Finite Group Databasebeta

← #6795008

Subgroups — #6795008

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#32953C211yesmaximal, characteristic, complemented
4#32426C214yesmaximal, complemented
5#32342C214yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#28527C212yesmaximal, complemented
8#27727C212yesmaximal, complemented
9#27582C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25449C212yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#24500C212yesmaximal
14#24298C211yesmaximal, characteristic, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21982C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C2×#552—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×C4×D4C2211yescharacteristic
23C2×#500C2212yescomplemented
24C2×#474C2212yes—
25C2×#474C2212yescomplemented
26C2×#447C2211yescharacteristic
27C2×C4×D8C2212yes—
28C2×#431C2212yes—
29#2857C2211yescharacteristic
30#2854C2212yes—
31#2552C2211yescharacteristic, complemented
32#2542—21no—
33#2542—21no—
34#2542—22no—
35#2542C2212yes—
36#2542C2214yes—
37#2542C2212yes—
38#2542C2211yescharacteristic, complemented
39#2519—22no—
40#2512C2212yescomplemented
41#2506C2212yescomplemented
42#2498—22no—
43#2489—21no—
44#2417—21no—
45C2×#399—22no—
46#2365C2211yescharacteristic, complemented
47#2362—21no—
48#2354C2211yescharacteristic, complemented
49C4×#121C2211yescharacteristic
50C2×#390C2212yes—
51#2285C2212yes—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2214yescomplemented
55#2093C2214yes—
56#2093C2214yescomplemented
57#2083C2214yescomplemented
58#2071C2214yes—
59#2071C2214yescomplemented
60#2067C2214yes—
61#2057C2212yescomplemented
62#2057C2214yescomplemented
63#2054C2214yes—
64#2023C2212yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1970C2214yes—
68#1947C2214yescomplemented
69#1947C2214yescomplemented
70#1939C2212yescomplemented
71#1934C2214yes—
72#1925C2214yes—
73#1925C2214yes—
74#1916C2214yescomplemented
75#1879C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1862C2212yes—
79#1862C2214yes—
80#1852C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1845C2214yescomplemented
86#1834C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99#552—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#126—22no—
107C2×#125—21no—
108C2×#125—22no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—48no—
120C2×C22⋊Q8—48no—
121C2×#121C2311yescharacteristic
122C2×C4×Q8—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—44no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—22no—
131C2×#117—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—22no—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#500—44no—
145#474—44no—
146#474—44no—
147#447—24no—
148C4×D8—44no—
149#431—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#107C2314yes—
154C2×#107C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2311yescharacteristic
166C2×C22⋊C8C2311yescharacteristic
167C2×C4×C8C2314yes—
168C2×C4×C8C2312yes—
169#400—22no—
170#399—24no—
171#399—24no—
172#399—24no—
173#399—24no—
174#399—22no—
175#398—22no—
176#398—22no—
177#397—22no—
178#396—22no—
179#391—21no—
180#391—21no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2314yes—
185#390C2314yes—
186#390C2312yes—
187#390C2311yescharacteristic
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2311yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#388—21no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#386—22no—
202#386—21no—
203#384—22no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2314yes—
208#384C2314yes—
209#384C2312yes—
210#382—22no—
211#380—21no—
212#380—22no—
213#380—21no—
214#380C2312yes—
215#380C2311yescharacteristic
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4C2411yescharacteristic
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—42no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#126—48no—
280#125—48no—
281#125—44no—
282#123—48no—
283C22⋊Q8—48no—
284C22⋊Q8—416no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288C22⋊Q8—416no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—48no—
292#121—22no—
293#121—22no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303#117—48no—
304#117—44no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—48no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4D818yescomplemented
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4—48no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377#107—42no—
378#107—44no—
379#107—44no—
380#107—44no—
381#107—44no—
382C4⋊C8—28no—
383C4⋊C8—24no—
384#102—28no—
385#102—24no—
386#102—24no—
387#102—42no—
388#102—42no—
389#102—44no—
390#102—44no—
391C22⋊C8—24no—
392C22⋊C8—22no—
393C22⋊C8#4114yes—
394C4×C8—28no—
395C4×C8—24no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—24no—
410C24—22no—
411C24—21no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24C22×D411yescharacteristic
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—816no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D42−1+414yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—41no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—816no—
482C22×C4C2×D818yes—
483C22×C4C2×D814yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496D8—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8C2×#4114yes—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—82no—
536C4⋊C4—81no—
537C4⋊C4C2×D814yes—
538C4⋊C4C2×D814yes—
539C4⋊C4C2×D818yes—
540C4⋊C4AGL(1,ℤ/8)14yes—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—81no—
556C22⋊C4—82no—
557C42—22no—
558C42—22no—
559C42—24no—
560C42—24no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—44no—
568C42—42no—
569C42H(5,𝔽2)14yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—24no—
575C23—48no—
576C23—48no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23#54712yes—
596C23C22×D812yes—
597C23#57812yes—
598C23C23×D411yescharacteristic
599Q8—88no—
600Q8—84no—
601Q8—88no—
602D4—24no—
603D4—24no—
604D4—48no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—84no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—216no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4#53512yes—
661C2×C4#53512yes—
662C2×C4#54712yes—
663C2×C4C22×D814yes—
664C2×C4C22×D814yes—
665C2×C4C22×D812yes—
666C2×C4C22×D812yes—
667C2×C4C2×AGL(1,ℤ/8)14yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)12yes—
671C2×C4C2×2−1+412yes—
672C2×C4C2×2−1+411yescharacteristic
673C8—24no—
674C8—44no—
675C8—42no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—816no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22#333414yes—
698C22C2×#53511yescharacteristic
699C22C2×#54711yescharacteristic
700C22#348611yescharacteristic
701C22C23×D811yescharacteristic
702C22C22×AGL(1,ℤ/8)11yescharacteristic
703C22C2×#57811yescharacteristic
704C22#358611yescharacteristic
705C4—22no—
706C4—24no—
707C4—24no—
708C4—24no—
709C4—22no—
710C4—22no—
711C4—42no—
712C4—41no—
713C4—81no—
714C4—81no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—81no—
719C4—82no—
720C4#320514yes—
721C4#352012yes—
722C4#353812yes—
723C2—24no—
724C2—84no—
725C2—161no—
726C2#3891711yescharacteristic
727C2C2×#320511yescharacteristic
728C2#6081411yescharacteristic
729C2C2×#333411yescharacteristic
730C2#6125211yescharacteristic
731C2#6181711yescharacteristic
732C2#6183711yescharacteristic
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×#2542C2semidirect
3. #32953C2semidirect
4. #32426C2semidirect
5. #32342C2semidirect
6. #29906C2semidirect
7. #28527C2semidirect
8. #27727C2semidirect
9. #27582C2semidirect
10. #25813C2semidirect
11. #25449C2semidirect
12. #24676C2semidirect
14. #24298C2semidirect
15. #22599C2semidirect
16. #21982C2semidirect
23. C2×#500C22semidirect
25. C2×#474C22semidirect
31. #2552C22semidirect
38. #2542C22semidirect
40. #2512C22semidirect
41. #2506C22semidirect
46. #2365C22semidirect
48. #2354C22semidirect
54. #2099C22semidirect
56. #2093C22semidirect
57. #2083C22semidirect
59. #2071C22semidirect
61. #2057C22semidirect
62. #2057C22semidirect
64. #2023C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1947C22semidirect
69. #1947C22semidirect
70. #1939C22semidirect
74. #1916C22semidirect
85. #1845C22semidirect
86. #1834C22semidirect
92. C8×C4⋊C4C22semidirect
330. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25422
#329532
#324262
#323422
#299062
#285272
#277272
#275822
#258132
#254492
#246762
#245002
#242982
#225992
#219822
C4×#4472