Finite Group Databasebeta

← #4418515

Subgroups — #4418515

Order 512.

2703 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic, complemented
2C2×#2637C211yesmaximal, characteristic, complemented
3C2×#2662C211yesmaximal, characteristic
4C2×#2351C211yesmaximal, characteristic, complemented
5#17069C212yesmaximal, complemented
6#16821C212yesmaximal, complemented
7#16566C212yesmaximal, complemented
8#16519C212yesmaximal, complemented
9#15620C214yesmaximal, complemented
10#15615C214yesmaximal, complemented
11#15157C212yesmaximal, complemented
12#14877C212yesmaximal, complemented
13#14872C212yesmaximal, complemented
14#14815C212yesmaximal
15#14662C211yesmaximal, characteristic, complemented
16#14291C211yesmaximal, characteristic, complemented
17#13954C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2662—28no—
23#2637—28no—
24#2592—28no—
25C2×#398C2212yes—
26C2×#398C2211yescharacteristic
27C2×#397C2211yescharacteristic
28C2×#397C2212yes—
29C2×#396C2212yes—
30C2×#396C2212yes—
31C2×#396C2214yes—
32C2×#396C2214yes—
33C2×#395C2211yescharacteristic
34C2×#395C2211yescharacteristic
35C2×#394C2211yescharacteristic
36C2×#394C2212yes—
37C2×#394C2214yes—
38C2×#393C2211yescharacteristic
39C2×#393C2212yes—
40C2×#393C2211yescharacteristic
41C2×#392C2211yescharacteristic
42#2351—28no—
43C2×#391C2212yes—
44C2×#390C2212yes—
45C2×#390C2212yes—
46C2×#390C2214yes—
47C2×#389C2214yes—
48C2×#389C2212yes—
49C2×#389C2214yes—
50C2×#388C2214yes—
51C2×#388C2212yes—
52C2×#388C2212yes—
53C2×#387C2212yes—
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#382C2212yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#1446C2212yes—
61#1444C2212yes—
62#1444C2212yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1438C2212yes—
80#1437C2214yes—
81#1437C2212yes—
82#1436C2212yes—
83#1436C2214yes—
84C4⋊#95C2214yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2312yes—
90C23×D4C2311yescharacteristic
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—24no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—48no—
101C2×C4×Q8D418yescomplemented
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#398—24no—
140#398—28no—
141#397—28no—
142#397—24no—
143#396—28no—
144#396—28no—
145#396—48no—
146#396—48no—
147#395—24no—
148#395—24no—
149#394—28no—
150#394—24no—
151#394—48no—
152#393—28no—
153#393—24no—
154#393—24no—
155#392—24no—
156#391—28no—
157#390—28no—
158#390—28no—
159#390—48no—
160#389—216no—
161#389—28no—
162#389—48no—
163#388—216no—
164#388—28no—
165#388—28no—
166#387—28no—
167#382—24no—
168#382—24no—
169#382—28no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—28no—
172C4×C4⋊C4D418yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C25C2411yescharacteristic
198C25C2412yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8C2×D414yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232#126—416no—
233#124—416no—
234C4×Q8—232no—
235C4×Q8—432no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
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×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
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⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D418yes—
332C2×C42C2×D418yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42#4114yes—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24C2511yescharacteristic
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—216no—
383C2×Q8—28no—
384C2×Q8—216no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—88no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+414yes—
476C22×C42−1+414yes—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—216no—
510C42—48no—
511C42—416no—
512C42—84no—
513C42#121116yes—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
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)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×2−1+412yes—
559C23C2×2−1+412yes—
560C23#58514yes—
561Q8—232no—
562Q8—88no—
563D4—816no—
564D4—88no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
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×C4—44no—
639C2×C4C2×#12114yes—
640C2×C4C2×#12114yes—
641C2×C4C2×#12114yes—
642C2×C4#53718yes—
643C2×C4#53718yes—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—81no—
655C22—84no—
656C22—82no—
657C22—84no—
658C22—82no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22C22×#12111yescharacteristic
664C22C2×#53712yes—
665C22C2×#53712yes—
666C22#348411yescharacteristic
667C22#348611yescharacteristic
668C22#348712yes—
669C22#349212yes—
670C22#349811yescharacteristic
671C22#349911yescharacteristic
672C22#350012yes—
673C22#351012yes—
674C22#351112yes—
675C22#354114yes—
676C22#355414yes—
677C22#355514yes—
678C22#355814yes—
679C4—28no—
680C4—24no—
681C4—24no—
682C4—28no—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—84no—
687C4—81no—
688C4—82no—
689C4—84no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—81no—
694C2—161no—
695C2—162no—
696C2#3522311yescharacteristic
697C2#3527311yescharacteristic
698C2#3567711yescharacteristic
699C2#4048912yes—
700C2#4279512yes—
701C2#4287812yes—
702C2#4288112yes—
703C2#5073414yes—
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
1. C2×#2592C2semidirect
2. C2×#2637C2semidirect
4. C2×#2351C2semidirect
5. #17069C2semidirect
6. #16821C2semidirect
7. #16566C2semidirect
8. #16519C2semidirect
9. #15620C2semidirect
10. #15615C2semidirect
11. #15157C2semidirect
12. #14877C2semidirect
13. #14872C2semidirect
15. #14662C2semidirect
16. #14291C2semidirect
17. #13954C2semidirect
101. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×#26372
C2×#26622
C2×#23512
#170692
#168212
#165662
#165192
#156202
#156152
#151572
#148772
#148722
#148152
#146622
#142912
#139542