Finite Group Databasebeta

← #195943

Subgroups — #195943

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic, complemented
4C2×#1460C212yesmaximal, complemented
5#9084C214yesmaximal, complemented
6#9083C214yesmaximal, complemented
7#7908C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C412yes—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#431—21no—
16C2×#431C2212yes—
17C2×#431C2212yes—
18C2×#423—21no—
19C2×#407C2212yes—
20C2×C4×M16—21no—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28C4×#121—21no—
29C2×#390C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#388C2212yes—
32C2×#386C2212yes—
33C2×#380C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C412yes—
36#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1779C2212yes—
40C4×C22⋊C8—21no—
41C4×C22⋊C8—21no—
42#1460—28no—
43C2×#323C2218yes—
44#1280C2218yes—
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C412yes—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C2×#121—24no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—24no—
68C2×C4×D4—48no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2312yes—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C412yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#432—28no—
94#432—28no—
95#431—28no—
96#431—24no—
97#431—24no—
98#431—24no—
99#431D418yescomplemented
100#423—28no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104#407—28no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C4×M16—28no—
111C2×#97—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C2×#95C2312yes—
145#336—22no—
146#336—22no—
147#323—216no—
148C25C22×C411yescharacteristic
149C25C22×C412yes—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—24no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4M1614yescomplemented
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—28no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—28no—
200C2×M16—44no—
201C2×M16—44no—
202C22×C8—24no—
203C22×C8—24no—
204#121—28no—
205#121—28no—
206#121—28no—
207#121—28no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—42no—
217C4×D4—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4M1614yescomplemented
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—44no—
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D414yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4114yes—
285C4⋊C8—28no—
286C4⋊C8—28no—
287C4⋊C8—44no—
288C22⋊C8—28no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C22⋊C8C2×D418yes—
295#97—28no—
296C4×C8—28no—
297C4×C8—44no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—41no—
315C24—41no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C24C2×M1614yes—
323C24C23×C411yescharacteristic
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—216no—
329C2×D4—44no—
330C2×D4—416no—
331C2×D4—44no—
332C2×D4—416no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—48no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
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—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
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—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—24no—
386C22×C4—24no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—416no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—816no—
415C22×C4#10014yes—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4C2×M1614yes—
429C22×C4C2×M1612yes—
430C22×C4C2×M1612yes—
431C22×C4#13114yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C2×#4112yes—
436M16—416no—
437C2×C8—28no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8—44no—
442C2×C8—44no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C42—21no—
463C42—21no—
464C42—22no—
465C42—22no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—44no—
477C42—44no—
478C42#12012yes—
479C42#12012yes—
480C42#12114yes—
481C42#12114yes—
482C42#12312yes—
483C42#12312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—24no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—24no—
503C23—28no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23#41012yes—
528C23C2×#10012yes—
529C23C22×C22⋊C411yescharacteristic
530C23C2×C4×D412yes—
531C23C22×M1611yescharacteristic
532C23C22×M1611yescharacteristic
533C23C2×#13112yes—
534D4—416no—
535D4—416no—
536D4—48no—
537D4—48no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4#38612yes—
597C2×C4#38612yes—
598C2×C4#38612yes—
599C2×C4#38612yes—
600C2×C4#40712yes—
601C2×C4#40712yes—
602C2×C4#40812yes—
603C2×C4#40812yes—
604C2×C4#41012yes—
605C2×C4C2×#10012yes—
606C2×C4C2×#10012yes—
607C2×C4#41312yes—
608C2×C4#43518yes—
609C2×C4#43618yes—
610C2×C4C2×#12011yescharacteristic
611C2×C4C2×#12112yes—
612C2×C4C2×#12311yescharacteristic
613C8—48no—
614C8—84no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—48no—
630C22—48no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—88no—
643C22—88no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C2×#38611yescharacteristic
649C22C2×#40711yescharacteristic
650C22C2×#40811yescharacteristic
651C22C2×#41011yescharacteristic
652C22C2×#41011yescharacteristic
653C22C22×#10011yescharacteristic
654C22C2×#41311yescharacteristic
655C22C2×#43512yes—
656C22C2×#43612yes—
657C4—21no—
658C4—22no—
659C4—21no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—44no—
666C4—41no—
667C4—41no—
668C4—82no—
669C4—82no—
670C4#187014yes—
671C4#187014yes—
672C4#190014yes—
673C2—22no—
674C2—22no—
675C2—42no—
676C2—42no—
677C2—84no—
678C2#1736411yescharacteristic
679C2#1736611yescharacteristic
680C2#1738311yescharacteristic
681C2#1738411yescharacteristic
682C2C2×#187011yescharacteristic
683C2#2098311yescharacteristic
684C2C2×#190011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20714C2semidirect
4. C2×#1460C2semidirect
5. #9084C2semidirect
6. #9083C2semidirect
7. #7908C2semidirect
99. #431D4semidirect
161. C22×D4M16semidirect
230. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×#1212
#207142
C2×#14602
#90842
#90832
#79082