Finite Group Databasebeta

← #6490578

Subgroups — #6490578

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33154C212yesmaximal, complemented
4#33113C212yesmaximal
5#30153C214yesmaximal, complemented
6#29356C212yesmaximal, complemented
7#29321C212yesmaximal, complemented
8#29287C211yesmaximal, characteristic, complemented
9#29286C211yesmaximal, characteristic, complemented
10#28280C211yesmaximal, characteristic, complemented
11#28279C211yesmaximal, characteristic, complemented
12#27646C212yesmaximal, complemented
13#24610C214yesmaximal, complemented
14#24300C212yesmaximal, complemented
15#22117C212yesmaximal, complemented
16#22116C212yesmaximal, complemented
17#20810C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529—21no—
20C2×#482C2214yes—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#432C2212yes—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2854—21no—
32#2854C2212yes—
33#2850—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#125C2212yes—
41C4×#125C2212yes—
42C4×#124—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×#121—21no—
48C4×#121C2212yes—
49#2296C2212yes—
50#2295—21no—
51#2295—21no—
52#2293—21no—
53#2293C2212yes—
54#2285C2211yescharacteristic
55C42×Q8—21no—
56C42×Q8—21no—
57C4×#117C2212yes—
58C2×C43C2211yescharacteristic
59#2088C2214yes—
60#2086C2214yes—
61#2085C2214yes—
62#2073C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2059C2214yes—
66#2025C2214yes—
67#1988C2212yes—
68#1988C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1982C2212yes—
72#1982C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1947C2212yes—
79#1944C2214yes—
80#1943C2214yes—
81#1936C2214yes—
82#1936C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1921C2214yes—
86#1920C2214yes—
87#1875C2214yes—
88#1854C2214yes—
89#1842C2211yescharacteristic
90#1842C2211yescharacteristic
91#1831C2212yes—
92#1825C2214yes—
93#1802C2212yes—
94#1802C2212yes—
95#1797C2212yes—
96#1797C2212yes—
97#1767C2212yes—
98#1767C2212yes—
99#533—24no—
100#533—24no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—48no—
106C2×#126—22no—
107C2×#125—21no—
108C2×#125C2312yes—
109C2×#124—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×#117—22no—
130C2×#117C2312yes—
131C2×#117C2312yes—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#482—48no—
140#476—28no—
141#474—28no—
142#432—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2314yes—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—21no—
212#382C2312yes—
213#380—21no—
214#380—21no—
215#380—21no—
216#380C2312yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239#376—22no—
240C43—22no—
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#336C2314yes—
258#336C2314yes—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C2412yes—
266C23×C4—22no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8#4114yes—
281C4⋊Q8#4114yes—
282#127—22no—
283#127—22no—
284#126—48no—
285#125—24no—
286#125—24no—
287#125—44no—
288#124—44no—
289#124—44no—
290#123—48no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—44no—
294#121—24no—
295#121—24no—
296#121—44no—
297#121—44no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—44no—
310#117—24no—
311#117—24no—
312#117—24no—
313#117—24no—
314#117—48no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—42no—
371C2×C42—42no—
372C2×C42—42no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—28no—
402#107—44no—
403#106—28no—
404#106—44no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—42no—
410C4⋊C8—42no—
411#102—28no—
412#102—28no—
413#102—44no—
414#102—44no—
415C22⋊C8—28no—
416#97—44no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433C24—22no—
434C24C22×D411yescharacteristic
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4C2×#4114yes—
547C4⋊C4C2×#4114yes—
548C22⋊C4—28no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C42—28no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—21no—
581C42—21no—
582C42—21no—
583C42—21no—
584C42—21no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42AGL(1,ℤ/8)14yes—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C422−1+412yes—
601C422−1+412yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23#53712yes—
616C23C23×D411yescharacteristic
617Q8—48no—
618Q8—48no—
619Q8—84no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—84no—
625D4—88no—
626D4—84no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—24no—
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—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
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—42no—
685C2×C4—42no—
686C2×C4—88no—
687C2×C4#52512yes—
688C2×C4#52512yes—
689C2×C4#53612yes—
690C2×C4#53612yes—
691C2×C4#53712yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×AGL(1,ℤ/8)12yes—
695C2×C4#57512yes—
696C2×C4#57512yes—
697C2×C4#57512yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)11yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C2×C4C2×2−1+411yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C8—44no—
705C8—44no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C2×#53611yescharacteristic
722C22C2×#53711yescharacteristic
723C22C22×AGL(1,ℤ/8)11yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—41no—
733C4—44no—
734C4—41no—
735C4—44no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#311514yes—
746C4#323514yes—
747C4#352211yescharacteristic
748C4#352211yescharacteristic
749C4#352311yescharacteristic
750C4#352311yescharacteristic
751C2—42no—
752C2—162no—
753C2#3524711yescharacteristic
754C2C2×#311511yescharacteristic
755C2#6055911yescharacteristic
756C2C2×#323511yescharacteristic
757C2#6086611yescharacteristic
758C2#6090711yescharacteristic
759C2#6091611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #33154C2semidirect
5. #30153C2semidirect
6. #29356C2semidirect
7. #29321C2semidirect
8. #29287C2semidirect
9. #29286C2semidirect
10. #28280C2semidirect
11. #28279C2semidirect
12. #27646C2semidirect
13. #24610C2semidirect
14. #24300C2semidirect
15. #22117C2semidirect
16. #22116C2semidirect
17. #20810C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#331542
#331132
#301532
#293562
#293212
#292872
#292862
#282802
#282792
#276462
#246102
#243002
#221172
#221162
#208102