Finite Group Databasebeta

← #416002

Subgroups — #416002

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#11292C212yesmaximal
6#11268C212yesmaximal
7#11067C212yesmaximal
8#11052C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2571—24no—
19#2571—24no—
20C2×#400—22no—
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382—21no—
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1803—24no—
31C4×#103—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#400—216no—
88#391—24no—
89#391—42no—
90#389—28no—
91#389—24no—
92#389—24no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384D414yes—
97#382—28no—
98#382—24no—
99#382—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—24no—
121#336—42no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8C2×Q812yes—
134C22×Q8#4112yes—
135C22×Q8#4112yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8#4112yes—
167C22×C8#4112yes—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—44no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—44no—
245#103—28no—
246#103—28no—
247#103—48no—
248C4×C8—44no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—44no—
255#95—44no—
256#95—82no—
257#95—82no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8#12514yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
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—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4#12412yes—
352C22×C4#12412yes—
353C22×C4#13512yes—
354C22×C4#13512yes—
355C22×C4#13712yes—
356C22×C4#13712yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C42−1+411yescharacteristic
370C22×C42−1+411yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—88no—
419C4⋊C4—84no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—48no—
425C42—48no—
426C42C22⋊Q814yes—
427C42C22⋊Q814yes—
428C42#12414yes—
429C42#12414yes—
430C23C2×#12011yescharacteristic
431C23C2×C22⋊Q811yescharacteristic
432C23C2×#12411yescharacteristic
433C23#54611yescharacteristic
434C23#54711yescharacteristic
435C23#54811yescharacteristic
436C23#55011yescharacteristic
437C23C2×#13511yescharacteristic
438C23C2×#13511yescharacteristic
439C23C2×#13711yescharacteristic
440C23C2×#13711yescharacteristic
441C23#57611yescharacteristic
442C23#57711yescharacteristic
443C23#57711yescharacteristic
444C23#57811yescharacteristic
445Q8—24no—
446Q8—24no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4#39314yes—
516C2×C4#39314yes—
517C2×C4#45214yes—
518C2×C4#45214yes—
519C2×C4#47914yes—
520C2×C4#47914yes—
521C2×C4#48814yes—
522C2×C4#49114yes—
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4#52511yescharacteristic
527C2×C4C2×#12411yescharacteristic
528C2×C4C2×#12411yescharacteristic
529C2×C4C2×#12511yescharacteristic
530C2×C4#52911yescharacteristic
531C2×C4#52911yescharacteristic
532C2×C4#53311yescharacteristic
533C2×C4#53711yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#54011yescharacteristic
536C2×C4#54111yescharacteristic
537C2×C4#54411yescharacteristic
538C2×C4#54411yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54711yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4#55011yescharacteristic
543C2×C4#55211yescharacteristic
544C2×C4#55211yescharacteristic
545C2×C4#55211yescharacteristic
546C2×C4#55211yescharacteristic
547C2×C4#55311yescharacteristic
548C2×C4#55311yescharacteristic
549C2×C4#55311yescharacteristic
550C2×C4#55311yescharacteristic
551C8—42no—
552C8—42no—
553C8—82no—
554C22C2×#39311yescharacteristic
555C22#240711yescharacteristic
556C22#240711yescharacteristic
557C22#244111yescharacteristic
558C22#244211yescharacteristic
559C22#248911yescharacteristic
560C22#249011yescharacteristic
561C22C2×#45211yescharacteristic
562C22C2×#45211yescharacteristic
563C22#301211yescharacteristic
564C22#301311yescharacteristic
565C22C2×#47911yescharacteristic
566C22C2×#47911yescharacteristic
567C22#308311yescharacteristic
568C22#308411yescharacteristic
569C22C2×#48811yescharacteristic
570C22C2×#49111yescharacteristic
571C22#313611yescharacteristic
572C22#313711yescharacteristic
573C22#330411yescharacteristic
574C22#330511yescharacteristic
575C22#332811yescharacteristic
576C22#332911yescharacteristic
577C22#334011yescharacteristic
578C22#334211yescharacteristic
579C22#335611yescharacteristic
580C22#335711yescharacteristic
581C22#336411yescharacteristic
582C22#336511yescharacteristic
583C22#336911yescharacteristic
584C22#337511yescharacteristic
585C22#339411yescharacteristic
586C22#339611yescharacteristic
587C22#339811yescharacteristic
588C22#340011yescharacteristic
589C4—24no—
590C4—44no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—84no—
596C4—82no—
597C4—162no—
598C4#202618yes—
599C4#244111yescharacteristic
600C4#244211yescharacteristic
601C4#248911yescharacteristic
602C4#249011yescharacteristic
603C4#256211yescharacteristic
604C4#256211yescharacteristic
605C4#262911yescharacteristic
606C4#285111yescharacteristic
607C2#1521611yescharacteristic
608C2C2×#202611yescharacteristic
609C2#2345811yescharacteristic
610C2#2412111yescharacteristic
611C2#2412311yescharacteristic
612C2#2418911yescharacteristic
613C2#2419011yescharacteristic
614C2#2608911yescharacteristic
615C2#2609311yescharacteristic
616C2#2610911yescharacteristic
617C2#2611311yescharacteristic
618C2#2773011yescharacteristic
619C2#2773811yescharacteristic
620C2#2775011yescharacteristic
621C2#2775711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#18032
C2×C4×#1032
#139262
#112922
#112682
#110672
#110522
#108492
#108412
#108342