Finite Group Databasebeta

← #4414504

Subgroups — #4414504

Order 512.

2688 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic, complemented
2C2×#2851C211yesmaximal, characteristic, complemented
3C2×#2850C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic, complemented
5#16664C212yesmaximal, complemented
6#16663C212yesmaximal, complemented
7#16562C214yesmaximal, complemented
8#16395C212yesmaximal, complemented
9#16392C212yesmaximal, complemented
10#14910C212yesmaximal, complemented
11#14894C212yesmaximal, complemented
12#14805C214yesmaximal, complemented
13#14764C212yesmaximal
14#14763C212yesmaximal, complemented
15#14677C211yesmaximal, characteristic, complemented
16#14664C211yesmaximal, characteristic, complemented
17#14662C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2851—28no—
24#2850—28no—
25#2820—28no—
26C2×#399C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#394C2212yes—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2214yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45#2283—28no—
46C2×#383C2212yes—
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#381C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2212yes—
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1444C2212yes—
63#1444C2212yes—
64#1443C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1439C2214yes—
82#1438C2214yes—
83#1438C2214yes—
84#1438C2212yes—
85#1437C2214yes—
86#1436C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#125—24no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#124—24no—
97C2×#124—24no—
98C2×#124—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—44no—
101C2×#121—24no—
102C2×#121—44no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42—24no—
132C22×C42—24no—
133C22×C42D418yes—
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#399—28no—
140#397—28no—
141#396—216no—
142#395—28no—
143#394—28no—
144#391—24no—
145#390—24no—
146#389—24no—
147#389—24no—
148#389—48no—
149#388—28no—
150#388—28no—
151#388—48no—
152#387—28no—
153#387—28no—
154#386—28no—
155#386—28no—
156#385—28no—
157#385—28no—
158#383—28no—
159#382—24no—
160#382—24no—
161#382—24no—
162#382D418yescomplemented
163#381—28no—
164#380—24no—
165#380—24no—
166#380—28no—
167C4×C4⋊C4—28no—
168C4×C22⋊C4—28no—
169#376—28no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×D4—24no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—42no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—41no—
224C23×C4—41no—
225C23×C4—41no—
226C23×C4—41no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#125—416no—
246#124—416no—
247C22⋊Q8—416no—
248#121—416no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
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⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—232no—
330C2×C42—28no—
331C2×C42—28no—
332C2×C42—44no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42#4114yes—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95C2×D418yes—
364#95C2×D418yes—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24C22×D412yes—
386C242−1+414yes—
387C24C2511yescharacteristic
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
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—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D418yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4H(5,𝔽2)14yes—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—416no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—88no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—416no—
537C42#121116yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
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—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23#53614yes—
584C23C23×D411yescharacteristic
585C23C23×D411yescharacteristic
586C23C22×#4111yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23C2×2−1+412yes—
589C23#58511yescharacteristic
590C23#58511yescharacteristic
591C23#58511yescharacteristic
592C23#58511yescharacteristic
593C23#58512yes—
594C23#58512yes—
595Q8—88no—
596D4—88no—
597D4—816no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4C2×#12114yes—
683C2×C4C2×#12114yes—
684C2×C4C2×#12114yes—
685C2×C4#53514yes—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—42no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22—82no—
703C22—84no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22C22×#12111yescharacteristic
709C22C2×#53511yescharacteristic
710C22C2×#53611yescharacteristic
711C22#344911yescharacteristic
712C22#344911yescharacteristic
713C22#344911yescharacteristic
714C22#344911yescharacteristic
715C22#348511yescharacteristic
716C22#348711yescharacteristic
717C22#348812yes—
718C22#348912yes—
719C22#349311yescharacteristic
720C22#349311yescharacteristic
721C22#349412yes—
722C22#349512yes—
723C22#352912yes—
724C22#352912yes—
725C22#355312yes—
726C22#355312yes—
727C22#356112yes—
728C22#356112yes—
729C22#356712yes—
730C22#356812yes—
731C4—24no—
732C4—24no—
733C4—24no—
734C4—24no—
735C4—42no—
736C4—48no—
737C4—42no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—84no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C2—44no—
747C2—162no—
748C2#3515011yescharacteristic
749C2#3525211yescharacteristic
750C2#3530611yescharacteristic
751C2#3668711yescharacteristic
752C2#3753711yescharacteristic
753C2#3754111yescharacteristic
754C2#3760711yescharacteristic
755C2#4597212yes—
756C2#4597512yes—
757C2#4598412yes—
758C2#4598712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2820C2semidirect
2. C2×#2851C2semidirect
4. C2×#2283C2semidirect
5. #16664C2semidirect
6. #16663C2semidirect
7. #16562C2semidirect
8. #16395C2semidirect
9. #16392C2semidirect
10. #14910C2semidirect
11. #14894C2semidirect
12. #14805C2semidirect
14. #14763C2semidirect
15. #14677C2semidirect
16. #14664C2semidirect
17. #14662C2semidirect
162. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#28512
C2×#28502
C2×#22832
#166642
#166632
#165622
#163952
#163922
#149102
#148942
#148052
#147642
#147632
#146772
#146642
#146622