Finite Group Databasebeta

← #10822331

Subgroups — #10822331

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
1C22×C3×#137C211yesmaximal, characteristic, complemented
2C2×#5832C211yesmaximal, characteristic
3C22×#942C211yesmaximal, characteristic, complemented
4C2×#5232C211yesmaximal, characteristic, complemented
5C2×#5229C211yesmaximal, characteristic, complemented
6C2×#5133C211yesmaximal, characteristic, complemented
7#72073C212yesmaximal, complemented
8#72072C212yesmaximal, complemented
9#72058C212yesmaximal, complemented
10#72056C212yesmaximal, complemented
11#71979C211yesmaximal, characteristic, complemented
12#23429—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×#137—24no—
17C2×C3×#137—24no—
18C2×C3×#137—24no—
19C22×C3×Q16C2212yes—
20C22×C3×SD16C2212yes—
21C22×C3×M16C2211yescharacteristic
22#5832—24no—
23#5832—24no—
24C2×#947C2211yescharacteristic
25C2×#944C2211yescharacteristic
26C22×#216C2212yes—
27C2×#942—24no—
28C2×#942—24no—
29C2×#942—24no—
30C22×#215C2212yes—
31C2×#940C2211yescharacteristic
32C2×#939—22no—
33C2×#936C2211yescharacteristic
34C2×C4×#67—22no—
35C22×C4⋊Dic3C2211yescharacteristic
36C22×C4×Dic3C2211yescharacteristic
37C22×#208C2211yescharacteristic
38#5232—24no—
39#5229—22no—
40#5229C2212yes—
41#5229C2212yes—
42C2×#835C2212yes—
43C2×#832C2212yes—
44C2×C3⋊#95—22no—
45#5133—24no—
46C2×#821C2212yes—
47C2×#816C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C22×#137—31no—
51C22×#137D311yescharacteristic, complemented
52C2×#532—31no—
53C2×#417—31no—
54C2×#417—31no—
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×Q8—21no—
68C22×C3×Q8—22no—
69C22×C3×Q8—22no—
70C22×C3×Q8—22no—
71C22×C3×Q8—22no—
72C22×C3×Q8—22no—
73C22×C3×Q8D414yescomplemented
74C22×C3×Q8C2312yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—22no—
83C23×Dic3—21no—
84C23×Dic3D412yes—
85C2×#235—22no—
86C2×#235—22no—
87C2×#235—22no—
88C2×#235—22no—
89C2×#235—22no—
90C2×#235D414yes—
91C22×D12C2311yescharacteristic
92C22×C4×D3—21no—
93C22×Dic6C2311yescharacteristic
94C3×#137—232no—
95C2×C3×Q16—24no—
96C2×C3×Q16—24no—
97C2×C3×Q16—24no—
98C2×C3×SD16—24no—
99C2×C3×SD16—24no—
100C2×C3×SD16—24no—
101C2×C3×M16—22no—
102C2×C3×M16—22no—
103C2×C3×M16D414yescomplemented
104C22×C3×C8C2312yes—
105C2×#217—22no—
106#947—22no—
107#947—22no—
108#944—22no—
109#944—22no—
110C2×#216—24no—
111C2×#216—24no—
112C2×#216—24no—
113#942—232no—
114C2×#215—24no—
115C2×#215—24no—
116C2×#215—24no—
117#940—22no—
118#940—22no—
119#939—216no—
120#936—22no—
121#936—22no—
122C4×#67—216no—
123C2×(C2×D3)⋊C4—22no—
124C2×C4⋊Dic3—22no—
125C2×C4⋊Dic3—22no—
126C2×C4⋊Dic3—21no—
127C2×C4⋊Dic3C2312yes—
128C2×Dic3⋊C4—22no—
129C2×C4×Dic3—22no—
130C2×C4×Dic3—22no—
131C2×C4×Dic3D412yes—
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2311yescharacteristic
134C2×#208—22no—
135C2×#208—22no—
136C2×#208D414yescomplemented
137C22×C3⋊C8C2312yes—
138#835—24no—
139#832—24no—
140C3⋊#95—24no—
141C3⋊#95—24no—
142#821—24no—
143#816—24no—
144C22×#41—31no—
145C22×#41C2×D311yescharacteristic
146C23×Q8C2×D311yescharacteristic
147C2×#137—24no—
148C2×#137—24no—
149C2×#137—24no—
150C2×#137—64no—
151C2×#137—64no—
152C2×#137—64no—
153C22×Q16—32no—
154C22×Q16C2×D312yes—
155C22×SD16—32no—
156C22×SD16C2×D312yes—
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⋊C4—31no—
168C22×C42—31no—
169#417—64no—
170#417—64no—
171C2×#103—32no—
172C2×#103—32no—
173C2×#102—32no—
174C2×#102—32no—
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—24no—
189C2×C3×Q8—21no—
190C2×C3×Q8—22no—
191C2×C3×Q8—28no—
192C2×C3×Q8—28no—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—42no—
202C2×C3×Q8—44no—
203C2×C3×Q8—44no—
204C2×C3×Q8—42no—
205C2×C3×Q8C2×D414yes—
206C2×C3×Q8C2×D412yes—
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—42no—
214C2×C3×D4—42no—
215C2×C3×D4C2×D412yes—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—24no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4—42no—
223C22×C3×C4—42no—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2×D412yes—
230C22×C3×C4#4112yes—
231C22×C3×C4C2411yescharacteristic
232C2×#67—28no—
233C2×#67—24no—
234C2×#67—44no—
235C2×#67—42no—
236C2×#67—42no—
237C2×#67—44no—
238C22×Dic3—24no—
239C22×Dic3—21no—
240C22×Dic3—24no—
241C22×Dic3—22no—
242C22×Dic3—42no—
243C22×Dic3—44no—
244C22×Dic3—42no—
245C22×Dic3C2×D412yes—
246C22×Dic3C2×D412yes—
247#235—28no—
248#235—28no—
249#235—44no—
250#235—44no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C2×D12—21no—
255C2×D12—21no—
256C2×D12C2×D412yes—
257C2×C4×D3—24no—
258C2×C4×D3—42no—
259C2×C4×D3—42no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6—21no—
264C2×Dic6—21no—
265C2×Dic6C2×D412yes—
266C3×Q16—216no—
267C3×SD16—216no—
268C3×M16—28no—
269C2×C3×C8—22no—
270C2×C3×C8—22no—
271C2×C3×C8C2×D414yes—
272#217—28no—
273#216—216no—
274#215—216no—
275(C2×D3)⋊C4—28no—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—42no—
279Dic3⋊C4—28no—
280C4×Dic3—24no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285#208—28no—
286C2×C3⋊C8—22no—
287C2×C3⋊C8—22no—
288C2×C3⋊C8C2×D414yes—
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×#41#6714yes—
301C22×Q8—21no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—22no—
305C22×Q8—22no—
306C22×Q8—22no—
307C22×Q8—31no—
308C22×Q8#6714yescomplemented
309C22×Q8C22×D311yescharacteristic
310C22×Q8C22×D312yes—
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—232no—
323#137—632no—
324C2×Q16—24no—
325C2×Q16—24no—
326C2×Q16—24no—
327C2×Q16—64no—
328C2×Q16—64no—
329C2×Q16—64no—
330C2×SD16—24no—
331C2×SD16—24no—
332C2×SD16—24no—
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—32no—
355C2×C4⋊C4—62no—
356C2×C4⋊C4—62no—
357C2×C4⋊C4—62no—
358C2×C4⋊C4—61no—
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—64no—
367#103—64no—
368#102—64no—
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—24no—
389C3×Q8—24no—
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—41no—
393C3×Q8—42no—
394C3×Q8—42no—
395C3×Q8—41no—
396C3×Q8—48no—
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—48no—
400C3×D4—41no—
401C3×D4—41no—
402C2×C3×C4—22no—
403C2×C3×C4—24no—
404C2×C3×C4—42no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—42no—
408C2×C3×C4—48no—
409C2×C3×C4—48no—
410C2×C3×C4#12014yes—
411C2×C3×C4#12012yes—
412C2×C3×C4#12014yes—
413C2×C3×C4#12114yes—
414C2×C3×C4#12114yes—
415C2×C3×C4#12112yes—
416C2×C3×C4#12712yes—
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423#67—416no—
424#67—48no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—48no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—42no—
435C2×Dic3—44no—
436C2×Dic3—84no—
437C2×Dic3#13714yescomplemented
438D12—22no—
439D12—22no—
440D12—41no—
441D12—41no—
442C4×D3—48no—
443Dic6—22no—
444Dic6—22no—
445Dic6—41no—
446Dic6—41no—
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—24no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—28no—
464C2×Q8—28no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—32no—
474C2×Q8—42no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—42no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8C2×#6712yes—
484C2×Q8C2×#6714yes—
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×#6712yes—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—32no—
511C22×C4—32no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—64no—
517C22×C4—64no—
518C22×C4—61no—
519C22×C4—64no—
520C22×C4—62no—
521C22×C4—122no—
522C22×C4—122no—
523C22×C4—122no—
524C22×C4—124no—
525C22×C4—122no—
526C22×C4D3×D412yes—
527C22×C4D3×D411yescharacteristic
528C22×C4D3×D412yes—
529C22×C4#23712yes—
530C22×C4C2×#6712yes—
531C22×C4C2×#6712yes—
532C22×C4C2×#6712yes—
533C22×C4C23×D311yescharacteristic
534Q16—216no—
535Q16—616no—
536SD16—216no—
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—68no—
547C4⋊C4—62no—
548C4⋊C4—62no—
549C4⋊C4—122no—
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—44no—
579C3×C4#39214yes—
580C3×C4C2×#12011yescharacteristic
581C3×C4C2×#12011yescharacteristic
582C3×C4C2×#12111yescharacteristic
583C3×C4C2×#12111yescharacteristic
584Dic3—24no—
585Dic3—42no—
586Dic3—42no—
587Dic3—82no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—42no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23—62no—
596C23—122no—
597C23—122no—
598C23—121no—
599C23—121no—
600C23#93712yes—
601C23#93912yes—
602C23#94012yes—
603C23C2×D3×D411yescharacteristic
604Q8—24no—
605Q8—22no—
606Q8—24no—
607Q8—22no—
608Q8—48no—
609Q8—41no—
610Q8—42no—
611Q8—41no—
612Q8—42no—
613Q8—62no—
614Q8—62no—
615Q8—121no—
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—24no—
630C2×C4—34no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—128no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—128no—
647C2×C4—124no—
648C2×C4—122no—
649C2×C4—244no—
650C2×C4#93712yes—
651C2×C4#93714yes—
652C2×C4#93814yes—
653C2×C4#93912yes—
654C2×C4#93914yes—
655C2×C4#94012yes—
656C2×C4#95314yes—
657C2×C4C2×D3×D411yescharacteristic
658C2×C4C2×D3×D411yescharacteristic
659C2×C4C2×#23711yescharacteristic
660C2×C4C22×#6711yescharacteristic
661C2×C4C22×#6711yescharacteristic
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—22no—
675C22—24no—
676C22—24no—
677C22—44no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—82no—
682C22—124no—
683C22—122no—
684C22—241no—
685C22—241no—
686C22—242no—
687C22#522314yes—
688C22D3×#13712yes—
689C22#577712yes—
690C22C2×#93711yescharacteristic
691C22C2×#93911yescharacteristic
692C22C2×#94011yescharacteristic
693C4—42no—
694C4—44no—
695C4—64no—
696C4—122no—
697C4—122no—
698C4—242no—
699C4#522314yes—
700C4C2×#93711yescharacteristic
701C4C2×#93811yescharacteristic
702C4C2×#93911yescharacteristic
703C4C2×#95311yescharacteristic
704C3#2342911yescharacteristic, complemented, Sylow
705C2—24no—
706C2—82no—
707C2—242no—
708C2#8396612yes—
709C2#8397211yescharacteristic
710C2#8397311yescharacteristic
711C2#8398112yes—
712C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC23×C3C22×D4
prime radicalC2×#5832C2
hypercenterC22×#137D3
hypocenterC3#23429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×C3×#137C2semidirect
3. C22×#942C2semidirect
4. C2×#5232C2semidirect
5. C2×#5229C2semidirect
6. C2×#5133C2semidirect
7. #72073C2semidirect
8. #72072C2semidirect
9. #72058C2semidirect
10. #72056C2semidirect
11. #71979C2semidirect
51. C22×#137D3semidirect
73. C22×C3×Q8D4semidirect
103. C2×C3×M16D4semidirect
136. C2×#208D4semidirect
308. C22×Q8#67semidirect
341. C2×M16#67semidirect
437. 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
C22×C3×#1372
C2×#58322
C22×#9422
C2×#52322
C2×#52292
C2×#51332
#720732
#720722
#720582
#720562
#719792
#234293