Finite Group Databasebeta

← #642306

Subgroups — #642306

Order 512.

2658 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#17079C211yesmaximal, characteristic, complemented
4#17078C211yesmaximal, characteristic, complemented
5#17075C211yesmaximal, characteristic, complemented
6#17074C211yesmaximal, characteristic, complemented
7#16900C212yesmaximal, complemented
8#16820C212yesmaximal, complemented
9#15166C212yesmaximal, complemented
10#15164C212yesmaximal, complemented
11#14826C212yesmaximal
12#14821C214yesmaximal
13#14758C212yesmaximal
14#14316C214yesmaximal, complemented
15#13958C212yesmaximal
16#13878C212yesmaximal
17C2×C4⋊#95C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2855—28no—
24C2×#400C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#395C2212yes—
28#2296—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—44no—
96C2×#117Q818yescomplemented
97C22×C4⋊C4—28no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42—44no—
124C22×C42D418yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#400—44no—
132#398—216no—
133#398—48no—
134#395—44no—
135#391—24no—
136#391—24no—
137#389—24no—
138#389—24no—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#387—28no—
144#387—28no—
145#385—28no—
146#385—28no—
147#385—28no—
148#381—28no—
149#380—24no—
150#380—48no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C2×#95—28no—
154C2×#95—44no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C22⋊Q8—416no—
231#117—232no—
232#117—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—432no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×Q814yes—
281C2×C4⋊C4C2×Q814yes—
282C2×C4⋊C4#4118yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4C2×Q818yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—416no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×Q818yes—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24C22×D412yes—
344C24C22×Q812yes—
345C24C2×#4112yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—84no—
353C2×Q8—84no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22⋊Q8116yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C42—216no—
486C42—44no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42—48no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23C2×C22⋊Q814yes—
510C23#53414yes—
511C23#53614yes—
512C23#53918yes—
513C23#54418yes—
514C23C23×D411yescharacteristic
515C23C23×Q811yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+412yes—
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23#58512yes—
525Q8—88no—
526C2×C4—232no—
527C2×C4—232no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—832no—
611C2×C4C2×C22⋊Q814yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4#53214yes—
614C2×C4#53714yes—
615C2×C4#53714yes—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22#262718yes—
627C22C22×C22⋊Q811yescharacteristic
628C22C2×#53211yescharacteristic
629C22C2×#53411yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#53711yescharacteristic
632C22C2×#53711yescharacteristic
633C22#344911yescharacteristic
634C22C2×#53912yes—
635C22C2×#54412yes—
636C22#347512yes—
637C22Q8×#4112yes—
638C22#349012yes—
639C22#349212yes—
640C22#350311yescharacteristic
641C22#350311yescharacteristic
642C22#350811yescharacteristic
643C22#350911yescharacteristic
644C22#352012yes—
645C22#352312yes—
646C22#353412yes—
647C22#353512yes—
648C22#356112yes—
649C22#356312yes—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—48no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C2—28no—
667C2C2×#262711yescharacteristic
668C2#3494711yescharacteristic
669C2#3505111yescharacteristic
670C2#3524111yescharacteristic
671C2#3528011yescharacteristic
672C2#3745211yescharacteristic
673C2#3755511yescharacteristic
674C2#3844612yes—
675C2#3844712yes—
676C2#3846412yes—
677C2#3846512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17079C2semidirect
4. #17078C2semidirect
5. #17075C2semidirect
6. #17074C2semidirect
7. #16900C2semidirect
8. #16820C2semidirect
9. #15166C2semidirect
10. #15164C2semidirect
14. #14316C2semidirect
96. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22962
#170792
#170782
#170752
#170742
#169002
#168202
#151662
#151642
#148262
#148212
#147582
#143162
#139582
#138782
C2×C4⋊#952