22×C4—22no—
548C22×C4—32no—
549C22×C4—42no—
550C22×C4—42no—
551C22×C4—64no—
552C22×C4—62no—
553C22×C4—62no—
554C22×C4—62no—
555C22×C4—62no—
556C22×C4—64no—
557C22×C4—64no—
558C22×C4—121no—
559C22×C4—121no—
560C22×C4—122no—
561C22×C4C2×D1211yescharacteristic
562C22×C4C2×D1211yescharacteristic
563C22×C4#23512yes—
564C22×C4D3×D411yescharacteristic
565C22×C4D3×D412yes—
566C22×C4D3×D412yes—
567C22×C4D3×D411yescharacteristic
568C22×C4D3×D411yescharacteristic
569C22×C4C2×#6712yes—
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—24no—
584C4⋊C4—34no—
585C4⋊C4—616no—
586C4⋊C4—68no—
587C4⋊C4—62no—
588C22⋊C4—24no—
589C22⋊C4—616no—
590C22⋊C4—68no—
591C22⋊C4—68no—
592C22⋊C4—62no—
593C22⋊C4—62no—
594C42—22no—
595C42—62no—
596C42—62no—
597C42C2×#6714yes—
598C22×C3—22no—
599C22×C3—22no—
600C22×C3—22no—
601C22×C3—42no—
602C22×C3—42no—
603C22×C3—42no—
604C22×C3—44no—
605C22×C3—84no—
606C22×C3#39212yes—
607C22×C3#39212yes—
608C22×C3C2×#12011yescharacteristic
609C22×C3C2×#12011yescharacteristic
610C22×C3C2×#12111yescharacteristic
611C2×D3—44no—
612C2×D3—42no—
613C2×D3—44no—
614C2×D3—82no—
615C3×C4—44no—
616C3×C4—42no—
617C3×C4#39212yes—
618C3×C4#39212yes—
619C3×C4C2×#12011yescharacteristic
620C3×C4C2×#12111yescharacteristic
621C3×C4C2×#12111yescharacteristic
622C3×C4C2×#12711yescharacteristic
623Dic3—44no—
624Dic3—42no—
625Dic3—44no—
626C23—22no—
627C23—42no—
628C23—42no—
629C23—62no—
630C23—62no—
631C23—62no—
632C23—82no—
633C23—124no—
634C23#88211yescharacteristic
635C23#88211yescharacteristic
636C23#88412yes—
637C23#93712yes—
638C23C2×D3×D411yescharacteristic
639Q8—22no—
640Q8—22no—
641Q8—42no—
642Q8—64no—
643Q8—64no—
644Q8—124no—
645Q8—124no—
646Q8—121no—
647Q8—121no—
648Q8—122no—
649Q8—121no—
650Q8—122no—
651Q8—121no—
652D4—22no—
653D4—22no—
654D4—44no—
655D4—42no—
656D4—42no—
657D4—42no—
658D4—124no—
659D4—124no—
660D4—121no—
661D4—121no—
662D4—121no—
663D4—121no—
664D4—248no—
665C2×C4—22no—
666C2×C4—24no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—64no—
673C2×C4—62no—
674C2×C4—62no—
675C2×C4—64no—
676C2×C4—62no—
677C2×C4—124no—
678C2×C4—124no—
679C2×C4—122no—
680C2×C4—122no—
681C2×C4—124no—
682C2×C4—124no—
683C2×C4—242no—
684C2×C4#88211yescharacteristic
685C2×C4#88211yescharacteristic
686C2×C4#88212yes—
687C2×C4#88212yes—
688C2×C4#88412yes—
689C2×C4#88412yes—
690C2×C4#88412yes—
691C2×C4#93012yes—
692C2×C4#93012yes—
693C2×C4#93712yes—
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×C3—82no—
709C2×C3C2×#39211yescharacteristic
710C2×C3#302212yes—
711D3—82no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—42no—
716C22—42no—
717C22—44no—
718C22—42no—
719C22—84no—
720C22—124no—
721C22—124no—
722C22—122no—
723C22—242no—
724C22#496012yes—
725C22#496012yes—
726C22C2×#88211yescharacteristic
727C22C2×#88411yescharacteristic
728C22C2×#93711yescharacteristic
729C4—44no—
730C4—42no—
731C4—124no—
732C4—122no—
733C4—124no—
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—82no—
746C2—242no—
747C2#7899512yes—
748C2C2×#496011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#421C2
socleC22×C3C2×#120
prime radical#87268C2
hypercenterC2×#421D3
hypocenterC3#23401
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. C2×#5526C2semidirect
4. C2×C3×#421C2semidirect
5. C2×#5120C2semidirect
6. C2×#5132C2semidirect
7. #71578C2semidirect
8. #70389C2semidirect
9. #70386C2semidirect
37. C2×#875C22semidirect
38. C2×#869C22semidirect
41. C2×#824C22semidirect
43. C2×#823C22semidirect
45. #5120C22semidirect
47. #4523C22semidirect
51. C2×#421D3semidirect
90. C22×Dic6D4semidirect
99. C3×#117D4semidirect
135. C2×#208D4semidirect
385. #117#67semidirect
740. C3#23401semidirect

Maximal subgroups

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

subgroupindex
#872682
C22×#9512
C2×#55262
C2×C3×#4212
C2×#51202
C2×#51322
#715782
#703892
#703862
#234013