Finite Group Databasebeta

← #4435094

Subgroups — #4435094

Order 512.

2649 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2604C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic, complemented
5C2×#2634C211yesmaximal, characteristic, complemented
6C2×#2664C211yesmaximal, characteristic
7#17062C212yesmaximal
8#16964C212yesmaximal, complemented
9#16813C212yesmaximal, complemented
10#16511C212yesmaximal
11#15804C214yesmaximal, complemented
12#15765C214yesmaximal
13#15145C212yesmaximal, complemented
14#15143C212yesmaximal, complemented
15#15065C212yesmaximal
16#15020C212yesmaximal
17#13806C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#126C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2821—28no—
24#2664—28no—
25#2660—28no—
26#2634—28no—
27#2604—28no—
28#2546—28no—
29C2×#400C2211yescharacteristic
30C2×#400C2212yes—
31C2×#400C2212yes—
32C2×#400C2211yescharacteristic
33C2×#400C2212yes—
34C2×#399C2211yescharacteristic
35C2×#399C2211yescharacteristic
36C2×#399C2212yes—
37C2×#399C2214yes—
38C2×#398C2211yescharacteristic
39C2×#398C2212yes—
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2212yes—
43C2×#398C2214yes—
44C2×#398C2212yes—
45C2×#397C2211yescharacteristic
46C2×#397C2212yes—
47C2×#396C2212yes—
48C2×#395C2211yescharacteristic
49C2×#395C2212yes—
50C2×#395C2212yes—
51C2×#395C2212yes—
52C2×#395C2212yes—
53C2×#393C2211yescharacteristic
54C2×#393C2212yes—
55C2×#391C2212yes—
56C2×#389C2212yes—
57C2×#389C2212yes—
58C2×#389C2214yes—
59C2×#388C2212yes—
60C2×#388C2212yes—
61C2×#388C2212yes—
62C2×#387C2214yes—
63C2×#387C2212yes—
64C2×#384C2212yes—
65C2×#384C2214yes—
66C2×#383C2211yescharacteristic
67C2×#382C2212yes—
68C2×C4×C4⋊C4C2211yescharacteristic
69C2×#376C2211yescharacteristic
70#1445C2214yes—
71#1445C2214yes—
72#1445C2214yes—
73#1445C2214yes—
74#1445C2214yes—
75#1444C2214yes—
76#1444C2214yes—
77#1444C2214yes—
78#1444C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—48no—
96C2×#126—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#400—28no—
141#400—24no—
142#400—28no—
143#400—28no—
144#400—24no—
145#399—28no—
146#399—24no—
147#399—24no—
148#399—48no—
149#398—28no—
150#398—28no—
151#398—28no—
152#398—216no—
153#398—24no—
154#398—28no—
155#398—28no—
156#397—28no—
157#397—24no—
158#396—28no—
159#395—28no—
160#395—24no—
161#395—28no—
162#395—28no—
163#395—28no—
164#393—28no—
165#393—24no—
166#391—28no—
167#389—216no—
168#389—28no—
169#389—28no—
170#388—28no—
171#388—28no—
172#388—28no—
173#387—28no—
174#387—216no—
175#384—28no—
176#384—48no—
177#383D418yes—
178#382—28no—
179C4×C4⋊C4D418yes—
180#376D418yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2312yes—
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C2×#95C2312yes—
205C2×#95C2311yescharacteristic
206C2×#95C2311yescharacteristic
207C2×#95C2312yes—
208C2×#95C2312yes—
209C25C2411yescharacteristic
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—42no—
218C22×Q8—44no—
219C22×Q8—44no—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C4⋊Q8—432no—
237C4⋊Q8—416no—
238#126—416no—
239#125—416no—
240C4×Q8—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D418yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—28no—
332C2×C42—28no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D418yes—
336C2×C42C2×D418yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95C2×D418yes—
361#95C2×D414yes—
362#95C2×D414yes—
363#95C2×D414yes—
364#95C2×D414yes—
365#95C2×D418yes—
366#95C2×D418yes—
367#95C2×D418yes—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24C2511yescharacteristic
377C2×Q8—28no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—88no—
386C2×Q8—88no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4C22×D418yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+414yes—
471C22×C42−1+418yes—
472C22×C42−1+414yes—
473C4⋊C4—216no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C42—216no—
504C42—48no—
505C42—416no—
506C42—48no—
507C42#127116yes—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23C2×2−1+414yes—
530C23C2×2−1+412yes—
531C23#58512yes—
532Q8—416no—
533Q8—816no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
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—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4C2×#12714yes—
616C2×C4C2×#12714yes—
617C2×C4C2×#12714yes—
618C2×C4#53618yes—
619C2×C4#53618yes—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C22×#12711yescharacteristic
629C22C2×#53612yes—
630C22C2×#53612yes—
631C22#348411yescharacteristic
632C22#348711yescharacteristic
633C22#348812yes—
634C22#348812yes—
635C22#349912yes—
636C22#349912yes—
637C22#349912yes—
638C22#349911yescharacteristic
639C22#349911yescharacteristic
640C22#353814yes—
641C22#355214yes—
642C22#355514yes—
643C22#356314yes—
644C4—28no—
645C4—28no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C2—161no—
661C2#3531711yescharacteristic
662C2#3531711yescharacteristic
663C2#3556511yescharacteristic
664C2#4049512yes—
665C2#4193912yes—
666C2#4287912yes—
667C2#4291812yes—
668C2#5133014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2546C2semidirect
4. C2×#2660C2semidirect
5. C2×#2634C2semidirect
8. #16964C2semidirect
9. #16813C2semidirect
11. #15804C2semidirect
13. #15145C2semidirect
14. #15143C2semidirect
17. #13806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#25462
C2×#26042
C2×#26602
C2×#26342
C2×#26642
#170622
#169642
#168132
#165112
#158042
#157652
#151452
#151432
#150652
#150202
#138062