Finite Group Databasebeta

← #762664

Subgroups — #762664

Order 512.

3822 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
1C2×#2280C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16698C214yesmaximal, complemented
4#15233C212yesmaximal, complemented
5#15150C212yesmaximal, complemented
6#15061C214yesmaximal, complemented
7#14727C214yesmaximal
8#14040C214yesmaximal, complemented
9#13939C212yesmaximal, complemented
10#13924C211yesmaximal, characteristic, complemented
11#13924C211yesmaximal, characteristic, complemented
12#13881C211yesmaximal, characteristic, complemented
13#13880C211yesmaximal, characteristic, complemented
14#13879C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C2×#397C2218yes—
23C2×#385C2214yes—
24C2×#384C2212yescomplemented
25C2×#382C2212yescomplemented
26#2280—28no—
27C2×#381C2212yes—
28C2×#380—28no—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2212yes—
46#1445C2218yescomplemented
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1444C2214yescomplemented
50#1443C2218yescomplemented
51#1442C2214yescomplemented
52#1440C2218yes—
53#1440C2218yes—
54#1439C2218yes—
55#1436C2218yescomplemented
56C4⋊#95C2212yescomplemented
57C4⋊#95C2212yescomplemented
58C4⋊#95C2212yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62#1434C2218yescomplemented
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—28no—
73C24×C4—44no—
74C24×C4D418yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—28no—
86C2×#117—44no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#397—416no—
120#385—216no—
121#384—44no—
122#382—44no—
123#381—28no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—24no—
129#380—24no—
130#380—24no—
131#380—24no—
132#379—28no—
133#379—28no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—44no—
140C2×#95—28no—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—22no—
162C25—22no—
163C25—24no—
164C25—42no—
165C25—42no—
166C25C2×D414yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—232no—
179C23×C4—232no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—416no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—432no—
200C23×C4—432no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4#4114yes—
213C23×C4#4118yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#117—416no—
227#117—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
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—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—432no—
280C2×C22⋊C4#4118yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—216no—
301#95—48no—
302#95—48no—
303#95—44no—
304#95—48no—
305#95—48no—
306#95—48no—
307#95—48no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—232no—
313C24—28no—
314C24—28no—
315C24—28no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—28no—
331C24—28no—
332C24—44no—
333C24—44no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—48no—
338C24—48no—
339C24—416no—
340C24—416no—
341C24—48no—
342C24—48no—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C2×#4114yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2×#4114yes—
350C24H(5,𝔽2)14yes—
351C242−1+414yes—
352C24C2511yescharacteristic
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—4128no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—816no—
439C22×C4#123116yes—
440C22×C4#123116yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4118yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)14yes—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—816no—
462C22⋊C4—216no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C42—48no—
475C42—48no—
476C42—48no—
477C42—48no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—232no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—416no—
512C23—416no—
513C23—416no—
514C23—416no—
515C23—464no—
516C23—464no—
517C23—416no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—416no—
529C23—44no—
530C23—44no—
531C23C2×#12314yes—
532C23C2×#12314yes—
533C23#53214yes—
534C23#53514yes—
535C23#53614yes—
536C23#55218yes—
537C23#55618yes—
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4112yes—
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×2−1+412yes—
546C23#58512yes—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—232no—
556C2×C4—232no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—816no—
607C2×C4—816no—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12314yes—
611C2×C4C2×#12314yes—
612C2×C4#53414yes—
613C2×C4#55118yes—
614C2×C4#55118yes—
615C22—24no—
616C22—216no—
617C22—216no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—464no—
630C22—42no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22#249918yes—
643C22C22×#12311yescharacteristic
644C22C22×#12311yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53411yescharacteristic
647C22C2×#53511yescharacteristic
648C22C2×#53611yescharacteristic
649C22#344911yescharacteristic
650C22C2×#55112yes—
651C22C2×#55112yes—
652C22C2×#55212yes—
653C22C2×#55612yes—
654C22#347514yes—
655C22#349214yes—
656C22#349412yes—
657C22#349412yes—
658C22#351914yes—
659C22#354114yes—
660C4—24no—
661C4—24no—
662C4—48no—
663C4—42no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4#248318yes—
672C2—28no—
673C2—44no—
674C2—44no—
675C2C2×#248311yescharacteristic
676C2C2×#249911yescharacteristic
677C2#3481011yescharacteristic
678C2#3507811yescharacteristic
679C2#3522711yescharacteristic
680C2#3526211yescharacteristic
681C2#3744911yescharacteristic
682C2#5920414yes—
683C2#5920514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16698C2semidirect
4. #15233C2semidirect
5. #15150C2semidirect
6. #15061C2semidirect
8. #14040C2semidirect
9. #13939C2semidirect
10. #13924C2semidirect
11. #13924C2semidirect
12. #13881C2semidirect
13. #13880C2semidirect
14. #13879C2semidirect
24. C2×#384C22semidirect
25. C2×#382C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
55. #1436C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
C22×#3802
#166982
#152332
#151502
#150612
#147272
#140402
#139392
#139242
#138812
#138802
#138792
C4×#3792