Finite Group Databasebeta

← #192757

Subgroups — #192757

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21365C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1488C211yesmaximal, characteristic
6#8897C211yesmaximal, characteristic, complemented
7#8895C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8891C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic, complemented
11#8887C211yesmaximal, characteristic, complemented
12#8885C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#7892C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2357—21no—
21#2356—21no—
22#2355C2211yescharacteristic
23#2355C2211yescharacteristic
24#2354C2211yescharacteristic
25#2354C2211yescharacteristic
26#2353—22no—
27#2352—21no—
28#2352—21no—
29#2298—22no—
30#2297—21no—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2283—21no—
34C4×#117—21no—
35C4×#117—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1854—22no—
43#1853—21no—
44#1852—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1825—21no—
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54#1788—21no—
55#1776—21no—
56#1776—21no—
57C4×#103—21no—
58C4×#102—21no—
59#1488C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C22×#41D411yescharacteristic
73C4×#41—22no—
74C4×#41—22no—
75C4×#41C2×C414yes—
76C4×#41Q814yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4D411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117—21no—
88C2×#117D411yescharacteristic
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#432—22no—
95#432C2×C414yes—
96#431—22no—
97#431—22no—
98#431—22no—
99#431C2×C414yes—
100C2×#107—21no—
101C2×#107—21no—
102C2×#107—22no—
103C2×#106—21no—
104C2×#106—21no—
105C2×#106—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#391—21no—
124#390—21no—
125#390—21no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#386—22no—
135#386—22no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#383—22no—
141#382—21no—
142#382—21no—
143#382—21no—
144#382—21no—
145#381—22no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380D412yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4D412yes—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C22⋊C4—22no—
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—22no—
173#376—22no—
174#376—22no—
175C2×#95—21no—
176C43—22no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—22no—
182#340—22no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—21no—
187Q8⋊C8C2×C414yescomplemented
188Q8⋊C8C2×C414yescomplemented
189D4⋊C8C2×C414yescomplemented
190D4⋊C8C2×C414yescomplemented
191C2×#41—22no—
192C2×#41—41no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—41no—
196C2×#41C4⋊C414yes—
197C22×Q8C2×D411yescharacteristic
198C22×D4—21no—
199C22×D4C2×D411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C4×Q8—22no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8C4⋊C414yes—
218C4×Q8C22×C412yes—
219C4×Q8C2×Q812yes—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4C4⋊C414yes—
228C4×D4C22×C412yes—
229C4×D4C2×Q812yes—
230#117—22no—
231#117—24no—
232#117—24no—
233#117—42no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—82no—
240#117C4⋊C414yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—41no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—41no—
293C2×C42C22×C412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×Q812yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C2×C42C2411yescharacteristic
301#107—42no—
302#107—42no—
303#107—82no—
304#106—42no—
305#106—42no—
306#106—82no—
307C4⋊C8—24no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8C22×C412yes—
315C4⋊C8C22×C412yes—
316#103—42no—
317#103—42no—
318#102—42no—
319#102—42no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—42no—
323#97—24no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—42no—
327C4×C8C22×C412yes—
328C4×C8C22×C412yes—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—21no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—41no—
344#95—41no—
345C24—21no—
346C24#12011yescharacteristic
347#41—44no—
348#41—44no—
349#41—44no—
350#41—44no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C2×C4⋊C412yes—
357C2×D4—24no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4C2×C4⋊C412yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4C2×C4⋊C412yes—
406C22×C4C4×D412yes—
407C22×C4C4×D412yes—
408C22×C4#12011yescharacteristic
409C22×C4#12011yescharacteristic
410C22×C4#12011yescharacteristic
411C22×C4C22⋊Q812yes—
412C22×C4C22⋊Q812yes—
413C22×C4#12312yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×C4⋊C412yes—
469C4⋊C4C2×C4⋊C412yes—
470C4⋊C4#13512yes—
471C4⋊C4#13512yes—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—42no—
487C42—42no—
488C42—44no—
489C42—44no—
490C42—44no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—82no—
496C42—82no—
497C42—82no—
498C42C2×C4⋊C412yes—
499C42C4×D412yes—
500C42C4×D412yes—
501C42C22⋊Q812yes—
502C42C22⋊Q812yes—
503C42#12312yes—
504C42C23×C411yescharacteristic
505C42C22×Q811yescharacteristic
506C42C2×#4111yescharacteristic
507C42C2×#4111yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23#38512yes—
522C23C2×#12011yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4#38512yes—
592C2×C4#38512yes—
593C2×C4#38512yes—
594C2×C4C22×C4⋊C411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4C2×#13511yescharacteristic
601C2×C4C2×#13511yescharacteristic
602C2×C4#57511yescharacteristic
603C2×C4#57511yescharacteristic
604C2×C4#57511yescharacteristic
605C2×C4#57511yescharacteristic
606C2×C4#57511yescharacteristic
607C2×C4#57511yescharacteristic
608C8—42no—
609C8—42no—
610C8—42no—
611C8—42no—
612C8—81no—
613C8—81no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C2×#38511yescharacteristic
629C22#300211yescharacteristic
630C22#300211yescharacteristic
631C22#300511yescharacteristic
632C22#300511yescharacteristic
633C22#302111yescharacteristic
634C22#302211yescharacteristic
635C4—22no—
636C4—21no—
637C4—22no—
638C4—21no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—161no—
650C4—161no—
651C4#291111yescharacteristic
652C4#291111yescharacteristic
653C4C4×#13511yescharacteristic
654C4#294311yescharacteristic
655C4#307611yescharacteristic
656C4#307611yescharacteristic
657C4#309311yescharacteristic
658C4#309311yescharacteristic
659C2—42no—
660C2—82no—
661C2#1879511yescharacteristic
662C2#1879611yescharacteristic
663C2#1880711yescharacteristic
664C2#1880811yescharacteristic
665C2#2084511yescharacteristic
666C2#2086511yescharacteristic
667C2#2087211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21984C2semidirect
3. #21365C2semidirect
6. #8897C2semidirect
7. #8895C2semidirect
8. #8893C2semidirect
9. #8891C2semidirect
10. #8889C2semidirect
11. #8887C2semidirect
12. #8885C2semidirect
13. #8883C2semidirect
14. #8046C2semidirect
15. #7892C2semidirect
59. #1488C4semidirect
187. Q8⋊C8C2×C4semidirect
188. Q8⋊C8C2×C4semidirect
189. D4⋊C8C2×C4semidirect
190. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341212
#219842
#213652
#207372
C2×#14882
#88972
#88952
#88932
#88912
#88892
#88872
#88852
#88832
#80462
#78922