Finite Group Databasebeta

← #554187

Subgroups — #554187

Order 512.

2992 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×#2542C211yesmaximal, characteristic
2C2×#2327C211yesmaximal, characteristic, complemented
3C2×#2290C211yesmaximal, characteristic
4#14702C212yesmaximal, complemented
5#14584C214yesmaximal, complemented
6#14578C214yesmaximal, complemented
7#14431C214yesmaximal
8#14425C214yesmaximal, complemented
9#14258C212yesmaximal, complemented
10#14155C211yesmaximal, characteristic, complemented
11#13969C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13955C211yesmaximal, characteristic, complemented
14#13905C211yesmaximal, characteristic
15#13797C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26#2327—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2212yes—
31#2290—28no—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#387C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#384C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1445C2214yes—
49#1445C2212yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1437C2212yes—
61#1437C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8C2×C418yescomplemented
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8C2×C4116yes—
91C2×C22⋊Q8C2×C4116yescomplemented
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4C2×C418yes—
98C2×#117—24no—
99C2×#117—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42—24no—
121C22×C42—44no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#399C2×C4116yes—
132#399C2×C4116yescomplemented
133#391—28no—
134#391—44no—
135#391—44no—
136#390C2×C4116yes—
137#388—28no—
138#387—216no—
139#387—216no—
140#387—28no—
141#386—28no—
142#385—28no—
143#384C2×C4116yescomplemented
144#382—24no—
145#380—44no—
146#380—44no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4C2×C418yescomplemented
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152#376D418yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8C22×C418yes—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—42no—
191C22×D4C22×C414yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—232no—
223C22⋊Q8—432no—
224C22⋊Q8—432no—
225C4×Q8—432no—
226C4×D4—232no—
227#117—416no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C4116yes—
260C2×C4⋊C4C22×C4116yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C4116yes—
287C2×C22⋊C4C22×C4116yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—416no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C22×C418yes—
305C2×C42C22×C414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4114yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C22×C418yes—
333#95C2×D418yes—
334#95C2×D418yes—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C23×C412yes—
351C24C23×C412yes—
352C242−1+414yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C23×C418yes—
447C22×C4C23×C418yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+414yes—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—28no—
485C42—28no—
486C42—48no—
487C42—48no—
488C42—416no—
489C42—44no—
490C42—44no—
491C42—84no—
492C42C4×D4116yes—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#51914yes—
535C23C24×C411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543C23#58512yes—
544Q8—432no—
545D4—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—432no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4C2×C4×D414yes—
624C2×C4C2×C4×D414yes—
625C2×C4C2×C4×D414yes—
626C2×C4#51818yes—
627C2×C4#53514yes—
628C2×C4#53514yes—
629C2×C4#54018yes—
630C22—24no—
631C22—28no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C22×C4×D411yescharacteristic
649C22C2×#51812yes—
650C22C2×#51911yescharacteristic
651C22#342811yescharacteristic
652C22C4×2−1+412yes—
653C22C2×#53511yescharacteristic
654C22C2×#53511yescharacteristic
655C22C2×#54012yes—
656C22#348912yes—
657C22#348912yes—
658C22#349512yes—
659C22#350911yescharacteristic
660C22#351011yescharacteristic
661C22#353812yes—
662C22#353912yes—
663C22#354112yes—
664C22#354112yes—
665C22#355212yes—
666C22#355412yes—
667C22#356112yes—
668C22#356212yes—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—48no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4#232818yes—
682C2—44no—
683C2—82no—
684C2C2×#232811yescharacteristic
685C2#3420311yescharacteristic
686C2#3426811yescharacteristic
687C2#3447312yes—
688C2#3447512yes—
689C2#3737512yes—
690C2#3737912yes—
691C2#3739912yes—
692C2#3740912yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2327C2semidirect
4. #14702C2semidirect
5. #14584C2semidirect
6. #14578C2semidirect
8. #14425C2semidirect
9. #14258C2semidirect
10. #14155C2semidirect
11. #13969C2semidirect
12. #13969C2semidirect
13. #13955C2semidirect
15. #13797C2semidirect
23. #2542C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
91. C2×C22⋊Q8C2×C4semidirect
132. #399C2×C4semidirect
143. #384C2×C4semidirect
149. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23272
C2×#22902
#147022
#145842
#145782
#144312
#144252
#142582
#141552
#139692
#139552
#139052
#137972