Finite Group Databasebeta

← #368152

Subgroups — #368152

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1728C211yesmaximal, characteristic, complemented
5C4×#386C211yesmaximal, characteristic
6#11255C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10877C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic, complemented
10#10714C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#435—24no—
18C2×#435—24no—
19C2×#435—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#386C412yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1870—24no—
31#1851—24no—
32#1728—24no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1437C411yescharacteristic
42#1437C411yescharacteristic
43#1436C411yescharacteristic
44#1436C411yescharacteristic
45C22⋊#95C412yes—
46C4×#95C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C412yes—
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—42no—
59C22×M16—42no—
60C22×M16D414yes—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2312yes—
71C23×C8C2311yescharacteristic
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4D414yes—
77C2×C4×D4Q814yes—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C412yes—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42—42no—
87C22×C42—42no—
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2312yes—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#435—232no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—42no—
111C2×C22⋊C8D414yes—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115#386—28no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—42no—
119C4×C22⋊C4—42no—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128#336—24no—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—42no—
134#336—42no—
135#336—42no—
136C25C22×C411yescharacteristic
137C25C22×C411yescharacteristic
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4C4⋊C414yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×Q812yes—
175C23×C4C2×Q812yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16—44no—
183C2×M16—48no—
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—48no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—28no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
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—48no—
219C22×C8—48no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8#4114yes—
227C22×C8#4114yes—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C4⋊C414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C4⋊C414yes—
248C2×C22⋊C4C4⋊C414yes—
249C2×C22⋊C4M1618yescomplemented
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—48no—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C4⋊C8—28no—
276C4⋊C8—48no—
277C22⋊C8—28no—
278C22⋊C8—28no—
279C22⋊C8—48no—
280C22⋊C8—44no—
281#97—28no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—42no—
288#95—42no—
289#95—42no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24C2×C4⋊C412yes—
304C24C2×C4⋊C412yes—
305C24C4×D412yes—
306C24C4×D412yes—
307C24C2×M1614yes—
308C24C23×C411yescharacteristic
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—44no—
312C2×D4—48no—
313C2×D4—48no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
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—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—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4C4×D412yes—
397C22×C4C4×D412yes—
398C22×C4#12014yes—
399C22×C4C22⋊Q814yes—
400C22×C4C22⋊Q814yes—
401C22×C4#12314yes—
402C22×C4C2×M1614yes—
403C22×C4C2×M1614yes—
404C22×C4#13114yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415M16—416no—
416M16—416no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—416no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
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#12314yes—
452C4⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—44no—
463C42—44no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
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—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23#38514yes—
513C23#42314yes—
514C23C22×C4⋊C411yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C2×C4×D411yescharacteristic
517C23C4×#4112yes—
518C23#51812yes—
519C23C22×M1612yes—
520C23C2×#13112yes—
521C23#56812yes—
522C23#56812yes—
523D4—416no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—432no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
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#38514yes—
585C2×C4C2×#12011yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×C22⋊Q811yescharacteristic
588C2×C4C2×#12311yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4#52511yescharacteristic
591C2×C4#52912yes—
592C2×C4#53411yescharacteristic
593C2×C4#53511yescharacteristic
594C2×C4#53812yes—
595C2×C4#54612yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#55312yes—
600C2×C4#55312yes—
601C2×C4#55412yes—
602C2×C4#55412yes—
603C8—24no—
604C8—42no—
605C8—44no—
606C8—42no—
607C8—42no—
608C8—82no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—28no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22C2×#38511yescharacteristic
628C22C4×#12311yescharacteristic
629C22#232111yescharacteristic
630C22#234912yes—
631C22#236412yes—
632C22C2×#42311yescharacteristic
633C22#290511yescharacteristic
634C22#290612yes—
635C22#293111yescharacteristic
636C22#293211yescharacteristic
637C22#293312yes—
638C22D4×M1612yes—
639C22D4×M1612yes—
640C22#296512yes—
641C22#296712yes—
642C22#296912yes—
643C22#297012yes—
644C22#297312yes—
645C22#297412yes—
646C22#297612yes—
647C22#297812yes—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4#240411yescharacteristic
657C4#240511yescharacteristic
658C4#244412yes—
659C4#245012yes—
660C4#248711yescharacteristic
661C4#257911yescharacteristic
662C2—44no—
663C2—44no—
664C2#1401611yescharacteristic
665C2#2083711yescharacteristic
666C2#2083911yescharacteristic
667C2#2156212yes—
668C2#2203612yes—
669C2#2269812yes—
670C2#2269912yes—
671C2#2311812yes—
672C2#2311912yes—
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
3. C2×#1851C2semidirect
4. C2×#1728C2semidirect
6. #11255C2semidirect
7. #11042C2semidirect
8. #10877C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10712C2semidirect
249. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#18702
C2×#18512
C2×#17282
C4×#3862
#112552
#110422
#108772
#107152
#107142
#107122