Finite Group Databasebeta

← #878970

Subgroups — #878970

Order 512.

3122 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2297C211yesmaximal, characteristic
3#17071C212yesmaximal, complemented
4#16599C212yesmaximal, complemented
5#15505C214yesmaximal, complemented
6#15355C214yesmaximal, complemented
7#14893C212yesmaximal, complemented
8#14771C214yesmaximal, complemented
9#14697C212yesmaximal, complemented
10#14467C211yesmaximal, characteristic, complemented
11#14467C212yesmaximal, complemented
12#14210C214yesmaximal, complemented
13#13937C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#2809—28no—
18C2×#400C2212yes—
19C2×#399C2212yes—
20C2×#399C2214yes—
21C2×#398C2214yes—
22C2×#396C2214yes—
23C2×#396C2214yes—
24#2297—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41Q818yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×#117—24no—
83C2×#117—42no—
84C2×#117—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42—24no—
109C22×C42—42no—
110C22×C42—42no—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#400—28no—
118#399—48no—
119#399—44no—
120#398—216no—
121#396—216no—
122#396—48no—
123#391—24no—
124#391—28no—
125#391—28no—
126#391D418yes—
127#390—24no—
128#390—24no—
129#390—44no—
130#390—44no—
131#389—28no—
132#389—44no—
133#389—44no—
134#387—216no—
135#386—216no—
136#385—48no—
137#380—28no—
138#380—44no—
139#380—44no—
140C4×C22⋊C4—216no—
141C4×C22⋊C4—44no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C2×#41—232no—
163C2×#41—416no—
164C2×#41—416no—
165C2×#41—416no—
166C2×#41—416no—
167C2×#41—416no—
168C2×#41—416no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8C2×D414yes—
174C22×Q8C2×Q814yes—
175C22×Q8#4114yes—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4C2×Q818yes—
189C22×D4C2×Q814yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×Q814yes—
210C23×C4C2×Q818yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222#117—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4#4118yescomplemented
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—41no—
300C2×C42—41no—
301C2×C42—42no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—416no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D414yes—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95C2×D4116yes—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24C22×Q814yes—
347C24C22×Q812yes—
348C24H(5,𝔽2)12yes—
349C24H(5,𝔽2)12yes—
350C24C2511yescharacteristic
351#41—4128no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8C22⋊Q8116yes—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—28no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+414yes—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—44no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—84no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23#54314yes—
542C23C23×D411yescharacteristic
543C23C23×Q811yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553Q8—416no—
554D4—432no—
555D4—416no—
556C2×C4—232no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—432no—
590C2×C4—432no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4C2×C22⋊Q814yes—
629C2×C4C2×C22⋊Q818yes—
630C2×C4#54314yes—
631C2×C4#55018yes—
632C22—28no—
633C22—24no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—48no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—44no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C22×C22⋊Q811yescharacteristic
655C22C2×#54311yescharacteristic
656C22C2×#54311yescharacteristic
657C22#345912yes—
658C22C2×#55012yes—
659C22Q8×#4112yes—
660C22#348212yes—
661C22#348311yescharacteristic
662C22#348411yescharacteristic
663C22#348411yescharacteristic
664C22#348912yes—
665C22#349011yescharacteristic
666C22#349011yescharacteristic
667C22#349212yes—
668C22#350312yes—
669C22#351712yes—
670C22#351912yes—
671C22#352911yescharacteristic
672C22#353412yes—
673C22#353612yes—
674C22#354911yescharacteristic
675C22#355011yescharacteristic
676C22#355912yes—
677C4—28no—
678C4—24no—
679C4—24no—
680C4—28no—
681C4—48no—
682C4—44no—
683C4—44no—
684C4—84no—
685C4—82no—
686C4—84no—
687C4—84no—
688C2—44no—
689C2—84no—
690C2#3498211yescharacteristic
691C2#3644512yes—
692C2#3730611yescharacteristic
693C2#3776812yes—
694C2#3844512yes—
695C2#3878012yes—
696C2#4622811yescharacteristic
697C2#4631811yescharacteristic
698C2#5229111yescharacteristic
699C2#5905611yescharacteristic
700C2#6004711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2809C2semidirect
3. #17071C2semidirect
4. #16599C2semidirect
5. #15505C2semidirect
6. #15355C2semidirect
7. #14893C2semidirect
8. #14771C2semidirect
9. #14697C2semidirect
10. #14467C2semidirect
11. #14467C2semidirect
12. #14210C2semidirect
263. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#22972
#170712
#165992
#155052
#153552
#148932
#147712
#146972
#144672
#142102
#139372