Finite Group Databasebeta

← #6019464

Subgroups — #6019464

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17306C211yesmaximal, characteristic, complemented
2#17298C211yesmaximal, characteristic, complemented
3#17220C212yesmaximal, complemented
4#17162C212yesmaximal, complemented
5#17155C212yesmaximal, complemented
6#17112C211yesmaximal, characteristic, complemented
7#16870C212yesmaximal, complemented
8#16710C212yesmaximal, complemented
9#16597C212yesmaximal, complemented
10#16403C212yesmaximal
11#16301C212yesmaximal
12#16301C212yesmaximal
13#15614C212yesmaximal
14#15521C211yesmaximal, characteristic, complemented
15#15484C212yesmaximal
16#15483C212yesmaximal
17#15015C211yesmaximal, characteristic
18#14936C211yesmaximal, characteristic
19#14934C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C2×#401C2212yes—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#396C2212yes—
28C2×#395C2212yes—
29C2×#389C2212yes—
30C2×#385C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C22×#95C2211yescharacteristic
36#1446C2211yescharacteristic
37#1446C2212yes—
38#1446C2211yescharacteristic
39#1445C2212yes—
40#1445C2212yes—
41#1445C2212yes—
42#1445C2211yescharacteristic
43#1445C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1445C2211yescharacteristic
47#1445C2211yescharacteristic
48#1445C2211yescharacteristic
49#1444C2212yes—
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1443C2212yes—
58#1443C2212yes—
59#1443C2212yes—
60#1443C2212yes—
61#1443C2212yes—
62#1443C2212yes—
63#1443C2212yes—
64#1443C2212yes—
65#1442C2212yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2212yes—
70#1442C2212yes—
71#1442C2212yes—
72#1442C2212yes—
73#1442C2211yescharacteristic
74#1442C2211yescharacteristic
75#1442C2211yescharacteristic
76#1442C2211yescharacteristic
77#1441C2212yes—
78#1441C2212yes—
79#1441C2212yes—
80#1441C2212yes—
81#1441C2212yes—
82#1441C2212yes—
83#1441C2212yes—
84#1441C2212yes—
85#1441C2212yes—
86#1441C2212yes—
87#1441C2212yes—
88#1441C2212yes—
89#1439C2212yes—
90#1439C2212yes—
91#1439C2212yes—
92#1439C2212yes—
93#1439C2212yes—
94#1439C2212yes—
95#1437C2212yes—
96#1436C2212yes—
97#1436C2212yes—
98#1436C2212yes—
99C4⋊#95C2212yes—
100#1434C2212yes—
101C22⋊#95C2211yescharacteristic
102C22⋊#95C2212yes—
103C22⋊#95C2212yes—
104C22⋊#95C2211yescharacteristic
105C22⋊#95C2212yes—
106C22⋊#95C2211yescharacteristic
107C23×Q8C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C24×C4C2311yescharacteristic
111C2×C22⋊Q8—42no—
112C2×C22⋊Q8—44no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2312yes—
137#401—44no—
138#400—44no—
139#400—44no—
140#400—44no—
141#400—44no—
142#398—44no—
143#396—44no—
144#395—44no—
145#389—44no—
146#385—44no—
147#381—44no—
148#381—44no—
149#380—42no—
150#380—42no—
151C2×#95—42no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2311yescharacteristic
208C2×#95C2311yescharacteristic
209C2×#95C2311yescharacteristic
210C2×#95C2311yescharacteristic
211C25C2411yescharacteristic
212C22×Q8—22no—
213C22×Q8—41no—
214C22×Q8—42no—
215C22×Q8—41no—
216C22×Q8—42no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—41no—
227C23×C4—41no—
228C23×C4—41no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C22⋊Q8—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—22no—
336#95—41no—
337#95—41no—
338#95—41no—
339#95—41no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—41no—
345#95—41no—
346#95—41no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95—42no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—42no—
378#95—42no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24H(5,𝔽2)12yes—
389C242−1+414yes—
390C24C2511yescharacteristic
391C2×Q8—44no—
392C2×Q8—82no—
393C2×Q8—84no—
394C2×Q8—82no—
395C2×Q8—84no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
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—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
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—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—84no—
514C22×C4—82no—
515C22×C4—84no—
516C22×C4—82no—
517C22×C4—82no—
518C22×C4—84no—
519C22×C4—84no—
520C22×C4H(5,𝔽2)14yes—
521C22×C4H(5,𝔽2)14yes—
522C22×C42−1+412yes—
523C22×C42−1+412yes—
524C22×C42−1+412yes—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—84no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23#56314yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)12yes—
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+412yes—
576C23C2×2−1+411yescharacteristic
577C23C2×2−1+411yescharacteristic
578C23C2×2−1+411yescharacteristic
579C23#58511yescharacteristic
580C23#58511yescharacteristic
581C23#58511yescharacteristic
582C23#58512yes—
583Q8—88no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—88no—
699C2×C4—88no—
700C2×C4—88no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—42no—
710C22—44no—
711C22—42no—
712C22C2×#56311yescharacteristic
713C22#353211yescharacteristic
714C22#353512yes—
715C22#353511yescharacteristic
716C22#353511yescharacteristic
717C22#353511yescharacteristic
718C22#353612yes—
719C22#353812yes—
720C22#353812yes—
721C22#354311yescharacteristic
722C22#354511yescharacteristic
723C22#354612yes—
724C22#355511yescharacteristic
725C22#355511yescharacteristic
726C22#355511yescharacteristic
727C22#355912yes—
728C22#355912yes—
729C22#356012yes—
730C22#356112yes—
731C22#356212yes—
732C22#356312yes—
733C22#356412yes—
734C22#356811yescharacteristic
735C4—44no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—42no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—81no—
745C4—82no—
746C4—82no—
747C4—81no—
748C4—82no—
749C4—81no—
750C4—82no—
751C4—82no—
752C4—82no—
753C2—44no—
754C2#5239611yescharacteristic
755C2#5256711yescharacteristic
756C2#5281711yescharacteristic
757C2#5330211yescharacteristic
758C2#5348811yescharacteristic
759C2#5377912yes—
760C2#5432612yes—
761C2#5772511yescharacteristic
762C2#5825811yescharacteristic
763C2#5883912yes—
764C2#5892012yes—
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
1. #17306C2semidirect
2. #17298C2semidirect
3. #17220C2semidirect
4. #17162C2semidirect
5. #17155C2semidirect
6. #17112C2semidirect
7. #16870C2semidirect
8. #16710C2semidirect
9. #16597C2semidirect
14. #15521C2semidirect

Maximal subgroups

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

subgroupindex
#173062
#172982
#172202
#171622
#171552
#171122
#168702
#167102
#165972
#164032
#163012
#156142
#155212
#154842
#154832
#150152
#149362
#149342