Finite Group Databasebeta

← #876773

Subgroups — #876773

Order 512.

2744 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C212yesmaximal, complemented
2C2×#2441C212yesmaximal, complemented
3#16323C214yesmaximal
4#15980C212yesmaximal, complemented
5#15968C212yesmaximal, complemented
6#15657C212yesmaximal, complemented
7#15649C212yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#14549C214yesmaximal, complemented
12#14006C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2851—216no—
21#2441—216no—
22C2×#399C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#386C2212yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#384C2212yes—
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1437C2211yescharacteristic
64#1437C2212yes—
65#1437C2211yescharacteristic
66#1437C2212yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2211yescharacteristic
71#1436C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—44no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C2×C4×D4Q818yes—
87C2×#117—28no—
88C2×#117—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C42—28no—
113C22×C42—42no—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#399—216no—
122#396—216no—
123#396—216no—
124#389—216no—
125#389—28no—
126#389—28no—
127#389—44no—
128#389D4116yescomplemented
129#387—216no—
130#387—216no—
131#387—216no—
132#386—28no—
133#385—216no—
134#385—216no—
135#384—28no—
136#382—28no—
137#382—42no—
138#380—28no—
139#380—28no—
140#380—28no—
141#380—42no—
142C4×C4⋊C4—42no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—42no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4C2×Q814yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C2×Q814yes—
193C23×C4C2×Q814yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#125—432no—
208C4×D4—416no—
209#117—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×D418yes—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D4116yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4C2×Q814yes—
275C2×C22⋊C4C2×Q814yes—
276C2×C22⋊C4#4118yes—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—416no—
288C2×C42—42no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D418yes—
294C2×C42C2×Q814yes—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—41no—
313#95—41no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95C2×D4116yes—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24C22×Q812yes—
336C24C22×Q812yes—
337C24C2×#4114yes—
338C24C2511yescharacteristic
339C2×D4—28no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4C22×D418yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D418yes—
431C22×C4C22×D414yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—416no—
445C4⋊C4—44no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C42—28no—
466C42—216no—
467C42—44no—
468C42—44no—
469C42—48no—
470C42—48no—
471C42—84no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23#53314yes—
514C23C23×D412yes—
515C23C23×Q811yescharacteristic
516C23C22×#4112yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+412yes—
520C23#58512yes—
521C23#58512yes—
522C23#58512yes—
523D4—416no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4#53718yes—
604C2×C4#54314yes—
605C2×C4D4214yes—
606C2×C4D4×Q818yes—
607C2×C4#56314yes—
608C22—28no—
609C22—24no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#53311yescharacteristic
627C22C2×#53712yes—
628C22C2×#54311yescharacteristic
629C22#345811yescharacteristic
630C22C2×D4211yescharacteristic
631C22C2×D4×Q812yes—
632C22D4×#4112yes—
633C22D4×#4112yes—
634C22#348512yes—
635C22#348712yes—
636C22#350412yes—
637C22#350512yes—
638C22#350612yes—
639C22#351512yes—
640C22#351512yes—
641C22C2×#56311yescharacteristic
642C22#353912yes—
643C22#355111yescharacteristic
644C22#355111yescharacteristic
645C22#356512yes—
646C22#356512yes—
647C4—42no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—48no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—84no—
658C4—84no—
659C4#268318yes—
660C2—44no—
661C2—82no—
662C2#3503311yescharacteristic
663C2#3613112yes—
664C2#3742511yescharacteristic
665C2#3769112yes—
666C2#3985612yes—
667C2#3988312yes—
668C2#4570912yes—
669C2C2×#268311yescharacteristic
670C2#5899912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2851C2semidirect
2. C2×#2441C2semidirect
4. #15980C2semidirect
5. #15968C2semidirect
6. #15657C2semidirect
7. #15649C2semidirect
8. #15228C2semidirect
9. #15220C2semidirect
10. #15131C2semidirect
11. #14549C2semidirect
12. #14006C2semidirect
128. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#24412
#163232
#159802
#159682
#156572
#156492
#152282
#152202
#151312
#145492
#140062
#139052
#138712
C4×#3862