Finite Group Databasebeta

← #628004

Subgroups — #628004

Order 512.

4022 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2639C211yesmaximal, characteristic
4C2×#2627C211yesmaximal, characteristic, complemented
5#15928C212yesmaximal, complemented
6#15910C212yesmaximal, complemented
7#15847C214yesmaximal, complemented
8#15465C212yesmaximal, complemented
9#15271C214yesmaximal, complemented
10#15265C212yesmaximal, complemented
11#14927C214yesmaximal, complemented
12#14698C211yesmaximal, characteristic
13#14281C211yesmaximal, characteristic, complemented
14#13975C214yesmaximal, complemented
15C2×#1436C211yesmaximal, characteristic, complemented
16C2×#539—28no—
17C2×#539—28no—
18C2×#539—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2212yes—
21C22×#125C2212yes—
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4C2212yes—
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2857—28no—
32#2639—28no—
33#2627—28no—
34C2×#400C2212yes—
35C2×#400C2212yes—
36C2×#398C2214yes—
37C2×#397C2212yes—
38C2×#395C2212yes—
39C2×#394C2212yes—
40C2×#393C2212yes—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2212yes—
44C2×#389C2214yes—
45C2×#386C2214yes—
46C2×#386C2214yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#384C2211yescharacteristic
50C2×#381C2212yes—
51C2×#380C2214yes—
52C2×#380C2211yescharacteristic
53C22×#95C2211yescharacteristic
54C22×#95C2212yes—
55C22×#95C2212yes—
56#1446C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1437C2212yes—
72#1437C2212yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2212yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#539—2128no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×#126—28no—
95C2×#126—28no—
96C2×#126—28no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—28no—
105C2×#123—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8D418yes—
115C2×#121—24no—
116C2×#121—24no—
117C2×#121D418yes—
118C2×C4×D4—28no—
119C2×C4×D4—28no—
120C2×C4×D4—28no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4D418yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2314yes—
149C22×C42—24no—
150C22×C42—44no—
151C22×C42C2311yescharacteristic
152C22×C42C2312yes—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155#400—28no—
156#400—28no—
157#398—216no—
158#397—28no—
159#395—28no—
160#394—28no—
161#393—28no—
162#391—24no—
163#391—24no—
164#390—44no—
165#389—216no—
166#386—216no—
167#386—216no—
168#385—28no—
169#385—48no—
170#384—24no—
171#381—28no—
172#380—24no—
173#380—216no—
174C2×#95—24no—
175C2×#95—28no—
176C2×#95—44no—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C25C2412yes—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—42no—
208C22×Q8C2×D414yes—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—28no—
212C22×D4—28no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4C2×D418yes—
218C22×D4C2×D414yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4#4118yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C4⋊Q8—416no—
267#126—264no—
268#125—264no—
269#123—264no—
270#123—264no—
271C22⋊Q8—264no—
272C22⋊Q8—232no—
273C22⋊Q8—416no—
274#121—232no—
275C4×D4—264no—
276C2×C4⋊C4—232no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2×D418yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×D414yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4C2×D418yes—
350C2×C22⋊C4C2×D418yes—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—416no—
360C2×C42—42no—
361C2×C42—42no—
362#95—24no—
363#95—24no—
364#95—28no—
365#95—28no—
366#95—24no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24C22×D414yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C2×#4114yes—
399C242−1+414yes—
400C24C2511yescharacteristic
401C2×Q8—28no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—84no—
408C2×Q8—84no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×D4—216no—
412C2×D4—232no—
413C2×D4—28no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—84no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—216no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—216no—
460C22×C4—28no—
461C22×C4—216no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4—88no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4—84no—
507C22×C4—88no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4#120116yes—
513C22×C4C22×D414yes—
514C22×C4C22×D414yes—
515C22×C4C22×D414yes—
516C22×C4C22×D414yes—
517C22×C4C22×D414yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C2×#4114yes—
525C22×C4C2×#4114yes—
526C22×C4H(5,𝔽2)14yes—
527C22×C42−1+414yes—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—216no—
532C4⋊C4—216no—
533C4⋊C4—28no—
534C4⋊C4—216no—
535C4⋊C4—28no—
536C4⋊C4—416no—
537C4⋊C4—416no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—84no—
541C4⋊C4—88no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—216no—
546C22⋊C4—216no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—416no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—28no—
554C42—216no—
555C42—48no—
556C42—44no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—28no—
575C23—28no—
576C23—24no—
577C23—24no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23C2×#12014yes—
595C23#53214yes—
596C23#53614yes—
597C23#54718yes—
598C23#54818yes—
599C23C23×D411yescharacteristic
600C23C23×D411yescharacteristic
601C23C23×D411yescharacteristic
602C23C22×#4112yes—
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+412yes—
605C23C2×2−1+412yes—
606C23#58512yes—
607C23#58512yes—
608Q8—88no—
609Q8—88no—
610D4—432no—
611D4—88no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—416no—
642C2×C4—432no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—416no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—88no—
686C2×C4—816no—
687C2×C4—816no—
688C2×C4C2×#12014yes—
689C2×C4C2×#12014yes—
690C2×C4#53918yescomplemented
691C22—24no—
692C22—28no—
693C22—24no—
694C22—28no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—48no—
703C22—48no—
704C22—84no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22#243218yes—
709C22C22×#12011yescharacteristic
710C22C2×#53211yescharacteristic
711C22C2×#53611yescharacteristic
712C22C2×#53912yes—
713C22#345512yes—
714C22C2×#54712yes—
715C22C2×#54812yes—
716C22#348411yescharacteristic
717C22#348511yescharacteristic
718C22#348912yes—
719C22#349612yes—
720C22#349612yes—
721C22#349712yes—
722C22#349911yescharacteristic
723C22#350011yescharacteristic
724C22#350512yes—
725C22#350712yes—
726C22#353512yes—
727C22#353812yes—
728C22#356312yes—
729C22#356412yes—
730C4—24no—
731C4—24no—
732C4—28no—
733C4—44no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—44no—
738C4—44no—
739C4—82no—
740C4—82no—
741C4—84no—
742C4—84no—
743C2—28no—
744C2—84no—
745C2C2×#243211yescharacteristic
746C2#3500612yes—
747C2#3534711yescharacteristic
748C2#3566011yescharacteristic
749C2#3674212yes—
750C2#3770212yes—
751C2#3770812yes—
752C2#3937112yes—
753C2#3937612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2857C2semidirect
4. C2×#2627C2semidirect
5. #15928C2semidirect
6. #15910C2semidirect
7. #15847C2semidirect
8. #15465C2semidirect
9. #15271C2semidirect
10. #15265C2semidirect
11. #14927C2semidirect
13. #14281C2semidirect
14. #13975C2semidirect
15. C2×#1436C2semidirect
690. C2×C4#539semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#28572
C2×#26392
C2×#26272
#159282
#159102
#158472
#154652
#152712
#152652
#149272
#146982
#142812
#139752
C2×#14362