Finite Group Databasebeta

← #6485005

Subgroups — #6485005

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#33096C212yesmaximal, complemented
5#29907C212yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#26318C212yesmaximal, complemented
8#26316C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26280C212yesmaximal, complemented
11#25068C214yesmaximal, complemented
12#25002C214yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic, complemented
16#21395C211yesmaximal, characteristic
17#21361C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2854C2212yes—
24#2854C2212yes—
25#2632—22no—
26#2580C2212yes—
27#2580C2212yes—
28#2567C2211yescharacteristic
29#2567C2211yescharacteristic
30#2548C2211yescharacteristic
31#2548C2211yescharacteristic
32#2501—21no—
33#2436—21no—
34#2406—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4—22no—
38Q8×C4⋊C4C2212yes—
39Q8×C4⋊C4C2214yes—
40Q8×C4⋊C4C2212yes—
41#2326—21no—
42#2326—21no—
43C2×#389—22no—
44C2×#389—21no—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47#2283C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2214yes—
51#2099C2214yes—
52#2094C2214yes—
53#2093C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2071C2214yes—
57#2071C2214yes—
58#1990C2212yes—
59#1987C2212yes—
60#1985C2214yes—
61#1984C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1957C2212yes—
65#1949C2212yes—
66#1949C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1926C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1853C2211yescharacteristic
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1852C2211yescharacteristic
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1847C2212yes—
88#1836C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91#1802C2214yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1762C2214yes—
95C22×#41—21no—
96C23×Q8D411yescharacteristic
97#532—24no—
98#532—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#124—21no—
103C2×#124—21no—
104C2×#123—22no—
105C2×#123—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8—48no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D412yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—22no—
132C22×C42—21no—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#432—24no—
136#431—24no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154#399—22no—
155#399—22no—
156#395—22no—
157#395—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—22no—
172#389—24no—
173#389—24no—
174#389—24no—
175#389—24no—
176#389—24no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2314yes—
199#384C2314yes—
200#384C2312yes—
201#383—21no—
202#382C2311yescharacteristic
203#382C2311yescharacteristic
204#381—21no—
205#380C2311yescharacteristic
206#380C2311yescharacteristic
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218#376—21no—
219C2×#95—22no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—44no—
236C2×#41—82no—
237C2×#41—82no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8C2×D412yes—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#127—22no—
264#127—22no—
265#124—44no—
266#124—44no—
267#124—44no—
268#123—84no—
269#123—84no—
270#121—24no—
271#121—24no—
272#121—44no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—28no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—416no—
283C4×Q8—48no—
284C4×Q8—48no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—44no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
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×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—48no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42#4114yes—
355C2×C42#4114yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—44no—
362#107—42no—
363#106—24no—
364#106—44no—
365#106—42no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375#97—24no—
376C4×C8—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—22no—
394C24C22×D411yescharacteristic
395#41—816no—
396C2×Q8—232no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8AGL(1,ℤ/8)14yes—
408C2×Q8#13714yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4#12314yes—
455C22×C4#12314yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2511yescharacteristic
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C42−1+414yes—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—44no—
531C42—48no—
532C42—48no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42#12314yes—
537C42#12314yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C2×#12312yes—
554C23#57612yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—216no—
558Q8—28no—
559Q8—28no—
560Q8—84no—
561D4—44no—
562D4—44no—
563D4—84no—
564D4—88no—
565D4—84no—
566C2×C4—216no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4C2×#12312yes—
623C2×C4C2×AGL(1,ℤ/8)14yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×2−1+412yes—
630C2×C4#58512yes—
631C8—42no—
632C8—42no—
633C8—44no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22#309614yes—
648C22C22×#12311yescharacteristic
649C22#348611yescharacteristic
650C22#348711yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57611yescharacteristic
654C22C2×#57711yescharacteristic
655C4—22no—
656C4—24no—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4—84no—
671C4#309214yes—
672C4#346112yes—
673C4#350712yes—
674C2—24no—
675C2—162no—
676C2#3518211yescharacteristic
677C2C2×#309211yescharacteristic
678C2C2×#309611yescharacteristic
679C2#6184311yescharacteristic
680C2#6184411yescharacteristic
681C2#6190611yescharacteristic
682C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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×Q8×C4⋊C4C2semidirect
3. #33096C2semidirect
4. #33096C2semidirect
5. #29907C2semidirect
6. #29906C2semidirect
7. #26318C2semidirect
8. #26316C2semidirect
9. #26282C2semidirect
10. #26280C2semidirect
11. #25068C2semidirect
12. #25002C2semidirect
13. #21987C2semidirect
15. #21820C2semidirect
17. #21361C2semidirect

Maximal subgroups

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

subgroupindex
#352492
C2×Q8×C4⋊C42
#330962
#299072
#299062
#263182
#263162
#262822
#262802
#250682
#250022
#219872
#218592
#218202
#213952
#213612