Finite Group Databasebeta

← #4457738

Subgroups — #4457738

Order 512.

4646 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#536C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2616C211yesmaximal, characteristic
5#16681C214yesmaximal, complemented
6#16657C218yesmaximal, complemented
7#16603C214yesmaximal, complemented
8#15236C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#15178C212yesmaximal, complemented
11#15178C212yesmaximal, complemented
12#13939C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14#13851C211yesmaximal, characteristic
15C23×#41C2211yescharacteristic
16C24×Q8C2211yescharacteristic
17C2×#536—28no—
18C2×#536—28no—
19C2×#536—28no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2212yes—
22C22×#124C2212yes—
23C22×#123C2214yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2212yes—
27C22×C22⋊Q8C2212yes—
28C22×#117C2211yescharacteristic
29#2616—28no—
30#2499—28no—
31#2434—28no—
32C2×#397C2214yes—
33C2×#395C2214yes—
34C2×#395C2212yes—
35C2×#394C2214yes—
36C2×#394C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#387C2214yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1446C2214yes—
54#1444C2218yes—
55#1444C2218yes—
56#1442C2218yes—
57#1441C2218yes—
58#1440C2214yes—
59#1440C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62#1439C2218yes—
63#1438C2214yes—
64#1438C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—44no—
72C22×#41—44no—
73C22×#41D418yes—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—44no—
77C23×Q8—44no—
78C23×Q8D418yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×Q8C2312yes—
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88#536—2128no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×#124—28no—
96C2×#124—28no—
97C2×#124—28no—
98C2×#123—216no—
99C2×#123—216no—
100C2×#123—216no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—44no—
112C2×C22⋊Q8—44no—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117D418yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2318yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#397—216no—
142#395—28no—
143#395—216no—
144#394—216no—
145#394—28no—
146#391—28no—
147#391—24no—
148#391—24no—
149#391—44no—
150#391—44no—
151#387—216no—
152#385—216no—
153#384—24no—
154#382—24no—
155#380—28no—
156#380—24no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C4×C22⋊C4—44no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C2×#41—232no—
178C2×#41—432no—
179C2×#41—432no—
180C2×#41—416no—
181C2×#41—416no—
182C22×Q8—232no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—432no—
203C22×Q8—432no—
204C22×Q8—416no—
205C22×Q8—416no—
206C22×Q8C2×D418yes—
207C22×Q8C2×D418yes—
208C22×Q8C2×D414yes—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4C2×D418yes—
220C22×D4C2×D414yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D418yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D418yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C4⋊Q8—264no—
251C4⋊Q8—432no—
252#124—264no—
253#123—2128no—
254C22⋊Q8—264no—
255C22⋊Q8—264no—
256C22⋊Q8—432no—
257C22⋊Q8—416no—
258#117—232no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4C2×D418yes—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D418yes—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—216no—
335#95—28no—
336#95—28no—
337#95—216no—
338#95—48no—
339#95—44no—
340#95—44no—
341#95—48no—
342#95—48no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C2511yescharacteristic
363#41—4128no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—28no—
367C2×Q8—216no—
368C2×Q8—216no—
369C2×Q8—416no—
370C2×Q8—48no—
371C2×Q8—416no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—4128no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×D4—216no—
388C2×D4—28no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4#120116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D418yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C42−1+418yes—
480C22×C42−1+414yes—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—432no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C22⋊C4—216no—
492C22⋊C4—232no—
493C22⋊C4—232no—
494C22⋊C4—216no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C42—216no—
503C42—48no—
504C42—44no—
505C42—44no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×#12014yes—
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+414yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553Q8—432no—
554Q8—416no—
555Q8—432no—
556Q8—88no—
557D4—416no—
558D4—432no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—232no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—432no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
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—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—432no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4C2×#12014yes—
622C2×C4C2×#12014yes—
623C2×C4#53418yes—
624C2×C4#53418yes—
625C2×C4#53514yes—
626C2×C4#53514yes—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—44no—
631C22—44no—
632C22—48no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22C22×#12011yescharacteristic
647C22C2×#53412yes—
648C22C2×#53412yes—
649C22C2×#53511yescharacteristic
650C22C2×#53511yescharacteristic
651C22#348612yes—
652C22#348912yes—
653C22#349712yes—
654C22#349911yescharacteristic
655C22#349912yes—
656C22#349912yes—
657C22#350011yescharacteristic
658C22#353814yes—
659C22#354114yes—
660C22#354714yes—
661C22#355014yes—
662C4—24no—
663C4—48no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—48no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#261318yes—
674C2—44no—
675C2—84no—
676C2C2×#261311yescharacteristic
677C2#3566011yescharacteristic
678C2#3566611yescharacteristic
679C2#3738612yes—
680C2#3738812yes—
681C2#4278614yes—
682C2#4291112yes—
683C2#4299412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2499C2semidirect
3. C2×#2434C2semidirect
5. #16681C2semidirect
6. #16657C2semidirect
7. #16603C2semidirect
8. #15236C2semidirect
9. #15182C2semidirect
10. #15178C2semidirect
11. #15178C2semidirect
12. #13939C2semidirect
13. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5362
C2×#24992
C2×#24342
C2×#26162
#166812
#166572
#166032
#152362
#151822
#151782
#139392
#139222
#138512