Finite Group Databasebeta

← #713264

Subgroups — #713264

Order 512.

3192 conjugacy classes of subgroups.

The subgroup lattice has 777 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic
3C2×#2346C211yesmaximal, characteristic, complemented
4C2×#2310C211yesmaximal, characteristic, complemented
5#16566C212yesmaximal, complemented
6#16565C212yesmaximal, complemented
7#15921C214yesmaximal
8#15915C214yesmaximal, complemented
9#15189C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#14902C212yesmaximal, complemented
12#14901C212yesmaximal, complemented
13#14264C211yesmaximal, characteristic, complemented
14#14142C212yesmaximal, complemented
15#13997C212yesmaximal, complemented
16#13922C211yesmaximal, characteristic, complemented
17#13839C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2485—28no—
24#2409—28no—
25C2×#397C2212yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28C2×#393C2212yes—
29#2346—28no—
30#2310—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2212yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#382C2212yes—
48C2×#381C2214yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2212yes—
57#1446C2212yes—
58#1446C2212yes—
59#1445C2214yes—
60#1444C2212yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2211yescharacteristic
67#1441C2211yescharacteristic
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2212yes—
73#1440C2214yes—
74#1439C2211yescharacteristic
75#1439C2211yescharacteristic
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1438C2212yes—
80#1437C2214yes—
81#1436C2214yes—
82C4⋊#95C2214yes—
83C4⋊#95C2212yes—
84#1434C2214yes—
85C22⋊#95C2214yes—
86C4×#95C2212yes—
87C23×Q8—28no—
88C23×Q8—24no—
89C23×Q8—24no—
90C23×Q8—24no—
91C23×Q8—44no—
92C23×Q8C2312yes—
93C23×Q8C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—44no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×#117—24no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117—44no—
108C2×#117D418yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2314yes—
138C22×C42C2312yes—
139#397—28no—
140#395—28no—
141#394—28no—
142#393—28no—
143#391—28no—
144#391—28no—
145#391—28no—
146#391—28no—
147#391—28no—
148#391—28no—
149#391—44no—
150#389—28no—
151#389—24no—
152#389—24no—
153#389—44no—
154#388—216no—
155#388—28no—
156#387—28no—
157#386—28no—
158#385—28no—
159#382—28no—
160#381—216no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380D418yes—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—44no—
168#376—28no—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—232no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—28no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—44no—
212C22×Q8—432no—
213C22×Q8—432no—
214C22×Q8—416no—
215C22×Q8—416no—
216C22×Q8#4118yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—41no—
223C22×D4—41no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—28no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C22⋊Q8—432no—
255C4×D4—416no—
256#117—232no—
257#117—416no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D418yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
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×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D418yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—28no—
339C2×C42—44no—
340C2×C42C2×D418yes—
341#95—22no—
342#95—28no—
343#95—22no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—24no—
360#95—42no—
361#95—44no—
362#95—44no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95C2×D418yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C2×#4112yes—
391C242−1+412yes—
392C24C2511yescharacteristic
393C2×Q8—216no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8—416no—
398C2×Q8—416no—
399C2×Q8—48no—
400C2×Q8—4128no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×D4—28no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
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—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4#121116yes—
505C22×C4C22×D414yes—
506C22×C4C22×D414yes—
507C22×C4C22×D418yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C2×#4118yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4114yes—
516C22×C4H(5,𝔽2)14yes—
517C22×C42−1+412yes—
518C22×C42−1+414yes—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—48no—
523C4⋊C4—416no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—88no—
530C4⋊C4—84no—
531C4⋊C4—88no—
532C22⋊C4—216no—
533C22⋊C4—48no—
534C22⋊C4—416no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—216no—
545C42—416no—
546C42—44no—
547C42—48no—
548C42—48no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23C2×#12114yes—
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C22×#4111yescharacteristic
598C23C22×#4112yes—
599C23C2×H(5,𝔽2)12yes—
600C23C2×2−1+412yes—
601C23C2×2−1+411yescharacteristic
602C23C2×2−1+411yescharacteristic
603C23#58512yes—
604C23#58511yescharacteristic
605C23#58511yescharacteristic
606Q8—432no—
607Q8—432no—
608D4—416no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—232no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—432no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—41no—
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—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—88no—
697C2×C4C2×#12114yes—
698C2×C4C2×#12114yes—
699C2×C4#52514yes—
700C2×C4#53514yes—
701C2×C4#54618yes—
702C2×C4#55118yes—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—44no—
717C22—42no—
718C22—42no—
719C22—82no—
720C22—82no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22C22×#12111yescharacteristic
726C22C2×#52511yescharacteristic
727C22C2×#53511yescharacteristic
728C22#346112yes—
729C22C2×#54612yes—
730C22C2×#55112yes—
731C22#348511yescharacteristic
732C22#348611yescharacteristic
733C22#348911yescharacteristic
734C22#348911yescharacteristic
735C22#348912yes—
736C22#349511yescharacteristic
737C22#349511yescharacteristic
738C22#349911yescharacteristic
739C22#350011yescharacteristic
740C22#351612yes—
741C22#351812yes—
742C22#352312yes—
743C22#352412yes—
744C22#353812yes—
745C22#354012yes—
746C22#354712yes—
747C22#354712yes—
748C4—24no—
749C4—44no—
750C4—42no—
751C4—48no—
752C4—44no—
753C4—48no—
754C4—44no—
755C4—82no—
756C4—82no—
757C4—81no—
758C4—81no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4#241118yes—
763C2—44no—
764C2—82no—
765C2C2×#241111yescharacteristic
766C2#3530811yescharacteristic
767C2#3563311yescharacteristic
768C2#3598511yescharacteristic
769C2#3600711yescharacteristic
770C2#3820312yes—
771C2#3904312yes—
772C2#4299411yescharacteristic
773C2#4299511yescharacteristic
774C2#4538512yes—
775C2#4541612yes—
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
3. C2×#2346C2semidirect
4. C2×#2310C2semidirect
5. #16566C2semidirect
6. #16565C2semidirect
8. #15915C2semidirect
9. #15189C2semidirect
10. #15179C2semidirect
11. #14902C2semidirect
12. #14901C2semidirect
13. #14264C2semidirect
14. #14142C2semidirect
15. #13997C2semidirect
16. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#24092
C2×#23462
C2×#23102
#165662
#165652
#159212
#159152
#151892
#151792
#149022
#149012
#142642
#141422
#139972
#139222
#138392