Finite Group Databasebeta

← #469999

Subgroups — #469999

Order 512.

1340 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×#2860C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2071C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic, complemented
5#11327C212yesmaximal, complemented
6#11293C212yesmaximal, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11121C212yesmaximal
9#11066C212yesmaximal
10#11049C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×#126—21no—
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2860—24no—
17#2860—24no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#388—21no—
28C2×#388—21no—
29C2×#388—21no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#2084—24no—
34#2071—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#126—28no—
51C2×#126—44no—
52C2×#126—44no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#106—24no—
80C2×#106—42no—
81C2×#106—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390D414yes—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—42no—
100#389—42no—
101#389—44no—
102#389—44no—
103#389D414yes—
104#388—28no—
105#388—28no—
106#388—28no—
107#384—24no—
108#384—22no—
109#384—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C43—22no—
117C43—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—42no—
127C25—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
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×C8#4114yes—
164#126—432no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—42no—
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—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42SD1614yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259#106—48no—
260#102—48no—
261#102—48no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—44no—
268C24—22no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24C22×D411yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
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—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
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—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
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—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
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#12012yes—
368C22×C4#12012yes—
369C22×C4C2×SD1612yes—
370C22×C4C2×SD1612yes—
371C22×C4C2×SD1612yes—
372C22×C4#13512yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4#13712yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+411yescharacteristic
385C22×C42−1+412yes—
386C22×C42−1+412yes—
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—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—84no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C42—24no—
432C42—24no—
433C42—28no—
434C42—44no—
435C42—44no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23C2×#12011yescharacteristic
479C23#52511yescharacteristic
480C23#53211yescharacteristic
481C23#53511yescharacteristic
482C23#53611yescharacteristic
483C23#53611yescharacteristic
484C23#53711yescharacteristic
485C23C22×SD1611yescharacteristic
486C23C2×#13511yescharacteristic
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23C2×AGL(1,ℤ/8)11yescharacteristic
490C23C2×#13711yescharacteristic
491C23#57511yescharacteristic
492C23#57511yescharacteristic
493D4—44no—
494D4—44no—
495D4—82no—
496D4—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4#44914yes—
573C2×C4#45014yes—
574C2×C4#53511yescharacteristic
575C2×C4#53511yescharacteristic
576C2×C4#53711yescharacteristic
577C2×C4#53711yescharacteristic
578C2×C4#53912yes—
579C2×C4#54112yes—
580C2×C4#54112yes—
581C2×C4#54212yes—
582C2×C4#54712yes—
583C2×C4#54712yes—
584C2×C4#54712yes—
585C2×C4#54812yes—
586C2×C4#55112yes—
587C2×C4#55312yes—
588C2×C4#56312yes—
589C2×C4#56312yes—
590C8—44no—
591C8—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—161no—
600C22—161no—
601C22—161no—
602C22—161no—
603C22—161no—
604C22—161no—
605C22—161no—
606C22—161no—
607C22#240611yescharacteristic
608C22#243211yescharacteristic
609C22#260311yescharacteristic
610C22#261511yescharacteristic
611C22#263711yescharacteristic
612C22#266411yescharacteristic
613C22#282111yescharacteristic
614C22C2×#45011yescharacteristic
615C22C2×#44911yescharacteristic
616C22#300611yescharacteristic
617C22#300711yescharacteristic
618C22#310311yescharacteristic
619C22#310811yescharacteristic
620C22#310911yescharacteristic
621C22#311411yescharacteristic
622C22#316711yescharacteristic
623C22#317111yescharacteristic
624C22#318111yescharacteristic
625C22#318411yescharacteristic
626C22#320611yescharacteristic
627C22#321211yescharacteristic
628C22#321411yescharacteristic
629C22#321711yescharacteristic
630C22#322711yescharacteristic
631C22#322711yescharacteristic
632C22#322811yescharacteristic
633C22#323211yescharacteristic
634C22#323411yescharacteristic
635C22#323811yescharacteristic
636C22#323911yescharacteristic
637C22#324011yescharacteristic
638C22#324611yescharacteristic
639C22#324711yescharacteristic
640C22#325111yescharacteristic
641C22#325411yescharacteristic
642C4—24no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—162no—
653C4#271812yes—
654C4#272212yes—
655C4#272312yes—
656C4#272512yes—
657C2—162no—
658C2#1667311yescharacteristic
659C2#2406111yescharacteristic
660C2#2406611yescharacteristic
661C2#2567111yescharacteristic
662C2#2567811yescharacteristic
663C2#2949411yescharacteristic
664C2#2949911yescharacteristic
665C2#2966211yescharacteristic
666C2#2966311yescharacteristic
667C2#3014911yescharacteristic
668C2#3015011yescharacteristic
669C2#3037911yescharacteristic
670C2#3038411yescharacteristic
671C2#3283511yescharacteristic
672C2#3283611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
4. #14712C2semidirect
5. #11327C2semidirect
6. #11293C2semidirect
7. #11207C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
C2×#20842
C2×#20712
#147122
#113272
#112932
#112072
#111212
#110662
#110492
#107952