Finite Group Databasebeta

← #769922

Subgroups — #769922

Order 512.

3082 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3C2×#2581C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic
5C2×#2399C211yesmaximal, characteristic, complemented
6C2×#2326C211yesmaximal, characteristic
7#15825C214yesmaximal, complemented
8#15765C214yesmaximal, complemented
9#15513C214yesmaximal, complemented
10#15430C214yesmaximal, complemented
11#15193C212yesmaximal, complemented
12#15148C212yesmaximal
13#14153C212yesmaximal
14#13910C212yesmaximal, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2582—28no—
23#2581—28no—
24#2580—28no—
25#2538—28no—
26#2399—28no—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#398C2214yes—
31C2×#398C2212yes—
32C2×#398C2212yes—
33C2×#396C2214yes—
34C2×#395C2212yes—
35C2×#395C2212yes—
36C2×#395C2212yes—
37C2×#393C2212yes—
38#2326—28no—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#390C2211yescharacteristic
44C2×#390C2212yes—
45C2×#390C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#388C2214yes—
52C2×#387C2214yes—
53C2×#385C2214yes—
54C2×#385C2214yes—
55C2×#384C2211yescharacteristic
56C2×#384C2214yes—
57C2×#384C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#380C2212yes—
60C2×C4×C4⋊C4C2212yes—
61C2×C4×C4⋊C4C2212yes—
62C2×C4×C4⋊C4C2212yes—
63C2×#376C2211yescharacteristic
64#1445C2214yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124Q818yescomplemented
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—28no—
138#399—28no—
139#399—48no—
140#398—216no—
141#398—28no—
142#398—28no—
143#396—216no—
144#395—28no—
145#395—28no—
146#395—28no—
147#393—28no—
148#391—28no—
149#391—28no—
150#391—28no—
151#391—24no—
152#390—28no—
153#390—28no—
154#390—24no—
155#389—28no—
156#389—28no—
157#389—28no—
158#389—28no—
159#389—28no—
160#388—216no—
161#387—216no—
162#385—216no—
163#385—216no—
164#384—24no—
165#384—48no—
166#384D418yes—
167#382D418yes—
168#380—28no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—28no—
172#376D418yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C25C2412yes—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8C2×Q814yes—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4C2×Q814yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237#124—232no—
238C22⋊Q8—432no—
239C4×Q8—416no—
240C4×Q8—416no—
241C4×Q8—416no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
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—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D418yes—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×Q8116yes—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360#95C2×D418yes—
361#95C2×D418yes—
362#95C2×D418yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24C22×Q814yes—
374C24C2511yescharacteristic
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
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—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×Q814yes—
486C22×C4C22×Q814yes—
487C22×C4C22×Q812yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C22⋊C4—232no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—28no—
522C42—416no—
523C42—48no—
524C42—48no—
525C42—48no—
526C42C22⋊Q8116yescomplemented
527C42#127116yes—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C23×Q811yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+411yescharacteristic
558C23C2×2−1+411yescharacteristic
559C23C2×2−1+412yes—
560C23#58512yes—
561C23#58512yes—
562Q8—416no—
563Q8—416no—
564Q8—416no—
565D4—416no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—816no—
647C2×C4C2×C22⋊Q818yes—
648C2×C4C2×C22⋊Q814yes—
649C2×C4C2×#12714yes—
650C2×C4C2×#12714yes—
651C2×C4C2×#12714yes—
652C2×C4#53614yes—
653C2×C4#54714yes—
654C2×C4#54714yes—
655C2×C4D4×Q814yes—
656C2×C4D4×Q814yes—
657C22—44no—
658C22—44no—
659C22—48no—
660C22—44no—
661C22—48no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—84no—
667C22—82no—
668C22C22×C22⋊Q811yescharacteristic
669C22C22×#12711yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#54711yescharacteristic
672C22C2×#54711yescharacteristic
673C22C2×D4×Q811yescharacteristic
674C22C2×D4×Q811yescharacteristic
675C22#348411yescharacteristic
676C22#348511yescharacteristic
677C22#348611yescharacteristic
678C22#348711yescharacteristic
679C22#348812yes—
680C22#348812yes—
681C22#348912yes—
682C22#348912yes—
683C22#350312yes—
684C22#350412yes—
685C22#350612yes—
686C22#350712yes—
687C22#355012yes—
688C22#355412yes—
689C22#355512yes—
690C22#355712yes—
691C4—24no—
692C4—28no—
693C4—24no—
694C4—48no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—48no—
699C4—44no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4#254218yes—
705C2—84no—
706C2C2×#254211yescharacteristic
707C2#3551912yes—
708C2#3556511yescharacteristic
709C2#3563411yescharacteristic
710C2#3727512yes—
711C2#4493411yescharacteristic
712C2#4496412yes—
713C2#4513911yescharacteristic
714C2#4535411yescharacteristic
715C2#4539411yescharacteristic
716C2#4542412yes—
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×#2582C2semidirect
3. C2×#2581C2semidirect
5. C2×#2399C2semidirect
7. #15825C2semidirect
8. #15765C2semidirect
9. #15513C2semidirect
10. #15430C2semidirect
11. #15193C2semidirect
14. #13910C2semidirect
15. #13779C2semidirect
88. C2×#124Q8semidirect
526. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25382
C2×#25812
C2×#25802
C2×#23992
C2×#23262
#158252
#157652
#155132
#154302
#151932
#151482
#141532
#139102
#137792