Finite Group Databasebeta

← #530337

Subgroups — #530337

Order 512.

3742 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2#14681C211yesmaximal, characteristic
3#14423C212yesmaximal, complemented
4#14422C212yesmaximal, complemented
5#14371C214yesmaximal, complemented
6#14297C212yesmaximal, complemented
7#14108C214yesmaximal, complemented
8#13923C212yesmaximal, complemented
9#13883C212yesmaximal, complemented
10#13882C212yesmaximal, complemented
11#13873C212yesmaximal
12#13873C212yesmaximal
13#13802C212yesmaximal, complemented
14#13780C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C2×#388C2214yes—
21C2×#388C2214yes—
22C2×#387C2214yescomplemented
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#380C4116yescomplemented
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#379C2212yes—
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44#1445C2214yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C24×C4—28no—
78C24×C4—44no—
79C24×C4—42no—
80C24×C4—42no—
81C24×C4C2×C418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C22×C4⋊C4—44no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2×C418yes—
94C22×C4⋊C4C2×C418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—48no—
104C22×C22⋊C4C2×C4116yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123#388—48no—
124#388—48no—
125#387—48no—
126#381—28no—
127#381—44no—
128#380—28no—
129#380—44no—
130#380C2×C4116yes—
131#380C2×C418yes—
132#380C2×C418yes—
133#380C2×C4116yes—
134#380C2×C418yes—
135#380C2×C418yes—
136#379—28no—
137#379—42no—
138#379—42no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—28no—
141C4×C22⋊C4—44no—
142C4×C22⋊C4—42no—
143C4×C22⋊C4—44no—
144C4×C22⋊C4—42no—
145C2×#95C2×C4116yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—22no—
175C25—22no—
176C25—42no—
177C25—42no—
178C25C22×C414yes—
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—232no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4—42no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—42no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4—42no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—432no—
217C23×C4—432no—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C418yes—
223C23×C4C22×C418yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4#4118yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C418yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C22×C418yes—
297C2×C22⋊C4C22×C418yes—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4#4118yes—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—44no—
322#95—42no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C22×C418yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—232no—
342C24—28no—
343C24—28no—
344C24—28no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—28no—
360C24—28no—
361C24—44no—
362C24—44no—
363C24—48no—
364C24—48no—
365C24—416no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—416no—
370C24—48no—
371C24—48no—
372C24C23×C412yes—
373C24C23×C412yes—
374C24C23×C412yes—
375C24C2×#4114yes—
376C24C2×#4114yes—
377C24H(5,𝔽2)14yes—
378C24H(5,𝔽2)12yes—
379C24H(5,𝔽2)12yes—
380C242−1+414yes—
381C24C2511yescharacteristic
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—4128no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4#117116yes—
479C22×C4C23×C412yes—
480C22×C4C23×C414yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C414yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C23×C412yes—
487C22×C4C2×#4118yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)14yes—
491C4⋊C4—216no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—84no—
514C42—84no—
515C42—88no—
516C42—88no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—232no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—416no—
554C23—464no—
555C23—464no—
556C23—416no—
557C23—416no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23C2×#11714yes—
571C23C4×#4114yes—
572C23#51814yes—
573C23#51814yes—
574C23#51914yes—
575C23#53918yes—
576C23#54218yes—
577C23C24×C411yescharacteristic
578C23C22×#4112yes—
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×H(5,𝔽2)12yes—
584C23C2×2−1+412yes—
585C23#58512yes—
586C2×C4—232no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4C2×#11714yes—
660C2×C4C2×#11714yes—
661C2×C4#51814yes—
662C22—216no—
663C22—216no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—464no—
677C22—48no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22#233218yes—
690C22C22×#11711yescharacteristic
691C22C2×C4×#4111yescharacteristic
692C22C2×#51811yescharacteristic
693C22C2×#51811yescharacteristic
694C22C2×#51811yescharacteristic
695C22C2×#51911yescharacteristic
696C22#342811yescharacteristic
697C22C2×#53912yes—
698C22C2×#54212yes—
699C22#347514yes—
700C22#349012yes—
701C22#349012yes—
702C22#349214yes—
703C22#351914yes—
704C22#353412yes—
705C22#353412yes—
706C22#354114yes—
707C4—24no—
708C4—24no—
709C4—24no—
710C4—48no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—44no—
715C4—44no—
716C4—84no—
717C4—84no—
718C4—84no—
719C2—28no—
720C2—44no—
721C2—44no—
722C2C2×#233211yescharacteristic
723C2#3410911yescharacteristic
724C2#3417611yescharacteristic
725C2#3417811yescharacteristic
726C2#3424611yescharacteristic
727C2#3435811yescharacteristic
728C2#3437011yescharacteristic
729C2#3790814yes—
730C2#3791014yes—
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
3. #14423C2semidirect
4. #14422C2semidirect
5. #14371C2semidirect
6. #14297C2semidirect
7. #14108C2semidirect
8. #13923C2semidirect
9. #13883C2semidirect
10. #13882C2semidirect
13. #13802C2semidirect
14. #13780C2semidirect
22. C2×#387C22semidirect
25. C2×#380C4semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
#146812
#144232
#144222
#143712
#142972
#141082
#139232
#138832
#138822
#138732
#138022
#137802
C4×#3792