Finite Group Databasebeta

← #556136

Subgroups — #556136

Order 512.

3954 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3C22×#385C211yesmaximal, characteristic
4#14693C212yesmaximal
5#14488C214yesmaximal, complemented
6#14442C214yesmaximal, complemented
7#14405C214yesmaximal, complemented
8#14403C214yesmaximal, complemented
9#14262C212yesmaximal
10#14149C212yesmaximal, complemented
11#14130C212yesmaximal, complemented
12#13966C211yesmaximal, characteristic, complemented
13#13965C211yesmaximal, characteristic, complemented
14#13881C211yesmaximal, characteristic, complemented
15#13880C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2312—28no—
22#2312C4116yescomplemented
23C2×#389C2214yescomplemented
24C2×#388C2212yes—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×#388C2212yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#385—28no—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#384C2212yescomplemented
40C2×#382C2212yescomplemented
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1444C2214yescomplemented
52#1444C2214yescomplemented
53#1444C2212yescomplemented
54#1444C2212yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2212yescomplemented
58#1442C2212yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70C4⋊#95C2214yescomplemented
71#1434C2214yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C26C2311yescharacteristic
79C24×C4—24no—
80C24×C4—44no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4C2×C418yes—
109C22×C22⋊C4D418yes—
110C22×C22⋊C4Q818yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#389—48no—
129#388—28no—
130#388—28no—
131#388—28no—
132#388—28no—
133#388C2×C4132yescomplemented
134#388C2×C4132yescomplemented
135#385—216no—
136#385—216no—
137#385—48no—
138#385—48no—
139#385—44no—
140#385—44no—
141#385—44no—
142#385—44no—
143#384—44no—
144#382—44no—
145#380—24no—
146#380—24no—
147#380C2×C4116yescomplemented
148#379—28no—
149#379C2×C4116yes—
150C4×C22⋊C4—28no—
151C4×C22⋊C4C2×C418yescomplemented
152C4×C22⋊C4C2×C418yescomplemented
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25—24no—
178C25—24no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25C22×C414yes—
184C25C2×D414yes—
185C25C2×Q814yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—416no—
216C23×C4—416no—
217C23×C4—416no—
218C23×C4—464no—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×Q814yes—
224C23×C4C2×Q814yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C22×C4116yescomplemented
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C4⋊C4164yescomplemented
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C4116yes—
301C2×C22⋊C4C22×C4116yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×Q818yes—
307C2×C22⋊C4C2×Q818yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C22×C418yescomplemented
317C2×C42C22×C418yescomplemented
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C418yescomplemented
338#95C22×C418yescomplemented
339#95C22×C4116yescomplemented
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—216no—
345C24—28no—
346C24—28no—
347C24—216no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—44no—
357C24—44no—
358C24—48no—
359C24—48no—
360C24—44no—
361C24—44no—
362C24—48no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—416no—
368C24C2×C4⋊C4116yes—
369C24C23×C414yes—
370C24C23×C412yes—
371C24C22×D414yes—
372C24C22×D412yes—
373C24C22×Q814yes—
374C24C22×Q812yes—
375C24H(5,𝔽2)14yes—
376C242−1+414yes—
377C24C2511yescharacteristic
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—4128no—
453C22×C4—416no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4C2×C4⋊C4116yes—
465C22×C4C2×C4⋊C4116yes—
466C22×C4C23×C418yes—
467C22×C4C23×C418yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q814yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)14yes—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4C2×C4⋊C4132yes—
503C22⋊C4C2×C4⋊C4132yes—
504C42—48no—
505C42—48no—
506C42—84no—
507C42—84no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—232no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—416no—
536C23—416no—
537C23—416no—
538C23—416no—
539C23—416no—
540C23—416no—
541C23—464no—
542C23—464no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C22×C4⋊C418yes—
552C23C22×C4⋊C414yes—
553C23#52018yes—
554C23#53514yes—
555C23#53614yes—
556C23#54418yes—
557C23C24×C411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×Q811yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23#58512yes—
565C23#58512yes—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—464no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
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—48no—
634C2×C4—48no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4C22×C4⋊C414yes—
638C2×C4C22×C4⋊C414yes—
639C2×C4C22×C4⋊C418yes—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—216no—
645C22—216no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—464no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22#231718yes—
661C22C23×C4⋊C411yescharacteristic
662C22C2×#52012yes—
663C22C4×H(5,𝔽2)12yes—
664C22C4×H(5,𝔽2)12yes—
665C22C2×#53511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#54412yes—
668C22#348411yescharacteristic
669C22#348411yescharacteristic
670C22#348511yescharacteristic
671C22#348511yescharacteristic
672C22#350112yes—
673C22#350112yes—
674C22#353214yes—
675C22#353314yes—
676C22#354114yes—
677C22#354114yes—
678C4—24no—
679C4—24no—
680C4—28no—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—48no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C2—28no—
692C2—48no—
693C2C2×#231711yescharacteristic
694C2#3424811yescharacteristic
695C2#3424811yescharacteristic
696C2#3441614yes—
697C2#3670311yescharacteristic
698C2#3670511yescharacteristic
699C2#3671611yescharacteristic
700C2#3674814yes—
701C2#3695611yescharacteristic
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
5. #14488C2semidirect
6. #14442C2semidirect
7. #14405C2semidirect
8. #14403C2semidirect
10. #14149C2semidirect
11. #14130C2semidirect
12. #13966C2semidirect
13. #13965C2semidirect
14. #13881C2semidirect
15. #13880C2semidirect
22. #2312C4semidirect
23. C2×#389C22semidirect
39. C2×#384C22semidirect
40. C2×#382C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
71. #1434C22semidirect
133. #388C2×C4semidirect
134. #388C2×C4semidirect
147. #380C2×C4semidirect
151. C4×C22⋊C4C2×C4semidirect
152. C4×C22⋊C4C2×C4semidirect
271. C2×C4⋊C4C22×C4semidirect
297. C2×C22⋊C4C4⋊C4semidirect
316. C2×C42C22×C4semidirect
317. C2×C42C22×C4semidirect
337. #95C22×C4semidirect
338. #95C22×C4semidirect
339. #95C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
C22×#3852
#146932
#144882
#144422
#144052
#144032
#142622
#141492
#141302
#139662
#139652
#138812
#138802