Finite Group Databasebeta

← #6986869

Subgroups — #6986869

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36444C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic, complemented
3#32790C212yesmaximal, complemented
4#32271C214yesmaximal, complemented
5#32012C214yesmaximal, complemented
6#30296C212yesmaximal, complemented
7#30172C212yesmaximal, complemented
8#29937C211yesmaximal, characteristic, complemented
9#29162C212yesmaximal, complemented
10#28712C212yesmaximal, complemented
11#26278C212yesmaximal, complemented
12#25838C212yesmaximal, complemented
13#25813C211yesmaximal, characteristic, complemented
14#25347C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23755C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#548—21no—
20C2×#536—21no—
21C22×C4⋊Q8C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#500C2212yes—
24C2×#474C2212yes—
25C2×#436C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2211yescharacteristic
30#2621—22no—
31#2547—21no—
32#2547C2212yes—
33#2547C2212yes—
34#2547C2211yescharacteristic
35#2546—21no—
36#2546C2212yes—
37#2537C2211yescharacteristic
38#2537C2212yes—
39#2486—22no—
40#2486C2214yes—
41#2427—22no—
42#2427—21no—
43#2415—21no—
44#2415—22no—
45Q8×C4⋊C4C2211yescharacteristic
46Q8×C22⋊C4—21no—
47#2310—21no—
48C4×#121C2211yescharacteristic
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51#2100C2214yes—
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2094C2211yescharacteristic
55#2084C2214yes—
56#2084C2211yescharacteristic
57#2084C2212yes—
58#2077C2214yes—
59#2077C2212yes—
60#2076C2214yes—
61#2076C2214yes—
62#2076C2214yes—
63#2071C2211yescharacteristic
64#2057C2214yes—
65#2057C2214yes—
66#2057C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2034C2214yes—
70#2034C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1988C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1919C2214yes—
78#1919C2214yes—
79#1895C2214yes—
80#1874C2212yes—
81#1870C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1852C2214yes—
85#1852C2212yes—
86#1817C2212yes—
87#1809C2214yes—
88#1809C2214yes—
89#1809C2214yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×Q8—21no—
98C22×M16C2314yes—
99D4×Q8—44no—
100#548—48no—
101#536—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#124—22no—
111C2×#124—21no—
112C2×#124—22no—
113C2×#123—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×#121—21no—
121C2×#121—22no—
122C2×#121C2311yescharacteristic
123C2×#121C2312yes—
124C4×#41—44no—
125C2×C4×Q8—21no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4D412yes—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2312yes—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117D411yescharacteristic
138C2×#117D412yes—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143#500—44no—
144#474—44no—
145#436—44no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160#398—24no—
161#398—24no—
162#397—24no—
163#397—24no—
164#397—22no—
165#397—22no—
166#397—22no—
167#394—22no—
168#394—22no—
169#394—22no—
170#391—22no—
171#391—22no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391—22no—
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#388—21no—
190#388—22no—
191#386—21no—
192#386—22no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#381—24no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2314yes—
210#380C2314yes—
211#380C2312yes—
212#380C2312yes—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—42no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8C2×D411yescharacteristic
250C22×D4—22no—
251C22×D4—21no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2×D412yes—
254C22×D4C2411yescharacteristic
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C2×M16—28no—
265C2×M16—44no—
266C2×M16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#124—48no—
279#124—48no—
280#124—44no—
281#123—416no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8—44no—
288#121—22no—
289#121—24no—
290#121—41no—
291#121—42no—
292#121—44no—
293#121—48no—
294C4×Q8—24no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—42no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—48no—
304C4×D4—44no—
305#117—44no—
306#117—42no—
307#117—44no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—44no—
369#107—44no—
370#107—44no—
371#106—24no—
372#106—42no—
373#106—44no—
374C4⋊C8—24no—
375#102—24no—
376#102—24no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—42no—
380C22⋊C8—44no—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397C24—21no—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400#41—416no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—816no—
416C2×Q8—816no—
417C2×Q8AGL(1,ℤ/8)14yes—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—81no—
432C2×D4—82no—
433C2×D42−1+412yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4AGL(1,ℤ/8)14yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480M16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C82−1+414yes—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—82no—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C2×AGL(1,ℤ/8)12yes—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—88no—
575Q8—88no—
576Q8—84no—
577Q8—88no—
578D4—24no—
579D4—48no—
580D4—44no—
581D4—42no—
582D4—84no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4#53412yes—
637C2×C4#53412yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×AGL(1,ℤ/8)14yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4#57812yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4C2×2−1+411yescharacteristic
645C2×C4C2×2−1+412yes—
646C2×C4#58511yescharacteristic
647C8—24no—
648C8—42no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22C2×#53411yescharacteristic
672C22#348911yescharacteristic
673C22#349911yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57811yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—24no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—84no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4#319214yes—
694C4#355212yes—
695C4#356112yes—
696C2—42no—
697C2—82no—
698C2—82no—
699C2#4291211yescharacteristic
700C2C2×#319211yescharacteristic
701C2#6075911yescharacteristic
702C2#6210911yescharacteristic
703C2#6214811yescharacteristic
704C2#6247911yescharacteristic
705C2#6250911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36444C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35316C2semidirect
3. #32790C2semidirect
4. #32271C2semidirect
5. #32012C2semidirect
6. #30296C2semidirect
7. #30172C2semidirect
8. #29937C2semidirect
9. #29162C2semidirect
10. #28712C2semidirect
11. #26278C2semidirect
12. #25838C2semidirect
13. #25813C2semidirect
14. #25347C2semidirect
15. #24298C2semidirect
16. #23755C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#364442
#353162
#327902
#322712
#320122
#302962
#301722
#299372
#291622
#287122
#262782
#258382
#258132
#253472
#242982
#237552
#224732