Finite Group Databasebeta

← #204232

Subgroups — #204232

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2321C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4C2×#1465C212yesmaximal, complemented
5C2×#1462C212yesmaximal, complemented
6#9653C212yesmaximal, complemented
7#9652C212yesmaximal, complemented
8#9651C212yesmaximal, complemented
9#9649C212yesmaximal, complemented
10C22×#120C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C22⋊C4C2211yescharacteristic
13C2×#407C2212yes—
14C2×#407C2212yes—
15#2321—24no—
16#2321—24no—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#388C411yescharacteristic
20C2×#388C411yescharacteristic
21C2×#388C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#386C411yescharacteristic
24C2×#386C411yescharacteristic
25C2×#386C411yescharacteristic
26C2×#385C411yescharacteristic
27C2×#379C411yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1851—24no—
30#1797C2212yescomplemented
31#1797C2212yescomplemented
32#1753—24no—
33#1749C2212yescomplemented
34C4×C22⋊C8C2212yescomplemented
35#1465—28no—
36#1462—28no—
37C2×#324C2214yescomplemented
38C2×#323C2214yes—
39C2×#323C2214yes—
40C2×#323C2214yescomplemented
41C26C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C2×#120—24no—
51C2×#120—24no—
52C2×#120—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C22⋊C4C2311yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—24no—
86#407—24no—
87#407—24no—
88#407D418yescomplemented
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×#97—22no—
94C2×C4×C8—22no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—24no—
104#388—24no—
105#386—24no—
106#386—24no—
107#386—24no—
108#385—24no—
109#379—24no—
110C4×C22⋊C4D414yes—
111C4×C22⋊C4D414yes—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#324—28no—
123#323—28no—
124#323—28no—
125#323—28no—
126C25—22no—
127C25—22no—
128C25—22no—
129C25—24no—
130C25—42no—
131C25—81no—
132C25—81no—
133C25—81no—
134C25—81no—
135C25C22×C411yescharacteristic
136C25C22×C411yescharacteristic
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4—81no—
150C22×D4—81no—
151C22×D4—81no—
152C22×D4—81no—
153C22×D4M1614yescomplemented
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—81no—
159C23×C4—81no—
160C23×C4—81no—
161C23×C4—81no—
162C23×C4—81no—
163C23×C4—81no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D414yes—
177C23×C4#4114yes—
178C23×C4C2411yescharacteristic
179C2×M16—28no—
180C2×M16—44no—
181C2×M16—82no—
182C2×M16—82no—
183C2×M16—82no—
184C2×M16—82no—
185C22×C8—24no—
186C22×C8—24no—
187#120—232no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
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⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—81no—
237C2×C22⋊C4—81no—
238C2×C22⋊C4—81no—
239C2×C22⋊C4—81no—
240C2×C22⋊C4—81no—
241C2×C22⋊C4—81no—
242C2×C22⋊C4C22⋊C414yes—
243C2×C22⋊C4C22⋊C414yes—
244C2×C22⋊C4M1614yescomplemented
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C4⋊C8—82no—
254C4⋊C8—82no—
255C22⋊C8—28no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—44no—
260C22⋊C8C2×D418yes—
261#97—82no—
262C4×C8—82no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊C414yes—
268#95C22⋊C414yes—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—28no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—48no—
290C24—48no—
291C24—44no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—44no—
305C24—82no—
306C24—82no—
307C24—82no—
308C24—82no—
309C24—84no—
310C24—84no—
311C24—84no—
312C24—84no—
313C24—88no—
314C24—88no—
315C24—88no—
316C24—88no—
317C24C2×C22⋊C412yes—
318C24C4×D414yes—
319C24C2×M1612yes—
320C24C2×M1612yes—
321C24C2×M1612yes—
322C24C23×C411yescharacteristic
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C4×D414yes—
394C22×C4#12114yes—
395C22×C4#12114yes—
396C22×C4#12414yes—
397C22×C4#12714yes—
398C22×C4C2×M1612yes—
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C2×#4112yes—
405M16—88no—
406M16—88no—
407C2×C8—28no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—44no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—416no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—88no—
518C23—88no—
519C23—88no—
520C23—88no—
521C23—832no—
522C23—832no—
523C23—88no—
524C23—88no—
525C23#39014yes—
526C23C22×C22⋊C411yescharacteristic
527C23C2×C4×D412yes—
528C23C22×M1611yescharacteristic
529C23C22×M1611yescharacteristic
530C23#56812yes—
531D4—416no—
532D4—88no—
533D4—88no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—88no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—88no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#39014yes—
588C2×C4#40714yes—
589C2×C4C2×#12112yes—
590C2×C4C2×#12411yescharacteristic
591C2×C4C2×#12711yescharacteristic
592C8—48no—
593C8—84no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—48no—
606C22—48no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—832no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C2×#39011yescharacteristic
640C22C2×#40711yescharacteristic
641C22#287911yescharacteristic
642C22#288311yescharacteristic
643C22#288311yescharacteristic
644C22#288811yescharacteristic
645C22#288811yescharacteristic
646C22D4×M1612yes—
647C22D4×M1612yes—
648C4—24no—
649C4—22no—
650C4—21no—
651C4—24no—
652C4—21no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C2—22no—
662C2—22no—
663C2—44no—
664C2—82no—
665C2—82no—
666C2—82no—
667C2#1736711yescharacteristic
668C2#1736911yescharacteristic
669C2#1740511yescharacteristic
670C2#1740511yescharacteristic
671C2#2119011yescharacteristic
672C2#2121311yescharacteristic
673C2#2121911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#120C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
3. #20524C2semidirect
4. C2×#1465C2semidirect
5. C2×#1462C2semidirect
6. #9653C2semidirect
7. #9652C2semidirect
8. #9651C2semidirect
9. #9649C2semidirect
30. #1797C22semidirect
31. #1797C22semidirect
33. #1749C22semidirect
34. C4×C22⋊C8C22semidirect
37. C2×#324C22semidirect
40. C2×#323C22semidirect
88. #407D4semidirect
153. C22×D4M16semidirect
244. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23212
#219342
#205242
C2×#14652
C2×#14622
#96532
#96522
#96512
#96492