Finite Group Databasebeta

← #719968

Subgroups — #719968

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic, complemented
2C2×#2442C212yesmaximal, complemented
3C2×#2440C212yesmaximal, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5#17075C212yesmaximal, complemented
6#16813C212yesmaximal, complemented
7#15513C214yesmaximal, complemented
8#15430C214yesmaximal, complemented
9#15223C212yesmaximal, complemented
10#15177C212yesmaximal
11#15135C212yesmaximal, complemented
12#14807C212yesmaximal
13#14354C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13905C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2214yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4C2212yes—
24#2550—28no—
25#2442—216no—
26#2440—216no—
27C2×#400C2214yes—
28C2×#399C2214yes—
29C2×#398C2214yes—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#395C2212yes—
33C2×#395C2214yes—
34D4×C4⋊C4—28no—
35C2×#391C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2214yes—
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
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#1445C2214yes—
65#1445C2212yes—
66#1445C2214yes—
67#1444C2214yes—
68#1444C2212yes—
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—
90C24×C4C2314yes—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×#125—28no—
94C2×#125—24no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—48no—
97C2×C22⋊Q8—48no—
98C2×C4×D4—24no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—48no—
102C2×C4×D4Q818yescomplemented
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#400—216no—
143#399—48no—
144#398—216no—
145#398—216no—
146#398—28no—
147#395—216no—
148#395—28no—
149#391—28no—
150#389—28no—
151#389—28no—
152#389—24no—
153#389—24no—
154#389—28no—
155#389—28no—
156#389D4116yescomplemented
157#385—216no—
158#385—216no—
159#385—216no—
160#384—24no—
161#384—28no—
162#384—44no—
163#382—28no—
164#382—24no—
165#380—28no—
166#380—28no—
167#380—216no—
168#380—28no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4D418yes—
172C4×C22⋊C4—28no—
173C4×C22⋊C4—28no—
174#376—24no—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C25C2412yes—
201C22×Q8—28no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—42no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×Q814yes—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×Q818yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C4⋊Q8—416no—
242#125—416no—
243C22⋊Q8—464no—
244C4×D4—232no—
245C4×D4—432no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—432no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D4116yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D418yes—
303C2×C4⋊C4C2×Q814yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
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—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4C2×Q818yes—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42C2×D418yes—
339C2×C42C2×D418yes—
340C2×C42C2×D414yes—
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367#95C2×D4116yes—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24C22×Q814yes—
381C24H(5,𝔽2)14yes—
382C242−1+414yes—
383C24C2511yescharacteristic
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—84no—
388C2×Q8—88no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—28no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—416no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D418yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×Q814yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+412yes—
499C22×C42−1+414yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—48no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C4⋊C4—88no—
521C4⋊C4#127116yes—
522C22⋊C4—216no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C42—216no—
531C42—48no—
532C42—48no—
533C42—44no—
534C42—44no—
535C42C22⋊Q8116yescomplemented
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23#54418yes—
571C23C23×D411yescharacteristic
572C23C23×D412yes—
573C23C23×Q811yescharacteristic
574C23C22×#4111yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×2−1+412yes—
578C23C2×2−1+412yes—
579C23C2×2−1+411yescharacteristic
580C23#58511yescharacteristic
581C23#58511yescharacteristic
582Q8—816no—
583D4—232no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—432no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—832no—
670C2×C4C2×C22⋊Q814yes—
671C2×C4C2×C22⋊Q814yes—
672C2×C4C2×C22⋊Q814yes—
673C2×C4C2×#12718yes—
674C2×C4C2×#12714yes—
675C2×C4#53614yes—
676C2×C4#54718yes—
677C2×C4D4×Q818yes—
678C22—28no—
679C22—28no—
680C22—28no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22C22×C22⋊Q811yescharacteristic
694C22C22×#12711yescharacteristic
695C22C2×#53611yescharacteristic
696C22C2×#54412yes—
697C22C2×#54712yes—
698C22C2×D4×Q812yes—
699C22#348412yes—
700C22#348411yescharacteristic
701C22#348612yes—
702C22#348611yescharacteristic
703C22#348812yes—
704C22#348811yescharacteristic
705C22#348912yes—
706C22#348911yescharacteristic
707C22#349212yes—
708C22#350311yescharacteristic
709C22#350411yescharacteristic
710C22#350611yescharacteristic
711C22#350711yescharacteristic
712C22#354412yes—
713C22#355212yes—
714C22#355912yes—
715C22#356312yes—
716C4—28no—
717C4—24no—
718C4—24no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—44no—
723C4—44no—
724C4—48no—
725C4—44no—
726C4—81no—
727C4—84no—
728C4—82no—
729C4—81no—
730C4—82no—
731C4#254218yes—
732C2—48no—
733C2C2×#254211yescharacteristic
734C2#3522411yescharacteristic
735C2#3551911yescharacteristic
736C2#3592111yescharacteristic
737C2#3592711yescharacteristic
738C2#3768612yes—
739C2#3768712yes—
740C2#4248211yescharacteristic
741C2#4281411yescharacteristic
742C2#4492112yes—
743C2#4525512yes—
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×#2442C2semidirect
3. C2×#2440C2semidirect
5. #17075C2semidirect
6. #16813C2semidirect
7. #15513C2semidirect
8. #15430C2semidirect
9. #15223C2semidirect
11. #15135C2semidirect
13. #14354C2semidirect
14. #13910C2semidirect
17. #13783C2semidirect
102. C2×C4×D4Q8semidirect
156. #389D4semidirect
535. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#24422
C2×#24402
C2×D4×C4⋊C42
#170752
#168132
#155132
#154302
#152232
#151772
#151352
#148072
#143542
#139102
#139052
#138472
#137832