Finite Group Databasebeta

← #726428

Subgroups — #726428

Order 512.

5058 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic, complemented
4C2×#2326C211yesmaximal, characteristic, complemented
5#16606C214yesmaximal, complemented
6#15397C218yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#14889C214yesmaximal, complemented
10#14282C211yesmaximal, characteristic, complemented
11#14139C212yesmaximal, complemented
12#13941C211yesmaximal, characteristic, complemented
13#13880C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic, complemented
15C23×#41C2212yes—
16C2×#547—28no—
17C2×#547—28no—
18C2×#547—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×#121C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29#2567—28no—
30#2536—28no—
31C2×#398C2212yes—
32C2×#395C2214yes—
33C2×#394C2212yes—
34C2×#394C2218yes—
35#2326—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#389C2214yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#385C2214yes—
44C2×#385C2218yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1445C2212yes—
55#1445C2214yes—
56#1444C2218yes—
57#1441C2218yes—
58#1439C2214yes—
59#1439C2218yes—
60#1439C2214yes—
61#1439C2218yes—
62#1438C2212yes—
63#1438C2218yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22⋊#95C2214yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—28no—
72C22×#41—416no—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2314yes—
78C24×C4C2312yes—
79C24×C4C2314yes—
80C24×C4C2312yes—
81#547—2128no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8D418yes—
85C2×#124—24no—
86C2×#124—24no—
87C2×#124—24no—
88C2×#123—216no—
89C2×#123—216no—
90C2×#123—216no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×#121—28no—
98C2×#121—28no—
99C2×#121—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4D418yes—
108C2×C4×D4D418yes—
109C22×C4⋊C4—28no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2318yes—
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2318yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#398—28no—
140#395—216no—
141#394—28no—
142#394—416no—
143#391—28no—
144#391—24no—
145#390—24no—
146#390—44no—
147#389—216no—
148#388—216no—
149#388—216no—
150#385—216no—
151#385—416no—
152#384—28no—
153#384—24no—
154#382—24no—
155#380—28no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—44no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2314yes—
173C25C2412yes—
174C25C2412yes—
175C2×#41—264no—
176C2×#41—464no—
177C2×#41—464no—
178C2×#41—464no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—48no—
188C22×Q8C2×D418yes—
189C22×Q8#4114yescomplemented
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—216no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—48no—
204C22×D4C2×D414yes—
205C22×D4C2×D414yes—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—216no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4C2×D418yes—
221C23×C4C2×D418yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235#124—232no—
236#123—2128no—
237C22⋊Q8—264no—
238C22⋊Q8—432no—
239#121—264no—
240C4×Q8—416no—
241C4×D4—232no—
242C4×D4—232no—
243C2×C4⋊C4—264no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yescomplemented
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—216no—
329#95—28no—
330#95—44no—
331#95—48no—
332#95—48no—
333#95—48no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—28no—
349C24—28no—
350C24—22no—
351C24C22×D414yes—
352C24C22×D414yes—
353C24H(5,𝔽2)18yes—
354C24C2511yescharacteristic
355#41—4256no—
356C2×Q8—216no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—416no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—232no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—432no—
376C2×D4—432no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—416no—
434C22×C4—432no—
435C22×C4—432no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4C22×D414yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+418yes—
462C22×C42−1+412yes—
463C4⋊C4—216no—
464C4⋊C4—232no—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—88no—
474C4⋊C4—816no—
475C22⋊C4—232no—
476C22⋊C4—232no—
477C22⋊C4—232no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—816no—
484C22⋊C4—88no—
485C42—28no—
486C42—48no—
487C42—44no—
488C42—48no—
489C42#120116yes—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23#53518yes—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)14yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+414yes—
539Q8—432no—
540Q8—416no—
541D4—232no—
542D4—464no—
543D4—416no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—216no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—432no—
566C2×C4—464no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—432no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4C2×#12014yes—
603C2×C4C2×#12014yes—
604C2×C4C2×#12014yes—
605C2×C4#52514yes—
606C2×C4#53714yes—
607C2×C4#54614yes—
608C2×C4#54714yes—
609C2×C4#54814yes—
610C2×C4#55014yes—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—48no—
615C22—44no—
616C22—48no—
617C22—48no—
618C22—44no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22C22×#12011yescharacteristic
632C22C2×#52511yescharacteristic
633C22C2×#53512yes—
634C22C2×#53711yescharacteristic
635C22C2×#54611yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×#54811yescharacteristic
638C22C2×#55011yescharacteristic
639C22#348412yes—
640C22#348412yes—
641C22#348612yes—
642C22#349214yes—
643C22#349912yes—
644C22#349912yes—
645C22#352214yes—
646C22#353614yes—
647C22#354414yes—
648C4—24no—
649C4—24no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4#240718yes—
660C2—48no—
661C2—84no—
662C2C2×#240711yescharacteristic
663C2#3522211yescharacteristic
664C2#3566611yescharacteristic
665C2#3569312yes—
666C2#3786312yes—
667C2#3807912yes—
668C2#4242812yes—
669C2#4493712yes—
670C2#4535612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#547C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. C2×#2536C2semidirect
4. C2×#2326C2semidirect
5. #16606C2semidirect
6. #15397C2semidirect
7. #15215C2semidirect
8. #15130C2semidirect
9. #14889C2semidirect
10. #14282C2semidirect
11. #14139C2semidirect
12. #13941C2semidirect
13. #13880C2semidirect
14. #13847C2semidirect
189. C22×Q8#41semidirect
320. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25672
C2×#25362
C2×#23262
#166062
#153972
#152152
#151302
#148892
#142822
#141392
#139412
#138802
#138472