Finite Group Databasebeta

← #719969

Subgroups — #719969

Order 512.

2976 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2441C212yesmaximal, complemented
3C2×#2442C212yesmaximal, complemented
4C2×#2352C211yesmaximal, characteristic
5#17064C212yesmaximal, complemented
6#16984C212yesmaximal, complemented
7#15804C214yesmaximal, complemented
8#15505C214yesmaximal
9#15223C212yesmaximal, complemented
10#15135C212yesmaximal, complemented
11#15135C212yesmaximal, complemented
12#14969C212yesmaximal, complemented
13#14354C211yesmaximal, characteristic, complemented
14#13954C211yesmaximal, characteristic, complemented
15#13906C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2552—28no—
23#2442—216no—
24#2441—216no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#399C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#396C2214yes—
31#2352—28no—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#387C2214yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#382C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2214yes—
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2212yes—
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic
61#1445C2212yes—
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1444C2212yes—
66#1444C2212yes—
67#1444C2214yes—
68#1444C2212yes—
69#1444C2214yes—
70#1444C2212yes—
71#1439C2211yescharacteristic
72#1439C2211yescharacteristic
73#1439C2211yescharacteristic
74#1439C2211yescharacteristic
75#1439C2212yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1438C2212yes—
81#1436C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2214yes—
86#1434C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#125—28no—
92C2×#125—28no—
93C2×#125—28no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4Q818yes—
97C2×#117—24no—
98C2×#117—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#400—216no—
139#399—216no—
140#399—48no—
141#398—216no—
142#398—216no—
143#396—216no—
144#390—44no—
145#389—216no—
146#389—28no—
147#389—28no—
148#389—28no—
149#389—28no—
150#389D4116yescomplemented
151#388—28no—
152#387—28no—
153#387—216no—
154#386—28no—
155#385—216no—
156#384—28no—
157#384—28no—
158#384—24no—
159#382—24no—
160#382—28no—
161#380—24no—
162#380—24no—
163#380—28no—
164#380—216no—
165#380—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4D418yes—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—28no—
171C4×C22⋊C4—28no—
172#376—24no—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C25C2411yescharacteristic
198C25C2411yescharacteristic
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4C2×Q814yes—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4C2×Q814yes—
220C23×C4C2×Q814yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#125—432no—
237C4×D4—232no—
238#117—416no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—416no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×Q814yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
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—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×Q814yes—
322C2×C22⋊C4C2×Q814yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42C2×D418yes—
335C2×C42C2×D418yes—
336C2×C42C2×D414yes—
337C2×C42C2×Q814yes—
338C2×C42#4114yes—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—44no—
360#95—44no—
361#95C2×D418yes—
362#95C2×D4116yes—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24C22×Q812yes—
382C24C22×Q812yes—
383C24H(5,𝔽2)12yes—
384C242−1+412yes—
385C24C2511yescharacteristic
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D418yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×Q812yes—
489C22×C4C22×Q812yes—
490C22×C4C22×Q812yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C42−1+414yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C4⋊C4—88no—
516C4⋊C4#127116yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—88no—
528C42—216no—
529C42—48no—
530C42—48no—
531C42—48no—
532C42—84no—
533C42C22⋊Q8116yes—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23#54414yes—
580C23C23×D411yescharacteristic
581C23C23×D412yes—
582C23C23×Q811yescharacteristic
583C23C22×#4111yescharacteristic
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×2−1+412yes—
586C23C2×2−1+411yescharacteristic
587C23#58511yescharacteristic
588C23#58511yescharacteristic
589C23#58512yes—
590C23#58512yes—
591D4—416no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—816no—
676C2×C4C2×C22⋊Q814yes—
677C2×C4C2×C22⋊Q814yes—
678C2×C4C2×C22⋊Q814yes—
679C2×C4C2×#12718yes—
680C2×C4C2×#12714yes—
681C2×C4#53614yes—
682C2×C4#54718yes—
683C2×C4D4×Q818yes—
684C22—28no—
685C22—24no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—82no—
701C22—81no—
702C22—81no—
703C22—82no—
704C22—81no—
705C22—81no—
706C22C22×C22⋊Q811yescharacteristic
707C22C22×#12711yescharacteristic
708C22C2×#53611yescharacteristic
709C22C2×#54411yescharacteristic
710C22#345811yescharacteristic
711C22C2×#54712yes—
712C22C2×D4×Q812yes—
713C22#348511yescharacteristic
714C22#348512yes—
715C22#348711yescharacteristic
716C22#348712yes—
717C22#348812yes—
718C22#348811yescharacteristic
719C22#348912yes—
720C22#348911yescharacteristic
721C22#349411yescharacteristic
722C22#349511yescharacteristic
723C22#350412yes—
724C22#351212yes—
725C22#355912yes—
726C22#356312yes—
727C22#356512yes—
728C22#356512yes—
729C4—24no—
730C4—24no—
731C4—28no—
732C4—44no—
733C4—48no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—84no—
743C4#254218yes—
744C2—44no—
745C2—82no—
746C2C2×#254211yescharacteristic
747C2#3525311yescharacteristic
748C2#3563911yescharacteristic
749C2#3606011yescharacteristic
750C2#3606611yescharacteristic
751C2#3768712yes—
752C2#3769112yes—
753C2#4268211yescharacteristic
754C2#4288811yescharacteristic
755C2#4513512yes—
756C2#4526412yes—
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×#2552C2semidirect
2. C2×#2441C2semidirect
3. C2×#2442C2semidirect
5. #17064C2semidirect
6. #16984C2semidirect
7. #15804C2semidirect
9. #15223C2semidirect
10. #15135C2semidirect
11. #15135C2semidirect
12. #14969C2semidirect
13. #14354C2semidirect
14. #13954C2semidirect
17. #13783C2semidirect
150. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24412
C2×#24422
C2×#23522
#170642
#169842
#158042
#155052
#152232
#151352
#149692
#143542
#139542
#139062
#138472
#137832