Finite Group Databasebeta

← #358115

Subgroups — #358115

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic, complemented
4C2×#1749C212yesmaximal, complemented
5#11202C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10964C212yesmaximal
9#10693C212yesmaximal, complemented
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C411yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#432—24no—
16C2×#432—28no—
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2214yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2212yes—
22C22×#97C2212yes—
23C42×D4—24no—
24C42×D4—24no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C411yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1749—24no—
36#1749—28no—
37C2×#336C2214yes—
38C2×#336C2212yes—
39C2×#336C2214yes—
40C2×#336C2214yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2314yes—
49C23×C8C2314yes—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C42—24no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—24no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D414yes—
84C22×C42D414yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—232no—
91C2×C4⋊C8—28no—
92C2×C4⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—28no—
96C2×C22⋊C8—28no—
97C2×C22⋊C8—28no—
98C2×C22⋊C8—28no—
99C2×C22⋊C8—28no—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97—24no—
103C2×#97—24no—
104C2×#97—24no—
105C2×#97D418yescomplemented
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C411yescharacteristic
124C43—22no—
125C43—22no—
126#336—28no—
127#336—28no—
128#336—24no—
129#336—28no—
130C25C22×C411yescharacteristic
131C25C22×C411yescharacteristic
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4M1614yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22⋊C414yes—
169C23×C4C22⋊C414yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4#4114yes—
185C23×C4C2411yescharacteristic
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8C2×D418yes—
199C22×C8#4118yes—
200C4×D4—232no—
201C4×D4—232no—
202C4×D4—216no—
203C4×D4—216no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4M1614yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4M1614yes—
230C2×C22⋊C4M1614yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—216no—
246C2×C42—28no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—48no—
264C2×C42—44no—
265C2×C42C22⋊C414yes—
266C2×C42C22⋊C414yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D414yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274C4⋊C8—216no—
275C22⋊C8—232no—
276C22⋊C8—216no—
277C22⋊C8—216no—
278#97—216no—
279#97—48no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24C2×C22⋊C412yes—
303C24#11712yes—
304C24C2×M1612yes—
305C24C2×M1612yes—
306C24C2×M1612yes—
307C24C2×M1612yes—
308C24C23×C411yescharacteristic
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
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—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4#11712yes—
408C22×C4#11712yes—
409C22×C4#11712yes—
410C22×C4C4×D414yes—
411C22×C4C4×D414yes—
412C22×C4C2×M1612yes—
413C22×C4C2×M1612yes—
414C22×C4C2×M1612yes—
415C22×C4C2×M1612yes—
416C22×C4C2×M1612yes—
417C22×C4C2×M1612yes—
418C22×C4C2×M1612yes—
419C22×C4C2×M1612yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C2×C8—28no—
428C2×C8—28no—
429C2×C8—28no—
430C2×C8—28no—
431C2×C8—28no—
432C2×C8—28no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C42—28no—
451C42—28no—
452C42—22no—
453C42—24no—
454C42—22no—
455C42—24no—
456C42—24no—
457C42—28no—
458C42—28no—
459C42—44no—
460C42—44no—
461C42—41no—
462C42—42no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—44no—
467C42—44no—
468C42#12014yes—
469C42#12118yes—
470C42#12314yes—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23#40714yes—
516C23#43214yescomplemented
517C23C22×C22⋊C411yescharacteristic
518C23C2×#11711yescharacteristic
519C23C2×#11711yescharacteristic
520C23C2×C4×D412yes—
521C23C4×#4112yes—
522C23C22×M1611yescharacteristic
523C23C22×M1611yescharacteristic
524C23C22×M1611yescharacteristic
525C23C22×M1611yescharacteristic
526C23#56812yes—
527C23#56812yes—
528D4—232no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4#38614yes—
598C2×C4#38614yes—
599C2×C4#40714yes—
600C2×C4#43214yescomplemented
601C2×C4C2×#12011yescharacteristic
602C2×C4C2×#12112yes—
603C2×C4C2×#12311yescharacteristic
604C2×C4#52514yes—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#52512yes—
608C2×C4#52914yes—
609C2×C4#52914yes—
610C2×C4#53214yes—
611C2×C4#53212yes—
612C8—48no—
613C8—48no—
614C22—28no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22#228011yescharacteristic
637C22C2×#38611yescharacteristic
638C22#229012yes—
639C22#229312yes—
640C22#229711yescharacteristic
641C22C2×#40711yescharacteristic
642C22C2×#40711yescharacteristic
643C22#287911yescharacteristic
644C22#287911yescharacteristic
645C22C2×#43211yescharacteristic
646C22C2×#43211yescharacteristic
647C22#292011yescharacteristic
648C22#292011yescharacteristic
649C22#292312yes—
650C22#292312yes—
651C22D4×M1612yes—
652C22D4×M1612yes—
653C22D4×M1612yes—
654C22D4×M1612yes—
655C22#297012yes—
656C22#297012yes—
657C22#297012yes—
658C22#297112yes—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4#238812yes—
672C4#239214yes—
673C4#239512yes—
674C2—44no—
675C2—44no—
676C2#1383911yescharacteristic
677C2#2071511yescharacteristic
678C2#2071911yescharacteristic
679C2#2089811yescharacteristic
680C2#2089811yescharacteristic
681C2#2110712yes—
682C2#2110812yes—
683C2#2127012yes—
684C2#2127112yes—
685C2#2137911yescharacteristic
686C2#2138011yescharacteristic
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
1. C22×#432C2semidirect
3. C2×#1749C2semidirect
4. C2×#1749C2semidirect
5. #11202C2semidirect
6. #11147C2semidirect
7. #11042C2semidirect
9. #10693C2semidirect
105. C2×#97D4semidirect
516. C23#432semidirect
600. C2×C4#432semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×C42×D42
C2×#17492
#112022
#111472
#110422
#109642
#106932