Finite Group Databasebeta

← #368000

Subgroups — #368000

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic, complemented
3C2×#1870C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic, complemented
5C2×#1850C212yesmaximal, complemented
6#13839C211yesmaximal, characteristic
7#11322C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10876C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×C8×D4—24no—
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C2×#424—28no—
21C2×#424—24no—
22C22×C4⋊C8C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2212yes—
25C22×C4⋊C8C2212yes—
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1870—24no—
34#1851—24no—
35#1850—28no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441C411yescharacteristic
43#1441C411yescharacteristic
44#1440C412yes—
45#1439C411yescharacteristic
46#1439C411yescharacteristic
47#1438C412yes—
48C4×#95C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—24no—
64C23×C8—42no—
65C23×C8—42no—
66C23×C8D414yes—
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C23×C8C2311yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D414yes—
78C2×C4×D4Q814yes—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—24no—
82C2×#117—24no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C412yes—
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C22⋊C4C2×C411yescharacteristic
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C22×C42C2312yes—
95C8×D4—232no—
96#424—232no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—24no—
106C2×C4⋊C8—24no—
107C2×C4⋊C8—42no—
108C2×C4⋊C8D418yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—24no—
119C2×C22⋊C8—42no—
120C2×C22⋊C8D414yes—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136#336—24no—
137#336—24no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—42no—
142C25C22×C411yescharacteristic
143C25C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C4⋊C414yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×Q812yes—
176C23×C4C2×Q812yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C2×M16—28no—
181C2×M16—48no—
182C2×M16—48no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—28no—
189C22×C8—28no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—28no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—44no—
217C22×C8—48no—
218C22×C8—48no—
219C22×C8—48no—
220C22×C8—48no—
221C22×C8—44no—
222C22×C8—44no—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D414yes—
226C22×C8C2×D412yes—
227C22×C8C2×D414yes—
228C22×C8C2×D412yes—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233#117—216no—
234#117—28no—
235#117—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4C4⋊C414yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4C4⋊C414yes—
260C2×C22⋊C4C4⋊C414yes—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42C2×D412yes—
276C2×C42C2×D414yes—
277C2×C42C2×Q812yes—
278C2×C42#4114yes—
279C4⋊C8—216no—
280C4⋊C8—28no—
281C4⋊C8—216no—
282C4⋊C8—48no—
283C22⋊C8—28no—
284C22⋊C8—28no—
285C22⋊C8—48no—
286C22⋊C8—44no—
287C4×C8—28no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—42no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24C2×C4⋊C412yes—
313C24C2×C4⋊C412yes—
314C24C4×D412yes—
315C24C4×D412yes—
316C24#13112yes—
317C24#13112yes—
318C24C23×C411yescharacteristic
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C4⋊C412yes—
398C22×C4C2×C4⋊C412yes—
399C22×C4C2×C4⋊C412yes—
400C22×C4C2×C4⋊C412yes—
401C22×C4C4×D414yes—
402C22×C4C4×D414yes—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12014yes—
406C22×C4C22⋊Q814yes—
407C22×C4C22⋊Q814yes—
408C22×C4#12314yes—
409C22×C4#13114yes—
410C22×C4#13112yes—
411C22×C4#13112yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422M16—416no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—416no—
438C2×C8—416no—
439C2×C8—41no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8#12014yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—22no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23#38514yes—
525C23#42414yescomplemented
526C23C22×C4⋊C411yescharacteristic
527C23C2×C4×D411yescharacteristic
528C23C2×C4×D412yes—
529C23C2×C4×D411yescharacteristic
530C23#52012yes—
531C23C2×#13111yescharacteristic
532C23C2×#13111yescharacteristic
533C23C2×#13111yescharacteristic
534C23C2×#13111yescharacteristic
535C23C2×#13112yes—
536C23#56812yes—
537D4—232no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4#38514yes—
599C2×C4#42414yescomplemented
600C2×C4C2×#12011yescharacteristic
601C2×C4C2×#12011yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×C22⋊Q811yescharacteristic
604C2×C4C2×#12311yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#52512yes—
607C2×C4#53411yescharacteristic
608C2×C4#53611yescharacteristic
609C2×C4#53912yes—
610C2×C4D4212yes—
611C2×C4D4212yes—
612C2×C4#54712yes—
613C2×C4#54712yes—
614C2×C4#54812yes—
615C2×C4D4×Q812yes—
616C2×C4#55312yes—
617C2×C4#55412yes—
618C8—24no—
619C8—42no—
620C8—42no—
621C8—44no—
622C8—42no—
623C8—82no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—28no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22C2×#38511yescharacteristic
647C22C4×#12011yescharacteristic
648C22#232211yescharacteristic
649C22D4×C4⋊C412yes—
650C22#236112yes—
651C22C2×#42411yescharacteristic
652C22C2×#42411yescharacteristic
653C22#290612yes—
654C22#292911yescharacteristic
655C22#292911yescharacteristic
656C22#292911yescharacteristic
657C22#292911yescharacteristic
658C22#292912yes—
659C22#293111yescharacteristic
660C22#293112yes—
661C22#293211yescharacteristic
662C22#293312yes—
663C22#293312yes—
664C22#293512yes—
665C22#299212yes—
666C22#299312yes—
667C22#299412yes—
668C22#299412yes—
669C4—24no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4#240311yescharacteristic
678C4#240611yescharacteristic
679C4#240911yescharacteristic
680C4#244312yes—
681C4#244912yes—
682C4#257911yescharacteristic
683C2—44no—
684C2—44no—
685C2#1401511yescharacteristic
686C2#2083411yescharacteristic
687C2#2083911yescharacteristic
688C2#2144211yescharacteristic
689C2#2144311yescharacteristic
690C2#2206411yescharacteristic
691C2#2206611yescharacteristic
692C2#2264712yes—
693C2#2265112yes—
694C2#2299112yes—
695C2#2299212yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#424C2semidirect
4. C2×#1851C2semidirect
5. C2×#1850C2semidirect
7. #11322C2semidirect
8. #11042C2semidirect
9. #10876C2semidirect
10. #10714C2semidirect
11. #10713C2semidirect
108. C2×C4⋊C8D4semidirect
525. C23#424semidirect
599. C2×C4#424semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C22×#4242
C2×#18702
C2×#18512
C2×#18502
#138392
#113222
#110422
#108762
#107142
#107132