Finite Group Databasebeta

← #6593936

Subgroups — #6593936

Order 512.

2214 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33002C212yesmaximal
4#32953C211yesmaximal, characteristic, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27644C211yesmaximal, characteristic, complemented
7#27610C212yesmaximal, complemented
8#27550C214yesmaximal, complemented
9#27294C214yesmaximal, complemented
10#26106C212yesmaximal, complemented
11#24811C212yesmaximal, complemented
12#24600C212yesmaximal, complemented
13#22962C212yesmaximal, complemented
14#22766C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#447C211yesmaximal, characteristic
17#20732C212yesmaximal, complemented
18C2×#552—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4—21no—
22C2×#480C2212yes—
23C2×#474C2212yes—
24C2×#447C2211yescharacteristic
25C2×C4×D8C2212yes—
26C2×#431C2212yes—
27#2857C2211yescharacteristic
28#2854C2212yes—
29#2552C2211yescharacteristic, complemented
30#2542C2211yescharacteristic, complemented
31#2519—22no—
32#2512C2212yescomplemented
33#2506C2212yescomplemented
34#2498—22no—
35#2489—21no—
36#2417—21no—
37#2365C2211yescharacteristic, complemented
38#2362—21no—
39#2354C2211yescharacteristic, complemented
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×#121C2211yescharacteristic
47C2×#389C2212yes—
48C2×#385—22no—
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2214yescomplemented
54#2099C2214yescomplemented
55#2086C2214yescomplemented
56#2083C2212yescomplemented
57#2083C2214yescomplemented
58#2071C2212yescomplemented
59#2067C2214yescomplemented
60#1990C2212yescomplemented
61#1947C2212yescomplemented
62#1942C2214yescomplemented
63#1939C2212yescomplemented
64#1934C2214yescomplemented
65#1918C2214yescomplemented
66#1918C2214yescomplemented
67#1916C2214yescomplemented
68#1879C2212yes—
69#1879C2214yes—
70#1879C2214yes—
71#1874C2211yescharacteristic
72#1862C2212yes—
73#1862C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1852C2212yes—
77#1852C2211yescharacteristic
78#1834C2212yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1802C2214yes—
84#1802C2214yes—
85#1802C2212yes—
86#1802C2212yescomplemented
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90C8×C4⋊C4C2212yescomplemented
91C4×#102C2214yes—
92C4×#102C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×D8C2312yes—
99#552—48no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#125—21no—
103C2×#125—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×D4—28no—
111C2×C4×D4—21no—
112C2×C4×D4—28no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—44no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C2×#117—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—48no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#480—44no—
141#474—28no—
142#447—24no—
143C4×D8—28no—
144#431—44no—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#397—22no—
167#396—22no—
168#391—21no—
169#391—21no—
170#390C2312yes—
171#390C2311yescharacteristic
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—21no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—22no—
191#386—21no—
192#385—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#380—21no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—24no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C25—21no—
239C22×Q8—21no—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4#4114yes—
246C22×D4C2411yescharacteristic
247C22×D4C2412yes—
248C23×C4—28no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—44no—
257C23×C4—48no—
258C23×C4—44no—
259C23×C4—48no—
260C23×C4—84no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×D8—24no—
267C2×D8—24no—
268C2×D8—24no—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273#126—48no—
274#125—48no—
275#125—44no—
276#123—48no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280#121—22no—
281#121—22no—
282C4×Q8—44no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4#4118yes—
297#117—48no—
298#117—44no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—416no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4D818yescomplemented
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—48no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372#107—24no—
373#107—28no—
374#107—44no—
375C4⋊C8—24no—
376C4⋊C8—44no—
377#102—24no—
378#102—24no—
379#102—28no—
380#102—28no—
381#102—24no—
382#102—24no—
383#102—44no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—24no—
387C4×C8—24no—
388C4×C8—44no—
389#95—24no—
390#95—24no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—22no—
408C24—21no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D411yescharacteristic
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×#4114yes—
433C2×D4C2×#4114yes—
434C2×D42−1+414yes—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—432no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—832no—
477C22×C4C2×D818yes—
478C22×C4C2×D814yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491D8—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4C2×D814yes—
533C4⋊C4C2×D814yes—
534C4⋊C4C2×D818yes—
535C4⋊C4C2×#4114yes—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—81no—
549C22⋊C4—82no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42C2×#4114yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—216no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—42no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23#54612yes—
589C23C22×D812yes—
590C23#57712yes—
591C23C23×D411yescharacteristic
592Q8—84no—
593D4—216no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—42no—
598D4—42no—
599D4—84no—
600D4#55218yescomplemented
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—216no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—416no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—416no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—816no—
650C2×C4—816no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4—84no—
654C2×C4#52512yes—
655C2×C4#52512yes—
656C2×C4#54612yes—
657C2×C4C22×D814yes—
658C2×C4C22×D812yes—
659C2×C4C22×D814yes—
660C2×C4C22×D812yes—
661C2×C4#57512yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4112yes—
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×2−1+412yes—
666C8—42no—
667C8—42no—
668C8—44no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—28no—
676C22—432no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#329414yes—
691C22C2×#52511yescharacteristic
692C22C2×#54611yescharacteristic
693C22#348611yescharacteristic
694C22C23×D811yescharacteristic
695C22C2×#57511yescharacteristic
696C22C2×#57711yescharacteristic
697C22#358611yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—24no—
702C4—24no—
703C4—44no—
704C4—41no—
705C4—42no—
706C4—42no—
707C4—81no—
708C4—82no—
709C4—81no—
710C4—82no—
711C4—84no—
712C4—81no—
713C4#310014yes—
714C4C2×#55212yes—
715C4#349212yes—
716C2—24no—
717C2—48no—
718C2—161no—
719C2#3616311yescharacteristic
720C2C2×#310011yescharacteristic
721C2#6058511yescharacteristic
722C2C2×#329411yescharacteristic
723C2#6115911yescharacteristic
724C2#6181711yescharacteristic
725C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36186C2semidirect
2. C2×D4×C4⋊C4C2semidirect
4. #32953C2semidirect
5. #27960C2semidirect
6. #27644C2semidirect
7. #27610C2semidirect
8. #27550C2semidirect
9. #27294C2semidirect
10. #26106C2semidirect
11. #24811C2semidirect
12. #24600C2semidirect
13. #22962C2semidirect
14. #22766C2semidirect
15. #22599C2semidirect
17. #20732C2semidirect
29. #2552C22semidirect
30. #2542C22semidirect
32. #2512C22semidirect
33. #2506C22semidirect
37. #2365C22semidirect
39. #2354C22semidirect
53. #2099C22semidirect
54. #2099C22semidirect
55. #2086C22semidirect
56. #2083C22semidirect
57. #2083C22semidirect
58. #2071C22semidirect
59. #2067C22semidirect
60. #1990C22semidirect
61. #1947C22semidirect
62. #1942C22semidirect
63. #1939C22semidirect
64. #1934C22semidirect
65. #1918C22semidirect
66. #1918C22semidirect
67. #1916C22semidirect
78. #1834C22semidirect
86. #1802C22semidirect
90. C8×C4⋊C4C22semidirect
328. C2×C4⋊C4D8semidirect
600. D4#552semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×D4×C4⋊C42
#330022
#329532
#279602
#276442
#276102
#275502
#272942
#261062
#248112
#246002
#229622
#227662
#225992
C4×#4472
#207322