Finite Group Databasebeta

← #6856360

Subgroups — #6856360

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3#32906C211yesmaximal, characteristic, complemented
4#32611C214yesmaximal, complemented
5#32553C214yesmaximal, complemented
6#29719C212yesmaximal
7#29330C212yesmaximal
8#29243C212yesmaximal, complemented
9#28186C212yesmaximal, complemented
10#27972C211yesmaximal, characteristic, complemented
11#27677C211yesmaximal, characteristic
12#25294C212yesmaximal, complemented
13#24995C212yesmaximal, complemented
14#24189C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22044C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#462C2212yes—
22C2×#451C2212yes—
23C2×#448C2211yescharacteristic
24C2×C8×Q8C2212yes—
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2850C2212yes—
28#2592C2212yes—
29#2582—21no—
30#2582—21no—
31#2582—22no—
32#2582C2212yes—
33#2582C2214yes—
34#2582C2212yes—
35#2581C2212yes—
36#2568C2211yescharacteristic
37#2538C2211yescharacteristic
38#2481—22no—
39#2469—22no—
40#2421—21no—
41#2409—21no—
42C2×#398—22no—
43#2362—21no—
44#2355C2211yescharacteristic
45#2352—21no—
46C4×C22⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389C2212yes—
52C2×#384C2211yescharacteristic
53#2101C2214yes—
54#2099C2212yes—
55#2093C2212yes—
56#2088C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2042C2214yes—
63#2015C2214yes—
64#1984C2214yes—
65#1984C2212yes—
66#1969C2214yes—
67#1956C2212yes—
68#1947C2212yes—
69#1934C2214yes—
70#1933C2214yes—
71#1933C2214yes—
72#1928C2214yes—
73#1918C2214yes—
74#1880C2212yes—
75#1878C2214yes—
76#1878C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2212yes—
83#1865C2214yes—
84#1853C2211yescharacteristic
85#1852C2212yes—
86#1834C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99#556—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#124C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D412yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D412yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#462—44no—
135#451—28no—
136#448—24no—
137C8×Q8—44no—
138C4×SD16—44no—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#400—22no—
158#400—22no—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#397—22no—
164#396—22no—
165#395—22no—
166#393—22no—
167#391—21no—
168#391—22no—
169#391—24no—
170#391—22no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2314yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—24no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389C2314yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2314yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—21no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#386—22no—
202#386—22no—
203#386—21no—
204#385—22no—
205#384—21no—
206#384—21no—
207#384C2312yes—
208#384C2311yescharacteristic
209#384C2312yes—
210#382—21no—
211#382C2312yes—
212#380—21no—
213#380—21no—
214#380—21no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8#4114yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×Q16—28no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×SD16—42no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270#126—48no—
271#124—44no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—22no—
277#121—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—44no—
293#117—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#107—44no—
360#106—42no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—24no—
364#103—28no—
365#103—44no—
366#103—42no—
367#103—44no—
368#103—42no—
369#103—44no—
370#102—44no—
371#102—42no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374C22⋊C8—22no—
375C4×C8—24no—
376C4×C8—44no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24—21no—
397C24C22×D411yescharacteristic
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—42no—
416C2×Q8—48no—
417C2×Q8—416no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—48no—
421C2×Q8#13714yes—
422C2×Q8C2×#4114yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4#13514yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476Q16—88no—
477SD16—48no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42H(5,𝔽2)14yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23#54812yes—
557C23C2×#13512yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—28no—
561Q8—24no—
562Q8—24no—
563Q8—48no—
564Q8—48no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—44no—
570D4—42no—
571D4—42no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4#53412yes—
628C2×C4#53412yes—
629C2×C4#54812yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13714yes—
633C2×C4C2×#13712yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58512yes—
638C8—42no—
639C8—44no—
640C8—42no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#336414yes—
655C22C2×#53411yescharacteristic
656C22C2×#54811yescharacteristic
657C22#348711yescharacteristic
658C22C22×#13511yescharacteristic
659C22C22×#13711yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—22no—
664C4—24no—
665C4—22no—
666C4—44no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—84no—
678C4#319014yes—
679C4#350512yes—
680C4#354112yes—
681C2—24no—
682C2—161no—
683C2#3972311yescharacteristic
684C2C2×#319011yescharacteristic
685C2#6076411yescharacteristic
686C2C2×#336411yescharacteristic
687C2#6131911yescharacteristic
688C2#6188511yescharacteristic
689C2#6190011yescharacteristic
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×#2582C2semidirect
3. #32906C2semidirect
4. #32611C2semidirect
5. #32553C2semidirect
8. #29243C2semidirect
9. #28186C2semidirect
10. #27972C2semidirect
12. #25294C2semidirect
13. #24995C2semidirect
14. #24189C2semidirect
15. Q8×C22⋊C8C2semidirect
16. #22044C2semidirect

Maximal subgroups

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

subgroupindex
#365792
C2×#25822
#329062
#326112
#325532
#297192
#293302
#292432
#281862
#279722
#276772
#252942
#249952
#241892
Q8×C22⋊C82
#220442
C4×#4482