Finite Group Databasebeta

← #413861

Subgroups — #413861

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#1933C211yesmaximal, characteristic, complemented
3C2×#1892C212yesmaximal, complemented
4#15181C211yesmaximal, characteristic
5#15177C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal, complemented
10#10820C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×C8×D4—24no—
16C2×C8×D4—28no—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28#1933—24no—
29#1892—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1444—21no—
38#1442—21no—
39#1439—22no—
40#1439—22no—
41#1438—21no—
42#1438—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8D418yes—
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42C2311yescharacteristic
82C8×D4—232no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—42no—
92C2×#103—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C4×C8—24no—
97C2×C4×C8D414yes—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—42no—
107#380—42no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—42no—
124#340—42no—
125C25C2×D412yes—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8#4112yes—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4Q1614yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
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—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D414yes—
162C23×C4C2×D411yescharacteristic
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—216no—
179C22×C8—48no—
180C22×C8—48no—
181C22×C8—48no—
182C22×C8—48no—
183C22×C8—48no—
184C22×C8—48no—
185C22×C8C2×D414yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D414yes—
189C22⋊Q8—216no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C4×D4—28no—
194C4×D4—216no—
195C4×D4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4#4114yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C4⋊C8—28no—
249#103—28no—
250#103—48no—
251#103—44no—
252C22⋊C8—216no—
253C4×C8—28no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—81no—
263#95—81no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24#12114yes—
277C24C2×Q1614yes—
278C24#13514yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—216no—
301C2×D4—216no—
302C2×D4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
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—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
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—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4#12112yes—
378C22×C4#12114yes—
379C22×C4#12112yes—
380C22×C4C2×Q1612yes—
381C22×C4#13512yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)11yescharacteristic
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—432no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8#12014yes—
410C4⋊C4—28no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—416no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—84no—
431C22⋊C4—216no—
432C22⋊C4—416no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—22no—
437C42—22no—
438C42—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—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—42no—
458C23—42no—
459C23—42no—
460C23—42no—
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#46718yes—
474C23C2×#12112yes—
475C23C2×#12111yescharacteristic
476C23#54611yescharacteristic
477C23#54612yes—
478C23#54611yescharacteristic
479C23C22×Q1611yescharacteristic
480C23C2×#13512yes—
481C23C2×#13511yescharacteristic
482C23#57711yescharacteristic
483C23#57712yes—
484C23#57711yescharacteristic
485Q8—42no—
486Q8—44no—
487Q8—42no—
488Q8—82no—
489Q8—81no—
490Q8—81no—
491D4—232no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
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—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4#46714yes—
551C2×C4C2×#12011yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52512yes—
555C2×C4#52512yes—
556C2×C4#53212yes—
557C2×C4#53211yescharacteristic
558C2×C4#53412yes—
559C2×C4#53412yes—
560C2×C4#53511yescharacteristic
561C2×C4#53511yescharacteristic
562C2×C4#55112yes—
563C2×C4#55112yes—
564C2×C4#55312yes—
565C2×C4#55311yescharacteristic
566C2×C4#55312yes—
567C2×C4#55311yescharacteristic
568C2×C4#55912yes—
569C8—24no—
570C8—44no—
571C8—42no—
572C22—28no—
573C22—28no—
574C22—28no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22#253412yes—
588C22#253511yescharacteristic
589C22#254312yes—
590C22#254411yescharacteristic
591C22#284711yescharacteristic
592C22C2×#46712yes—
593C22C2×#46711yescharacteristic
594C22#305012yes—
595C22#305011yescharacteristic
596C22#305712yes—
597C22#305711yescharacteristic
598C22#306012yes—
599C22#306011yescharacteristic
600C22#329612yes—
601C22#329612yes—
602C22#329911yescharacteristic
603C22#329912yes—
604C22#329911yescharacteristic
605C22#330112yes—
606C22#330112yes—
607C22#330312yes—
608C22#330411yescharacteristic
609C22#330411yescharacteristic
610C4—24no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—84no—
615C4—82no—
616C4—162no—
617C4—161no—
618C4—161no—
619C4#238811yescharacteristic
620C4#259811yescharacteristic
621C4#260012yes—
622C4#260212yes—
623C4#260311yescharacteristic
624C4#261311yescharacteristic
625C2—48no—
626C2#1515911yescharacteristic
627C2#2809512yes—
628C2#2809912yes—
629C2#2814211yescharacteristic
630C2#2814411yescharacteristic
631C2#2855312yes—
632C2#2855612yes—
633C2#2859811yescharacteristic
634C2#2860211yescharacteristic
635C2#3293111yescharacteristic
636C2#3293211yescharacteristic
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×#1933C2semidirect
3. C2×#1892C2semidirect
6. #11228C2semidirect
7. #11200C2semidirect
8. #11077C2semidirect
9. #10970C2semidirect
10. #10820C2semidirect
11. #10721C2semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#19332
C2×#18922
#151812
#151772
#112282
#112002
#110772
#109702
#108202
#107212