Finite Group Databasebeta

← #358989

Subgroups — #358989

Order 512.

2348 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic, complemented
2C22×#435C212yesmaximal, complemented
3C2×#1753C212yesmaximal, complemented
4C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
5C4×#392C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic, complemented
7#11042C211yesmaximal, characteristic, complemented
8#11042C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic, complemented
10#10964C212yesmaximal, complemented
11#10696C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C24×C8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#436—24no—
17C2×#436—28no—
18C2×#435—28no—
19C2×#435—28no—
20C2×#435—28no—
21C22×C4⋊C8C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2212yes—
28C22×#97C2212yes—
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#392C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2212yes—
34#1753—28no—
35C8×C22⋊C4—24no—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1444C411yescharacteristic
43#1439C412yes—
44#1439C411yescharacteristic
45#1438C412yes—
46#1438C411yescharacteristic
47C4×#95C2211yescharacteristic
48C23×D4C2×C412yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2312yes—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16D418yescomplemented
59C23×C8—24no—
60C23×C8—24no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8D414yescomplemented
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4—24no—
79C2×C4×D4—22no—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92#436—232no—
93#435—264no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—24no—
114C2×#97—24no—
115C2×#97—24no—
116C2×#97D418yescomplemented
117C2×C4×C8—24no—
118C2×C4×C8—22no—
119C2×C4×C8—24no—
120C2×C4×C8D414yescomplemented
121#392—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—24no—
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C412yes—
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133#336—22no—
134#336—24no—
135#336—22no—
136#336—24no—
137#336—24no—
138#336—22no—
139C25C22×C412yes—
140C25C22×C411yescharacteristic
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D414yes—
183C23×C4#4114yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C2×M16—216no—
187C2×M16—216no—
188C2×M16—216no—
189C2×M16—48no—
190C2×M16—48no—
191C2×M16—48no—
192C2×M16—48no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—216no—
218C22×C8—28no—
219C22×C8—44no—
220C22×C8—48no—
221C22×C8—44no—
222C22×C8C2×D412yes—
223C22×C8C2×D414yes—
224C22×C8C2×D412yes—
225C22×C8C2×D414yes—
226C22×C8C2×D414yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D414yes—
230C22×C8#4112yes—
231C22×C8#4112yes—
232C4×D4—216no—
233C4×D4—28no—
234C4×D4—216no—
235C4×D4—216no—
236C4×D4—216no—
237C4×D4—216no—
238C4×D4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4M1614yescomplemented
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42C2×D412yes—
276C2×C42C2×D414yes—
277C2×C42#4114yes—
278C2×C42#4112yes—
279C4⋊C8—216no—
280C4⋊C8—28no—
281C22⋊C8—28no—
282C22⋊C8—216no—
283C22⋊C8—216no—
284C22⋊C8—216no—
285C22⋊C8—216no—
286#97—216no—
287C4×C8—28no—
288C4×C8—44no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—42no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C4×D414yes—
319C24C4×D412yes—
320C24C2×M1612yes—
321C24#13112yes—
322C24C23×C411yescharacteristic
323C2×D4—28no—
324C2×D4—216no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
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—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C4×D412yes—
418C22×C4C4×D414yes—
419C22×C4C4×D414yes—
420C22×C4C4×D412yes—
421C22×C4C4×D414yes—
422C22×C4C4×D412yes—
423C22×C4C2×M1612yes—
424C22×C4C2×M1612yes—
425C22×C4#13112yes—
426C22×C4#13112yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4112yes—
437M16—432no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—41no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—42no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—416no—
468C2×C8#12014yes—
469C2×C8#12018yes—
470C2×C8#12114yes—
471C2×C8#12118yes—
472C2×C8#12714yes—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C22⋊C4—28no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C42—22no—
483C42—24no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—41no—
490C42—41no—
491C42—42no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23#43614yescomplemented
550C23C2×C4×D411yescharacteristic
551C23C2×C4×D412yes—
552C23C2×C4×D411yescharacteristic
553C23C2×C4×D412yes—
554C23C4×#4111yescharacteristic
555C23C22×M1611yescharacteristic
556C23C2×#13111yescharacteristic
557C23C2×#13111yescharacteristic
558C23C2×#13111yescharacteristic
559C23#56812yes—
560C23#56812yes—
561D4—416no—
562D4—432no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4#43514yescomplemented
623C2×C4C2×#12011yescharacteristic
624C2×C4C2×#12012yes—
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12112yes—
627C2×C4#52512yes—
628C2×C4#52512yes—
629C2×C4#52512yes—
630C2×C4#52511yescharacteristic
631C2×C4#52512yes—
632C2×C4#52511yescharacteristic
633C2×C4#52512yes—
634C2×C4#52911yescharacteristic
635C2×C4#52912yes—
636C2×C4C2×#12711yescharacteristic
637C2×C4#53211yescharacteristic
638C2×C4#53212yes—
639C2×C4#53211yescharacteristic
640C2×C4#53212yes—
641C8—42no—
642C8—44no—
643C8—42no—
644C8—44no—
645C8—42no—
646C8#39218yescomplemented
647C22—28no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22C4×#12011yescharacteristic
674C22C4×#12012yes—
675C22C4×#12111yescharacteristic
676C22C4×#12112yes—
677C22C4×#12711yescharacteristic
678C22C2×#43511yescharacteristic
679C22C2×#43611yescharacteristic
680C22#292911yescharacteristic
681C22#293011yescharacteristic
682C22#293012yes—
683C22#293111yescharacteristic
684C22#293211yescharacteristic
685C22#293312yes—
686C22#293312yes—
687C22#293312yes—
688C22#293312yes—
689C22D4×M1612yes—
690C22D4×M1612yes—
691C22#293512yes—
692C22#293512yes—
693C22#296511yescharacteristic
694C22#296611yescharacteristic
695C22#296712yes—
696C4—24no—
697C4—42no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—44no—
704C4C2×#39211yescharacteristic
705C4#238811yescharacteristic
706C4#238812yes—
707C4#239211yescharacteristic
708C4#239212yes—
709C4#239711yescharacteristic
710C2—44no—
711C2—84no—
712C2C4×#39211yescharacteristic
713C2#2143911yescharacteristic
714C2#2144112yes—
715C2#2144311yescharacteristic
716C2#2144512yes—
717C2#2148711yescharacteristic
718C2#2149012yes—
719C2#2149311yescharacteristic
720C2#2149612yes—
721C2#2168211yescharacteristic
722C2#2168511yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C22×#436C2semidirect
2. C22×#435C2semidirect
3. C2×#1753C2semidirect
4. C2×C8×C22⋊C4C2semidirect
6. #11322C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10964C2semidirect
10. #10964C2semidirect
11. #10696C2semidirect
58. C22×M16D4semidirect
63. C23×C8D4semidirect
116. C2×#97D4semidirect
120. C2×C4×C8D4semidirect
264. C2×C22⋊C4M16semidirect
549. C23#436semidirect
622. C2×C4#435semidirect
646. C8#392semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C22×#4352
C2×#17532
C2×C8×C22⋊C42
C4×#3922
#113222
#110422
#109642
#106962