Finite Group Databasebeta

← #405894

Subgroups — #405894

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1892C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#13907C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11227C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1892—24no—
27#1872—24no—
28#1850—24no—
29C2×#340—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1441—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—42no—
52C23×C8—42no—
53C23×C8D414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8Q814yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42C2312yes—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8—42no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98#391—24no—
99#391—42no—
100#389—24no—
101#389—42no—
102#380—22no—
103#380—22no—
104#380—44no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—28no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—48no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—48no—
186C22×C8C2×D414yes—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q812yes—
231C2×C4⋊C4#4114yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4SD1614yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C4⋊C8—48no—
252#103—44no—
253#103—48no—
254#103—44no—
255C22⋊C8—44no—
256#95—24no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—82no—
263#95—82no—
264#95—82no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24#12112yes—
270C24#12112yes—
271C24C22⋊Q812yes—
272C24C2×SD1612yes—
273C24#13512yes—
274C24#13712yes—
275C24#13712yes—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—48no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—84no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4#12114yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q812yes—
374C22×C4#12614yes—
375C22×C4C2×SD1614yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)12yes—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—416no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—88no—
425C4⋊C4#12314yes—
426C22⋊C4—28no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23#39614yes—
451C23#46114yes—
452C23#46414yes—
453C23#46514yes—
454C23#47014yes—
455C23#47514yes—
456C23#47914yes—
457C23C2×#12111yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×C22⋊Q811yescharacteristic
460C23#54612yes—
461C23#54612yes—
462C23C22×SD1611yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×#13711yescharacteristic
465C23C2×#13711yescharacteristic
466C23#57612yes—
467C23#57812yes—
468Q8—44no—
469Q8—42no—
470Q8—42no—
471Q8—81no—
472Q8—81no—
473Q8—82no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—816no—
514C2×C4—816no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4#39614yes—
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4C2×#12611yescharacteristic
541C2×C4#52912yes—
542C2×C4#53212yes—
543C2×C4#53412yes—
544C2×C4#53411yescharacteristic
545C2×C4#53511yescharacteristic
546C2×C4#53812yes—
547C2×C4#55112yes—
548C2×C4#55112yes—
549C2×C4#55312yes—
550C2×C4#55312yes—
551C2×C4#55412yes—
552C2×C4#55412yes—
553C8—44no—
554C8—81no—
555C8—81no—
556C22—28no—
557C22—28no—
558C22—24no—
559C22—24no—
560C22—28no—
561C22—28no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22#206618yes—
567C22C2×#39611yescharacteristic
568C22#241111yescharacteristic
569C22#241211yescharacteristic
570C22#245012yes—
571C22#254312yes—
572C22C2×#46111yescharacteristic
573C22C2×#46411yescharacteristic
574C22#304111yescharacteristic
575C22#304411yescharacteristic
576C22C2×#46511yescharacteristic
577C22C2×#47011yescharacteristic
578C22#306112yes—
579C22C2×#47511yescharacteristic
580C22C2×#47911yescharacteristic
581C22#308312yes—
582C22#329512yes—
583C22#329712yes—
584C22#329812yes—
585C22#330012yes—
586C22#331212yes—
587C22#331212yes—
588C22#331412yes—
589C22#331412yes—
590C4—24no—
591C4—44no—
592C4—44no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—162no—
599C4#248711yescharacteristic
600C4#251012yes—
601C4#256412yes—
602C4#257911yescharacteristic
603C4#261711yescharacteristic
604C4#285411yescharacteristic
605C2—28no—
606C2#1492111yescharacteristic
607C2C2×#206611yescharacteristic
608C2#2361111yescharacteristic
609C2#2447911yescharacteristic
610C2#2448011yescharacteristic
611C2#2452911yescharacteristic
612C2#2453011yescharacteristic
613C2#2641212yes—
614C2#2641812yes—
615C2#2855512yes—
616C2#2856312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11315C2semidirect
8. #11227C2semidirect
9. #11077C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18922
C2×#18722
C2×#18502
C22×#3402
#141392
#139072
#113152
#112272
#110772
#110672
#107302