Finite Group Databasebeta

← #364650

Subgroups — #364650

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic
2C2×#2395C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1818C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10874C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic, complemented
10#10715C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#436—24no—
18C2×#436—24no—
19C2×#436—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2395—24no—
27#2395—24no—
28C2×#400C411yescharacteristic
29C2×#399C411yescharacteristic
30C2×#398C411yescharacteristic
31C2×#397C412yes—
32C2×#397C411yescharacteristic
33C2×#394C412yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1869—24no—
38#1818—24no—
39#1729—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—24no—
59C22×M16—42no—
60C22×M16—42no—
61C22×M16D414yes—
62C23×C8—22no—
63C23×C8—24no—
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—42no—
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×D4D414yes—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C412yes—
92C22×C22⋊C4C2×C412yes—
93C22×C22⋊C4C2×C411yescharacteristic
94C22×C22⋊C4C2×C411yescharacteristic
95C22×C22⋊C4C2×C412yes—
96C22×C42C2312yes—
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99#436—232no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—42no—
114C2×C22⋊C8D414yes—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118#400—24no—
119#399—24no—
120#398—24no—
121#397—28no—
122#397—24no—
123#394—28no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136#336—24no—
137#336—24no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—42no—
144#336—42no—
145C25C22×C411yescharacteristic
146C25C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—24no—
150C22×D4—42no—
151C22×D4C22⋊C414yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—44no—
185C2×M16—48no—
186C2×M16—48no—
187C2×M16—48no—
188C2×M16—48no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—28no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—48no—
221C22×C8—48no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8#4114yes—
229C22×C8#4114yes—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234#117—44no—
235#117—44no—
236#117—44no—
237#117—44no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22⋊C414yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4C22⋊C414yes—
270C2×C22⋊C4C22⋊C414yes—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C4⋊C8—28no—
290C22⋊C8—28no—
291C22⋊C8—28no—
292C22⋊C8—48no—
293C22⋊C8—44no—
294C4×C8—28no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C2×C22⋊C412yes—
316C24C2×C22⋊C412yes—
317C24C4×D412yes—
318C24C4×D412yes—
319C24C23×C411yescharacteristic
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—48no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
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—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
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×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4#12014yes—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4#12314yes—
407C22×C4#13114yes—
408C22×C4#13114yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+412yes—
419M16—416no—
420M16—416no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—416no—
436C2×C8—42no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—42no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
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#12314yes—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23#38614yes—
521C23C22×C22⋊C411yescharacteristic
522C23C2×C4×D411yescharacteristic
523C23C2×C4×D411yescharacteristic
524C23C4×#4112yes—
525C23#52012yes—
526C23C2×#13112yes—
527C23C2×#13112yes—
528C23#56812yes—
529C23#56812yes—
530D4—416no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4#38614yes—
590C2×C4#40814yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52912yes—
598C2×C4#53411yescharacteristic
599C2×C4#53611yescharacteristic
600C2×C4#53912yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54712yes—
605C2×C4#54712yes—
606C2×C4#54712yes—
607C2×C4#55112yes—
608C2×C4#55212yes—
609C8—24no—
610C8—42no—
611C8—42no—
612C8—42no—
613C8—44no—
614C8—82no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#38611yescharacteristic
634C22C4×#12311yescharacteristic
635C22#232211yescharacteristic
636C22#233912yes—
637C22#235912yes—
638C22C2×#40811yescharacteristic
639C22#287911yescharacteristic
640C22#288012yes—
641C22#293111yescharacteristic
642C22#293211yescharacteristic
643C22#293312yes—
644C22#293312yes—
645C22#293512yes—
646C22#296612yes—
647C22#296712yes—
648C22#296912yes—
649C22#296912yes—
650C22#299312yes—
651C22#299312yes—
652C22#299412yes—
653C22#299512yes—
654C4—24no—
655C4—24no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#240511yescharacteristic
663C4#240611yescharacteristic
664C4#241912yes—
665C4#242512yes—
666C4#248611yescharacteristic
667C4#253411yescharacteristic
668C2—44no—
669C2—82no—
670C2#1396811yescharacteristic
671C2#2089711yescharacteristic
672C2#2090011yescharacteristic
673C2#2157112yes—
674C2#2207412yes—
675C2#2247212yes—
676C2#2247412yes—
677C2#2290712yes—
678C2#2291512yes—
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
4. C2×#1818C2semidirect
5. C2×#1729C2semidirect
6. #11202C2semidirect
7. #11147C2semidirect
8. #10874C2semidirect
9. #10715C2semidirect
10. #10715C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C2×#23952
C2×#18692
C2×#18182
C2×#17292
#112022
#111472
#108742
#107152
#107142