Finite Group Databasebeta

← #400678

Subgroups — #400678

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2036C212yesmaximal, complemented
3C2×#1933C211yesmaximal, characteristic, complemented
4C2×#1892C212yesmaximal, complemented
5#15181C211yesmaximal, characteristic
6#15177C211yesmaximal, characteristic
7#11228C211yesmaximal, characteristic, complemented
8#11096C212yesmaximal, complemented
9#11077C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×C8×D4—28no—
16C2×C8×D4—24no—
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2036—28no—
30#1933—24no—
31#1892—28no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1444—21no—
39#1442—21no—
40#1439—22no—
41#1439—22no—
42#1438—21no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8—42no—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42C2311yescharacteristic
83C8×D4—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103D414yescomplemented
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8D418yes—
100C2×C4×C8—24no—
101C2×C4×C8D414yes—
102#391—22no—
103#391—22no—
104#389—22no—
105#389—22no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—24no—
110#380—42no—
111#380—42no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#340—42no—
128C25C2×D412yes—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8C2×D412yes—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4Q1614yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D414yes—
165C23×C4C2×D411yescharacteristic
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—216no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—48no—
181C22×C8—48no—
182C22×C8—48no—
183C22×C8—48no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8C2×D414yes—
187C22×C8C2×D412yes—
188C22×C8C2×D414yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8#4112yes—
192C22⋊Q8—216no—
193C22⋊Q8—48no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C4×D4—28no—
197C4×D4—216no—
198C4×D4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251#107—48no—
252C4⋊C8—28no—
253#103—28no—
254#103—48no—
255#103—44no—
256C22⋊C8—216no—
257C4×C8—28no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—81no—
267#95—81no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24#12114yes—
281C24C2×Q1614yes—
282C24#13514yes—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—216no—
305C2×D4—216no—
306C2×D4—28no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
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—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4#12112yes—
382C22×C4#12114yes—
383C22×C4#12112yes—
384C22×C4C2×Q1612yes—
385C22×C4#13512yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—432no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8#12014yes—
413C2×C8#12114yes—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—416no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C22⋊C4—216no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C42—22no—
441C42—22no—
442C42—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23#46718yes—
478C23C2×#12112yes—
479C23C2×#12111yescharacteristic
480C23D4211yescharacteristic
481C23#54612yes—
482C23#54711yescharacteristic
483C23C22×Q1611yescharacteristic
484C23C2×#13512yes—
485C23C2×#13511yescharacteristic
486C23#57611yescharacteristic
487C23#57712yes—
488C23#57811yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—44no—
492Q8—81no—
493Q8—81no—
494Q8—82no—
495D4—232no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#46714yes—
555C2×C4C2×#12011yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4#52511yescharacteristic
558C2×C4#52512yes—
559C2×C4#52512yes—
560C2×C4#52912yes—
561C2×C4#53211yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53711yescharacteristic
564C2×C4#53812yes—
565C2×C4#53912yes—
566C2×C4D4212yes—
567C2×C4#54612yes—
568C2×C4#54612yes—
569C2×C4#54712yes—
570C2×C4#54711yescharacteristic
571C2×C4#55511yescharacteristic
572C2×C4#55912yes—
573C8—24no—
574C8—24no—
575C8—44no—
576C22—28no—
577C22—28no—
578C22—28no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22#242412yes—
592C22#242512yes—
593C22#253711yescharacteristic
594C22#254411yescharacteristic
595C22#284711yescharacteristic
596C22C2×#46712yes—
597C22C2×#46711yescharacteristic
598C22#305012yes—
599C22#305011yescharacteristic
600C22#305711yescharacteristic
601C22#306111yescharacteristic
602C22#306212yes—
603C22#306312yes—
604C22#328212yes—
605C22D4×Q1611yescharacteristic
606C22#328711yescharacteristic
607C22#329712yes—
608C22#329912yes—
609C22#330012yes—
610C22#330312yes—
611C22#333712yes—
612C22#333811yescharacteristic
613C22#334211yescharacteristic
614C4—24no—
615C4—42no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4—162no—
621C4—161no—
622C4—161no—
623C4#239211yescharacteristic
624C4#240811yescharacteristic
625C4#243011yescharacteristic
626C4#245512yes—
627C4#245912yes—
628C4#263611yescharacteristic
629C2—48no—
630C2#1480611yescharacteristic
631C2#2511012yes—
632C2#2511512yes—
633C2#2523412yes—
634C2#2523912yes—
635C2#2822411yescharacteristic
636C2#2822811yescharacteristic
637C2#2859811yescharacteristic
638C2#2860111yescharacteristic
639C2#3293111yescharacteristic
640C2#3293811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2036C2semidirect
3. C2×#1933C2semidirect
4. C2×#1892C2semidirect
7. #11228C2semidirect
8. #11096C2semidirect
9. #11077C2semidirect
10. #10862C2semidirect
11. #10721C2semidirect
96. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#20362
C2×#19332
C2×#18922
#151812
#151772
#112282
#110962
#110772
#108622
#107212