Finite Group Databasebeta

← #4929102

Subgroups — #4929102

Order 512.

2910 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic
3C2×#2398C211yesmaximal, characteristic
4C2×#2276C211yesmaximal, characteristic, complemented
5#14902C212yesmaximal, complemented
6#14894C212yesmaximal, complemented
7#14866C212yesmaximal, complemented
8#14846C214yesmaximal, complemented
9#14808C212yesmaximal, complemented
10#14780C214yesmaximal
11#14348C211yesmaximal, characteristic, complemented
12#13994C212yesmaximal, complemented
13#13993C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13969C212yesmaximal, complemented
16#13925C211yesmaximal, characteristic, complemented
17#13906C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2582—28no—
25#2536—28no—
26#2398—28no—
27C2×#400C2212yes—
28C2×#398C2212yes—
29C2×#398C2214yes—
30C2×#397C2211yescharacteristic
31C2×#397C2211yescharacteristic
32C2×#394C2212yes—
33C2×#394C2212yes—
34C2×#393C2211yescharacteristic
35C2×#393C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50#2276—28no—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2214yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2212yes—
58C2×#376C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1443C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
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×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8D418yes—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2314yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#400—28no—
143#398—216no—
144#398—28no—
145#397—24no—
146#397—24no—
147#394—28no—
148#394—28no—
149#393—24no—
150#393—24no—
151#391—28no—
152#391—28no—
153#390—44no—
154#389—28no—
155#389—28no—
156#388—28no—
157#388—28no—
158#388—48no—
159#384—24no—
160#382—24no—
161#380—24no—
162#380—24no—
163#380—28no—
164#380—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—216no—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—28no—
171#376—28no—
172#376D418yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8C2×D414yes—
210C22×Q8#4118yes—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—41no—
217C22×D4—41no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C4×Q8—264no—
246C4×Q8—232no—
247C4×Q8—416no—
248C4×D4—416no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
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—28no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4#4118yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42C2×D418yes—
340C2×C42C2×D418yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42#4114yes—
344#95—28no—
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—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95C2×D414yes—
368#95C2×D414yes—
369#95C2×D418yes—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—21no—
390C24H(5,𝔽2)12yes—
391C242−1+412yes—
392C24C2511yescharacteristic
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8—28no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8#124116yes—
403C2×D4—28no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C2×#4118yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4118yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C42−1+414yes—
511C4⋊C4—232no—
512C4⋊C4—216no—
513C4⋊C4—28no—
514C4⋊C4—216no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—88no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—88no—
538C42—232no—
539C42—216no—
540C42—216no—
541C42—48no—
542C42—84no—
543C42—88no—
544C42#121116yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23C23×D411yescharacteristic
592C23C23×D411yescharacteristic
593C23C22×#4111yescharacteristic
594C23C22×#4112yes—
595C23C22×#4112yes—
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×2−1+412yes—
598C23C2×2−1+411yescharacteristic
599C23#58512yes—
600C23#58511yescharacteristic
601C23#58511yescharacteristic
602Q8—232no—
603Q8—416no—
604D4—416no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
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—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
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—88no—
689C2×C4—88no—
690C2×C4C2×#12114yes—
691C2×C4C2×#12114yes—
692C2×C4C2×#12114yes—
693C2×C4#52518yes—
694C2×C4C2×#12418yes—
695C2×C4C2×#12414yes—
696C2×C4#52918yes—
697C2×C4#53214yes—
698C22—42no—
699C22—42no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—82no—
715C22—81no—
716C22—81no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C22×#12111yescharacteristic
721C22C2×#52512yes—
722C22C22×#12411yescharacteristic
723C22C2×#52912yes—
724C22C2×#53211yescharacteristic
725C22#348611yescharacteristic
726C22#348711yescharacteristic
727C22#348811yescharacteristic
728C22#348811yescharacteristic
729C22#348911yescharacteristic
730C22#348912yes—
731C22#348911yescharacteristic
732C22#349211yescharacteristic
733C22#349311yescharacteristic
734C22#349512yes—
735C22#351312yes—
736C22#351512yes—
737C22#351612yes—
738C22#351812yes—
739C22#352212yes—
740C22#352312yes—
741C22#352412yes—
742C22#352412yes—
743C4—28no—
744C4—24no—
745C4—24no—
746C4—44no—
747C4—44no—
748C4—44no—
749C4—44no—
750C4—48no—
751C4—81no—
752C4—81no—
753C4—84no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4#285018yes—
758C2—82no—
759C2—82no—
760C2#3551811yescharacteristic
761C2#3552012yes—
762C2#3566111yescharacteristic
763C2#3566312yes—
764C2#3585311yescharacteristic
765C2#3585411yescharacteristic
766C2#3591112yes—
767C2#3594811yescharacteristic
768C2#3594911yescharacteristic
769C2#3605012yes—
770C2C2×#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
1. C2×#2582C2semidirect
4. C2×#2276C2semidirect
5. #14902C2semidirect
6. #14894C2semidirect
7. #14866C2semidirect
8. #14846C2semidirect
9. #14808C2semidirect
11. #14348C2semidirect
12. #13994C2semidirect
13. #13993C2semidirect
14. #13969C2semidirect
15. #13969C2semidirect
16. #13925C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25362
C2×#23982
C2×#22762
#149022
#148942
#148662
#148462
#148082
#147802
#143482
#139942
#139932
#139692
#139252
#139062