Finite Group Databasebeta

← #6468209

Subgroups — #6468209

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#29144C211yesmaximal, characteristic, complemented
4#29143C211yesmaximal, characteristic, complemented
5#29134C214yesmaximal, complemented
6#29119C214yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28716C211yesmaximal, characteristic, complemented
9#27961C211yesmaximal, characteristic, complemented
10#27960C211yesmaximal, characteristic, complemented
11#25857C211yesmaximal, characteristic
12#25857C211yesmaximal, characteristic
13#23832C214yesmaximal, complemented
14#21647C211yesmaximal, characteristic, complemented
15#21646C211yesmaximal, characteristic, complemented
16C4×#487C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18#20746C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#520C2212yes—
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#433C2212yes—
27#2868C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2647C2212yes—
32#2629C2212yes—
33#2507—21no—
34#2507—21no—
35#2497—21no—
36#2497—21no—
37#2446—21no—
38#2446—21no—
39#2429—21no—
40#2429—21no—
41#2361—21no—
42#2361—21no—
43#2361—21no—
44#2361—21no—
45#2354—21no—
46#2354—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49#2330C2212yes—
50#2326C2212yes—
51#2326C2212yes—
52#2314C2212yes—
53#2311C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#2100C2214yes—
63#2099C2214yes—
64#2072C2214yes—
65#2071C2214yes—
66#2042C2214yes—
67#2042C2214yes—
68#2026C2214yes—
69#2025C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1936C2214yes—
79#1935C2214yes—
80#1934C2214yes—
81#1933C2214yes—
82#1834C2211yescharacteristic
83#1834C2211yescharacteristic
84#1834C2211yescharacteristic
85#1834C2211yescharacteristic
86#1824C2214yes—
87#1823C2214yes—
88#1805C2214yes—
89#1804C2214yes—
90#1803C2214yes—
91#1802C2214yes—
92#1788C2214yes—
93#1788C2214yes—
94#1768C2212yes—
95C4×C4⋊C8C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97#1763C2214yes—
98#1762C2214yes—
99C4×#103C2214yes—
100C4×#102C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×#41C2312yes—
108#532—24no—
109#532—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×#127C2311yescharacteristic
116C2×#125C2312yes—
117C2×#124C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×#121—21no—
121C2×#121—21no—
122#520—216no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4C2312yes—
144C22×C42C2311yescharacteristic
145#487—24no—
146#487—24no—
147#486—24no—
148#486—24no—
149#433—28no—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#400—22no—
171#400—22no—
172#398—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#396—22no—
177#396—22no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391C2312yes—
183#391C2312yes—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#386—22no—
219#386—22no—
220#385—22no—
221#385—22no—
222#385—22no—
223#385—22no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2312yes—
233#384C2312yes—
234#384C2312yes—
235#382C2312yes—
236#380—21no—
237#380—21no—
238#380—22no—
239#380—22no—
240#380C2312yes—
241#380C2312yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258#376C2311yescharacteristic
259#376C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C2×#41—24no—
275C2×#41—24no—
276C2×#41—28no—
277C2×#41—44no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8C2412yes—
281C22×D4—22no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303C4⋊Q8#4114yes—
304#127—22no—
305#127#4114yes—
306#125—44no—
307#124—24no—
308#124—24no—
309C22⋊Q8—82no—
310C22⋊Q8—82no—
311#121—28no—
312#121—28no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—82no—
316C4×Q8—82no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—48no—
322C4×D4—48no—
323C4×D4—82no—
324C4×D4—82no—
325C4×D4—82no—
326C4×D4—82no—
327#117—42no—
328#117—48no—
329#117#4118yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—48no—
359C2×C4⋊C4—82no—
360C2×C4⋊C4—82no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C4⋊C8—24no—
408C4⋊C8—24no—
409#103—28no—
410#103—28no—
411#103—44no—
412#103—44no—
413#102—28no—
414#102—28no—
415#102—44no—
416#102—44no—
417C22⋊C8—24no—
418C22⋊C8—24no—
419#97—24no—
420#97—42no—
421C4×C8—24no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—42no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449C24—22no—
450C24C22×D411yescharacteristic
451#41—416no—
452#41—416no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8C2×#4114yes—
463C2×D4—28no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—24no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×#4114yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C2×#4114yes—
518C22×C4C2511yescharacteristic
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4#12414yes—
569C4⋊C4#12414yes—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4#12414yes—
583C22⋊C4#12414yes—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—21no—
595C42—21no—
596C42—22no—
597C42—22no—
598C42—44no—
599C42—44no—
600C42—82no—
601C42—82no—
602C42C2×#4112yes—
603C42C2×#4112yes—
604C42C2×#4114yes—
605C42C2×#4114yes—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—24no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23C2×#12412yes—
620C23C23×D411yescharacteristic
621Q8—28no—
622Q8—44no—
623Q8—84no—
624Q8—84no—
625D4—28no—
626D4—416no—
627D4—48no—
628D4—48no—
629D4—44no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—416no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—164no—
685C2×C4—164no—
686C2×C4—164no—
687C2×C4#52512yes—
688C2×C4#52512yes—
689C2×C4C2×#12412yes—
690C2×C4C2×#12412yes—
691C2×C4C2×#12412yes—
692C2×C4C2×#12412yes—
693C2×C4C2×#12412yes—
694C2×C4#57612yes—
695C2×C4#57712yes—
696C2×C4#57712yes—
697C2×C4#57812yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C22×#4112yes—
701C2×C4#58512yes—
702C8—42no—
703C8—42no—
704C8—42no—
705C8—42no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C22×#12411yescharacteristic
722C22#344911yescharacteristic
723C22C2×#57611yescharacteristic
724C22C2×#57711yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57811yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—44no—
737C4—41no—
738C4—41no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#311914yes—
746C4#312014yes—
747C4#347512yes—
748C4#347512yes—
749C2—42no—
750C2—84no—
751C2#3494611yescharacteristic
752C2C2×#311911yescharacteristic
753C2C2×#312011yescharacteristic
754C2#6062211yescharacteristic
755C2#6062311yescharacteristic
756C2#6101411yescharacteristic
757C2#6101511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #29144C2semidirect
4. #29143C2semidirect
5. #29134C2semidirect
6. #29119C2semidirect
7. #28717C2semidirect
8. #28716C2semidirect
9. #27961C2semidirect
10. #27960C2semidirect
13. #23832C2semidirect
14. #21647C2semidirect
15. #21646C2semidirect
16. C4×#487C2semidirect
17. C4×#486C2semidirect
18. #20746C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#342072
#291442
#291432
#291342
#291192
#287172
#287162
#279612
#279602
#258572
#238322
#216472
#216462
C4×#4872
C4×#4862
#207462
C4×#4332