Finite Group Databasebeta

← #706060

Subgroups — #706060

Order 512.

2528 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#17060C212yesmaximal, complemented
4#16963C212yesmaximal, complemented
5#15554C214yesmaximal
6#15544C214yesmaximal, complemented
7#15197C212yesmaximal, complemented
8#15186C212yesmaximal, complemented
9#15132C212yesmaximal, complemented
10#15025C212yesmaximal, complemented
11#14258C211yesmaximal, characteristic, complemented
12#14139C211yesmaximal, characteristic, complemented
13#13997C212yesmaximal, complemented
14#13933C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16#13839C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2415—28no—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#394C2212yes—
28#2317—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#382C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2211yescharacteristic
57#1441C2211yescharacteristic
58#1441C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1440C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2211yescharacteristic
67#1439C2211yescharacteristic
68#1439C2212yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×#117—24no—
99C2×#117—28no—
100C2×#117—44no—
101C2×#117D418yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135#397—28no—
136#397—48no—
137#394—28no—
138#391—24no—
139#391—44no—
140#391—44no—
141#389—28no—
142#389—28no—
143#389—24no—
144#389—28no—
145#388—28no—
146#386—44no—
147#385—216no—
148#384—28no—
149#382—28no—
150#380—24no—
151#380—24no—
152#380—44no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—41no—
198C22×D4—41no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C22⋊Q8—416no—
230C22⋊Q8—416no—
231C4×D4—416no—
232#117—232no—
233#117—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
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—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—416no—
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⋊C4#4118yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4C2×D418yes—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D418yes—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24C22×D412yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×D4—28no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4#124116yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4118yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C22×C42−1+412yes—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C4⋊C4#123116yes—
503C22⋊C4—216no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C42—216no—
516C42—44no—
517C42—48no—
518C42—48no—
519C42—88no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×#12414yes—
566C23C23×D411yescharacteristic
567C23C22×#4112yes—
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×2−1+412yes—
573C23C2×2−1+411yescharacteristic
574C23#58512yes—
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577Q8—88no—
578D4—416no—
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—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4—88no—
666C2×C4C2×#12318yes—
667C2×C4C2×#12314yes—
668C2×C4C2×#12414yes—
669C2×C4C2×#12414yes—
670C2×C4#53514yes—
671C2×C4#55318yes—
672C2×C4#55318yes—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—82no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22C22×#12311yescharacteristic
696C22C22×#12411yescharacteristic
697C22C2×#53511yescharacteristic
698C22#346112yes—
699C22C2×#55312yes—
700C22C2×#55312yes—
701C22#348811yescharacteristic
702C22#348911yescharacteristic
703C22#349211yescharacteristic
704C22#349311yescharacteristic
705C22#349411yescharacteristic
706C22#349511yescharacteristic
707C22#349512yes—
708C22#349911yescharacteristic
709C22#350011yescharacteristic
710C22#351512yes—
711C22#351612yes—
712C22#352112yes—
713C22#352212yes—
714C22#354112yes—
715C22#354212yes—
716C22#355012yes—
717C22#355112yes—
718C4—24no—
719C4—28no—
720C4—44no—
721C4—44no—
722C4—42no—
723C4—84no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—81no—
730C4—82no—
731C4—81no—
732C4#250418yes—
733C2—44no—
734C2—82no—
735C2C2×#250411yescharacteristic
736C2#3531211yescharacteristic
737C2#3554211yescharacteristic
738C2#3599611yescharacteristic
739C2#3606711yescharacteristic
740C2#3821712yes—
741C2#3905612yes—
742C2#4291111yescharacteristic
743C2#4296211yescharacteristic
744C2#4497812yes—
745C2#4516212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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. #17060C2semidirect
4. #16963C2semidirect
6. #15544C2semidirect
7. #15197C2semidirect
8. #15186C2semidirect
9. #15132C2semidirect
10. #15025C2semidirect
11. #14258C2semidirect
12. #14139C2semidirect
13. #13997C2semidirect
14. #13933C2semidirect
17. #13801C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
C2×#23172
#170602
#169632
#155542
#155442
#151972
#151862
#151322
#150252
#142582
#141392
#139972
#139332
#138722
#138392
#138012