Finite Group Databasebeta

← #719967

Subgroups — #719967

Order 512.

2954 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic, complemented
2C2×#2441C212yesmaximal, complemented
3C2×#2439C212yesmaximal, complemented
4C2×#2354C211yesmaximal, characteristic
5#17071C212yesmaximal
6#16814C212yesmaximal
7#15825C214yesmaximal, complemented
8#15765C214yesmaximal, complemented
9#15223C212yesmaximal, complemented
10#15193C212yesmaximal, complemented
11#15135C212yesmaximal, complemented
12#15082C212yesmaximal, complemented
13#14354C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13905C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2550—28no—
23#2441—216no—
24#2439—216no—
25C2×#399C2214yes—
26C2×#399C2214yes—
27C2×#398C2212yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#395C2212yes—
31C2×#393C2214yes—
32#2354—28no—
33C2×#391C2212yes—
34C2×#391C2214yes—
35C2×#390C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#387C2214yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic
61#1445C2211yescharacteristic
62#1445C2211yescharacteristic
63#1445C2214yes—
64#1445C2212yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2212yes—
68#1444C2214yes—
69#1444C2212yes—
70#1444C2214yes—
71#1444C2212yes—
72#1444C2212yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1438C2211yescharacteristic
79#1438C2211yescharacteristic
80#1438C2212yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—24no—
92C2×#125—28no—
93C2×#125—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4Q818yescomplemented
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—28no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#399—216no—
138#399—48no—
139#398—28no—
140#396—216no—
141#396—216no—
142#395—28no—
143#393—216no—
144#391—216no—
145#391—28no—
146#390—28no—
147#389—28no—
148#389—216no—
149#389—28no—
150#389—24no—
151#389—24no—
152#389—28no—
153#389D4116yescomplemented
154#388—216no—
155#387—216no—
156#387—216no—
157#384—24no—
158#384—28no—
159#384—44no—
160#382—28no—
161#382—24no—
162#380—28no—
163#380—28no—
164#380—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4D418yes—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—28no—
171#376—24no—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C25C2412yes—
198C22×Q8—28no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—44no—
202C22×Q8—42no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4C2×Q814yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4C2×Q818yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232#125—416no—
233C4×D4—232no—
234#117—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D4116yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×Q818yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42—42no—
329C2×C42C2×D418yes—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332C2×C42C2×Q814yes—
333C2×C42#4114yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—22no—
340#95—22no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—44no—
356#95—44no—
357#95C2×D418yes—
358#95C2×D4116yes—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24C22×Q814yes—
374C24C2511yescharacteristic
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—84no—
379C2×Q8—88no—
380C2×D4—28no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D418yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C42−1+414yes—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—216no—
487C4⋊C4—416no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C4⋊C4#127116yes—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—216no—
515C42—48no—
516C42—48no—
517C42—416no—
518C42—44no—
519C42—44no—
520C42C22⋊Q8116yes—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
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—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23C23×D411yescharacteristic
557C23C23×D412yes—
558C23C23×Q811yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×2−1+412yes—
562C23C2×2−1+411yescharacteristic
563C23#58512yes—
564C23#58512yes—
565C23#58511yescharacteristic
566C23#58511yescharacteristic
567Q8—816no—
568D4—416no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—432no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4C2×C22⋊Q814yes—
655C2×C4C2×C22⋊Q814yes—
656C2×C4C2×C22⋊Q814yes—
657C2×C4C2×#12718yes—
658C2×C4C2×#12714yes—
659C2×C4#53614yes—
660C2×C4#54718yes—
661C2×C4D4×Q818yes—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22C22×C22⋊Q811yescharacteristic
678C22C22×#12711yescharacteristic
679C22C2×#53611yescharacteristic
680C22#345812yes—
681C22C2×#54712yes—
682C22C2×D4×Q812yes—
683C22#348512yes—
684C22#348511yescharacteristic
685C22#348712yes—
686C22#348711yescharacteristic
687C22#348812yes—
688C22#348811yescharacteristic
689C22#348912yes—
690C22#348911yescharacteristic
691C22#349312yes—
692C22#350311yescharacteristic
693C22#350411yescharacteristic
694C22#350611yescharacteristic
695C22#350711yescharacteristic
696C22#354712yes—
697C22#355512yes—
698C22#356512yes—
699C22#356512yes—
700C4—24no—
701C4—24no—
702C4—28no—
703C4—44no—
704C4—48no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—81no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4—82no—
715C4#254218yes—
716C2—84no—
717C2C2×#254211yescharacteristic
718C2#3515111yescharacteristic
719C2#3551911yescharacteristic
720C2#3606011yescharacteristic
721C2#3606611yescharacteristic
722C2#3769012yes—
723C2#3769112yes—
724C2#4268111yescharacteristic
725C2#4289111yescharacteristic
726C2#4490612yes—
727C2#4526112yes—
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
1. C2×#2550C2semidirect
2. C2×#2441C2semidirect
3. C2×#2439C2semidirect
7. #15825C2semidirect
8. #15765C2semidirect
9. #15223C2semidirect
10. #15193C2semidirect
11. #15135C2semidirect
12. #15082C2semidirect
13. #14354C2semidirect
14. #13910C2semidirect
17. #13783C2semidirect
96. C2×C4×D4Q8semidirect
153. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#24412
C2×#24392
C2×#23542
#170712
#168142
#158252
#157652
#152232
#151932
#151352
#150822
#143542
#139102
#139052
#138472
#137832