Finite Group Databasebeta

← #4419016

Subgroups — #4419016

Order 512.

3570 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
1C22×#552C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2659C211yesmaximal, characteristic, complemented
4C2×#2638C211yesmaximal, characteristic
5#16683C212yesmaximal, complemented
6#16608C212yesmaximal, complemented
7#16607C212yesmaximal, complemented
8#16601C212yesmaximal, complemented
9#15911C214yesmaximal, complemented
10#15269C214yesmaximal
11#14907C212yesmaximal, complemented
12#14887C212yesmaximal, complemented
13#14879C212yesmaximal, complemented
14#14786C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic, complemented
16#13960C211yesmaximal, characteristic
17#13958C211yesmaximal, characteristic, complemented
18C2×#552—28no—
19C2×#552—28no—
20C2×#552—28no—
21C22×#126C2212yes—
22C22×#125C2211yescharacteristic
23C22×#125C2212yes—
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2212yes—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2659—28no—
34#2638—28no—
35#2625—28no—
36C2×#400C2212yes—
37C2×#398C2211yescharacteristic
38C2×#398C2212yes—
39C2×#398C2212yes—
40C2×#398C2212yes—
41C2×#397C2212yes—
42C2×#397C2211yescharacteristic
43C2×#397C2214yes—
44C2×#395C2211yescharacteristic
45C2×#395C2212yes—
46C2×#395C2212yes—
47C2×#394C2212yes—
48C2×#393C2211yescharacteristic
49C2×#391C2212yes—
50C2×#389C2212yes—
51C2×#387C2212yes—
52C2×#386C2212yes—
53C2×#385C2212yes—
54C2×#385C2214yes—
55C2×#381C2211yescharacteristic
56C2×#380C2214yes—
57C2×#380C2211yescharacteristic
58C22×#95C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2212yes—
76#1440C2214yes—
77#1440C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1434C2212yes—
83#1434C2212yes—
84#1434C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2212yes—
87C22⋊#95C2214yes—
88C22⋊#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96#552—2128no—
97C2×#126—28no—
98C2×#126—28no—
99C2×#126—28no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—28no—
105C2×#125—24no—
106C2×#123—28no—
107C2×#123—28no—
108C2×#123—28no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8D418yes—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4D418yes—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117—28no—
127C2×#117—28no—
128C2×#117—28no—
129C2×#117D418yes—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4—24no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2314yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2311yescharacteristic
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2311yescharacteristic
158C22×C22⋊C4C2312yes—
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2312yes—
161C22×C22⋊C4C2312yes—
162C22×C42C2312yes—
163C22×C42C2312yes—
164C22×C42C2311yescharacteristic
165#400—28no—
166#398—28no—
167#398—24no—
168#398—28no—
169#398—28no—
170#397—216no—
171#397—24no—
172#397—28no—
173#395—28no—
174#395—24no—
175#395—28no—
176#394—28no—
177#393—24no—
178#391—28no—
179#389—28no—
180#387—28no—
181#386—44no—
182#385—28no—
183#385—48no—
184#381—24no—
185#380—24no—
186#380—216no—
187C2×#95—24no—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2312yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C25C2411yescharacteristic
215C25C2411yescharacteristic
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2×D414yes—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4C2×D414yes—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—28no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—42no—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C23×C4C2414yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266#126—264no—
267#125—264no—
268#125—232no—
269#123—264no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—416no—
273C4×Q8—232no—
274C4×D4—232no—
275#117—264no—
276#117—232no—
277C2×C4⋊C4—232no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D418yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D414yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4C2×D414yes—
358C2×C22⋊C4C2×D418yes—
359C2×C22⋊C4C2×D414yes—
360C2×C22⋊C4C2×D418yes—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42C2×D418yes—
369C2×C42C2×D414yes—
370#95—24no—
371#95—24no—
372#95—28no—
373#95—28no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—28no—
383#95—28no—
384#95—28no—
385#95—28no—
386#95—28no—
387#95—28no—
388#95—44no—
389#95—44no—
390#95—44no—
391#95—44no—
392#95—44no—
393#95—44no—
394#95—44no—
395#95—44no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—24no—
400C24—24no—
401C24—21no—
402C24—24no—
403C24—21no—
404C24—24no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C22×D412yes—
411C24C22×D412yes—
412C24H(5,𝔽2)12yes—
413C242−1+412yes—
414C24C2511yescharacteristic
415C2×Q8—28no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—44no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—216no—
472C22×C4—28no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—88no—
515C22×C4—88no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4#120116yes—
519C22×C4C22×D412yes—
520C22×C4C22×D414yes—
521C22×C4C22×D414yes—
522C22×C4C22×D414yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D414yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C42−1+418yes—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—216no—
536C4⋊C4—216no—
537C4⋊C4—216no—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—216no—
542C4⋊C4—28no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—84no—
548C4⋊C4—88no—
549C4⋊C4—88no—
550C4⋊C4—88no—
551C22⋊C4—216no—
552C22⋊C4—216no—
553C22⋊C4—28no—
554C22⋊C4—216no—
555C22⋊C4—28no—
556C22⋊C4—416no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C42—216no—
563C42—216no—
564C42—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—28no—
586C23—28no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23C2×#12014yes—
598C23#53714yes—
599C23C23×D411yescharacteristic
600C23C23×D411yescharacteristic
601C23C23×D411yescharacteristic
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+414yes—
605C23#58514yes—
606C23#58511yescharacteristic
607C23#58511yescharacteristic
608Q8—416no—
609Q8—88no—
610D4—416no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—416no—
639C2×C4—416no—
640C2×C4—432no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—816no—
696C2×C4C2×#12014yes—
697C2×C4C2×#12014yes—
698C22—24no—
699C22—24no—
700C22—28no—
701C22—24no—
702C22—24no—
703C22—42no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22C22×#12011yescharacteristic
717C22C2×#53711yescharacteristic
718C22#344912yes—
719C22#344911yescharacteristic
720C22#348512yes—
721C22#348512yes—
722C22#348612yes—
723C22#348811yescharacteristic
724C22#348811yescharacteristic
725C22#348911yescharacteristic
726C22#348911yescharacteristic
727C22#349912yes—
728C22#349912yes—
729C22#353514yes—
730C22#356314yes—
731C22#356414yes—
732C22#356514yes—
733C4—24no—
734C4—24no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—44no—
739C4—48no—
740C4—44no—
741C4—44no—
742C4—42no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—84no—
748C2—44no—
749C2—44no—
750C2#3508811yescharacteristic
751C2#3525111yescharacteristic
752C2#3566611yescharacteristic
753C2#4137312yes—
754C2#4194812yes—
755C2#4230612yes—
756C2#4231012yes—
757C2#4877912yes—
758C2#5016912yes—
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
2. C2×#2625C2semidirect
3. C2×#2659C2semidirect
5. #16683C2semidirect
6. #16608C2semidirect
7. #16607C2semidirect
8. #16601C2semidirect
9. #15911C2semidirect
11. #14907C2semidirect
12. #14887C2semidirect
13. #14879C2semidirect
14. #14786C2semidirect
15. #14660C2semidirect
17. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#26252
C2×#26592
C2×#26382
#166832
#166082
#166072
#166012
#159112
#152692
#149072
#148872
#148792
#147862
#146602
#139602
#139582