Finite Group Databasebeta

← #6852156

Subgroups — #6852156

Order 512.

1711 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#32690C214yesmaximal, complemented
5#32584C214yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29303C212yesmaximal, complemented
8#29235C212yesmaximal
9#28684C212yesmaximal, complemented
10#26283C212yesmaximal, complemented
11#25835C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25478C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#498C2212yes—
23C2×#480C2211yescharacteristic
24C2×#474C2212yes—
25C2×#446C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2584C2212yes—
29#2580C2212yes—
30#2567C2211yescharacteristic
31#2546—21no—
32#2546—21no—
33#2546—22no—
34#2546C2212yes—
35#2546C2214yes—
36#2546C2212yes—
37#2538C2211yescharacteristic
38#2465—22no—
39#2449—22no—
40#2423—21no—
41#2409—21no—
42C2×#398—22no—
43#2360—21no—
44#2355C2211yescharacteristic
45D4×C4⋊C4—21no—
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2088C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2085C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2058C2214yes—
61#2057C2212yes—
62#2023C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1948C2212yes—
69#1944C2214yes—
70#1943C2214yes—
71#1943C2214yes—
72#1936C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1918C2212yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1864C2214yes—
81#1864C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2212yes—
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1823C2211yescharacteristic
88#1823C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#554—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D412yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#498—44no—
138#480Q818yescomplemented
139#474—44no—
140#446—44no—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#399—22no—
158#398—22no—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#396—22no—
164#395—22no—
165#395—22no—
166#393—22no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391—24no—
172#391—21no—
173#390C2312yes—
174#390C2311yescharacteristic
175#389—22no—
176#389—22no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2312yes—
190#389C2314yes—
191#388—22no—
192#388—21no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—21no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#384—21no—
203#384C2311yescharacteristic
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380C2311yescharacteristic
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95—22no—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—416no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272#125—44no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—82no—
277#121—22no—
278#121C2×Q814yes—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×Q814yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—44no—
359#107C2×Q818yescomplemented
360#106—28no—
361#106—44no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#102—24no—
366#102—44no—
367#102C2×Q818yescomplemented
368C22⋊C8—24no—
369C22⋊C8—22no—
370C22⋊C8C2×Q814yes—
371#97—44no—
372C4×C8—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24—21no—
391C24C22×D411yescharacteristic
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—416no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×Q8—816no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C22×Q812yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4AGL(1,ℤ/8)14yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q812yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8C22×Q814yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4C22×Q814yes—
513C4⋊C4C22×Q812yes—
514C4⋊C42−1+414yes—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—42no—
534C42—82no—
535C42H(5,𝔽2)14yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23D4×Q812yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23#57712yes—
550C23C23×D411yescharacteristic
551Q8—48no—
552Q8—44no—
553Q8—44no—
554Q8—88no—
555Q8—88no—
556Q8—162no—
557D4—24no—
558D4—48no—
559D4—42no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—164no—
616C2×C4#53412yes—
617C2×C4#53412yes—
618C2×C4D4×Q812yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4#57512yes—
622C2×C4C23×Q811yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×2−1+412yes—
625C2×C4C2×2−1+411yescharacteristic
626C8—24no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22#338414yes—
643C22C2×#53411yescharacteristic
644C22C2×D4×Q811yescharacteristic
645C22#348611yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—81no—
662C4—84no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4#319414yes—
667C4#350312yes—
668C4#355212yes—
669C2—24no—
670C2—82no—
671C2#3959311yescharacteristic
672C2C2×#319411yescharacteristic
673C2#6075811yescharacteristic
674C2C2×#338411yescharacteristic
675C2#6136911yescharacteristic
676C2#6182711yescharacteristic
677C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #33148C2semidirect
4. #32690C2semidirect
5. #32584C2semidirect
6. #29972C2semidirect
7. #29303C2semidirect
9. #28684C2semidirect
10. #26283C2semidirect
11. #25835C2semidirect
12. #25811C2semidirect
13. #25478C2semidirect
15. #22510C2semidirect
138. #480Q8semidirect
359. #107C2×Q8semidirect
367. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365442
C2×#25462
#331482
#326902
#325842
#299722
#293032
#292352
#286842
#262832
#258352
#258112
#254782
#242982
#225102
#220812
C4×#4802