Finite Group Databasebeta

← #706053

Subgroups — #706053

Order 512.

2786 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2363C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#17066C212yesmaximal, complemented
4#16704C212yesmaximal, complemented
5#15958C214yesmaximal, complemented
6#15358C214yesmaximal, complemented
7#15187C212yesmaximal, complemented
8#15148C212yesmaximal, complemented
9#15135C212yesmaximal, complemented
10#15017C212yesmaximal, complemented
11#14257C211yesmaximal, characteristic
12#14137C211yesmaximal, characteristic
13#14130C211yesmaximal, characteristic, complemented
14#14046C212yesmaximal, complemented
15#13951C211yesmaximal, characteristic, complemented
16#13920C212yesmaximal, complemented
17#13844C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#399C2214yes—
24#2363—28no—
25#2317—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2214yes—
51#1444C2211yescharacteristic
52#1444C2211yescharacteristic
53#1444C2212yes—
54#1443C2214yes—
55#1442C2211yescharacteristic
56#1442C2211yescharacteristic
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1438C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121Q818yes—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×#117—24no—
96C2×#117—28no—
97C2×#117—44no—
98C2×#117D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136#399—48no—
137#390—24no—
138#390—28no—
139#390—44no—
140#390—44no—
141#389—28no—
142#389—28no—
143#389—24no—
144#389—28no—
145#389—48no—
146#389—44no—
147#388—28no—
148#386—28no—
149#386—48no—
150#385—216no—
151#385—44no—
152#384—28no—
153#382—28no—
154#380—24no—
155#380—48no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4C2×Q814yes—
197C22×D4C2×Q818yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×Q814yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228#121—232no—
229C4×D4—416no—
230#117—232no—
231#117—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C4⋊C4#4118yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
310C2×C22⋊C4C2×Q818yes—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42C2×D418yes—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24C22×D412yes—
369C24C22×Q814yes—
370C24C22×Q812yes—
371C24C2×#4112yes—
372C24C2511yescharacteristic
373C2×D4—216no—
374C2×D4—28no—
375C2×D4—216no—
376C2×D4—416no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—84no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
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—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—84no—
464C22×C4—88no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4C22⋊Q8116yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4118yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4#123116yes—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—216no—
515C42—44no—
516C42—44no—
517C42—48no—
518C42—48no—
519C42—84no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23C2×C22⋊Q814yes—
566C23C23×D411yescharacteristic
567C23C23×Q811yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×2−1+411yescharacteristic
574C23#58511yescharacteristic
575C23#58511yescharacteristic
576C23#58512yes—
577D4—416no—
578D4—816no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—232no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—816no—
661C2×C4—816no—
662C2×C4C2×C22⋊Q814yes—
663C2×C4C2×C22⋊Q814yes—
664C2×C4C2×#12318yes—
665C2×C4C2×#12314yes—
666C2×C4#53514yes—
667C2×C4#55318yes—
668C2×C4#55418yes—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—84no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22C22×C22⋊Q811yescharacteristic
692C22C22×#12311yescharacteristic
693C22C2×#53511yescharacteristic
694C22#346011yescharacteristic
695C22#346111yescharacteristic
696C22C2×#55312yes—
697C22C2×#55412yes—
698C22#348811yescharacteristic
699C22#348911yescharacteristic
700C22#349412yes—
701C22#349811yescharacteristic
702C22#350011yescharacteristic
703C22#350112yes—
704C22#350311yescharacteristic
705C22#350411yescharacteristic
706C22#351312yes—
707C22#351712yes—
708C22#352012yes—
709C22#352412yes—
710C22#353712yes—
711C22#354012yes—
712C22#354612yes—
713C22#354712yes—
714C4—28no—
715C4—24no—
716C4—44no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—44no—
721C4—82no—
722C4—84no—
723C4—82no—
724C4—84no—
725C4—81no—
726C4—82no—
727C4—81no—
728C4#248718yes—
729C2—44no—
730C2—84no—
731C2C2×#248711yescharacteristic
732C2#3525711yescharacteristic
733C2#3552911yescharacteristic
734C2#3598711yescharacteristic
735C2#3607611yescharacteristic
736C2#3821212yes—
737C2#3905212yes—
738C2#4298611yescharacteristic
739C2#4299511yescharacteristic
740C2#4535912yes—
741C2#4536112yes—
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×#2317C2semidirect
3. #17066C2semidirect
4. #16704C2semidirect
5. #15958C2semidirect
6. #15358C2semidirect
7. #15187C2semidirect
8. #15148C2semidirect
9. #15135C2semidirect
10. #15017C2semidirect
13. #14130C2semidirect
14. #14046C2semidirect
15. #13951C2semidirect
16. #13920C2semidirect
17. #13844C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23632
C2×#23172
#170662
#167042
#159582
#153582
#151872
#151482
#151352
#150172
#142572
#141372
#141302
#140462
#139512
#139202
#138442