Finite Group Databasebeta

← #5955340

Subgroups — #5955340

Order 512.

5746 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic, complemented
2C2×#2308C211yesmaximal, characteristic
3C2×#2308C211yesmaximal, characteristic
4C22×#386C211yesmaximal, characteristic
5C22×#385C211yesmaximal, characteristic, complemented
6C22×#380C211yesmaximal, characteristic, complemented
7#14935C214yesmaximal, complemented
8#14921C214yesmaximal, complemented
9#14921C214yesmaximal, complemented
10#14729C214yesmaximal, complemented
11#14729C214yesmaximal, complemented
12#14726C214yesmaximal, complemented
13#13794C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2212yescomplemented
17C23×C4⋊C4C2212yescomplemented
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C2×#396C2218yescomplemented
25#2315—28no—
26#2308—28no—
27#2308—28no—
28C2×#386—28no—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385—28no—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yescomplemented
41C2×#385C2212yescomplemented
42C2×#385C2212yescomplemented
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#380—28no—
46C2×#380C2214yes—
47C2×#380C2214yescomplemented
48C2×#380C2214yescomplemented
49C2×#379C2212yes—
50C2×#379C2212yes—
51C22×#95C2212yes—
52C22×#95C2212yescomplemented
53C22×#95C2212yescomplemented
54#1443C2218yescomplemented
55#1441C2218yescomplemented
56#1441C2218yescomplemented
57#1441C2218yescomplemented
58#1439C2218yescomplemented
59#1439C2218yescomplemented
60#1439C2218yescomplemented
61C22⋊#95C2214yes—
62C22⋊#95C2212yescomplemented
63C22⋊#95C2214yescomplemented
64C22⋊#95C2212yescomplemented
65C26C2311yescharacteristic
66C26C2311yescharacteristic
67C23×D4—24no—
68C23×D4—24no—
69C23×D4—24no—
70C23×D4—24no—
71C23×D4—24no—
72C23×D4—24no—
73C23×D4Q818yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C24×C4—24no—
77C24×C4—24no—
78C24×C4—48no—
79C24×C4D418yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—48no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—48no—
114C22×C22⋊C4D418yes—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2318yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131#396—416no—
132#386—216no—
133#386—216no—
134#386—28no—
135#386—216no—
136#386—28no—
137#386—28no—
138#386—216no—
139#386—28no—
140#385—216no—
141#385—216no—
142#385—28no—
143#385—28no—
144#385—216no—
145#381—28no—
146#381—28no—
147#380—216no—
148#380—216no—
149#380—216no—
150#379—28no—
151#379—28no—
152C2×#95—28no—
153C2×#95—28no—
154C2×#95—28no—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C25—22no—
167C25—22no—
168C25—22no—
169C25—22no—
170C25—22no—
171C25—22no—
172C25—44no—
173C25—44no—
174C25—44no—
175C25—44no—
176C25C2×D414yes—
177C25C2×D414yes—
178C25C2×D414yes—
179C25C2×Q814yes—
180C25C2×Q814yes—
181C25#4114yes—
182C25#4114yes—
183C25#4114yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×D4—232no—
189C22×D4—232no—
190C22×D4—232no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—416no—
204C22×D4—416no—
205C22×D4—416no—
206C22×D4—416no—
207C22×D4C2×Q818yes—
208C22×D4C2×Q818yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—28no—
212C23×C4—232no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—28no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—432no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—432no—
258C23×C4—432no—
259C23×C4C2×D414yes—
260C23×C4C2×D418yes—
261C23×C4C2×D418yes—
262C23×C4C2×D418yes—
263C23×C4C2×D418yes—
264C23×C4C2×D414yes—
265C23×C4C2×Q814yes—
266C23×C4C2412yes—
267C23×C4C2414yes—
268C23×C4C2414yes—
269C23×C4C2412yes—
270C23×C4C2414yes—
271C23×C4C2414yes—
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2412yes—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—432no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C22⋊C4—232no—
288C2×C22⋊C4—232no—
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—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—432no—
323C2×C22⋊C4—416no—
324C2×C22⋊C4—416no—
325C2×C22⋊C4—416no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4C2×D418yes—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4C2×D418yes—
334#95—216no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—216no—
340#95—28no—
341#95—216no—
342#95—28no—
343#95—48no—
344#95—48no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—28no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—28no—
359C24—28no—
360C24—28no—
361C24—28no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—416no—
368C24—416no—
369C24—48no—
370C24—416no—
371C24—416no—
372C24—416no—
373C24—48no—
374C24—48no—
375C24—416no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—84no—
381C24—84no—
382C24—84no—
383C24—84no—
384C24—88no—
385C24—88no—
386C24#121116yes—
387C24#121116yes—
388C24#121116yes—
389C24C22⋊Q8116yes—
390C24C22⋊Q8116yes—
391C24C22⋊Q8116yes—
392C24#126116yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C22×Q814yes—
403C24C22×Q812yes—
404C24C22×Q812yes—
405C24C22×Q812yes—
406C24C2×#4112yes—
407C24C2×#4112yes—
408C24C2×#4112yes—
409C24C2×#4112yes—
410C24C2×#4112yes—
411C24C2×#4112yes—
412C24C2×#4112yes—
413C24C2×#4112yes—
414C24C2×#4112yes—
415C24C2511yescharacteristic
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—4128no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—216no—
464C22×C4—216no—
465C22×C4—216no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—416no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—416no—
491C22×C4—416no—
492C22×C4—416no—
493C22×C4—416no—
494C22×C4—4128no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—416no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—44no—
512C22×C4—48no—
513C22×C4—44no—
514C22×C4—88no—
515C22×C4—88no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—84no—
519C22×C4—84no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4—88no—
523C22×C4—88no—
524C22×C4C22×D418yes—
525C22×C4C22×D418yes—
526C22×C4C22×D414yes—
527C22×C4C22×D414yes—
528C22×C4C22×Q814yes—
529C22×C4H(5,𝔽2)18yes—
530C4⋊C4—432no—
531C4⋊C4—416no—
532C4⋊C4—416no—
533C4⋊C4—816no—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—816no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—28no—
578C23—28no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—216no—
583C23—416no—
584C23—416no—
585C23—432no—
586C23—432no—
587C23—432no—
588C23—432no—
589C23—432no—
590C23—432no—
591C23—432no—
592C23—416no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—48no—
604C23—48no—
605C23—48no—
606C23—816no—
607C23—816no—
608C23—88no—
609C23—816no—
610C23—816no—
611C23—88no—
612C23—816no—
613C23#396164yescomplemented
614C23C2×#12114yes—
615C23C2×#12114yes—
616C23C2×#12114yes—
617C23C2×#12114yes—
618C23C2×#12114yes—
619C23C2×#12114yes—
620C23C2×#12114yes—
621C23C2×#12114yes—
622C23C2×#12114yes—
623C23C2×C22⋊Q814yes—
624C23C2×C22⋊Q814yes—
625C23C2×C22⋊Q814yes—
626C23C2×C22⋊Q814yes—
627C23C2×C22⋊Q814yes—
628C23C2×C22⋊Q814yes—
629C23C2×C22⋊Q814yes—
630C23C2×C22⋊Q814yes—
631C23C2×C22⋊Q814yes—
632C23C2×#12614yes—
633C23C2×#12614yes—
634C23C2×#12614yes—
635C23C23×D411yescharacteristic
636C23C23×D411yescharacteristic
637C23C23×D411yescharacteristic
638C23C23×Q811yescharacteristic
639C23C22×#4111yescharacteristic
640C23C22×#4111yescharacteristic
641C23C22×#4111yescharacteristic
642C23C2×H(5,𝔽2)14yes—
643C23#58514yes—
644D4—432no—
645D4—432no—
646C2×C4—232no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—28no—
660C2×C4—28no—
661C2×C4—216no—
662C2×C4—432no—
663C2×C4—432no—
664C2×C4—432no—
665C2×C4—416no—
666C2×C4—416no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—48no—
676C2×C4—48no—
677C2×C4—48no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—48no—
696C2×C4—816no—
697C2×C4—88no—
698C2×C4—816no—
699C2×C4—88no—
700C2×C4—816no—
701C2×C4#53418yes—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—28no—
706C22—28no—
707C22—28no—
708C22—28no—
709C22—28no—
710C22—28no—
711C22—28no—
712C22—28no—
713C22—28no—
714C22—28no—
715C22—28no—
716C22—28no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—28no—
721C22—28no—
722C22—28no—
723C22—28no—
724C22—28no—
725C22—28no—
726C22—464no—
727C22—48no—
728C22—48no—
729C22—48no—
730C22—44no—
731C22—832no—
732C22—82no—
733C22—84no—
734C22—84no—
735C22—82no—
736C22—84no—
737C22C2×#39618yes—
738C22C2×#39618yes—
739C22C2×#39618yes—
740C22C22×#12111yescharacteristic
741C22C22×#12111yescharacteristic
742C22C22×#12111yescharacteristic
743C22C22×C22⋊Q811yescharacteristic
744C22C22×C22⋊Q811yescharacteristic
745C22C22×C22⋊Q811yescharacteristic
746C22C22×#12611yescharacteristic
747C22C2×#53412yes—
748C22#344912yes—
749C22#348412yes—
750C22#348512yes—
751C22#349012yes—
752C22#349112yes—
753C22#349614yes—
754C22#350114yes—
755C22#350514yes—
756C22#350514yes—
757C4—28no—
758C4—48no—
759C4—44no—
760C4—48no—
761C4—44no—
762C4—82no—
763C4—84no—
764C4—82no—
765C4—84no—
766C2—28no—
767C2—28no—
768C2—28no—
769C2—84no—
770C2C22×#39611yescharacteristic
771C2#3498111yescharacteristic
772C2#3500012yes—
773C2#3501012yes—
774C2#3508911yescharacteristic
775C2#3534912yes—
776C2#3535812yes—
777C2#3535912yes—
778C2#3537012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2315C2semidirect
5. C22×#385C2semidirect
6. C22×#380C2semidirect
7. #14935C2semidirect
8. #14921C2semidirect
9. #14921C2semidirect
10. #14729C2semidirect
11. #14729C2semidirect
12. #14726C2semidirect
13. #13794C2semidirect
16. C23×C4⋊C4C22semidirect
17. C23×C4⋊C4C22semidirect
24. C2×#396C22semidirect
40. C2×#385C22semidirect
41. C2×#385C22semidirect
42. C2×#385C22semidirect
47. C2×#380C22semidirect
48. C2×#380C22semidirect
52. C22×#95C22semidirect
53. C22×#95C22semidirect
54. #1443C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
62. C22⋊#95C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
613. C23#396semidirect

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#23082
C22×#3862
C22×#3852
C22×#3802
#149352
#149212
#147292
#147262
#137942