Finite Group Databasebeta

← #195923

Subgroups — #195923

Order 512.

3112 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic, complemented
2C2×C4×#120C211yesmaximal, characteristic
3C2×#1779C211yesmaximal, characteristic, complemented
4C2×#1459C212yesmaximal, complemented
5#9081C214yesmaximal, complemented
6#9079C214yesmaximal, complemented
7#7907C212yesmaximal, complemented
8C25×C4C2211yescharacteristic
9C24×C8—21no—
10C22×#120C411yescharacteristic
11C22×C4×D4C412yes—
12C22×C4×D4C2211yescharacteristic
13C22×C22⋊C8—28no—
14C22×C22⋊C8—22no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8—22no—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2214yes—
19C22×C22⋊C8C2212yes—
20C4×#120—21no—
21C4×#120—22no—
22C4×#120—22no—
23C4×#120—21no—
24C4×#120—21no—
25C4×#120—21no—
26C2×#386C411yescharacteristic
27C2×#386C2212yes—
28C2×#385C411yescharacteristic
29C2×#385C2212yes—
30C2×#379C411yescharacteristic
31C2×C4×C22⋊C4C412yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1779—21no—
34#1779—21no—
35#1779C2214yes—
36#1459—28no—
37C2×#324C2214yescomplemented
38C2×#323C2214yescomplemented
39#1280C2218yescomplemented
40C26C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C412yes—
43C24×C4—24no—
44C24×C4—24no—
45C24×C4—48no—
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D412yes—
51C24×C4D411yescharacteristic
52C24×C4D412yes—
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C23×C8—28no—
58C23×C8—24no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—48no—
62C2×#120—21no—
63C2×#120—22no—
64C2×#120—21no—
65C2×#120—24no—
66C2×#120C818yescomplemented
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—24no—
77C2×C4×D4—48no—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C412yes—
90C2×C22⋊C8—28no—
91C2×C22⋊C8—216no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—216no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—48no—
102C2×C22⋊C8D418yescomplemented
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105#386—21no—
106#386—22no—
107#386—21no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—21no—
113#385—22no—
114#385—21no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#379—21no—
120#379—22no—
121#379—21no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2312yes—
133#324—28no—
134#323—28no—
135C25—24no—
136C25—24no—
137C25—44no—
138C25—44no—
139C25C22⋊C411yescharacteristic
140C25C22⋊C412yes—
141C25C22⋊C411yescharacteristic
142C25C2×C814yes—
143C25M1614yes—
144C25C22×C412yes—
145C25C22×C411yescharacteristic
146C25C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4C2×C814yescomplemented
158C22×D4C2×C818yescomplemented
159C23×C4—28no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—432no—
190C23×C4—432no—
191C23×C4—44no—
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C412yes—
194C23×C4C22⋊C412yes—
195C23×C4C22⋊C412yes—
196C23×C4C22⋊C412yes—
197C23×C4C22⋊C412yes—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C412yes—
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C412yes—
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D414yes—
215C23×C4#4114yes—
216C23×C4C2411yescharacteristic
217C22×C8—216no—
218C22×C8—28no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—432no—
222C22×C8—48no—
223C22×C8—48no—
224C22×C8—48no—
225C22×C8—48no—
226C22×C8—44no—
227#120—28no—
228#120—216no—
229#120—28no—
230C4×D4—48no—
231C4×D4—48no—
232C4×D4—48no—
233C4×D4—48no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—42no—
239C4×D4—42no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×C818yescomplemented
273C2×C22⋊C4C2×C814yescomplemented
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—44no—
280C22⋊C8—216no—
281C22⋊C8—28no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—48no—
286C22⋊C8—48no—
287C22⋊C8—44no—
288C22⋊C8C2×D418yes—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—216no—
310C24—216no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—416no—
322C24—416no—
323C24—42no—
324C24—42no—
325C24—416no—
326C24—416no—
327C24—432no—
328C24—42no—
329C24—42no—
330C24C22⋊C814yes—
331C24C22⋊C818yes—
332C24C22⋊C814yes—
333C24C2×C22⋊C412yes—
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C412yes—
337C24C4×D414yes—
338C24C22×C814yes—
339C24C22×C812yes—
340C24C22×C812yescomplemented
341C24C2×M1614yes—
342C24C2×M1612yes—
343C24#13114yes—
344C24C23×C411yescharacteristic
345C2×D4—28no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—28no—
349C2×D4—216no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—416no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—432no—
419C22×C4—432no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—416no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—88no—
440C22×C4#9918yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C4×D414yes—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4#12114yes—
449C22×C4#12114yes—
450C22×C4#12312yes—
451C22×C4#12312yes—
452C22×C4C22×C812yescomplemented
453C22×C4C22×C814yescomplemented
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C2×#4112yes—
458C2×C8—28no—
459C2×C8—432no—
460C2×C8—48no—
461C2×C8—416no—
462C2×C8—48no—
463C2×C8—44no—
464C2×C8—88no—
465C4⋊C4—44no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—41no—
469C4⋊C4—44no—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—28no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—41no—
486C22⋊C4—44no—
487C42—44no—
488C42—42no—
489C42—42no—
490C42—44no—
491C23—28no—
492C23—216no—
493C23—28no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—216no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—216no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—216no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—44no—
539C23—464no—
540C23—464no—
541C23—48no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—82no—
551C23—84no—
552C23—82no—
553C23#38612yes—
554C23#38612yes—
555C23C2×C22⋊C814yes—
556C23C2×C22⋊C812yes—
557C23C2×C22⋊C812yes—
558C23C2×C22⋊C814yes—
559C23#40812yes—
560C23#40812yes—
561C23C2×#9914yes—
562C23#41314yes—
563C23C8×D418yes—
564C23#43518yes—
565C23C22×C22⋊C411yescharacteristic
566C23C2×C4×D412yes—
567C23C23×C811yescharacteristic
568C23C22×M1611yescharacteristic
569C23C2×#13112yes—
570D4—416no—
571D4—48no—
572D4—48no—
573D4—416no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—216no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4#38612yes—
628C2×C4#38612yes—
629C2×C4C2×#9914yes—
630C2×C4C2×#9914yes—
631C2×C4C2×#12011yescharacteristic
632C2×C4C2×#12112yes—
633C2×C4C2×#12311yescharacteristic
634C8—48no—
635C8—84no—
636C22—232no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—22no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—24no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—48no—
660C22—48no—
661C22—416no—
662C22—416no—
663C22—432no—
664C22—416no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—84no—
672C22—84no—
673C22—82no—
674C22—81no—
675C22—81no—
676C22#145918yescomplemented
677C22#186914yes—
678C22#186914yes—
679C22C2×#38611yescharacteristic
680C22C22×C22⋊C811yescharacteristic
681C22C2×#40811yescharacteristic
682C22C22×#9912yes—
683C22C2×#41312yes—
684C22C2×C8×D412yes—
685C22C2×#43512yes—
686C4—21no—
687C4—22no—
688C4—21no—
689C4—24no—
690C4—22no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—82no—
695C4—81no—
696C4—81no—
697C4#189814yes—
698C2—24no—
699C2—28no—
700C2—22no—
701C2—22no—
702C2—24no—
703C2—44no—
704C2—82no—
705C2C2×#145912yes—
706C2#1738112yes—
707C2C2×#186911yescharacteristic
708C2C2×#189811yescharacteristic
709C2#2098911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#120C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊C8C2semidirect
3. C2×#1779C2semidirect
4. C2×#1459C2semidirect
5. #9081C2semidirect
6. #9079C2semidirect
7. #7907C2semidirect
37. C2×#324C22semidirect
38. C2×#323C22semidirect
39. #1280C22semidirect
66. C2×#120C8semidirect
102. C2×C22⋊C8D4semidirect
157. C22×D4C2×C8semidirect
158. C22×D4C2×C8semidirect
272. C2×C22⋊C4C2×C8semidirect
273. C2×C22⋊C4C2×C8semidirect
340. C24C22×C8semidirect
452. C22×C4C22×C8semidirect
453. C22×C4C22×C8semidirect
676. C22#1459semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C2×C4×#1202
C2×#17792
C2×#14592
#90812
#90792
#79072