Finite Group Databasebeta

← #4420165

Subgroups — #4420165

Order 512.

2770 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2610C211yesmaximal, characteristic
2C2×#2658C211yesmaximal, characteristic, complemented
3C2×#2628C211yesmaximal, characteristic, complemented
4C2×#2346C211yesmaximal, characteristic, complemented
5#16564C212yesmaximal, complemented
6#16506C212yesmaximal, complemented
7#16504C212yesmaximal
8#16398C212yesmaximal, complemented
9#15696C214yesmaximal, complemented
10#15675C214yesmaximal, complemented
11#15128C212yesmaximal, complemented
12#15070C212yesmaximal, complemented
13#14910C212yesmaximal, complemented
14#14839C212yesmaximal, complemented
15#14662C211yesmaximal, characteristic, complemented
16#14282C211yesmaximal, characteristic, complemented
17#13963C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#124C2211yescharacteristic
21C22×#124C2212yes—
22C22×#117C2211yescharacteristic
23#2658—28no—
24#2628—28no—
25#2610—28no—
26C2×#400C2212yes—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#399C2212yes—
31C2×#397C2214yes—
32C2×#395C2211yescharacteristic
33C2×#395C2212yes—
34C2×#394C2211yescharacteristic
35C2×#394C2214yes—
36C2×#393C2212yes—
37C2×#393C2211yescharacteristic
38C2×#392C2211yescharacteristic
39#2346—28no—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#388C2214yes—
49C2×#386C2212yes—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×#376C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60#1446C2212yes—
61#1446C2212yes—
62#1446C2212yes—
63#1445C2212yes—
64#1445C2214yes—
65#1445C2212yes—
66#1444C2214yes—
67#1444C2212yes—
68#1442C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1438C2214yes—
80#1438C2212yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85C4⋊#95C2214yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×D4C2311yescharacteristic
90C23×D4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—48no—
97C2×#124—24no—
98C2×#124—28no—
99C2×#124—24no—
100C2×#124—24no—
101C2×#124—48no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117D418yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—28no—
140#399—28no—
141#399—28no—
142#399—28no—
143#399—216no—
144#397—48no—
145#395—24no—
146#395—28no—
147#394—24no—
148#394—48no—
149#393—24no—
150#393—28no—
151#392—24no—
152#390—28no—
153#389—28no—
154#389—28no—
155#389—48no—
156#388—28no—
157#388—28no—
158#388—28no—
159#388—28no—
160#388—48no—
161#386—28no—
162#382—24no—
163#382—24no—
164#382—24no—
165#380—28no—
166#380—24no—
167C4×C4⋊C4—28no—
168C4×C22⋊C4—24no—
169#376—24no—
170#376—24no—
171#376D418yescomplemented
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—42no—
217C22×D4—44no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—42no—
222C23×C4C2×D414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C4⋊Q8—432no—
239C4⋊Q8—416no—
240#124—432no—
241#124—416no—
242#117—232no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D414yes—
358#95C2×D414yes—
359#95C2×D418yes—
360C24—21no—
361C24—21no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24C22×D412yes—
379C24C2511yescharacteristic
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—84no—
386C2×Q8—88no—
387C2×Q8—84no—
388C2×Q8—88no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—88no—
396C2×D4—84no—
397C2×D4—88no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)18yes—
478C22×C42−1+414yes—
479C22×C42−1+414yes—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—28no—
509C42—216no—
510C42—48no—
511C42—48no—
512C42—48no—
513C42#121116yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C23×D411yescharacteristic
554C23C23×D411yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)14yes—
557C23C2×2−1+412yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560C23#58512yes—
561Q8—816no—
562Q8—88no—
563D4—88no—
564D4—816no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4C2×#12114yes—
639C2×C4C2×#12114yes—
640C2×C4C2×#12114yes—
641C2×C4#53518yes—
642C2×C4#53614yes—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—82no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—84no—
662C22C22×#12111yescharacteristic
663C22C2×#53512yes—
664C22C2×#53611yescharacteristic
665C22#344911yescharacteristic
666C22#348912yes—
667C22#348912yes—
668C22#349512yes—
669C22#349812yes—
670C22#349911yescharacteristic
671C22#350011yescharacteristic
672C22#350712yes—
673C22#350812yes—
674C22#353514yes—
675C22#354014yes—
676C22#354414yes—
677C22#354714yes—
678C4—24no—
679C4—28no—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—82no—
693C2—82no—
694C2—162no—
695C2#3527211yescharacteristic
696C2#3530611yescharacteristic
697C2#3562111yescharacteristic
698C2#4196012yes—
699C2#4229712yes—
700C2#4296812yes—
701C2#4299512yes—
702C2#5162014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2658C2semidirect
3. C2×#2628C2semidirect
4. C2×#2346C2semidirect
5. #16564C2semidirect
6. #16506C2semidirect
8. #16398C2semidirect
9. #15696C2semidirect
10. #15675C2semidirect
11. #15128C2semidirect
12. #15070C2semidirect
13. #14910C2semidirect
14. #14839C2semidirect
15. #14662C2semidirect
16. #14282C2semidirect
17. #13963C2semidirect
171. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26102
C2×#26582
C2×#26282
C2×#23462
#165642
#165062
#165042
#163982
#156962
#156752
#151282
#150702
#149102
#148392
#146622
#142822
#139632