Finite Group Databasebeta

← #535488

Subgroups — #535488

Order 512.

4852 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic
2C2×#2366C212yesmaximal, complemented
3#14505C214yesmaximal, complemented
4#14363C214yesmaximal, complemented
5#14160C212yesmaximal, complemented
6#14023C214yesmaximal, complemented
7#13970C212yesmaximal, complemented
8#13970C212yesmaximal, complemented
9#13968C214yesmaximal, complemented
10#13916C212yesmaximal, complemented
11#13852C211yesmaximal, characteristic, complemented
12D4×#95C212yesmaximal, complemented
13#13777C211yesmaximal, characteristic, complemented
14C2×#540—216no—
15C2×#540C4116yescomplemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#123C2214yes—
19C22×#121C2214yes—
20C22×#121C2212yes—
21C22×C4×D4C2212yes—
22C22×C4×D4C2212yes—
23C22×C4×D4C2212yes—
24C22×C4×D4C2214yes—
25C22×#117C2211yescharacteristic
26#2366—216no—
27C2×#390C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#381C2214yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2214yes—
41#1441C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1438C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C4×#95C2212yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2314yes—
76C24×C4C2314yes—
77#540—2128no—
78C2×#127C2×C418yes—
79C2×#127D4116yes—
80C2×#125—28no—
81C2×#125C2×C418yescomplemented
82C2×#123—216no—
83C2×#123—216no—
84C2×#123C2×C4132yescomplemented
85C2×#121—216no—
86C2×#121—216no—
87C2×#121—28no—
88C2×#121—28no—
89C2×#121C2×C4132yescomplemented
90C2×#121C2×C4116yescomplemented
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—216no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—44no—
99C2×C4×D4—44no—
100C2×C4×D4—48no—
101C2×C4×D4—48no—
102C2×C4×D4C2×C4116yescomplemented
103C2×#117—28no—
104C2×#117C2×C418yescomplemented
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130#390—28no—
131#390—28no—
132#390—216no—
133#386—216no—
134#386—216no—
135#386—48no—
136#386—48no—
137#381—48no—
138#380—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—28no—
141C4×C22⋊C4—42no—
142C4×C22⋊C4—42no—
143C2×#95—48no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C25C2411yescharacteristic
166C25C2414yes—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4C22×C418yes—
182C22×D4C22×C4116yes—
183C22×D4C22×C4116yes—
184C22×D4C2×D418yes—
185C22×D4C2×D418yes—
186C22×D4C2×D4116yes—
187C22×D4C2×D4116yes—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C22×C414yes—
213C23×C4C22×C4116yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225#127C22⋊C4164yescomplemented
226#125—232no—
227#123—2128no—
228#121—2128no—
229#121—264no—
230C4×D4—264no—
231C4×D4—432no—
232C4×D4—432no—
233C4×D4—464no—
234#117—232no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
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—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C4116yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C4116yes—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C414yes—
304C2×C42C22×C414yes—
305C2×C42C2×D418yes—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—41no—
321#95—42no—
322#95—41no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24C23×C412yes—
348C24C23×C418yes—
349C24C22×D418yes—
350C24C22×D418yes—
351C24C2511yescharacteristic
352C2×D4—232no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4C2×C22⋊C4132yes—
372C2×D4C2×C22⋊C4164yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4C23×C414yes—
459C22×C4C23×C412yes—
460C22×C4C23×C418yes—
461C22×C4C23×C414yes—
462C22×C4C23×C412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—232no—
476C4⋊C4—216no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C22⋊C4—232no—
480C22⋊C4—232no—
481C22⋊C4—216no—
482C22⋊C4—44no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—28no—
493C42—416no—
494C42—48no—
495C42—48no—
496C42C2×C22⋊C4116yes—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23C22×C22⋊C4116yes—
553C23C24×C411yescharacteristic
554C23C23×D412yes—
555C23C22×#4112yes—
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×2−1+411yescharacteristic
560C23#58512yes—
561C23#58512yes—
562D4—264no—
563D4—464no—
564D4—832no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—832no—
634C2×C4—832no—
635C2×C4—832no—
636C2×C4C22×C22⋊C418yes—
637C2×C4C22×C22⋊C414yes—
638C2×C4C4×#4114yes—
639C2×C4#52014yes—
640C2×C4#54618yes—
641C2×C4#54718yes—
642C22—216no—
643C22—216no—
644C22—216no—
645C22—48no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22C23×C22⋊C411yescharacteristic
667C22C2×C4×#4111yescharacteristic
668C22C2×#52011yescharacteristic
669C22C4×H(5,𝔽2)12yes—
670C22C4×H(5,𝔽2)12yes—
671C22C2×#54612yes—
672C22C2×#54712yes—
673C22#348412yes—
674C22#348512yes—
675C22#349612yes—
676C22#349612yes—
677C22#351312yes—
678C22#351712yes—
679C22#351912yes—
680C22#352412yes—
681C22#353711yescharacteristic
682C22#353911yescharacteristic
683C22#354212yes—
684C22#354811yescharacteristic
685C22#355412yes—
686C22#355711yescharacteristic
687C4—28no—
688C4—42no—
689C4—48no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—48no—
694C4—84no—
695C4—82no—
696C4—82no—
697C4#233918yes—
698C2—416no—
699C2—82no—
700C2C2×#233911yescharacteristic
701C2#3425612yes—
702C2#3428712yes—
703C2#3441811yescharacteristic
704C2#3442311yescharacteristic
705C2#4488012yes—
706C2#4488112yes—
707C2#4489512yes—
708C2#4510012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#540C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2366C2semidirect
3. #14505C2semidirect
4. #14363C2semidirect
5. #14160C2semidirect
6. #14023C2semidirect
7. #13970C2semidirect
8. #13970C2semidirect
9. #13968C2semidirect
10. #13916C2semidirect
11. #13852C2semidirect
12. D4×#95C2semidirect
13. #13777C2semidirect
15. C2×#540C4semidirect
81. C2×#125C2×C4semidirect
84. C2×#123C2×C4semidirect
89. C2×#121C2×C4semidirect
90. C2×#121C2×C4semidirect
102. C2×C4×D4C2×C4semidirect
104. C2×#117C2×C4semidirect
225. #127C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
C2×#23662
#145052
#143632
#141602
#140232
#139702
#139682
#139162
#138522
D4×#952
#137772