Finite Group Databasebeta

← #627069

Subgroups — #627069

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic
3C2×#2316C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic
5#15678C214yesmaximal, complemented
6#15676C214yesmaximal, complemented
7#15623C214yesmaximal, complemented
8#15618C214yesmaximal, complemented
9#14935C212yesmaximal
10#14728C212yesmaximal
11#14702C211yesmaximal, characteristic
12#14697C211yesmaximal, characteristic
13#14339C211yesmaximal, characteristic
14#13962C212yesmaximal
15#13870C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2627—28no—
24#2627—28no—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33#2316—28no—
34C2×#391C2212yes—
35C2×#389—28no—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1444C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4D418yes—
103C22×C4⋊C4Q818yescomplemented
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42—24no—
127C22×C42—44no—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131#400—28no—
132#400—28no—
133#398—216no—
134#398—28no—
135#398—28no—
136#398—48no—
137#395—28no—
138#395—28no—
139#391—44no—
140#389—216no—
141#389—216no—
142#389—216no—
143#389—216no—
144#389—44no—
145#385—28no—
146#385—28no—
147#385—44no—
148#381—28no—
149#381—28no—
150#381—44no—
151#380—24no—
152#380—24no—
153#380—28no—
154#380D418yes—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—42no—
188C22×Q8—42no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×Q814yes—
224C23×C4C2×Q814yes—
225C23×C4C2×Q814yes—
226C23×C4#4114yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2412yes—
242C22⋊Q8—416no—
243#117—432no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×Q8116yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4C2×D418yes—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—416no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C22×Q812yes—
353C24C2×#4112yes—
354C24C2×#4112yes—
355C24H(5,𝔽2)12yes—
356C242−1+412yes—
357C24C2511yescharacteristic
358C2×Q8—48no—
359C2×Q8—44no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8—84no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4#121116yes—
467C22×C4C22⋊Q8116yes—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q814yes—
477C22×C4C22×Q814yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+414yes—
483C22×C42−1+414yes—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—232no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C42—48no—
507C42—48no—
508C42—416no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23C2×#12114yes—
526C23C2×C22⋊Q814yes—
527C23#53714yes—
528C23#54614yes—
529C23D4×Q814yes—
530C23#55214yes—
531C23#55414yes—
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+412yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543Q8—88no—
544C2×C4—232no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—42no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4C2×#12114yes—
630C2×C4C2×#12114yes—
631C2×C4C2×C22⋊Q818yes—
632C2×C4#54418yes—
633C22—24no—
634C22—28no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22#241718yes—
644C22C22×#12111yescharacteristic
645C22C22×C22⋊Q811yescharacteristic
646C22C2×#53711yescharacteristic
647C22C2×#54412yes—
648C22#346112yes—
649C22C2×#54611yescharacteristic
650C22C2×D4×Q811yescharacteristic
651C22C2×#55211yescharacteristic
652C22C2×#55411yescharacteristic
653C22#348411yescharacteristic
654C22#348511yescharacteristic
655C22#348611yescharacteristic
656C22#348611yescharacteristic
657C22#349212yes—
658C22#349612yes—
659C22#349912yes—
660C22#350212yes—
661C22#350512yes—
662C22#350912yes—
663C22#353612yes—
664C22#353812yes—
665C22#355912yes—
666C22#356312yes—
667C4—24no—
668C4—24no—
669C4—28no—
670C4—28no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—84no—
681C2—28no—
682C2C2×#241711yescharacteristic
683C2#3515411yescharacteristic
684C2#3522411yescharacteristic
685C2#3535811yescharacteristic
686C2#3566711yescharacteristic
687C2#3671911yescharacteristic
688C2#3696811yescharacteristic
689C2#3936812yes—
690C2#3941312yes—
691C2#3949612yes—
692C2#3954112yes—
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
5. #15678C2semidirect
6. #15676C2semidirect
7. #15623C2semidirect
8. #15618C2semidirect
103. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23162
C22×#3892
#156782
#156762
#156232
#156182
#149352
#147282
#147022
#146972
#143392
#139622
#138702