Finite Group Databasebeta

← #617535

Subgroups — #617535

Order 512.

5478 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C2×#2561C211yesmaximal, characteristic, complemented
3C2×#2503C211yesmaximal, characteristic, complemented
4C2×#2626C211yesmaximal, characteristic, complemented
5#15669C214yesmaximal, complemented
6#15668C214yesmaximal, complemented
7#15667C218yesmaximal, complemented
8#15073C212yesmaximal, complemented
9#14918C212yesmaximal, complemented
10#14681C212yesmaximal, complemented
11#14335C212yesmaximal
12#13904C211yesmaximal, characteristic, complemented
13#13903C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#123—28no—
18C22×#123—28no—
19C22×#123—28no—
20C22×#123C2214yes—
21C22×#123C2214yes—
22C22×C22⋊Q8C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28#2626—28no—
29#2561—28no—
30#2503—28no—
31C2×#400C2214yescomplemented
32C2×#399C2212yescomplemented
33C2×#397C2212yescomplemented
34C2×#396C2214yescomplemented
35C2×#393C2212yescomplemented
36C2×#393C2212yescomplemented
37C2×#391C2212yescomplemented
38C2×#388C2214yescomplemented
39C2×#385C2212yes—
40C2×#382C2212yescomplemented
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2214yescomplemented
55#1442C2218yescomplemented
56#1441C2218yescomplemented
57#1440C2218yescomplemented
58#1439C2214yescomplemented
59#1439C2218yescomplemented
60#1438C2218yes—
61#1438C2218yes—
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2218yescomplemented
68C22⋊#95C2218yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—28no—
75C23×D4—24no—
76C23×D4—44no—
77C23×D4C2314yes—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—28no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—2128no—
90C2×#123—216no—
91C2×#123—216no—
92C2×#123—216no—
93C2×#123—216no—
94C2×#123—216no—
95C2×#123—216no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4D418yes—
120C22×C22⋊C4D418yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2318yes—
135C22×C22⋊C4C2318yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—216no—
139#399—28no—
140#397—28no—
141#396—216no—
142#393—28no—
143#393—28no—
144#391—28no—
145#388—216no—
146#385—28no—
147#382—28no—
148#381—28no—
149#381—28no—
150#381—28no—
151#381—44no—
152#381—44no—
153#381—44no—
154#380—24no—
155#379—28no—
156#379—28no—
157#379—28no—
158#379—44no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—22no—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—22no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25—44no—
186C25C2×D414yes—
187C25C2×D414yes—
188C25#4118yes—
189C25C2412yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—44no—
196C22×D4—232no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—216no—
200C22×D4—28no—
201C22×D4—48no—
202C22×D4—432no—
203C22×D4—432no—
204C22×D4—416no—
205C22×D4—416no—
206C23×C4—232no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—216no—
225C23×C4—48no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—432no—
229C23×C4—432no—
230C23×C4—416no—
231C23×C4—416no—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250#123—2128no—
251#123—2128no—
252C22⋊Q8—416no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—232no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—232no—
277C2×C22⋊C4—232no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
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—28no—
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—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D418yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42—44no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—48no—
333#95—48no—
334#95—48no—
335#95—48no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—28no—
340C24—216no—
341C24—216no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—28no—
346C24—28no—
347C24—28no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—28no—
355C24—216no—
356C24—48no—
357C24—48no—
358C24—432no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—432no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—44no—
371C24—44no—
372C24—48no—
373C24—816no—
374C24—816no—
375C24#124116yes—
376C24C22×D414yes—
377C24C22×D414yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C2×#4118yes—
382C24C2×#4114yes—
383C24H(5,𝔽2)14yes—
384C24C2511yescharacteristic
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×D4—232no—
390C2×D4—432no—
391C2×D4—432no—
392C2×D4—416no—
393C2×D4—4128no—
394C2×D4—416no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—232no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
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—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—4128no—
453C22×C4—416no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4#120116yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C22⋊C4—232no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—216no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—816no—
495C22⋊C4—816no—
496C42—48no—
497C42—84no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—216no—
505C23—216no—
506C23—24no—
507C23—24no—
508C23—232no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—216no—
514C23—216no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—432no—
521C23—416no—
522C23—416no—
523C23—432no—
524C23—464no—
525C23—464no—
526C23—416no—
527C23—416no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—416no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—416no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—832no—
546C23—816no—
547C23—816no—
548C23—832no—
549C23C2×#12014yes—
550C23C2×#12418yes—
551C23C2×#12414yes—
552C23#53514yes—
553C23#54618yes—
554C23#54618yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4112yes—
559C23C2×H(5,𝔽2)12yes—
560C23#58514yes—
561C23#58514yes—
562Q8—88no—
563D4—464no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—464no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
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—48no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4C2×#12014yes—
619C2×C4C2×#12014yes—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—216no—
625C22—216no—
626C22—216no—
627C22—28no—
628C22—464no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—416no—
633C22—416no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—832no—
640C22—84no—
641C22—84no—
642C22—88no—
643C22#243518yes—
644C22C22×#12011yescharacteristic
645C22C22×#12411yescharacteristic
646C22C2×#53511yescharacteristic
647C22#344912yes—
648C22#344912yes—
649C22C2×#54612yes—
650C22C2×#54612yes—
651C22#348512yes—
652C22#348512yes—
653C22#348512yes—
654C22#348712yes—
655C22#349118yes—
656C22#353318yes—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4—84no—
668C2—28no—
669C2—48no—
670C2—84no—
671C2C2×#243511yescharacteristic
672C2#3508811yescharacteristic
673C2#3514911yescharacteristic
674C2#3545512yes—
675C2#3695612yes—
676C2#3784214yes—
677C2#3784314yes—
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
2. C2×#2561C2semidirect
3. C2×#2503C2semidirect
4. C2×#2626C2semidirect
5. #15669C2semidirect
6. #15668C2semidirect
7. #15667C2semidirect
8. #15073C2semidirect
9. #14918C2semidirect
10. #14681C2semidirect
12. #13904C2semidirect
13. #13903C2semidirect
14. #13775C2semidirect
31. C2×#400C22semidirect
32. C2×#399C22semidirect
33. C2×#397C22semidirect
34. C2×#396C22semidirect
35. C2×#393C22semidirect
36. C2×#393C22semidirect
37. C2×#391C22semidirect
38. C2×#388C22semidirect
40. C2×#382C22semidirect
54. #1445C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. C4⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1232
C2×#25612
C2×#25032
C2×#26262
#156692
#156682
#156672
#150732
#149182
#146812
#143352
#139042
#139032
#137752