Finite Group Databasebeta

← #4929113

Subgroups — #4929113

Order 512.

2910 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic
4C2×#2276C211yesmaximal, characteristic, complemented
5#14908C212yesmaximal, complemented
6#14898C212yesmaximal
7#14869C212yesmaximal, complemented
8#14854C214yesmaximal, complemented
9#14813C212yesmaximal
10#14789C214yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#13994C212yesmaximal, complemented
13#13993C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13969C212yesmaximal, complemented
16#13926C211yesmaximal, characteristic, complemented
17#13905C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2581—28no—
24#2538—28no—
25#2398—28no—
26C2×#400C2212yes—
27C2×#400C2211yescharacteristic
28C2×#400C2211yescharacteristic
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#395C2212yes—
32C2×#394C2211yescharacteristic
33C2×#394C2211yescharacteristic
34C2×#393C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388C2214yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50#2276—28no—
51C2×C4×C4⋊C4C2214yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2212yes—
58C2×#376C2211yescharacteristic
59#1445C2212yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2212yes—
74#1437C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8D418yes—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—28no—
140#400—24no—
141#400—24no—
142#397—28no—
143#396—216no—
144#395—28no—
145#394—24no—
146#394—24no—
147#393—28no—
148#391—24no—
149#391—24no—
150#391—44no—
151#390—28no—
152#389—24no—
153#389—24no—
154#389—28no—
155#388—48no—
156#387—28no—
157#387—28no—
158#384—24no—
159#382—24no—
160#380—28no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—216no—
166C4×C4⋊C4—28no—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—28no—
169#376—28no—
170#376D418yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C25C2412yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—41no—
209C22×Q8—41no—
210C22×Q8C2×D414yes—
211C22×Q8#4118yes—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—42no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22⋊Q8—432no—
241C4×Q8—264no—
242C4×Q8—232no—
243C4×Q8—416no—
244C4×D4—416no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4#4118yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
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×C42—28no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336C2×C42C2×D418yes—
337C2×C42C2×D418yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42#4114yes—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D414yes—
367#95C2×D418yes—
368#95C2×D414yes—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—216no—
388C2×Q8—216no—
389C2×Q8—28no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8#124116yes—
397C2×D4—28no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—44no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4118yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4118yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4114yes—
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C4⋊C4—232no—
508C4⋊C4—216no—
509C4⋊C4—28no—
510C4⋊C4—216no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—84no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—416no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—232no—
533C42—216no—
534C42—216no—
535C42—44no—
536C42—44no—
537C42—84no—
538C42—88no—
539C42#121116yes—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
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—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×2−1+412yes—
582C23C2×2−1+411yescharacteristic
583C23#58512yes—
584C23#58511yescharacteristic
585C23#58511yescharacteristic
586Q8—232no—
587Q8—416no—
588D4—416no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
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—816no—
680C2×C4C2×#12114yes—
681C2×C4C2×#12114yes—
682C2×C4C2×#12114yes—
683C2×C4#52518yes—
684C2×C4C2×#12418yes—
685C2×C4C2×#12414yes—
686C2×C4#52918yes—
687C2×C4#53214yes—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C22×#12111yescharacteristic
704C22C2×#52512yes—
705C22C22×#12411yescharacteristic
706C22C2×#52912yes—
707C22C2×#53211yescharacteristic
708C22#348611yescharacteristic
709C22#348711yescharacteristic
710C22#348811yescharacteristic
711C22#348811yescharacteristic
712C22#348911yescharacteristic
713C22#348912yes—
714C22#348911yescharacteristic
715C22#349211yescharacteristic
716C22#349311yescharacteristic
717C22#349512yes—
718C22#351612yes—
719C22#351612yes—
720C22#351712yes—
721C22#351812yes—
722C22#352012yes—
723C22#352112yes—
724C22#352312yes—
725C22#352412yes—
726C4—28no—
727C4—24no—
728C4—24no—
729C4—44no—
730C4—44no—
731C4—42no—
732C4—42no—
733C4—44no—
734C4—48no—
735C4—82no—
736C4—82no—
737C4—81no—
738C4—81no—
739C4—82no—
740C4—84no—
741C4#285018yes—
742C2—84no—
743C2#3551811yescharacteristic
744C2#3552112yes—
745C2#3566111yescharacteristic
746C2#3566312yes—
747C2#3585512yes—
748C2#3590911yescharacteristic
749C2#3591011yescharacteristic
750C2#3594512yes—
751C2#3604811yescharacteristic
752C2#3604911yescharacteristic
753C2C2×#285011yescharacteristic
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×#2581C2semidirect
4. C2×#2276C2semidirect
5. #14908C2semidirect
7. #14869C2semidirect
8. #14854C2semidirect
10. #14789C2semidirect
11. #14348C2semidirect
12. #13994C2semidirect
13. #13993C2semidirect
14. #13969C2semidirect
15. #13969C2semidirect
16. #13926C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25812
C2×#23982
C2×#22762
#149082
#148982
#148692
#148542
#148132
#147892
#143482
#139942
#139932
#139692
#139262
#139052