Finite Group Databasebeta

← #535951

Subgroups — #535951

Order 512.

3346 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic, complemented
3C2×#2312C211yesmaximal, characteristic
4#14578C214yesmaximal, complemented
5#14529C214yesmaximal, complemented
6#14231C214yesmaximal, complemented
7#14194C214yesmaximal, complemented
8#13924C211yesmaximal, characteristic, complemented
9#13908C211yesmaximal, characteristic, complemented
10#13882C212yesmaximal, complemented
11#13873C212yesmaximal
12#13873C212yesmaximal
13#13802C211yesmaximal, characteristic, complemented
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15#13775C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C2×#399C4116yescomplemented
21C2×#399C2212yes—
22C2×#399C2212yes—
23C2×#399C2212yes—
24C2×#399C2212yes—
25#2345—28no—
26#2312—28no—
27C2×#389C2212yescomplemented
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yescomplemented
33C2×#387C2214yescomplemented
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#379C2211yescharacteristic
39C2×#379C2211yescharacteristic
40C2×#379C2212yes—
41C2×C4×C4⋊C4C2212yescomplemented
42C2×C4×C4⋊C4C2214yescomplemented
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1445C2214yescomplemented
50#1445C2212yescomplemented
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1442C2212yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2214yescomplemented
69C4⋊#95C2214yescomplemented
70#1434C2212yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C4×#95C2212yescomplemented
78C26C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—48no—
86C22×C4⋊C4C2×C418yes—
87C22×C4⋊C4C2×C4116yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2×C418yes—
103C22×C22⋊C4C2×C4116yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2314yes—
125#399C2×C4116yes—
126#399C2×C4116yes—
127#399C2×C4116yes—
128#399C2×C4116yes—
129#389—28no—
130#388—216no—
131#388—216no—
132#388—48no—
133#388—48no—
134#387—216no—
135#387—48no—
136#380—24no—
137#380—28no—
138#380—28no—
139#380D418yes—
140#379—28no—
141#379—24no—
142#379—24no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—48no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C2×#95C2×C418yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—22no—
179C25—44no—
180C25—44no—
181C25C22×C414yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C22×C418yes—
197C23×C4C22×C414yes—
198C23×C4C22×C418yes—
199C23×C4C22×C414yes—
200C23×C4C2×D414yes—
201C23×C4#4114yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C2×C4⋊C4—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—432no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C418yes—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C2×D418yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—232no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—432no—
271C2×C22⋊C4—44no—
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⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4#4114yes—
289C2×C22⋊C4#4114yes—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—44no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95C22×C414yes—
322#95C22×C418yes—
323#95C22×C414yes—
324#95C2×D418yes—
325C24—24no—
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—28no—
340C24—48no—
341C24—48no—
342C24—416no—
343C24—416no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—84no—
349C24—84no—
350C24—84no—
351C24—84no—
352C24—88no—
353C24—88no—
354C24C23×C414yes—
355C24C23×C412yes—
356C24C22×D412yes—
357C24C2×#4112yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C242−1+412yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4C4×D4116yes—
433C22×C4C23×C414yes—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C414yes—
437C22×C4C23×C412yes—
438C22×C4C23×C412yes—
439C22×C4C23×C414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)18yes—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—416no—
476C42—48no—
477C42—84no—
478C42—88no—
479C42—88no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—432no—
498C23—416no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—88no—
516C23—88no—
517C23—816no—
518C23—816no—
519C23—816no—
520C23—816no—
521C23—816no—
522C23C2×C4×D414yes—
523C23C4×#4114yes—
524C23#51914yes—
525C23#54714yes—
526C23#54814yes—
527C23#55214yes—
528C23#55614yes—
529C23C24×C411yescharacteristic
530C23C23×D411yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)14yes—
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23#58514yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—416no—
568C2×C4—432no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4—816no—
606C2×C4C2×C4×D414yes—
607C2×C4C2×C4×D414yes—
608C2×C4#51818yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—24no—
615C22—24no—
616C22—28no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—832no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22#236218yes—
630C22C22×C4×D411yescharacteristic
631C22C2×C4×#4111yescharacteristic
632C22C2×#51812yes—
633C22C2×#51911yescharacteristic
634C22#342812yes—
635C22C2×#54711yescharacteristic
636C22C2×#54811yescharacteristic
637C22C2×#55211yescharacteristic
638C22C2×#55611yescharacteristic
639C22#348512yes—
640C22#349112yes—
641C22#349812yes—
642C22#350812yes—
643C22#351314yes—
644C22#351314yes—
645C22#354114yes—
646C22#354114yes—
647C4—24no—
648C4—24no—
649C4—28no—
650C4—44no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—84no—
660C2—28no—
661C2—84no—
662C2C2×#236211yescharacteristic
663C2#3419911yescharacteristic
664C2#3421011yescharacteristic
665C2#3423812yes—
666C2#3437012yes—
667C2#4408112yes—
668C2#4408212yes—
669C2#4410212yes—
670C2#4410312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
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×#2345C2semidirect
4. #14578C2semidirect
5. #14529C2semidirect
6. #14231C2semidirect
7. #14194C2semidirect
8. #13924C2semidirect
9. #13908C2semidirect
10. #13882C2semidirect
13. #13802C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
20. C2×#399C4semidirect
27. C2×#389C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
41. C2×C4×C4⋊C4C22semidirect
42. C2×C4×C4⋊C4C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. #1434C22semidirect
77. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23452
C2×#23122
#145782
#145292
#142312
#141942
#139242
#139082
#138822
#138732
#138022
C22⋊C4×C4⋊C42
#137752