Finite Group Databasebeta

← #6560042

Subgroups — #6560042

Order 512.

1905 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36415C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30200C212yesmaximal, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29139C211yesmaximal, characteristic, complemented
7#29113C212yesmaximal, complemented
8#27821C212yesmaximal, complemented
9#27776C212yesmaximal, complemented
10#27706C214yesmaximal, complemented
11#27282C214yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23452C212yesmaximal, complemented
15#22503C212yesmaximal, complemented
16#22431C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×#547C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C2×#489C2214yes—
23C2×#436C2212yes—
24C2×#417C2214yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2563C2211yescharacteristic
30#2557—22no—
31#2557—22no—
32#2551—21no—
33#2551C2212yes—
34#2547C2211yescharacteristic
35#2546—21no—
36#2535—21no—
37#2535—21no—
38#2499C2212yes—
39#2440—22no—
40#2440C2214yes—
41#2434—22no—
42#2431C2212yes—
43#2352—21no—
44#2352—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49#2297C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2072C2214yes—
54#2072C2212yes—
55#2071C2211yescharacteristic
56#2071C2211yescharacteristic
57#2042C2214yes—
58#2041C2212yes—
59#2024C2212yes—
60#2024C2212yes—
61#2024C2214yes—
62#2024C2212yes—
63#2023C2211yescharacteristic
64#2023C2211yescharacteristic
65#2023C2212yes—
66#2023C2214yes—
67#2023C2212yes—
68#1993C2214yes—
69#1990C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1957C2212yes—
73#1956C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1933C2214yes—
77#1895C2214yes—
78#1894C2214yes—
79#1874C2214yes—
80#1872C2214yes—
81#1834C2212yes—
82#1825C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#102C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C23×Q8—22no—
97C22×M16C2314yes—
98#555—28no—
99D4×Q8—48no—
100#547—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#124—22no—
114C2×#124C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2312yes—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121C2312yes—
122C2×C4×Q8—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117C2312yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2314yes—
136C22×C42C2312yes—
137#489—48no—
138#436—28no—
139C2×C4⋊C8C2312yes—
140#417—48no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#397—22no—
157#397—22no—
158#397—22no—
159#395—24no—
160#395—24no—
161#393—22no—
162#392—22no—
163#392—22no—
164#391—21no—
165#391—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#390—21no—
173#390C2311yescharacteristic
174#390C2311yescharacteristic
175#390C2312yes—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389C2312yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—21no—
202#382C2312yes—
203#380—22no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380C2314yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—28no—
237C2×#41—42no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—21no—
243C22×Q8—48no—
244C22×Q8—48no—
245C22×Q8—84no—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—24no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×M16—28no—
262C2×M16—44no—
263C2×M16—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—28no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274C4⋊Q8—48no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—82no—
278#127—22no—
279#127—41no—
280#124—44no—
281#124—48no—
282#123—48no—
283C22⋊Q8—28no—
284C22⋊Q8—416no—
285C22⋊Q8—48no—
286#121—28no—
287#121—44no—
288#121—44no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4#4118yes—
298#117—28no—
299#117—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—24no—
362#103—28no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—42no—
370#102—44no—
371#102—44no—
372C22⋊C8—28no—
373#97—44no—
374C4×C8—24no—
375C4×C8—42no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24—21no—
393C24C22×D412yes—
394#41—416no—
395#41—88no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—84no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8—832no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4#12414yes—
435C2×D42−1+412yes—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4118yes—
474C22×C4C2511yescharacteristic
475M16—88no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4#12414yes—
514C4⋊C4H(5,𝔽2)14yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4#12418yes—
532C42—24no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42—42no—
538C42—41no—
539C42—41no—
540C42—44no—
541C42—44no—
542C42AGL(1,ℤ/8)14yes—
543C42#13714yes—
544C42C2×#4114yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×#12414yes—
566C23C23×D411yescharacteristic
567Q8—48no—
568Q8—44no—
569Q8—44no—
570Q8—88no—
571Q8—816no—
572Q8—162no—
573D4—28no—
574D4—24no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—84no—
579D4—84no—
580D4—88no—
581D4—164no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—88no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4C2×#12414yes—
630C2×C4C2×#12412yes—
631C2×C4C2×#12412yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×#13712yes—
636C2×C4C2×#13712yes—
637C2×C4C2×#13712yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—44no—
643C8—42no—
644C8—42no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22C22×#12411yescharacteristic
667C22#348411yescharacteristic
668C22#348911yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C22×#13711yescharacteristic
671C22#358511yescharacteristic
672C22#358611yescharacteristic
673C4—24no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4#314114yes—
689C4#351412yes—
690C4#352012yes—
691C2—44no—
692C2—82no—
693C2—161no—
694C2#3593911yescharacteristic
695C2C2×#314111yescharacteristic
696C2#6062511yescharacteristic
697C2#6158211yescharacteristic
698C2#6158411yescharacteristic
699C2#6210711yescharacteristic
700C2#6210811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36415C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Q8C2semidirect
3. #30205C2semidirect
4. #30200C2semidirect
5. #29347C2semidirect
6. #29139C2semidirect
7. #29113C2semidirect
8. #27821C2semidirect
9. #27776C2semidirect
10. #27706C2semidirect
11. #27282C2semidirect
12. #25908C2semidirect
13. #23815C2semidirect
14. #23452C2semidirect
15. #22503C2semidirect
16. #22431C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364152
D4×C4⋊Q82
#302052
#302002
#293472
#291392
#291132
#278212
#277762
#277062
#272822
#259082
#238152
#234522
#225032
#224312
#205682