Finite Group Databasebeta

← #6856088

Subgroups — #6856088

Order 512.

1743 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#32639C214yesmaximal, complemented
5#32584C214yesmaximal, complemented
6#29760C212yesmaximal, complemented
7#29336C212yesmaximal, complemented
8#29235C212yesmaximal, complemented
9#28648C212yesmaximal, complemented
10#26281C212yesmaximal, complemented
11#25903C211yesmaximal, characteristic
12#25652C212yesmaximal, complemented
13#25479C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#486C2212yes—
23C2×#483C2211yescharacteristic
24C2×#480C2212yes—
25C2×#475C2212yes—
26C2×#446C2212yes—
27#2856C2211yescharacteristic
28#2850C2212yes—
29#2592C2212yes—
30#2582C2211yescharacteristic
31#2580C2212yes—
32#2548C2211yescharacteristic
33#2546—21no—
34#2546—21no—
35#2546—22no—
36#2546C2212yes—
37#2546C2214yes—
38#2546C2212yes—
39#2458—22no—
40#2448—21no—
41#2443—22no—
42#2436—21no—
43C2×#398—22no—
44#2367—21no—
45#2361—21no—
46#2355C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#391—21no—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2088C2214yes—
55#2085C2214yes—
56#2068C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2058C2212yes—
60#2042C2214yes—
61#2042C2214yes—
62#2024C2212yes—
63#2023C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1982C2212yes—
69#1947C2212yes—
70#1943C2214yes—
71#1936C2214yes—
72#1936C2214yes—
73#1935C2214yes—
74#1927C2212yes—
75#1925C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1920C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1874C2212yes—
82#1872C2212yes—
83#1872C2211yescharacteristic
84#1864C2214yes—
85#1864C2214yes—
86#1853C2211yescharacteristic
87#1853C2212yes—
88#1823C2211yescharacteristic
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#559—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⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#124—21no—
109C2×#124—21no—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—44no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D412yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D412yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#486—44no—
139#483—24no—
140#480—28no—
141#475—44no—
142#446—44no—
143C2×#107C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#399—22no—
162#398—22no—
163#398—24no—
164#398—24no—
165#398—24no—
166#398—24no—
167#397—22no—
168#394—22no—
169#393—22no—
170#393—22no—
171#391—21no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—24no—
176#391C2312yes—
177#391C2311yescharacteristic
178#390—21no—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2311yescharacteristic
190#389C2314yes—
191#389C2314yes—
192#389C2312yes—
193#389C2314yes—
194#389C2314yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—21no—
202#386—22no—
203#386—22no—
204#386—22no—
205#385—21no—
206#385—22no—
207#385—22no—
208#384C2311yescharacteristic
209#382C2312yes—
210#380C2311yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—22no—
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×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—416no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273#127—24no—
274#124—44no—
275#124—44no—
276#124—44no—
277#123—48no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280#121—24no—
281#121—24no—
282#121—44no—
283#121—82no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—84no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
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⋊C4#4114yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—28no—
361#106—24no—
362#106—24no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—24no—
368#103—44no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373C22⋊C8—24no—
374C22⋊C8—22no—
375C22⋊C8—22no—
376#97—44no—
377C4×C8—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24C22×D411yescharacteristic
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—416no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—816no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4AGL(1,ℤ/8)14yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4C2×#4114yes—
519C4⋊C42−1+414yes—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42H(5,𝔽2)14yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23#54812yes—
557C23C2×AGL(1,ℤ/8)12yes—
558C23#57712yes—
559C23C23×D411yescharacteristic
560Q8—48no—
561Q8—44no—
562Q8—44no—
563Q8—42no—
564Q8—42no—
565Q8—88no—
566Q8—88no—
567D4—44no—
568D4—44no—
569D4—88no—
570D4—84no—
571D4—162no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—164no—
627C2×C4#53412yes—
628C2×C4#53412yes—
629C2×C4#54812yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4#57512yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×2−1+412yes—
636C2×C4C2×2−1+411yescharacteristic
637C8—44no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#337314yes—
654C22C2×#53411yescharacteristic
655C22C2×#54811yescharacteristic
656C22#348611yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—44no—
671C4—81no—
672C4—84no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4#319414yes—
678C4#351112yes—
679C4#355212yes—
680C2—24no—
681C2—162no—
682C2#3972111yescharacteristic
683C2C2×#319411yescharacteristic
684C2#6075811yescharacteristic
685C2C2×#337311yescharacteristic
686C2#6132011yescharacteristic
687C2#6182711yescharacteristic
688C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
4. #32639C2semidirect
5. #32584C2semidirect
6. #29760C2semidirect
7. #29336C2semidirect
8. #29235C2semidirect
9. #28648C2semidirect
10. #26281C2semidirect
12. #25652C2semidirect
13. #25479C2semidirect
14. #24301C2semidirect
15. #22510C2semidirect
16. #22081C2semidirect
17. C4×#483C2semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25462
#331062
#326392
#325842
#297602
#293362
#292352
#286482
#262812
#259032
#256522
#254792
#243012
#225102
#220812
C4×#4832