Finite Group Databasebeta

← #4414341

Subgroups — #4414341

Order 512.

2872 conjugacy classes of subgroups.

The subgroup lattice has 771 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×#2285C211yesmaximal, characteristic, complemented
5#16515C214yesmaximal, complemented
6#16401C212yesmaximal, complemented
7#16398C212yesmaximal, complemented
8#16264C212yesmaximal, complemented
9#16261C212yesmaximal
10#14890C214yesmaximal, complemented
11#14872C212yesmaximal, complemented
12#14870C212yesmaximal, complemented
13#14813C212yesmaximal, complemented
14#14808C212yesmaximal, complemented
15#14677C211yesmaximal, characteristic, complemented
16#14664C211yesmaximal, characteristic, complemented
17#14662C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24#2851—28no—
25#2850—28no—
26#2820—28no—
27C2×#399C2212yes—
28C2×#398C2214yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#395C2212yes—
33C2×#394C2212yes—
34C2×#394C2214yes—
35C2×#391C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48#2285—28no—
49C2×#384C2212yes—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×#380C2212yes—
57C2×#380C2212yes—
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C4⋊C4C2212yes—
60#1446C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1444C2212yes—
64#1444C2212yes—
65#1442C2212yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2214yes—
81#1439C2214yes—
82#1438C2214yes—
83#1438C2212yes—
84#1437C2214yes—
85#1436C2214yes—
86C4⋊#95C2214yes—
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—
103C2×#117—28no—
104C2×#117—44no—
105C2×#117—44no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42—24no—
137C22×C42—24no—
138C22×C42D418yes—
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#399—28no—
145#398—48no—
146#397—28no—
147#396—216no—
148#396—48no—
149#395—28no—
150#394—28no—
151#394—48no—
152#391—24no—
153#390—24no—
154#389—24no—
155#389—24no—
156#388—28no—
157#388—28no—
158#388—48no—
159#387—28no—
160#387—28no—
161#386—28no—
162#386—28no—
163#385—28no—
164#385—28no—
165#384—28no—
166#382—24no—
167#382—24no—
168#382—24no—
169#382D418yescomplemented
170#380—24no—
171#380—24no—
172#380—28no—
173#380—28no—
174C4×C4⋊C4—28no—
175C4×C4⋊C4—28no—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C25C2412yes—
201C25C2411yescharacteristic
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4—44no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#125—416no—
250#124—416no—
251C22⋊Q8—416no—
252#121—416no—
253#117—432no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
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—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D418yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
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—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—232no—
341C2×C42—28no—
342C2×C42—28no—
343C2×C42—44no—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—28no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95C2×D418yes—
373#95C2×D418yes—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—24no—
393C24C22×D412yes—
394C24H(5,𝔽2)12yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—88no—
411C2×D4—88no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
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—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4C22×D418yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C42−1+414yes—
513C22×C42−1+412yes—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—88no—
530C4⋊C4—88no—
531C22⋊C4—416no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—416no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C42—28no—
545C42—28no—
546C42—28no—
547C42—416no—
548C42#121116yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23#53714yes—
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C22×#4111yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×2−1+412yes—
602C23C2×2−1+411yescharacteristic
603C23C2×2−1+411yescharacteristic
604C23#58512yes—
605C23#58512yes—
606Q8—88no—
607D4—88no—
608D4—816no—
609C2×C4—232no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
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—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4C2×#12114yes—
692C2×C4C2×#12114yes—
693C2×C4C2×#12114yes—
694C2×C4#53514yes—
695C2×C4#53614yes—
696C2×C4#53714yes—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—42no—
701C22—42no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—42no—
708C22—82no—
709C22—82no—
710C22—84no—
711C22—84no—
712C22—84no—
713C22—82no—
714C22—84no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22C22×#12111yescharacteristic
720C22C2×#53511yescharacteristic
721C22C2×#53611yescharacteristic
722C22C2×#53711yescharacteristic
723C22C2×#53711yescharacteristic
724C22#344911yescharacteristic
725C22#344911yescharacteristic
726C22#348411yescharacteristic
727C22#348611yescharacteristic
728C22#348812yes—
729C22#348912yes—
730C22#349211yescharacteristic
731C22#349211yescharacteristic
732C22#349412yes—
733C22#349512yes—
734C22#352912yes—
735C22#353112yes—
736C22#354812yes—
737C22#355012yes—
738C22#355812yes—
739C22#355912yes—
740C22#356112yes—
741C22#356312yes—
742C4—24no—
743C4—24no—
744C4—24no—
745C4—24no—
746C4—24no—
747C4—24no—
748C4—48no—
749C4—82no—
750C4—82no—
751C4—84no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—82no—
757C2—44no—
758C2—162no—
759C2#3522311yescharacteristic
760C2#3525211yescharacteristic
761C2#3530611yescharacteristic
762C2#3668411yescharacteristic
763C2#3754611yescharacteristic
764C2#3754911yescharacteristic
765C2#3764011yescharacteristic
766C2#4622412yes—
767C2#4622712yes—
768C2#4623012yes—
769C2#4623312yes—
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×#2285C2semidirect
5. #16515C2semidirect
6. #16401C2semidirect
7. #16398C2semidirect
8. #16264C2semidirect
10. #14890C2semidirect
11. #14872C2semidirect
12. #14870C2semidirect
13. #14813C2semidirect
14. #14808C2semidirect
15. #14677C2semidirect
16. #14664C2semidirect
17. #14662C2semidirect
169. #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×#22852
#165152
#164012
#163982
#162642
#162612
#148902
#148722
#148702
#148132
#148082
#146772
#146642
#146622