Finite Group Databasebeta

← #711896

Subgroups — #711896

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic
3C2×#2343C211yesmaximal, characteristic, complemented
4C2×#2317C211yesmaximal, characteristic, complemented
5#16676C212yesmaximal, complemented
6#16564C212yesmaximal, complemented
7#15765C214yesmaximal, complemented
8#15739C214yesmaximal, complemented
9#14910C212yesmaximal, complemented
10#14870C212yesmaximal, complemented
11#14786C212yesmaximal, complemented
12#14726C212yesmaximal
13#14342C211yesmaximal, characteristic, complemented
14#14130C212yesmaximal, complemented
15#13991C212yesmaximal
16#13878C211yesmaximal, characteristic, complemented
17#13868C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24#2482—28no—
25C2×#399—28no—
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#399C2214yes—
29C2×#399C2212yes—
30C2×#398C2214yescomplemented
31C2×#397C2212yes—
32C2×#396C2214yes—
33#2343—28no—
34#2317—28no—
35C2×#389C2212yescomplemented
36C2×#389C2212yescomplemented
37C2×#389C2214yescomplemented
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yescomplemented
42C2×#387C2212yescomplemented
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2211yescharacteristic
46C2×#385C2212yes—
47C2×#385C2211yescharacteristic
48C2×#384C2212yescomplemented
49C2×#383C2211yescharacteristic, complemented
50C2×#382C2212yescomplemented
51C2×#382C2211yescharacteristic, complemented
52C2×#381C2211yescharacteristic
53C2×#381C2211yescharacteristic
54C2×#380C2212yes—
55C2×#379C2212yes—
56C2×#379C2211yescharacteristic
57C2×#379C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic, complemented
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic, complemented
61C22×#95C2211yescharacteristic
62#1445C2212yescomplemented
63#1445C2212yescomplemented
64#1445C2214yescomplemented
65#1445C2214yescomplemented
66#1444C2212yescomplemented
67#1444C2214yescomplemented
68#1444C2214yescomplemented
69#1442C2214yescomplemented
70#1442C2212yescomplemented
71#1441C2214yescomplemented
72#1440C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1437C2214yes—
80#1437C2214yes—
81#1436C2214yescomplemented
82C4⋊#95C2214yescomplemented
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C26C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—28no—
96C2×#117D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—48no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4—28no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—48no—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#399—28no—
143#399—28no—
144#399—28no—
145#399—216no—
146#398—48no—
147#397—28no—
148#396—48no—
149#389—28no—
150#389—28no—
151#389—48no—
152#388—28no—
153#388—28no—
154#388—48no—
155#387—28no—
156#387—28no—
157#385—28no—
158#385—28no—
159#385—28no—
160#385—24no—
161#385—24no—
162#384—28no—
163#383—24no—
164#382—28no—
165#382—24no—
166#381—24no—
167#381—24no—
168#380—28no—
169#379—24no—
170#379—24no—
171#379—28no—
172C4×C4⋊C4—24no—
173C4×C22⋊C4—28no—
174#376—24no—
175C2×#95D418yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C25—22no—
202C25—24no—
203C25—22no—
204C25—22no—
205C25—24no—
206C25—24no—
207C25—44no—
208C25—42no—
209C25C2412yes—
210C25C2411yescharacteristic
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D418yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245#117—232no—
246#117—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—432no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—432no—
324C2×C22⋊C4—416no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4#4118yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—28no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D414yes—
367#95C2×D414yes—
368#95C2×D418yes—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—28no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—48no—
384C24—48no—
385C24—416no—
386C24—416no—
387C24—48no—
388C24—44no—
389C24—44no—
390C24—48no—
391C24—44no—
392C24—44no—
393C24—48no—
394C24—84no—
395C24—84no—
396C24—88no—
397C24—88no—
398C24C22×D412yes—
399C24C22×D412yes—
400C24C2×#4112yes—
401C24C2×#4114yes—
402C242−1+414yes—
403C242−1+414yes—
404C24C2511yescharacteristic
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—216no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—216no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—84no—
483C22×C4—88no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4#121116yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)18yes—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—416no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—416no—
510C4⋊C4—84no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—416no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—88no—
531C42—216no—
532C42—28no—
533C42—28no—
534C42—48no—
535C42—84no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—432no—
553C23—416no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—816no—
573C23—816no—
574C23—832no—
575C23—816no—
576C23—88no—
577C23—88no—
578C23C2×#12114yes—
579C23#52514yes—
580C23#53614yes—
581C23#54718yes—
582C23#55218yes—
583C23C23×D411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C22×#4112yes—
587C23C2×H(5,𝔽2)14yes—
588C23C2×2−1+412yes—
589C23C2×2−1+412yes—
590C23#58512yes—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—232no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—432no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—816no—
668C2×C4—88no—
669C2×C4C2×#12114yes—
670C2×C4C2×#12114yes—
671C2×C4#53518yes—
672C22—28no—
673C22—24no—
674C22—28no—
675C22—28no—
676C22—24no—
677C22—28no—
678C22—216no—
679C22—48no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—48no—
684C22—832no—
685C22—82no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22#241318yes—
692C22C22×#12111yescharacteristic
693C22C2×#52511yescharacteristic
694C22C2×#53512yes—
695C22C2×#53611yescharacteristic
696C22C2×#54712yes—
697C22C2×#55212yes—
698C22#348511yescharacteristic
699C22#348611yescharacteristic
700C22#348912yes—
701C22#349512yes—
702C22#349812yes—
703C22#350812yes—
704C22#351314yes—
705C22#352014yes—
706C22#353514yes—
707C22#353814yes—
708C4—28no—
709C4—24no—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—48no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—84no—
722C4—82no—
723C2—28no—
724C2—84no—
725C2C2×#241311yescharacteristic
726C2#3527211yescharacteristic
727C2#3563311yescharacteristic
728C2#3599012yes—
729C2#3803912yes—
730C2#3891712yes—
731C2#4195312yes—
732C2#4402114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2343C2semidirect
4. C2×#2317C2semidirect
5. #16676C2semidirect
6. #16564C2semidirect
7. #15765C2semidirect
8. #15739C2semidirect
9. #14910C2semidirect
10. #14870C2semidirect
11. #14786C2semidirect
13. #14342C2semidirect
14. #14130C2semidirect
16. #13878C2semidirect
30. C2×#398C22semidirect
35. C2×#389C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
41. C2×#387C22semidirect
42. C2×#387C22semidirect
48. C2×#384C22semidirect
49. C2×#383C22semidirect
50. C2×#382C22semidirect
51. C2×#382C22semidirect
58. C2×C4×C4⋊C4C22semidirect
60. C2×#376C22semidirect
62. #1445C22semidirect
63. #1445C22semidirect
64. #1445C22semidirect
65. #1445C22semidirect
66. #1444C22semidirect
67. #1444C22semidirect
68. #1444C22semidirect
69. #1442C22semidirect
70. #1442C22semidirect
71. #1441C22semidirect
81. #1436C22semidirect
82. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C22×#3992
C2×#23432
C2×#23172
#166762
#165642
#157652
#157392
#149102
#148702
#147862
#147262
#143422
#141302
#139912
#138782
#138682