Finite Group Databasebeta

← #1006139

Subgroups — #1006139

Order 512.

2706 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal, complemented
2#17159C212yesmaximal, complemented
3#16472C214yesmaximal, complemented
4#16288C212yesmaximal
5#16081C212yesmaximal, complemented
6#15556C214yesmaximal, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15139C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#14514C212yesmaximal, complemented
11#14130C211yesmaximal, characteristic, complemented
12#13951C212yesmaximal, complemented
13#13872C212yesmaximal, complemented
14#13871C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2485—216no—
21C2×#400C2214yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#394C2214yes—
25C2×#392C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2214yes—
32C2×#385C2212yes—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1444C2211yescharacteristic
42#1444C2212yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2212yes—
46#1442C2212yes—
47#1442C2214yes—
48#1442C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2211yescharacteristic
58#1439C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1438C2211yescharacteristic
62#1438C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2211yescharacteristic
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×C4×D4—28no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—44no—
85C2×C4×D4—42no—
86C2×#117—28no—
87C2×#117—28no—
88C2×#117—44no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120#400—48no—
121#397—216no—
122#396—48no—
123#394—216no—
124#392—24no—
125#389—28no—
126#389—28no—
127#389—44no—
128#388—216no—
129#387—216no—
130#386—48no—
131#385—28no—
132#384—42no—
133#380—28no—
134#380—28no—
135#380—44no—
136#380—44no—
137#380D418yescomplemented
138C4×C22⋊C4—42no—
139C4×C22⋊C4—44no—
140C2×#95C2311yescharacteristic
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—42no—
175C22×D4—41no—
176C22×D4—42no—
177C22×D4—41no—
178C22×D4—42no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4#4118yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4×D4—432no—
210C4×D4—416no—
211#117—432no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D418yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4#4114yes—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—44no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—22no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—42no—
305#95—44no—
306#95—41no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C2×D418yes—
319C24—22no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C22×D412yes—
342C24C2×#4114yes—
343C24C2×#4112yes—
344C242−1+412yes—
345C24C2511yescharacteristic
346C2×D4—216no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—48no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4C22×D412yes—
443C22×C4C22×D418yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C4⋊C4—216no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C22⋊C4—216no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—416no—
480C42—48no—
481C42—44no—
482C42—44no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23#54714yes—
541C23C23×D411yescharacteristic
542C23C22×#4112yes—
543C23C22×#4111yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23#58512yes—
551C23#58511yescharacteristic
552D4—432no—
553D4—416no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—416no—
596C2×C4—432no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—816no—
631C2×C4#53714yes—
632C2×C4#53918yes—
633C2×C4#54618yes—
634C2×C4#55114yes—
635C2×C4#56014yes—
636C22—24no—
637C22—28no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22C2×#53711yescharacteristic
663C22C2×#53912yes—
664C22C2×#54612yes—
665C22C2×#54711yescharacteristic
666C22C2×#55111yescharacteristic
667C22#347512yes—
668C22#348812yes—
669C22#349212yes—
670C22#349312yes—
671C22#349711yescharacteristic
672C22#350511yescharacteristic
673C22#350911yescharacteristic
674C22#351112yes—
675C22#351912yes—
676C22C2×#56011yescharacteristic
677C22#353811yescharacteristic
678C22#354011yescharacteristic
679C22#354112yes—
680C22#354112yes—
681C22#354612yes—
682C22#354712yes—
683C22#354911yescharacteristic
684C22#356311yescharacteristic
685C4—42no—
686C4—42no—
687C4—48no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—84no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4—84no—
697C4#269318yes—
698C2—44no—
699C2—84no—
700C2#3948711yescharacteristic
701C2#4000512yes—
702C2#4250712yes—
703C2#4297111yescharacteristic
704C2#4491411yescharacteristic
705C2#4525812yes—
706C2#4633811yescharacteristic
707C2C2×#269311yescharacteristic
708C2#5313711yescharacteristic
709C2#5902112yes—
710C2#5920111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2485C2semidirect
2. #17159C2semidirect
3. #16472C2semidirect
5. #16081C2semidirect
6. #15556C2semidirect
7. #15195C2semidirect
8. #15139C2semidirect
9. #15131C2semidirect
10. #14514C2semidirect
11. #14130C2semidirect
12. #13951C2semidirect
13. #13872C2semidirect
14. #13871C2semidirect
137. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
#171592
#164722
#162882
#160812
#155562
#151952
#151392
#151312
#145142
#141302
#139512
#138722
#138712
C4×#3922