Finite Group Databasebeta

← #10822444

Subgroups — #10822444

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 794 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3C2×#5525C211yesmaximal, characteristic, complemented
4C2×C3×#410C211yesmaximal, characteristic, complemented
5#82269C211yesmaximal, characteristic, complemented
6C2×#5006C211yesmaximal, characteristic, complemented
7#71593C211yesmaximal, characteristic, complemented
8#70037C218yesmaximal, complemented
9#23445—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5833—22no—
16#5832—22no—
17#5830—24no—
18#5830—21no—
19#5830—24no—
20#5828—22no—
21#5825—21no—
22#5816—21no—
23C2×#947—21no—
24#5805—21no—
25#5804C2214yes—
26C2×#940—21no—
27C2×#937C2214yes—
28#5791—22no—
29C2×#930—22no—
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32#5525—24no—
33#5525—24no—
34C2×#895—22no—
35C2×#882C2214yes—
36C2×#875—21no—
37C2×#874—21no—
38C3×#410—24no—
39C2×C3×#99C2214yes—
40#5006—24no—
41#4988—22no—
42#4955C2214yes—
43C2×#806C2214yes—
44#4939—22no—
45#4527C2214yes—
46#3449—31no—
47C2×#535—31no—
48C2×#535—31no—
49C2×#410—31no—
50C2×#410D311yescharacteristic, complemented
51#2309—31no—
52#2021—38no—
53#1740—31no—
54C24×D3C2312yes—
55C2×C3×#41—22no—
56C2×C3×#41—21no—
57C2×C3×#41—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—21no—
61C2×C3×#41—21no—
62C2×C3×#41—22no—
63C22×C3×Q8D411yescharacteristic, complemented
64C22×C3×D4D411yescharacteristic, complemented
65C22×C3×D4C2312yes—
66C23×C3×C4D412yescomplemented
67C23×C3×C4C2311yescharacteristic
68C22×#67—24no—
69C22×#67C2312yes—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—21no—
74C2×#235—21no—
75C2×#235—21no—
76C2×#235—21no—
77C2×#235—22no—
78C22×D12—21no—
79C22×D12—21no—
80C22×D12—21no—
81C22×D12—21no—
82C22×D12—22no—
83C22×D12—21no—
84C22×D12—21no—
85C22×D12—22no—
86C22×D12—22no—
87C22×D12—24no—
88C22×D12D414yescomplemented
89C22×D12D411yescharacteristic, complemented
90C22×C4×D3D412yescomplemented
91C22×Dic6D411yescharacteristic, complemented
92C3×#117—22no—
93C3×#117—22no—
94C3×#117—22no—
95C3×#117D414yescomplemented
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C22⋊C4C2312yes—
99C2×C3×C22⋊C4C2312yes—
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102#953—24no—
103C2×#217C2312yes—
104C2×#217C2312yes—
105#947—28no—
106#947—22no—
107#946—24no—
108#944—22no—
109#940—28no—
110#940—22no—
111#939—28no—
112#938—24no—
113#937—28no—
114#937—24no—
115#937—28no—
116#936—22no—
117#935—24no—
118#930—216no—
119#930—24no—
120#929—24no—
121C4×#67—28no—
122C2×(C2×D3)⋊C4—24no—
123C2×(C2×D3)⋊C4—24no—
124C2×(C2×D3)⋊C4—28no—
125#926—21no—
126#926—21no—
127#926—21no—
128#926—21no—
129#926—22no—
130#926D414yescomplemented
131C2×C4⋊Dic3—21no—
132C2×C4⋊Dic3D412yescomplemented
133#924—24no—
134C2×C4×Dic3—21no—
135C2×C4×Dic3D412yescomplemented
136#895—216no—
137#882—28no—
138#882—28no—
139#875—28no—
140#874—28no—
141C3×#99—28no—
142#806—28no—
143C22×#41—31no—
144C22×#41C2×D311yescharacteristic
145C23×D4—31no—
146#537—62no—
147#536—61no—
148#536—61no—
149#535—64no—
150#535—64no—
151#535—64no—
152#535—61no—
153#535—61no—
154#535—64no—
155#534—34no—
156#532—62no—
157C2×#127—61no—
158C2×#127—61no—
159C2×#124—61no—
160C2×#124—61no—
161#525—62no—
162#525—62no—
163C2×#121—62no—
164C2×#121—62no—
165C2×#120—34no—
166C2×#120—34no—
167C2×#117—31no—
168C2×#117—31no—
169C2×#117C2×D311yescharacteristic
170C2×#117C2×D311yescharacteristic
171#410—24no—
172#410—64no—
173C2×#99—34no—
174C2×#99C2×D314yes—
175#390—62no—
176#390—62no—
177#379—34no—
178#342—34no—
179C24×C3C2×D412yes—
180C24×C3C2411yescharacteristic
181C23×D3—22no—
182C23×D3—22no—
183C23×D3—22no—
184C23×D3—22no—
185C23×D3—22no—
186C23×D3—22no—
187C23×D3—84no—
188C23×D3C2×D412yes—
189C23×D3C2×D414yes—
190C3×#41—42no—
191C3×#41—42no—
192C3×#41—42no—
193C3×#41—42no—
194C3×#41—42no—
195C3×#41—42no—
196C3×#41—42no—
197C3×#41—42no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4#4112yes—
229C2×#67—22no—
230C2×#67—22no—
231C2×#67—22no—
232C2×#67—22no—
233C2×#67—22no—
234C2×#67—22no—
235C2×#67—28no—
236C2×#67—216no—
237C2×#67—88no—
238C22×Dic3—22no—
239C22×Dic3C2×D412yes—
240C22×Dic3C2×D414yes—
241#235—42no—
242#235—42no—
243#235—42no—
244#235—42no—
245#235—42no—
246#235—42no—
247#235—42no—
248#235—42no—
249C2×D12—22no—
250C2×D12—24no—
251C2×D12—24no—
252C2×D12—24no—
253C2×D12—21no—
254C2×D12—24no—
255C2×D12—21no—
256C2×D12—24no—
257C2×D12—24no—
258C2×D12—22no—
259C2×D12—216no—
260C2×D12—41no—
261C2×D12—41no—
262C2×D12—41no—
263C2×D12—41no—
264C2×D12—42no—
265C2×D12—42no—
266C2×D12—42no—
267C2×D12—42no—
268C2×D12—42no—
269C2×D12—42no—
270C2×D12—88no—
271C2×C4×D3—24no—
272C2×C4×D3—22no—
273C2×C4×D3—22no—
274C2×C4×D3—24no—
275C2×Dic6—22no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—22no—
279C3×C4⋊C4—24no—
280C3×C4⋊C4—42no—
281C3×C22⋊C4—22no—
282C3×C22⋊C4—22no—
283C3×C22⋊C4—22no—
284C3×C22⋊C4C2×D414yes—
285C3×C42—24no—
286C3×C42—42no—
287#217—28no—
288#217—22no—
289#217—22no—
290#217—22no—
291#217C2×D414yes—
292(C2×D3)⋊C4—216no—
293(C2×D3)⋊C4—216no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—88no—
296C4⋊Dic3—22no—
297C4⋊Dic3—24no—
298C4⋊Dic3—22no—
299Dic3⋊C4—28no—
300C4×Dic3—24no—
301C4×Dic3—22no—
302C4×Dic3—22no—
303C25—32no—
304C2×#41—22no—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—22no—
312C2×#41—61no—
313C2×#41—61no—
314C2×#41—62no—
315C2×#41—61no—
316C2×#41—62no—
317C2×#41—61no—
318C2×#41—62no—
319C2×#41—62no—
320C22×Q8—31no—
321C22×Q8D1211yescharacteristic, complemented
322C22×D4—32no—
323C22×D4—31no—
324C22×D4—34no—
325C22×D4—64no—
326C22×D4—64no—
327C22×D4—61no—
328C22×D4—61no—
329C22×D4—62no—
330C22×D4—61no—
331C22×D4—61no—
332C22×D4—62no—
333C22×D4—61no—
334C22×D4—61no—
335C22×D4—62no—
336C22×D4D1211yescharacteristic, complemented
337C22×D4C22×D312yes—
338C23×C4—32no—
339C23×C4D1212yescomplemented
340C23×C4C22×D311yescharacteristic
341C4⋊Q8—62no—
342#127—68no—
343#127—68no—
344#127—62no—
345#124—68no—
346#124—62no—
347#124—68no—
348#124—62no—
349#123—64no—
350#123—64no—
351C22⋊Q8—64no—
352C22⋊Q8—64no—
353#121—616no—
354#121—616no—
355#121—64no—
356#121—68no—
357#121—64no—
358#120—68no—
359#120—68no—
360#120—68no—
361#120—64no—
362#120—64no—
363#120—68no—
364C4×D4—68no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117—34no—
369#117—61no—
370#117—61no—
371#117—61no—
372#117—62no—
373#117—61no—
374#117#6714yescomplemented
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—61no—
379C2×C22⋊C4—32no—
380C2×C22⋊C4—32no—
381C2×C22⋊C4—64no—
382C2×C22⋊C4—64no—
383C2×C22⋊C4—68no—
384C2×C22⋊C4C22×D312yes—
385C2×C22⋊C4C22×D312yes—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—32no—
389C2×C42—61no—
390#99—28no—
391#99—68no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—42no—
396C23×C3—44no—
397C23×C3C22×D411yescharacteristic
398C23×C3C22×D411yescharacteristic
399C23×C3C22×D411yescharacteristic
400C23×C3C22×D411yescharacteristic
401C23×C3C22×D411yescharacteristic
402C23×C3C22×D411yescharacteristic
403C23×C3C2×#4111yescharacteristic
404C22×D3—22no—
405C22×D3—24no—
406C22×D3—24no—
407C22×D3—22no—
408C22×D3—24no—
409C22×D3—22no—
410C22×D3—44no—
411C22×D3—48no—
412C22×D3—48no—
413C22×D3—48no—
414C22×D3—42no—
415C22×D3—42no—
416C22×D3—42no—
417C22×D3—42no—
418C22×D3—84no—
419C22×D3—84no—
420C22×D3—84no—
421C22×D3—84no—
422C3×Q8—42no—
423C3×Q8—42no—
424C3×D4—42no—
425C3×D4—42no—
426C3×D4—42no—
427C3×D4—42no—
428C3×D4—42no—
429C3×D4—42no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—24no—
433C2×C3×C4—22no—
434C2×C3×C4—44no—
435C2×C3×C4—44no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4—42no—
439C2×C3×C4#12011yescharacteristic
440C2×C3×C4#12011yescharacteristic
441C2×C3×C4#12012yes—
442C2×C3×C4#12011yescharacteristic
443C2×C3×C4#12011yescharacteristic
444C2×C3×C4#12012yes—
445C2×C3×C4#12012yes—
446C2×C3×C4#12012yes—
447C2×C3×C4#12112yes—
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—
455#67—42no—
456#67—42no—
457#67—42no—
458#67—42no—
459#67—88no—
460#67—816no—
461C2×Dic3—24no—
462C2×Dic3—24no—
463C2×Dic3—22no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—44no—
467C2×Dic3—44no—
468D12—44no—
469D12—44no—
470D12—44no—
471D12—44no—
472D12—42no—
473D12—42no—
474D12—816no—
475C4×D3—44no—
476C4×D3—44no—
477Dic6—42no—
478Dic6—42no—
479C24—32no—
480C24—34no—
481C24—62no—
482C24—62no—
483C24—62no—
484C24—62no—
485C24—62no—
486C24—62no—
487C24—244no—
488C24C2×D1212yes—
489C24C23×D311yescharacteristic
490#41—42no—
491#41—42no—
492#41—42no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—42no—
497#41—42no—
498#41—122no—
499#41—122no—
500#41—122no—
501#41—122no—
502#41—122no—
503#41—122no—
504#41—122no—
505#41—122no—
506C2×Q8—21no—
507C2×Q8—22no—
508C2×Q8—21no—
509C2×Q8—22no—
510C2×Q8—61no—
511C2×Q8—61no—
512C2×Q8—62no—
513C2×Q8—62no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—616no—
525C2×D4—62no—
526C2×D4—62no—
527C2×D4—62no—
528C2×D4—62no—
529C2×D4—62no—
530C2×D4—62no—
531C2×D4—68no—
532C2×D4—616no—
533C2×D4—64no—
534C2×D4—61no—
535C2×D4—64no—
536C2×D4—61no—
537C2×D4—64no—
538C2×D4—64no—
539C2×D4—62no—
540C2×D4—64no—
541C2×D4—64no—
542C2×D4—62no—
543C2×D4—121no—
544C2×D4—121no—
545C2×D4—121no—
546C2×D4—121no—
547C2×D4—122no—
548C2×D4—122no—
549C2×D4—122no—
550C2×D4—122no—
551C2×D4—122no—
552C2×D4—122no—
553C2×D4—248no—
554C2×D4—248no—
555C22×C4—22no—
556C22×C4—22no—
557C22×C4—24no—
558C22×C4—22no—
559C22×C4—24no—
560C22×C4—22no—
561C22×C4—32no—
562C22×C4—34no—
563C22×C4—62no—
564C22×C4—62no—
565C22×C4—62no—
566C22×C4—64no—
567C22×C4—64no—
568C22×C4C2×D1212yes—
569C22×C4C2×D1211yescharacteristic
570C22×C4C2×D1211yescharacteristic
571C22×C4#23512yes—
572C22×C4D3×D411yescharacteristic
573C22×C4D3×D411yescharacteristic
574C22×C4D3×D412yes—
575C22×C4D3×D411yescharacteristic
576C22×C4D3×D411yescharacteristic
577C22×C4D3×D412yes—
578C22×C4C2×#6712yes—
579C4⋊C4—24no—
580C4⋊C4—42no—
581C4⋊C4—68no—
582C4⋊C4—62no—
583C4⋊C4—62no—
584C4⋊C4—64no—
585C22⋊C4—22no—
586C22⋊C4—22no—
587C22⋊C4—22no—
588C22⋊C4—34no—
589C22⋊C4—616no—
590C22⋊C4—616no—
591C22⋊C4—68no—
592C22⋊C4—68no—
593C22⋊C4—62no—
594C22⋊C4—62no—
595C22⋊C4—62no—
596C22⋊C4—248no—
597C22⋊C4C2×#6714yes—
598C42—24no—
599C42—42no—
600C42—62no—
601C42—62no—
602C42—64no—
603C22×C3—22no—
604C22×C3—22no—
605C22×C3—22no—
606C22×C3—41no—
607C22×C3—41no—
608C22×C3—41no—
609C22×C3—41no—
610C22×C3—44no—
611C22×C3—42no—
612C22×C3—44no—
613C22×C3—82no—
614C22×C3C2×#12011yescharacteristic
615C22×C3C2×#12011yescharacteristic
616C22×C3C2×#12011yescharacteristic
617C22×C3C2×#12111yescharacteristic
618C22×C3C2×#12111yescharacteristic
619C22×C3C2×#12111yescharacteristic
620C22×C3C2×#12711yescharacteristic
621C2×D3—44no—
622C2×D3—48no—
623C2×D3—48no—
624C2×D3—48no—
625C2×D3—42no—
626C2×D3—44no—
627C2×D3—44no—
628C2×D3—84no—
629C2×D3—84no—
630C2×D3—84no—
631C2×D3—84no—
632C2×D3—82no—
633C3×C4—44no—
634C3×C4—42no—
635C3×C4—82no—
636C3×C4#39212yes—
637C3×C4#39212yes—
638C3×C4#39212yes—
639C3×C4#39212yes—
640Dic3—44no—
641Dic3—42no—
642Dic3—44no—
643C23—22no—
644C23—22no—
645C23—22no—
646C23—42no—
647C23—44no—
648C23—64no—
649C23—62no—
650C23—64no—
651C23—64no—
652C23—62no—
653C23—62no—
654C23—124no—
655C23—128no—
656C23—128no—
657C23—128no—
658C23—122no—
659C23—122no—
660C23—122no—
661C23—122no—
662C23—244no—
663C23—244no—
664C23—244no—
665C23—244no—
666C23C22×D1211yescharacteristic
667C23C2×#23511yescharacteristic
668C23C2×D3×D411yescharacteristic
669C23C2×D3×D411yescharacteristic
670C23C2×D3×D411yescharacteristic
671C23C2×D3×D411yescharacteristic
672C23C22×#6711yescharacteristic
673Q8—42no—
674Q8—42no—
675Q8—122no—
676Q8—122no—
677D4—42no—
678D4—42no—
679D4—42no—
680D4—42no—
681D4—42no—
682D4—42no—
683D4—122no—
684D4—122no—
685D4—122no—
686D4—122no—
687D4—124no—
688D4—124no—
689D4—124no—
690D4—124no—
691D4—122no—
692D4—122no—
693D4—2416no—
694D4—2416no—
695D4—248no—
696C2×C4—22no—
697C2×C4—22no—
698C2×C4—22no—
699C2×C4—24no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—42no—
705C2×C4—64no—
706C2×C4—64no—
707C2×C4—62no—
708C2×C4—62no—
709C2×C4—62no—
710C2×C4—124no—
711C2×C4—124no—
712C2×C4—124no—
713C2×C4—124no—
714C2×C4#88212yes—
715C2×C4#88212yes—
716C2×C4#88211yescharacteristic
717C2×C4#88211yescharacteristic
718C2×C4#88211yescharacteristic
719C2×C4#88211yescharacteristic
720C2×C4#88412yes—
721C2×C4#88412yes—
722C2×C4#88412yes—
723C2×C4#88412yes—
724C2×C4#93012yes—
725C2×C4#93012yes—
726C2×C4#93712yes—
727C2×C4#93712yes—
728C2×C4#94012yes—
729C2×C4#94012yes—
730C2×C3—21no—
731C2×C3—21no—
732C2×C3—21no—
733C2×C3—21no—
734C2×C3—21no—
735C2×C3—21no—
736C2×C3—82no—
737C2×C3C2×#39211yescharacteristic
738C2×C3#302712yes—
739D3—84no—
740D3—82no—
741C22—22no—
742C22—22no—
743C22—22no—
744C22—41no—
745C22—41no—
746C22—41no—
747C22—41no—
748C22—44no—
749C22—44no—
750C22—42no—
751C22—82no—
752C22—124no—
753C22—128no—
754C22—128no—
755C22—128no—
756C22—124no—
757C22—122no—
758C22—124no—
759C22—242no—
760C22—244no—
761C22—244no—
762C22—244no—
763C22—244no—
764C22C2×#88211yescharacteristic
765C22C2×#88211yescharacteristic
766C22C2×#88411yescharacteristic
767C22C2×#88411yescharacteristic
768C22C2×#93011yescharacteristic
769C22C2×#93711yescharacteristic
770C22C2×#94011yescharacteristic
771C4—44no—
772C4—42no—
773C4—82no—
774C4—124no—
775C4—124no—
776C4—122no—
777C4#496012yes—
778C4#496012yes—
779C4#496012yes—
780C4#496012yes—
781C3#2344511yescharacteristic, complemented, Sylow
782C2—21no—
783C2—21no—
784C2—21no—
785C2—21no—
786C2—21no—
787C2—21no—
788C2—82no—
789C2—244no—
790C2—242no—
791C2#7869112yes—
792C2C2×#496011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#410C2
socleC22×C3C2×#120
hypercenterC2×#410D3
hypocenterC3#23445
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #87268C2semidirect
2. C2×#5830C2semidirect
3. C2×#5525C2semidirect
4. C2×C3×#410C2semidirect
5. #82269C2semidirect
6. C2×#5006C2semidirect
7. #71593C2semidirect
8. #70037C2semidirect
50. C2×#410D3semidirect
63. C22×C3×Q8D4semidirect
64. C22×C3×D4D4semidirect
66. C23×C3×C4D4semidirect
88. C22×D12D4semidirect
89. C22×D12D4semidirect
90. C22×C4×D3D4semidirect
91. C22×Dic6D4semidirect
95. C3×#117D4semidirect
130. #926D4semidirect
132. C2×C4⋊Dic3D4semidirect
135. C2×C4×Dic3D4semidirect
321. C22×Q8D12semidirect
336. C22×D4D12semidirect
339. C23×C4D12semidirect
374. #117#67semidirect
781. C3#23445semidirect

Maximal subgroups

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

subgroupindex
#872682
C2×#58302
C2×#55252
C2×C3×#4102
#822692
C2×#50062
#715932
#700372
#234453