Finite Group Databasebeta

← #355094

Subgroups — #355094

Order 512.

2734 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
1C22×#407C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3C2×#1753C212yesmaximal, complemented
4C4×#379C211yesmaximal, characteristic
5#10922C212yesmaximal, complemented
6#10922C212yesmaximal, complemented
7#10921C212yesmaximal, complemented
8#10921C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C2×#407—28no—
14C2×#407C418yescomplemented
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2212yescomplemented
22C22×C4×C8C2212yescomplemented
23C2×#379C412yes—
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1779C418yescomplemented
29#1753—28no—
30C2×#336C2212yescomplemented
31C2×#336C2212yescomplemented
32C2×#336C2212yescomplemented
33C2×#336C2212yescomplemented
34C2×#336C2212yescomplemented
35C2×#336C2212yescomplemented
36C22⋊#95C412yes—
37C22⋊#95C412yes—
38C22⋊#95C412yes—
39C26C2×C411yescharacteristic
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—42no—
43C24×C4—41no—
44C24×C4—41no—
45C24×C4—42no—
46C24×C4—41no—
47C24×C4—41no—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C414yes—
52C24×C4C2×C414yes—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C22×M16—24no—
57C22×M16—41no—
58C22×M16—42no—
59C22×M16—41no—
60C22×M16—41no—
61C22×M16—42no—
62C22×M16—41no—
63C22×M16C2×C414yes—
64C22×M16C2×C414yes—
65C22×M16D418yes—
66C23×C8C2312yes—
67C23×C8C2312yes—
68C23×C8C2312yes—
69C23×C8C2312yes—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2×C414yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C42C2312yes—
75C22×C42C2312yes—
76#407—232no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—41no—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—41no—
89C2×C22⋊C8C2×C418yes—
90C2×C22⋊C8C2×C418yes—
91C2×C22⋊C8C2×C418yes—
92C2×C22⋊C8C2×C414yes—
93C2×C22⋊C8C2×C418yes—
94C2×C22⋊C8C2×C414yes—
95C2×#97—24no—
96C2×#97—42no—
97C2×#97—42no—
98C2×C4×C8—24no—
99C2×C4×C8—42no—
100C2×C4×C8—42no—
101#379—28no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—41no—
105C4×C22⋊C4—42no—
106C4×C22⋊C4—41no—
107C4×C22⋊C4—41no—
108C4×C22⋊C4—42no—
109C4×C22⋊C4—41no—
110C2×#95C2×C414yes—
111C2×#95C2×C412yes—
112C2×#95C2×C414yes—
113C2×#95C2×C412yes—
114#336—24no—
115#336—24no—
116#336—42no—
117#336—42no—
118#336—42no—
119#336—42no—
120C25—24no—
121C25—24no—
122C25—24no—
123C25—42no—
124C25—41no—
125C25—41no—
126C25—42no—
127C25—42no—
128C25C4214yes—
129C25C22×C411yescharacteristic
130C25C22×C411yescharacteristic
131C25C22×C411yescharacteristic
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—216no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—48no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—41no—
168C23×C4—42no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—42no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—42no—
175C23×C4—41no—
176C23×C4—48no—
177C23×C4—48no—
178C23×C4—48no—
179C23×C4—48no—
180C23×C4—48no—
181C23×C4—48no—
182C23×C4—48no—
183C23×C4—48no—
184C23×C4C4214yes—
185C23×C4C4214yes—
186C23×C4C4214yes—
187C23×C4C22×C414yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C22×C414yes—
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C412yes—
197C23×C4C22×C412yes—
198C23×C4C2×D414yes—
199C23×C4#4114yes—
200C23×C4C2411yescharacteristic
201C2×M16—28no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—44no—
205C2×M16—44no—
206C2×M16—48no—
207C2×M16—48no—
208C2×M16—48no—
209C2×M16—48no—
210C2×M16C22⋊C4116yescomplemented
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8C22×C414yes—
234C22×C8C22×C414yes—
235C22×C8C22×C414yes—
236C22×C8C22×C414yes—
237C22×C8C2×D414yes—
238C22×C8C2×D414yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4M1618yescomplemented
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
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—
263C22⋊C8—216no—
264C22⋊C8—216no—
265C22⋊C8—216no—
266C22⋊C8—44no—
267C22⋊C8—48no—
268C22⋊C8—44no—
269#97—48no—
270C4×C8—48no—
271#95—28no—
272#95—42no—
273#95—44no—
274#95—42no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—28no—
297C24—28no—
298C24—28no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—416no—
304C24—416no—
305C24—416no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—416no—
313C24C2×C4212yes—
314C24C2×C4212yes—
315C24C2×C4212yes—
316C24C4×D414yes—
317C24C2×M1614yes—
318C24#13114yes—
319C24C23×C411yescharacteristic
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
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—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—432no—
397C22×C4—432no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
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×C4C2×C4212yes—
417C22×C4C2×C4212yes—
418C22×C4C2×C4212yes—
419C22×C4C2×C22⋊C414yes—
420C22×C4#11714yes—
421C22×C4C4×D418yes—
422C22×C4C4×D414yes—
423C22×C4C2×M1618yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)11yescharacteristic
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)11yescharacteristic
434M16—416no—
435M16—416no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8C2×C22⋊C418yes—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—416no—
507C23—416no—
508C23—416no—
509C23—416no—
510C23—416no—
511C23—416no—
512C23—464no—
513C23—464no—
514C23—416no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23C4×C22⋊C414yes—
534C23C4×M1614yes—
535C23#40514yes—
536C23C22×C4211yescharacteristic
537C23C2×C4×D412yes—
538C23#51811yescharacteristic
539C23#51812yes—
540C23#51811yescharacteristic
541C23C22×M1612yes—
542C23C2×#13112yes—
543C23#56812yes—
544C23#56812yes—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4C4×C22⋊C414yes—
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C2×C4×D412yes—
598C2×C4#51811yescharacteristic
599C2×C4#51811yescharacteristic
600C2×C4#51811yescharacteristic
601C2×C4#51811yescharacteristic
602C2×C4#51811yescharacteristic
603C2×C4#51811yescharacteristic
604C2×C4#51812yes—
605C2×C4#53412yes—
606C2×C4#53412yes—
607C2×C4#53512yes—
608C2×C4#53512yes—
609C2×C4#53812yes—
610C2×C4#53812yes—
611C2×C4#53812yes—
612C2×C4#53812yes—
613C8—28no—
614C8—44no—
615C8—44no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—464no—
627C22—48no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22C2×C4×C22⋊C411yescharacteristic
646C22#227311yescharacteristic
647C22#227311yescharacteristic
648C22#232112yes—
649C22#232112yes—
650C22C2×C4×M1611yescharacteristic
651C22C2×#40511yescharacteristic
652C22#287311yescharacteristic
653C22#287311yescharacteristic
654C22#293314yes—
655C22D4×M1614yes—
656C22#297312yes—
657C22#297414yes—
658C22#297512yes—
659C22#297612yes—
660C22#297612yes—
661C22#297814yes—
662C4—24no—
663C4—48no—
664C4—42no—
665C4—42no—
666C4—84no—
667C4#230811yescharacteristic
668C4#230911yescharacteristic
669C4#231211yescharacteristic
670C4#231211yescharacteristic
671C4#232112yes—
672C4#232112yes—
673C2—44no—
674C2—44no—
675C2—44no—
676C2#1377511yescharacteristic
677C2C22⋊C4×M1611yescharacteristic
678C2#2038911yescharacteristic
679C2#2052411yescharacteristic
680C2#2052511yescharacteristic
681C2#2052711yescharacteristic
682C2#2052911yescharacteristic
683C2#2203614yes—
684C2#2203814yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ 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×#1753C2semidirect
5. #10922C2semidirect
6. #10922C2semidirect
7. #10921C2semidirect
8. #10921C2semidirect
9. #10715C2semidirect
14. C2×#407C4semidirect
21. C22×#97C22semidirect
22. C22×C4×C8C22semidirect
28. #1779C4semidirect
30. C2×#336C22semidirect
31. C2×#336C22semidirect
32. C2×#336C22semidirect
33. C2×#336C22semidirect
34. C2×#336C22semidirect
35. C2×#336C22semidirect
210. C2×M16C22⋊C4semidirect
250. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4072
C2×#17792
C2×#17532
C4×#3792
#109222
#109212
#107152