Finite Group Databasebeta

← #409260

Subgroups — #409260

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal
2#14686C211yesmaximal, characteristic
3#13756C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11257C212yesmaximal, complemented
9#11256C212yesmaximal, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#380—21no—
21C2×#380—21no—
22C2×#380C2212yes—
23#1854—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22⋊#95—21no—
41C22⋊#95—21no—
42C4×#95—21no—
43C4×#95—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117—42no—
56C2×#117D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—24no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—42no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—42no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#106—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8Q818yes—
89#384—22no—
90#384—22no—
91#384—42no—
92#380—24no—
93#380—24no—
94#380—44no—
95#380—44no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#336—22no—
114#336—22no—
115#336—42no—
116C25C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×Q814yes—
150C23×C4#4112yes—
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×Q814yes—
166C22×C8C2×Q814yes—
167#117—48no—
168#117—48no—
169#117—48no—
170#117—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4112yes—
210C2×C4⋊C4#4112yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—48no—
233C2×C42—48no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42#4112yes—
241C2×C42#4112yes—
242#107—48no—
243#106—48no—
244C22⋊C8—216no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—42no—
255#95—42no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24#12112yes—
269C24#12312yes—
270C24C4⋊Q812yes—
271C24C22×D411yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4#12112yes—
367C22×C4C22⋊Q818yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4#12518yes—
372C22×C4C4⋊Q812yes—
373C22×C4#13512yes—
374C22×C4#13512yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4#13712yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q812yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—88no—
426C42—24no—
427C42—24no—
428C42—44no—
429C42—44no—
430C42—48no—
431C42—48no—
432C42—42no—
433C42—42no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#39814yes—
469C23C2×#12111yescharacteristic
470C23C2×#12311yescharacteristic
471C23C2×#12311yescharacteristic
472C23#52511yescharacteristic
473C23C2×C4⋊Q811yescharacteristic
474C23#53411yescharacteristic
475C23#53511yescharacteristic
476C23C2×#13511yescharacteristic
477C23C2×#13511yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×#13711yescharacteristic
481C23C2×#13711yescharacteristic
482C23#57511yescharacteristic
483C23#57511yescharacteristic
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#39814yes—
565C2×C4#48214yes—
566C2×C4#48514yes—
567C2×C4C2×C22⋊Q812yes—
568C2×C4C2×#12512yes—
569C2×C4#52911yescharacteristic
570C2×C4#52911yescharacteristic
571C2×C4#53211yescharacteristic
572C2×C4#53211yescharacteristic
573C2×C4#53412yes—
574C2×C4#53812yes—
575C2×C4#55112yes—
576C2×C4#55112yes—
577C2×C4#55112yes—
578C2×C4#55112yes—
579C2×C4#55412yes—
580C2×C4#55412yes—
581C2×C4#55412yes—
582C2×C4#55412yes—
583C8—44no—
584C8—44no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22C2×#39811yescharacteristic
601C22#248211yescharacteristic
602C22#248311yescharacteristic
603C22#250211yescharacteristic
604C22#258311yescharacteristic
605C22#261711yescharacteristic
606C22#285411yescharacteristic
607C22#303511yescharacteristic
608C22#303511yescharacteristic
609C22#303811yescharacteristic
610C22#303911yescharacteristic
611C22C2×#48211yescharacteristic
612C22C2×#48511yescharacteristic
613C22#309111yescharacteristic
614C22#309111yescharacteristic
615C22#309211yescharacteristic
616C22#309211yescharacteristic
617C22#309411yescharacteristic
618C22#309511yescharacteristic
619C22#310611yescharacteristic
620C22#310711yescharacteristic
621C22#311511yescharacteristic
622C22#311611yescharacteristic
623C22#316011yescharacteristic
624C22#316211yescharacteristic
625C22#316511yescharacteristic
626C22#316511yescharacteristic
627C22#318811yescharacteristic
628C22#319011yescharacteristic
629C22#319411yescharacteristic
630C22#319611yescharacteristic
631C22#320711yescharacteristic
632C22#321011yescharacteristic
633C22#321411yescharacteristic
634C22#321611yescharacteristic
635C4—24no—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—162no—
645C4—162no—
646C4#250512yes—
647C4#250512yes—
648C4#256412yes—
649C4#256412yes—
650C2—44no—
651C2#1501411yescharacteristic
652C2#2365611yescharacteristic
653C2#2365811yescharacteristic
654C2#2758411yescharacteristic
655C2#2758711yescharacteristic
656C2#2760211yescharacteristic
657C2#2760411yescharacteristic
658C2#2794911yescharacteristic
659C2#2795511yescharacteristic
660C2#2929911yescharacteristic
661C2#2930411yescharacteristic
662C2#2971611yescharacteristic
663C2#2971711yescharacteristic
664C2#3311311yescharacteristic
665C2#3311411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11357C2semidirect
5. #11357C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11257C2semidirect
9. #11256C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
#146862
#137562
#113572
#113242
#113232
#112572
#112562
#107312
#107122