Finite Group Databasebeta

← #656408

Subgroups — #656408

Order 512.

12196 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#551C211yesmaximal, characteristic, complemented
2C2×#2846C211yesmaximal, characteristic
3C22×#379C212yesmaximal, complemented
4#15986C214yesmaximal, complemented
5#15948C212yesmaximal, complemented
6#15259C212yesmaximal, complemented
7#14911C214yesmaximal, complemented
8#14717C214yesmaximal, complemented
9#14134C211yesmaximal, characteristic, complemented
10#13967C214yesmaximal, complemented
11#13869C212yesmaximal, complemented
12#13868C214yesmaximal, complemented
13C27C2211yescharacteristic
14C2×#551—28no—
15C2×#551—216no—
16C22×#126C2212yescomplemented
17C22×#124C2211yescharacteristic, complemented
18C22×#123C2211yescharacteristic, complemented
19C22×#123C2212yescomplemented
20C22×C22⋊Q8C2212yescomplemented
21C22×#120C2214yes—
22C22×#120C2211yescharacteristic
23C22×#120C2212yes—
24C22×C4×D4C2212yescomplemented
25C22×#117C2212yescomplemented
26C23×C22⋊C4C2214yes—
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2214yes—
30#2846—28no—
31C2×#393C2214yescomplemented
32C2×#386C2214yescomplemented
33C2×#386C2214yescomplemented
34C2×#381C2214yescomplemented
35C2×#379—216no—
36C2×#379C2214yes—
37C2×#379C2211yescharacteristic
38C2×#379C2212yes—
39C2×#379C2214yes—
40C2×#379C2214yes—
41C2×#379C2214yes—
42C2×#379C2214yes—
43#1440C2214yescomplemented
44#1440C2214yescomplemented
45#1440C2212yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1439C2218yescomplemented
49#1439C2218yescomplemented
50#1438C2212yescomplemented
51#1438C2214yescomplemented
52#1438C2218yescomplemented
53#1437C2214yescomplemented
54#1437C2218yescomplemented
55C22⋊#95C2214yescomplemented
56C22⋊#95C2214yescomplemented
57C22⋊#95C2214yescomplemented
58C22⋊#95C2214yescomplemented
59C26—28no—
60C26—216no—
61C26—24no—
62C26—48no—
63C26—48no—
64C26C2314yes—
65C26C2312yes—
66C26C2311yescharacteristic
67C23×Q8C2311yescharacteristic, complemented
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2314yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic, complemented
75#551—2128no—
76C2×#126—28no—
77C2×#126—28no—
78C2×#126—28no—
79C2×#124—24no—
80C2×#124—28no—
81C2×#123—28no—
82C2×#123—24no—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×#120—28no—
90C2×#120—216no—
91C2×#120—216no—
92C2×#120—28no—
93C2×#120—48no—
94C2×#120—48no—
95C2×#120D418yes—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—28no—
102C22×C4⋊C4C2312yescomplemented
103C22×C4⋊C4C2312yescomplemented
104C22×C4⋊C4C2312yescomplemented
105C22×C4⋊C4C2312yescomplemented
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—216no—
110C22×C22⋊C4—216no—
111C22×C22⋊C4—216no—
112C22×C22⋊C4—48no—
113C22×C22⋊C4—416no—
114C22×C22⋊C4—48no—
115C22×C22⋊C4D4116yescomplemented
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yescomplemented
133C22×C22⋊C4C2311yescharacteristic, complemented
134C22×C22⋊C4C2314yescomplemented
135C22×C42C2311yescharacteristic, complemented
136C22×C42C2312yescomplemented
137#393—48no—
138#386—48no—
139#386—48no—
140#381—48no—
141#379—24no—
142#379—216no—
143#379—28no—
144#379—216no—
145#379—216no—
146#379—216no—
147#379—216no—
148C2×#95C2314yescomplemented
149C2×#95C2318yescomplemented
150C2×#95C2318yescomplemented
151C2×#95C2318yescomplemented
152C2×#95C2318yescomplemented
153C2×#95C2318yescomplemented
154C25—28no—
155C25—216no—
156C25—28no—
157C25—216no—
158C25—28no—
159C25—28no—
160C25—264no—
161C25—24no—
162C25—28no—
163C25—28no—
164C25—28no—
165C25—216no—
166C25—28no—
167C25—28no—
168C25—28no—
169C25—44no—
170C25—48no—
171C25—48no—
172C25—48no—
173C25—44no—
174C25—464no—
175C25—464no—
176C25—464no—
177C25—464no—
178C25—4128no—
179C25—432no—
180C25—432no—
181C25—464no—
182C25—416no—
183C25C2×D418yes—
184C25C2×D414yes—
185C25#41116yes—
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C25C2414yes—
189C25C2412yes—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—216no—
200C22×D4—28no—
201C22×D4—44no—
202C22×D4—48no—
203C22×D4—44no—
204C22×D4C2×D418yes—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—48no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—44no—
221C23×C4C2×D418yes—
222C23×C4C2×D418yes—
223C23×C4C2418yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231#126—264no—
232#124—232no—
233#123—264no—
234#123—232no—
235C22⋊Q8—264no—
236#120—232no—
237#120—464no—
238#120—432no—
239C4×D4—264no—
240#117—264no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C22⋊C4—264no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—464no—
299C2×C22⋊C4—464no—
300C2×C22⋊C4—432no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4C2×D418yes—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×D418yes—
308C2×C22⋊C4C2×D414yes—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314#95—48no—
315#95—48no—
316#95—48no—
317#95—48no—
318#95—48no—
319#95—44no—
320C24—24no—
321C24—24no—
322C24—28no—
323C24—232no—
324C24—232no—
325C24—264no—
326C24—216no—
327C24—216no—
328C24—232no—
329C24—232no—
330C24—232no—
331C24—264no—
332C24—28no—
333C24—24no—
334C24—24no—
335C24—28no—
336C24—24no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—24no—
341C24—24no—
342C24—216no—
343C24—232no—
344C24—232no—
345C24—232no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—416no—
351C24—48no—
352C24—432no—
353C24—432no—
354C24—432no—
355C24—416no—
356C24—432no—
357C24—432no—
358C24—432no—
359C24—432no—
360C24—432no—
361C24—432no—
362C24—416no—
363C24—432no—
364C24—432no—
365C24—432no—
366C24—416no—
367C24—432no—
368C24—432no—
369C24—416no—
370C24—416no—
371C24—432no—
372C24—4512no—
373C24—4256no—
374C24—416no—
375C24—48no—
376C24—432no—
377C24—416no—
378C24—4128no—
379C24—4512no—
380C24—4512no—
381C24—416no—
382C24—48no—
383C24—416no—
384C24—416no—
385C24—416no—
386C24—416no—
387C24—416no—
388C24#120116yes—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C22×D414yes—
392C24C22×D414yes—
393C24C22×D414yes—
394C24C22×D414yes—
395C24C2×#41116yes—
396C24C2×#4118yes—
397C24H(5,𝔽2)18yes—
398C24H(5,𝔽2)18yes—
399C24C2511yescharacteristic
400C2×Q8—28no—
401C2×Q8—88no—
402C2×Q8—84no—
403C2×D4—216no—
404C2×D4—232no—
405C2×D4—28no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—432no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—88no—
414C2×D4—816no—
415C2×D4—88no—
416C22×C4—216no—
417C22×C4—232no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—816no—
464C22×C4—816no—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—28no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—28no—
483C22⋊C4—432no—
484C22⋊C4—432no—
485C22⋊C4—432no—
486C22⋊C4—432no—
487C22⋊C4—432no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—88no—
494C42—28no—
495C42—216no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—216no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—216no—
505C23—216no—
506C23—216no—
507C23—2128no—
508C23—264no—
509C23—24no—
510C23—216no—
511C23—24no—
512C23—216no—
513C23—216no—
514C23—28no—
515C23—28no—
516C23—216no—
517C23—216no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—216no—
522C23—216no—
523C23—216no—
524C23—28no—
525C23—22no—
526C23—28no—
527C23—24no—
528C23—416no—
529C23—464no—
530C23—4128no—
531C23—464no—
532C23—4128no—
533C23—464no—
534C23—464no—
535C23—464no—
536C23—4128no—
537C23—464no—
538C23—464no—
539C23—464no—
540C23—4128no—
541C23—432no—
542C23—4128no—
543C23—464no—
544C23—464no—
545C23—41024no—
546C23—464no—
547C23—4128no—
548C23—464no—
549C23—44no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—44no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—48no—
568C23—44no—
569C23C2×#12014yes—
570C23C2×#12018yes—
571C23#52518yes—
572C23#53518yes—
573C23#546116yes—
574C23#546116yes—
575C23#551116yescomplemented
576C23C23×D412yes—
577C23C23×D411yescharacteristic
578C23C22×#4114yes—
579C23C2×H(5,𝔽2)14yes—
580C23C2×H(5,𝔽2)14yes—
581Q8—88no—
582D4—432no—
583D4—88no—
584D4—832no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—432no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—816no—
625C2×C4—88no—
626C2×C4—832no—
627C22—28no—
628C22—216no—
629C22—216no—
630C22—232no—
631C22—28no—
632C22—216no—
633C22—216no—
634C22—24no—
635C22—232no—
636C22—216no—
637C22—28no—
638C22—464no—
639C22—4256no—
640C22—4128no—
641C22—48no—
642C22—48no—
643C22—416no—
644C22—416no—
645C22—48no—
646C22—48no—
647C22—416no—
648C22—48no—
649C22—416no—
650C22—416no—
651C22—42no—
652C22—48no—
653C22—44no—
654C22—84no—
655C22—84no—
656C22#2404116yes—
657C22C22×#12011yescharacteristic
658C22C2×#52512yes—
659C22C2×#53512yes—
660C22C2×#54614yes—
661C22C2×#54614yes—
662C22C2×#55114yes—
663C22#348414yes—
664C22#348412yes—
665C22#349018yes—
666C22#353214yes—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—84no—
674C4—88no—
675C2—216no—
676C2—44no—
677C2—416no—
678C2—82no—
679C2C2×#240412yes—
680C2#3534611yescharacteristic
681C2#3615814yes—
682C2#3783314yes—
683C2#5900614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2846C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#551C2semidirect
3. C22×#379C2semidirect
4. #15986C2semidirect
5. #15948C2semidirect
6. #15259C2semidirect
7. #14911C2semidirect
8. #14717C2semidirect
9. #14134C2semidirect
10. #13967C2semidirect
11. #13869C2semidirect
12. #13868C2semidirect
16. C22×#126C22semidirect
17. C22×#124C22semidirect
18. C22×#123C22semidirect
19. C22×#123C22semidirect
20. C22×C22⋊Q8C22semidirect
24. C22×C4×D4C22semidirect
25. C22×#117C22semidirect
31. C2×#393C22semidirect
32. C2×#386C22semidirect
33. C2×#386C22semidirect
34. C2×#381C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
45. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1438C22semidirect
51. #1438C22semidirect
52. #1438C22semidirect
53. #1437C22semidirect
54. #1437C22semidirect
55. C22⋊#95C22semidirect
56. C22⋊#95C22semidirect
57. C22⋊#95C22semidirect
58. C22⋊#95C22semidirect
67. C23×Q8C23semidirect
74. C24×C4C23semidirect
102. C22×C4⋊C4C23semidirect
103. C22×C4⋊C4C23semidirect
104. C22×C4⋊C4C23semidirect
105. C22×C4⋊C4C23semidirect
115. C22×C22⋊C4D4semidirect
132. C22×C22⋊C4C23semidirect
133. C22×C22⋊C4C23semidirect
134. C22×C22⋊C4C23semidirect
135. C22×C42C23semidirect
136. C22×C42C23semidirect
148. C2×#95C23semidirect
149. C2×#95C23semidirect
150. C2×#95C23semidirect
151. C2×#95C23semidirect
152. C2×#95C23semidirect
153. C2×#95C23semidirect
575. C23#551semidirect

Maximal subgroups

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

subgroupindex
C22×#5512
C2×#28462
C22×#3792
#159862
#159482
#152592
#149112
#147172
#141342
#139672
#138692
#138682