Finite Group Databasebeta

← #407006

Subgroups — #407006

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2C22×#340C211yesmaximal, characteristic, complemented
3#13872C211yesmaximal, characteristic
4#13872C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#386—21no—
20C2×#386—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25#1869—24no—
26C2×#340—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1441—21no—
38#1441—21no—
39#1439—21no—
40#1439—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94#386—44no—
95#386—44no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—24no—
101#380—24no—
102#380—42no—
103#380—42no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336—22no—
122C25C2×D411yescharacteristic
123C25C2×D411yescharacteristic
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4SD1614yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—82no—
158C23×C4—82no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—28no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—44no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
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—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4#4114yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262C22⋊C8—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—42no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24#12112yes—
286C24#12312yes—
287C24#12312yes—
288C24C2×SD1612yes—
289C24C2×SD1612yes—
290C24#13512yes—
291C24AGL(1,ℤ/8)12yes—
292C24#13712yes—
293C24C22×D411yescharacteristic
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—48no—
336C22×C4—48no—
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—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4#12112yes—
390C22×C4#12314yes—
391C22×C4#12312yes—
392C22×C4#12312yes—
393C22×C4#12414yes—
394C22×C4#12614yes—
395C22×C4#12614yes—
396C22×C4C2×SD1612yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—416no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23#39714yes—
497C23#46514yes—
498C23#46814yes—
499C23#48114yes—
500C23#48214yes—
501C23#48314yes—
502C23#48514yes—
503C23C2×#12111yescharacteristic
504C23C2×#12311yescharacteristic
505C23C2×#12311yescharacteristic
506C23#54612yes—
507C23#54612yes—
508C23C22×SD1611yescharacteristic
509C23C2×#13511yescharacteristic
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×#13711yescharacteristic
512C23#57612yes—
513C23#57812yes—
514D4—416no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—816no—
549C2×C4—816no—
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—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#39714yes—
579C2×C4C2×#12311yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#52511yescharacteristic
582C2×C4C2×#12411yescharacteristic
583C2×C4C2×#12611yescharacteristic
584C2×C4C2×#12611yescharacteristic
585C2×C4#52912yes—
586C2×C4#52912yes—
587C2×C4#53411yescharacteristic
588C2×C4#53411yescharacteristic
589C2×C4#53812yes—
590C2×C4#53812yes—
591C2×C4#55112yes—
592C2×C4#55112yes—
593C2×C4#55312yes—
594C2×C4#55312yes—
595C2×C4#55312yes—
596C2×C4#55312yes—
597C8—42no—
598C8—42no—
599C8—81no—
600C8—81no—
601C22—24no—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—24no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—82no—
620C22—81no—
621C22—81no—
622C22#207718yes—
623C22C2×#39711yescharacteristic
624C22#244412yes—
625C22#244412yes—
626C22#253411yescharacteristic
627C22#253411yescharacteristic
628C22C2×#46511yescharacteristic
629C22C2×#46811yescharacteristic
630C22#305811yescharacteristic
631C22#305911yescharacteristic
632C22C2×#48311yescharacteristic
633C22C2×#48211yescharacteristic
634C22C2×#48511yescharacteristic
635C22C2×#48111yescharacteristic
636C22#309812yes—
637C22#309812yes—
638C22#329512yes—
639C22#329712yes—
640C22#329812yes—
641C22#330012yes—
642C22#330812yes—
643C22#331212yes—
644C22#331412yes—
645C22#331812yes—
646C4—24no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—84no—
651C4—84no—
652C4—162no—
653C4—162no—
654C4#251012yes—
655C4#251012yes—
656C4#259111yescharacteristic
657C4#259111yescharacteristic
658C4#262211yescharacteristic
659C4#285311yescharacteristic
660C2—28no—
661C2—82no—
662C2#1494511yescharacteristic
663C2C2×#207711yescharacteristic
664C2#2363911yescharacteristic
665C2#2620012yes—
666C2#2620212yes—
667C2#2620812yes—
668C2#2621012yes—
669C2#2809711yescharacteristic
670C2#2809811yescharacteristic
671C2#2810411yescharacteristic
672C2#2810511yescharacteristic
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. C22×#340C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C22×#3402
#138722
#113162
#113152
#112002
#111992
#107302
#107212