Finite Group Databasebeta

← #920676

Subgroups — #920676

Order 512.

4204 conjugacy classes of subgroups.

The subgroup lattice has 812 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C211yesmaximal, characteristic
2C22×#393C211yesmaximal, characteristic, complemented
3#17198C212yesmaximal, complemented
4#17105C212yesmaximal, complemented
5#16823C212yesmaximal, complemented
6#16368C214yesmaximal, complemented
7#16357C212yesmaximal, complemented
8#15506C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14873C212yesmaximal, complemented
11#14718C212yesmaximal, complemented
12#14259C212yesmaximal, complemented
13#14134C212yesmaximal, complemented
14#13956C212yesmaximal, complemented
15#13869C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C22×#120C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2404—28no—
23C2×#394C2214yes—
24C2×#393—28no—
25C2×#393C2211yescharacteristic
26C2×#393C2212yes—
27C2×#393C2212yes—
28C2×#393C2211yescharacteristic
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#393C2214yescomplemented
32C2×#393C2214yescomplemented
33C2×#392C2211yescharacteristic
34C2×#392C2211yescharacteristic
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#385C2212yes—
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#379C2211yescharacteristic
47C22×#95C2212yes—
48C22×#95C2211yescharacteristic
49#1446C2214yescomplemented
50#1446C2214yescomplemented
51#1446C2214yescomplemented
52#1445C2214yescomplemented
53#1443C2214yescomplemented
54#1443C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1440C2212yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yescomplemented
63#1439C2214yes—
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yescomplemented
76C22⋊#95C2214yescomplemented
77C26C2311yescharacteristic
78C22×#41—28no—
79C22×#41—24no—
80C22×#41—42no—
81C22×#41—42no—
82C22×#41—44no—
83C23×Q8—24no—
84C23×Q8—42no—
85C23×Q8—42no—
86C23×Q8—48no—
87C23×Q8C2312yes—
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90C23×D4C2312yes—
91C23×D4C2312yes—
92C23×D4C2311yescharacteristic
93C23×D4C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2312yes—
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C24×C4C2312yes—
100C2×#120—28no—
101C2×#120—28no—
102C2×#120—44no—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—42no—
108C22×C22⋊C4—42no—
109C22×C22⋊C4—42no—
110C22×C22⋊C4D418yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131#394—48no—
132#393—28no—
133#393—28no—
134#393—28no—
135#393—28no—
136#393—24no—
137#393—24no—
138#393—48no—
139#393—48no—
140#392—24no—
141#392—24no—
142#386—28no—
143#386—28no—
144#386—28no—
145#386—28no—
146#386—48no—
147#386—44no—
148#386—44no—
149#386—44no—
150#385—44no—
151#381—24no—
152#381—44no—
153#379—42no—
154C2×#95—28no—
155C2×#95—24no—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C25—24no—
181C25—24no—
182C25—22no—
183C25—24no—
184C25—22no—
185C25—44no—
186C25—42no—
187C25C2×D414yes—
188C25C2411yescharacteristic
189C25C2412yes—
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C2×#41—432no—
193C2×#41—416no—
194C2×#41—416no—
195C2×#41—432no—
196C2×#41—416no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—28no—
202C22×Q8—28no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—42no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8—432no—
213C22×Q8—416no—
214C22×Q8—432no—
215C22×Q8—432no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—28no—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4#4118yes—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—28no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—42no—
258C23×C4—42no—
259C23×C4—42no—
260C23×C4—42no—
261C23×C4—44no—
262C23×C4—42no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—42no—
266C23×C4—44no—
267C23×C4C2×D414yes—
268C23×C4C2×D414yes—
269C23×C4C2414yes—
270C23×C4C2414yes—
271C23×C4C2414yes—
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2412yes—
279C23×C4C2414yes—
280#120—432no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—41no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—432no—
323C2×C22⋊C4—416no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—28no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—28no—
395C24—216no—
396C24—48no—
397C24—44no—
398C24—416no—
399C24—48no—
400C24—48no—
401C24—416no—
402C24—48no—
403C24—44no—
404C24—44no—
405C24—88no—
406C24—84no—
407C24—84no—
408C24—88no—
409C24C22×D412yes—
410C24C22×D412yes—
411C24C22×D412yes—
412C24C2×#4114yes—
413C24C2×#4114yes—
414C24H(5,𝔽2)12yes—
415C24H(5,𝔽2)12yes—
416C24H(5,𝔽2)14yes—
417C242−1+414yes—
418C24C2511yescharacteristic
419#41—4128no—
420C2×Q8—416no—
421C2×Q8—416no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—44no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—416no—
429C2×Q8—416no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—4128no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×D4—216no—
437C2×D4—216no—
438C2×D4—216no—
439C2×D4—48no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—48no—
443C2×D4—44no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—48no—
447C2×D4—48no—
448C2×D4—48no—
449C2×D4—48no—
450C2×D4—416no—
451C2×D4—48no—
452C2×D4—48no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4—88no—
456C2×D4—88no—
457C2×D4—88no—
458C22×C4—216no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—216no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—48no—
520C22×C4—48no—
521C22×C4—416no—
522C22×C4—416no—
523C22×C4—48no—
524C22×C4—48no—
525C22×C4—48no—
526C22×C4—48no—
527C22×C4—48no—
528C22×C4—48no—
529C22×C4—48no—
530C22×C4—48no—
531C22×C4—48no—
532C22×C4—48no—
533C22×C4—48no—
534C22×C4—44no—
535C22×C4—44no—
536C22×C4—48no—
537C22×C4—48no—
538C22×C4—48no—
539C22×C4—48no—
540C22×C4—88no—
541C22×C4—84no—
542C22×C4—88no—
543C22×C4—88no—
544C22×C4—84no—
545C22×C4—88no—
546C22×C4C22×D412yes—
547C22×C4C22×D412yes—
548C22×C4C22×D412yes—
549C22×C4C22×D412yes—
550C22×C4C2×#4114yes—
551C22×C4H(5,𝔽2)18yes—
552C4⋊C4—88no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—416no—
558C22⋊C4—416no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4—48no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C22⋊C4—88no—
568C22⋊C4—88no—
569C22⋊C4—88no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4—88no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—28no—
577C23—28no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—28no—
583C23—28no—
584C23—22no—
585C23—22no—
586C23—24no—
587C23—24no—
588C23—28no—
589C23—24no—
590C23—24no—
591C23—28no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—24no—
596C23—24no—
597C23—24no—
598C23—24no—
599C23—24no—
600C23—28no—
601C23—28no—
602C23—416no—
603C23—432no—
604C23—42no—
605C23—41no—
606C23—44no—
607C23—41no—
608C23—42no—
609C23—42no—
610C23—44no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—416no—
617C23—42no—
618C23—41no—
619C23—42no—
620C23—42no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—41no—
626C23—42no—
627C23—44no—
628C23—44no—
629C23—42no—
630C23—42no—
631C23—44no—
632C23—44no—
633C23—42no—
634C23—42no—
635C23—44no—
636C23—44no—
637C23—44no—
638C23—44no—
639C23—44no—
640C23—44no—
641C23—88no—
642C23—816no—
643C23—816no—
644C23—832no—
645C23—816no—
646C23#53414yes—
647C23#53614yes—
648C23#53818yes—
649C23#54618yes—
650C23#55114yes—
651C23H(3,𝔽4)14yes—
652C23C23×D411yescharacteristic
653C23C22×#4112yes—
654C23C2×H(5,𝔽2)11yescharacteristic
655C23C2×H(5,𝔽2)11yescharacteristic
656C23C2×H(5,𝔽2)12yes—
657C23C2×H(5,𝔽2)14yes—
658C23C2×2−1+412yes—
659C23#58512yes—
660Q8—432no—
661Q8—432no—
662Q8—416no—
663D4—416no—
664D4—432no—
665D4—816no—
666D4—816no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—24no—
672C2×C4—24no—
673C2×C4—24no—
674C2×C4—24no—
675C2×C4—24no—
676C2×C4—28no—
677C2×C4—24no—
678C2×C4—24no—
679C2×C4—28no—
680C2×C4—28no—
681C2×C4—432no—
682C2×C4—416no—
683C2×C4—432no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—416no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—44no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—44no—
705C2×C4—42no—
706C2×C4—44no—
707C2×C4—42no—
708C2×C4—42no—
709C2×C4—44no—
710C2×C4—44no—
711C2×C4—42no—
712C2×C4—42no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—44no—
716C2×C4—44no—
717C2×C4—44no—
718C2×C4—44no—
719C2×C4—44no—
720C2×C4—44no—
721C2×C4—44no—
722C2×C4—44no—
723C2×C4—44no—
724C2×C4—44no—
725C2×C4—44no—
726C2×C4—44no—
727C2×C4—44no—
728C2×C4—44no—
729C2×C4—44no—
730C2×C4—44no—
731C2×C4—44no—
732C2×C4—44no—
733C2×C4—44no—
734C2×C4—816no—
735C2×C4—816no—
736C22—24no—
737C22—24no—
738C22—28no—
739C22—28no—
740C22—216no—
741C22—28no—
742C22—28no—
743C22—24no—
744C22—28no—
745C22—42no—
746C22—44no—
747C22—42no—
748C22—48no—
749C22—44no—
750C22—44no—
751C22—42no—
752C22—42no—
753C22—48no—
754C22—44no—
755C22—44no—
756C22—44no—
757C22—42no—
758C22—48no—
759C22—44no—
760C22—44no—
761C22—832no—
762C22—82no—
763C22—82no—
764C22—82no—
765C22—82no—
766C22—84no—
767C22—84no—
768C22—82no—
769C22—84no—
770C22—82no—
771C22#268418yes—
772C22C2×#53411yescharacteristic
773C22C2×#53611yescharacteristic
774C22#344911yescharacteristic
775C22C2×#53812yes—
776C22C2×#54612yes—
777C22C2×#55111yescharacteristic
778C22#348412yes—
779C22#351414yes—
780C22#352014yes—
781C22C2×H(3,𝔽4)11yescharacteristic
782C22#353212yes—
783C22#353212yes—
784C22#353512yes—
785C22#353512yes—
786C22#353614yes—
787C22#355814yes—
788C4—42no—
789C4—42no—
790C4—48no—
791C4—42no—
792C4—42no—
793C4—44no—
794C4—82no—
795C4—84no—
796C4—84no—
797C4—82no—
798C4—84no—
799C2—28no—
800C2—44no—
801C2—84no—
802C2—82no—
803C2#3670211yescharacteristic
804C2#3744711yescharacteristic
805C2#3802112yes—
806C2#3802812yes—
807C2#4084214yes—
808C2#4364312yes—
809C2C2×#268411yescharacteristic
810C2#5239512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#393C2semidirect
3. #17198C2semidirect
4. #17105C2semidirect
5. #16823C2semidirect
6. #16368C2semidirect
7. #16357C2semidirect
8. #15506C2semidirect
9. #14919C2semidirect
10. #14873C2semidirect
11. #14718C2semidirect
12. #14259C2semidirect
13. #14134C2semidirect
14. #13956C2semidirect
15. #13869C2semidirect
31. C2×#393C22semidirect
32. C2×#393C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1446C22semidirect
52. #1445C22semidirect
53. #1443C22semidirect
54. #1443C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
62. #1440C22semidirect
64. #1439C22semidirect
67. #1438C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
C22×#3932
#171982
#171052
#168232
#163682
#163572
#155062
#149192
#148732
#147182
#142592
#141342
#139562
#138692