Finite Group Databasebeta

← #402406

Subgroups — #402406

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic, complemented
2C2×#1749C211yesmaximal, characteristic
3#14697C211yesmaximal, characteristic
4#14686C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal
9#11200C212yesmaximal
10#10881C212yesmaximal, complemented
11#10821C212yesmaximal, complemented
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#385—21no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#380—21no—
21C2×#380—21no—
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#2099—24no—
26#1749—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—21no—
38#1436—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—24no—
68C22×C42—42no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×#97—24no—
77C2×#97—24no—
78C2×#97—42no—
79#389—22no—
80#389—22no—
81#389—24no—
82#385—44no—
83#384—22no—
84#384—22no—
85#384—42no—
86#380—22no—
87#380—22no—
88#380—24no—
89#380—24no—
90#380—42no—
91#380—44no—
92#380—44no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336D414yes—
114C25C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4114yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
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×C8C2×D414yes—
164C22×C8#4114yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
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⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4112yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—48no—
232C2×C42—48no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C2×D412yes—
236C2×C42#4112yes—
237C2×C42#4114yes—
238C22⋊C8—216no—
239#97—48no—
240#95—24no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—42no—
250#95—42no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24#12412yes—
264C24#13712yes—
265C24#13712yes—
266C24C22×D411yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
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—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12312yes—
368C22×C4#12312yes—
369C22×C4#12412yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4#13712yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
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—84no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—48no—
423C22⋊C4—88no—
424C22⋊C4—84no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—44no—
430C42—44no—
431C23—22no—
432C23—22no—
433C23—22no—
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—41no—
450C23—41no—
451C23—41no—
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#49114yes—
466C23C2×#12111yescharacteristic
467C23C2×#12311yescharacteristic
468C23#52511yescharacteristic
469C23C2×#12411yescharacteristic
470C23#53211yescharacteristic
471C23#53411yescharacteristic
472C23#53611yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×AGL(1,ℤ/8)11yescharacteristic
475C23C2×AGL(1,ℤ/8)11yescharacteristic
476C23C2×#13711yescharacteristic
477C23C2×#13711yescharacteristic
478C23C2×#13711yescharacteristic
479C23#57511yescharacteristic
480C23#57511yescharacteristic
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
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—416no—
503C2×C4—416no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4#46814yes—
563C2×C4#46914yes—
564C2×C4#52512yes—
565C2×C4#52912yes—
566C2×C4#52912yes—
567C2×C4#52912yes—
568C2×C4#53912yes—
569C2×C4#54212yes—
570C2×C4#54612yes—
571C2×C4#54711yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4#55112yes—
574C2×C4#55212yes—
575C2×C4#55212yes—
576C2×C4#55211yescharacteristic
577C2×C4#55312yes—
578C2×C4#55312yes—
579C2×C4#55611yescharacteristic
580C8—44no—
581C8—44no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22#239511yescharacteristic
598C22#241211yescharacteristic
599C22#241511yescharacteristic
600C22#250311yescharacteristic
601C22#259111yescharacteristic
602C22#266311yescharacteristic
603C22#285011yescharacteristic
604C22C2×#46811yescharacteristic
605C22C2×#46911yescharacteristic
606C22#305611yescharacteristic
607C22#305611yescharacteristic
608C22#309211yescharacteristic
609C22#309211yescharacteristic
610C22#309411yescharacteristic
611C22#309511yescharacteristic
612C22#311011yescharacteristic
613C22#311311yescharacteristic
614C22#311511yescharacteristic
615C22#311511yescharacteristic
616C22C2×#49111yescharacteristic
617C22#313511yescharacteristic
618C22#313811yescharacteristic
619C22#314111yescharacteristic
620C22#318111yescharacteristic
621C22#318111yescharacteristic
622C22#318211yescharacteristic
623C22#318411yescharacteristic
624C22#319311yescharacteristic
625C22#319411yescharacteristic
626C22#319411yescharacteristic
627C22#319511yescharacteristic
628C22#323511yescharacteristic
629C22#323711yescharacteristic
630C22#323911yescharacteristic
631C22#323911yescharacteristic
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—162no—
640C4—162no—
641C4—161no—
642C4—161no—
643C4#246012yes—
644C4#246812yes—
645C4#253012yes—
646C4#257712yes—
647C2—44no—
648C2#1484111yescharacteristic
649C2#2376111yescharacteristic
650C2#2376311yescharacteristic
651C2#2455811yescharacteristic
652C2#2455911yescharacteristic
653C2#2470911yescharacteristic
654C2#2471011yescharacteristic
655C2#2796511yescharacteristic
656C2#2796811yescharacteristic
657C2#2933511yescharacteristic
658C2#2933711yescharacteristic
659C2#3036511yescharacteristic
660C2#3036611yescharacteristic
661C2#3304811yescharacteristic
662C2#3305311yescharacteristic
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
1. C2×#2099C2semidirect
5. #11373C2semidirect
6. #11267C2semidirect
7. #11207C2semidirect
10. #10881C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
C2×#17492
#146972
#146862
#113732
#112672
#112072
#112002
#108812
#108212