Finite Group Databasebeta

← #10822336

Subgroups — #10822336

Order 768.

2066 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic, complemented
3C22×#909C211yesmaximal, characteristic, complemented
4C2×C3×#417C211yesmaximal, characteristic, complemented
5C2×#4973C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70283C212yesmaximal, complemented
9#70278C212yesmaximal, complemented
10#70215C212yesmaximal, complemented
11#70210C212yesmaximal, complemented
12#23429—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5832—24no—
19#5832—24no—
20C2×#951—24no—
21C2×#951—24no—
22C2×#951—24no—
23C2×#947C2211yescharacteristic
24C2×#944C2211yescharacteristic
25C22×#216C2212yes—
26C2×#940C2211yescharacteristic
27C2×#939—22no—
28C2×#936C2211yescharacteristic
29C22×#214C2212yes—
30C2×C4×#67—22no—
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33C2×#909—24no—
34C2×#909—24no—
35C2×#909—24no—
36C22×Dic12C2212yes—
37C22×#204C2212yes—
38C3×#417—24no—
39C2×C3×#103C2212yes—
40C2×C3×#102C2212yes—
41C2×C3⋊#95—22no—
42#4973—22no—
43#4973C2212yes—
44#4973C2212yes—
45C2×#810C2212yes—
46#4966—24no—
47C2×#809C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C22×#137—31no—
51C22×#137—31no—
52C2×#532—31no—
53C2×#417—31no—
54C2×#417D311yescharacteristic, complemented
55C2×#391—31no—
56#2016—32no—
57#2015—32no—
58#2015—32no—
59#2014—32no—
60#1728—31no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41D414yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C22×#67—22no—
75C23×Dic3—21no—
76C23×Dic3D412yes—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235D414yes—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6—22no—
86C22×Dic6—22no—
87C22×Dic6—22no—
88C22×Dic6—22no—
89C22×Dic6—21no—
90C22×Dic6—22no—
91C22×Dic6D414yescomplemented
92C22×Dic6C2312yes—
93C22×Dic6C2311yescharacteristic
94C2×C3×M16—22no—
95C2×C3×M16—22no—
96C2×C3×M16D414yescomplemented
97C22×C3×C8C2312yes—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2312yes—
102C2×#217—22no—
103#951—232no—
104#947—22no—
105#947—22no—
106#944—22no—
107#944—22no—
108C2×#216—24no—
109C2×#216—24no—
110C2×#216—24no—
111#940—22no—
112#940—22no—
113#939—216no—
114#936—22no—
115#936—22no—
116C2×#214—24no—
117C2×#214—24no—
118C2×#214—24no—
119C4×#67—216no—
120C2×(C2×D3)⋊C4—22no—
121C2×Dic3⋊C4—22no—
122C2×C4×Dic3—22no—
123C2×C4×Dic3—22no—
124C2×C4×Dic3D412yes—
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C2×#208—22no—
128C2×#208—22no—
129C2×#208D414yescomplemented
130C22×C3⋊C8C2312yes—
131#909—232no—
132C2×Dic12—24no—
133C2×Dic12—24no—
134C2×Dic12—24no—
135C2×#204—24no—
136C2×#204—24no—
137C2×#204—24no—
138C3×#103—24no—
139C3×#102—24no—
140C3⋊#95—24no—
141C3⋊#95—24no—
142#810—24no—
143#809—24no—
144C22×#41—31no—
145C22×#41C2×D311yescharacteristic
146C23×Q8—31no—
147C2×#137—64no—
148C2×#137—64no—
149C2×#137—64no—
150C2×#137—64no—
151C2×#137—64no—
152C2×#137—64no—
153C22×Q16—32no—
154C22×Q16—32no—
155C22×SD16—32no—
156C22×SD16—32no—
157C22×M16—31no—
158C22×M16C2×D311yescharacteristic
159#532—64no—
160#532—64no—
161C2×C4⋊Q8—31no—
162C2×#127—31no—
163C2×#124—31no—
164C2×#124—31no—
165C2×#121—62no—
166C2×C4×D4—62no—
167C22×C4⋊C4C2×D311yescharacteristic
168C22×C42—31no—
169#417—24no—
170#417—64no—
171C2×#103—32no—
172C2×#103C2×D312yes—
173C2×#102—32no—
174C2×#102C2×D312yes—
175#391—32no—
176#391—32no—
177#391—62no—
178C2×#95—62no—
179#336—32no—
180#336—32no—
181C24×C3—21no—
182C24×C3C2×D411yescharacteristic
183C23×D3—21no—
184C3×#41—28no—
185C3×#41—28no—
186C3×#41—44no—
187C3×#41—44no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8C2×D412yes—
194C2×C3×D4—24no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—21no—
204C22×C3×C4—24no—
205C22×C3×C4—22no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—44no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4#4112yes—
217C22×C3×C4C2411yescharacteristic
218C2×#67—28no—
219C2×#67—24no—
220C2×#67—44no—
221C2×#67—42no—
222C2×#67—42no—
223C2×#67—44no—
224C22×Dic3—24no—
225C22×Dic3—24no—
226C22×Dic3—22no—
227C22×Dic3—21no—
228C22×Dic3—24no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3C2×D412yes—
233C22×Dic3C2×D412yes—
234#235—28no—
235#235—28no—
236#235—44no—
237#235—44no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12C2×D412yes—
244C2×C4×D3—24no—
245C2×C4×D3—42no—
246C2×C4×D3—42no—
247C2×Dic6—28no—
248C2×Dic6—28no—
249C2×Dic6—21no—
250C2×Dic6—22no—
251C2×Dic6—22no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—22no—
255C2×Dic6—22no—
256C2×Dic6—21no—
257C2×Dic6—24no—
258C2×Dic6—21no—
259C2×Dic6—22no—
260C2×Dic6—42no—
261C2×Dic6—44no—
262C2×Dic6—44no—
263C2×Dic6—42no—
264C2×Dic6C2×D414yes—
265C2×Dic6C2×D412yes—
266C3×M16—28no—
267C2×C3×C8—22no—
268C2×C3×C8—22no—
269C2×C3×C8C2×D414yes—
270C3×C4⋊C4—22no—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—42no—
273#217—28no—
274#216—216no—
275#214—216no—
276(C2×D3)⋊C4—28no—
277Dic3⋊C4—28no—
278C4×Dic3—24no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283#208—28no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8—22no—
286C2×C3⋊C8C2×D414yes—
287Dic12—216no—
288#204—216no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—34no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—62no—
298C2×#41—62no—
299C2×#41—62no—
300C2×#41D1214yes—
301C22×Q8—34no—
302C22×Q8—31no—
303C22×Q8—32no—
304C22×Q8—62no—
305C22×Q8—61no—
306C22×Q8—62no—
307C22×Q8—62no—
308C22×Q8—62no—
309C22×Q8—62no—
310C22×Q8C22×D311yescharacteristic
311C22×D4—21no—
312C22×D4—31no—
313C22×D4—62no—
314C22×D4—61no—
315C22×D4C22×D311yescharacteristic
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—32no—
319C23×C4—61no—
320C23×C4—61no—
321C23×C4C22×D311yescharacteristic
322#137—632no—
323#137—632no—
324C2×Q16—64no—
325C2×Q16—64no—
326C2×Q16—64no—
327C2×Q16—64no—
328C2×Q16—64no—
329C2×Q16—64no—
330C2×SD16—64no—
331C2×SD16—64no—
332C2×SD16—64no—
333C2×SD16—64no—
334C2×SD16—64no—
335C2×SD16—64no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—34no—
339C2×M16—62no—
340C2×M16—62no—
341C2×M16#6714yescomplemented
342C22×C8—32no—
343C22×C8C22×D312yes—
344C4⋊Q8—62no—
345C4⋊Q8—62no—
346#127—62no—
347#127—62no—
348#124—62no—
349#124—62no—
350#124—62no—
351#124—62no—
352#121—616no—
353C4×D4—616no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—62no—
358C2×C4⋊C4C22×D312yes—
359C2×C22⋊C4—62no—
360C2×C22⋊C4—62no—
361C2×C42—32no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—62no—
365C2×C42—62no—
366#103—24no—
367#103—64no—
368#102—24no—
369#102—64no—
370#95—64no—
371#95—64no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3#12012yes—
380C23×C3#12112yes—
381C23×C3#12712yes—
382C23×C3C22×D411yescharacteristic
383C22×D3—22no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—42no—
387C22×D3—42no—
388C3×Q8—22no—
389C3×Q8—22no—
390C3×Q8—41no—
391C3×Q8—41no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—48no—
395C3×D4—41no—
396C3×D4—41no—
397C2×C3×C4—22no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—48no—
401C2×C3×C4—42no—
402C2×C3×C4—44no—
403C2×C3×C4—84no—
404C2×C3×C4#12014yes—
405C2×C3×C4#12012yes—
406C2×C3×C4#12014yes—
407C2×C3×C4#12114yes—
408C2×C3×C4#12112yes—
409C2×C3×C4#12114yes—
410C2×C3×C4#12712yes—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417#67—416no—
418#67—48no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—48no—
427C2×Dic3—42no—
428C2×Dic3—41no—
429C2×Dic3—42no—
430C2×Dic3—41no—
431C2×Dic3—48no—
432C2×Dic3#13714yescomplemented
433D12—22no—
434D12—22no—
435D12—41no—
436D12—41no—
437C4×D3—48no—
438Dic6—24no—
439Dic6—22no—
440Dic6—24no—
441Dic6—22no—
442Dic6—41no—
443Dic6—42no—
444Dic6—42no—
445Dic6—41no—
446Dic6—48no—
447C3×C8—24no—
448C3⋊C8—24no—
449C24—21no—
450C24—61no—
451C24D3×D411yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456#41—68no—
457#41—68no—
458#41—124no—
459#41—124no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—32no—
466C2×Q8—34no—
467C2×Q8—64no—
468C2×Q8—61no—
469C2×Q8—62no—
470C2×Q8—68no—
471C2×Q8—68no—
472C2×Q8—62no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—62no—
476C2×Q8—62no—
477C2×Q8—61no—
478C2×Q8—62no—
479C2×Q8—61no—
480C2×Q8—122no—
481C2×Q8—124no—
482C2×Q8—124no—
483C2×Q8—122no—
484C2×Q8C2×D1212yes—
485C2×D4—24no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—32no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—68no—
495C2×D4—64no—
496C2×D4—61no—
497C2×D4—61no—
498C2×D4—61no—
499C2×D4—61no—
500C2×D4—61no—
501C2×D4—124no—
502C2×D4—124no—
503C2×D4—122no—
504C2×D4—122no—
505C2×D4C2×D1212yes—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—32no—
510C22×C4—32no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—44no—
514C22×C4—42no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—64no—
518C22×C4—62no—
519C22×C4—61no—
520C22×C4—64no—
521C22×C4—122no—
522C22×C4—122no—
523C22×C4—122no—
524C22×C4—122no—
525C22×C4—122no—
526C22×C4C2×D1212yes—
527C22×C4#23512yes—
528C22×C4D3×D411yescharacteristic
529C22×C4D3×D412yes—
530C22×C4D3×D412yes—
531C22×C4D3×D412yes—
532C22×C4C2×#6712yes—
533C22×C4C23×D311yescharacteristic
534Q16—616no—
535Q16—616no—
536SD16—616no—
537SD16—616no—
538M16—28no—
539M16—68no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—34no—
543C2×C8—62no—
544C2×C8—62no—
545C2×C8C2×#6714yes—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—42no—
549C4⋊C4—68no—
550C22⋊C4—68no—
551C22⋊C4—68no—
552C42—64no—
553C42—61no—
554C42—61no—
555C42—61no—
556C42—61no—
557C22×C3—22no—
558C22×C3—22no—
559C22×C3—24no—
560C22×C3—24no—
561C22×C3—42no—
562C22×C3—44no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—82no—
566C22×C3#39214yes—
567C22×C3C2×#12011yescharacteristic
568C22×C3C2×#12111yescharacteristic
569C22×C3C2×#12711yescharacteristic
570C22×C3C2×#13712yes—
571C22×C3#57512yes—
572C2×D3—42no—
573C2×D3—44no—
574C2×D3—81no—
575C2×D3—81no—
576C2×D3—82no—
577C3×C4—42no—
578C3×C4—82no—
579C3×C4#39214yes—
580C3×C4C2×#12011yescharacteristic
581C3×C4C2×#12011yescharacteristic
582C3×C4C2×#12111yescharacteristic
583C3×C4C2×#12111yescharacteristic
584Dic3—24no—
585Dic3—44no—
586Dic3—42no—
587Dic3—42no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—41no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—62no—
596C23—121no—
597C23—122no—
598C23—122no—
599C23—121no—
600C23#88212yes—
601C23#88412yes—
602C23#94012yes—
603C23C2×D3×D411yescharacteristic
604Q8—22no—
605Q8—22no—
606Q8—41no—
607Q8—41no—
608Q8—64no—
609Q8—62no—
610Q8—64no—
611Q8—62no—
612Q8—128no—
613Q8—121no—
614Q8—122no—
615Q8—122no—
616Q8—121no—
617D4—22no—
618D4—22no—
619D4—48no—
620D4—41no—
621D4—41no—
622D4—62no—
623D4—62no—
624D4—1216no—
625D4—128no—
626D4—121no—
627D4—121no—
628C2×C4—22no—
629C2×C4—34no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—84no—
643C2×C4—128no—
644C2×C4—128no—
645C2×C4—122no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—122no—
649C2×C4—128no—
650C2×C4#88212yes—
651C2×C4#88214yes—
652C2×C4#88412yes—
653C2×C4#88414yes—
654C2×C4#93014yes—
655C2×C4#93714yes—
656C2×C4#94012yes—
657C2×C4C22×D1211yescharacteristic
658C2×C4C2×#23511yescharacteristic
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×D3×D411yescharacteristic
662C2×C4C22×#6711yescharacteristic
663C8—24no—
664C8—64no—
665C2×C3—24no—
666C2×C3—82no—
667C2×C3C2×#39211yescharacteristic
668C2×C3#300712yes—
669C2×C3#303712yes—
670C2×C3#315211yescharacteristic
671C2×C3#315311yescharacteristic
672D3—82no—
673C22—22no—
674C22—24no—
675C22—24no—
676C22—22no—
677C22—44no—
678C22—42no—
679C22—81no—
680C22—82no—
681C22—81no—
682C22—124no—
683C22—122no—
684C22—241no—
685C22—242no—
686C22—241no—
687C22#496014yes—
688C22C2×#88211yescharacteristic
689C22C2×#88411yescharacteristic
690C22#577312yes—
691C22D3×#13712yes—
692C22C2×#94011yescharacteristic
693C4—42no—
694C4—64no—
695C4—82no—
696C4—122no—
697C4—122no—
698C4—124no—
699C4#496014yes—
700C4C2×#88211yescharacteristic
701C4C2×#88411yescharacteristic
702C4C2×#93011yescharacteristic
703C4C2×#93711yescharacteristic
704C3#2342911yescharacteristic, complemented, Sylow
705C2—24no—
706C2—82no—
707C2—242no—
708C2#7888512yes—
709C2#7891312yes—
710C2C2×#496011yescharacteristic
711C2#8393811yescharacteristic
712C2#8397211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#417C2
socleC23×C3C22×D4
prime radicalC2×#5832C2
hypercenterC2×#417D3
hypocenterC3#23429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C22×#909C2semidirect
4. C2×C3×#417C2semidirect
5. C2×#4973C2semidirect
6. C2×#4966C2semidirect
7. #71979C2semidirect
8. #70283C2semidirect
9. #70278C2semidirect
10. #70215C2semidirect
11. #70210C2semidirect
54. C2×#417D3semidirect
91. C22×Dic6D4semidirect
96. C2×C3×M16D4semidirect
129. C2×#208D4semidirect
341. C2×M16#67semidirect
432. C2×Dic3#137semidirect
704. C3#23429semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C22×#9512
C22×#9092
C2×C3×#4172
C2×#49732
C2×#49662
#719792
#702832
#702782
#702152
#702102
#234293