Finite Group Databasebeta

← #515612

Subgroups — #515612

Order 512.

3754 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2C22×#388C211yesmaximal, characteristic
3C2×#2273C211yesmaximal, characteristic
4#14149C212yesmaximal, complemented
5#14130C212yesmaximal, complemented
6#14127C211yesmaximal, characteristic
7#14127C211yesmaximal, characteristic
8#14127C211yesmaximal, characteristic
9#14127C211yesmaximal, characteristic
10#14118C214yesmaximal, complemented
11#14116C214yesmaximal, complemented
12#14114C214yesmaximal, complemented
13#14112C214yesmaximal, complemented
14#13782C212yesmaximal, complemented
15C4×#385C212yesmaximal, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2312C4116yescomplemented
24C2×#389C2214yescomplemented
25C2×#388—28no—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#380C2212yes—
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41#2273—28no—
42C2×C4×C4⋊C4C2214yescomplemented
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1444C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C4⋊#95C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2214yescomplemented
78C26C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4C2×C418yes—
106C22×C22⋊C4D418yes—
107C22×C22⋊C4Q818yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
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×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42—24no—
125C22×C42—44no—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132#389—48no—
133#388—216no—
134#388—216no—
135#388C2×C4132yescomplemented
136#388C2×C4132yescomplemented
137#385—28no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—44no—
142#385—44no—
143#385—44no—
144#385—44no—
145#380C2×C4116yescomplemented
146#379C2×C418yes—
147#379C2×C418yes—
148C4×C4⋊C4—48no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4C2×C4116yescomplemented
156C2×#95—24no—
157C2×#95—24no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C25—24no—
179C25—24no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25—42no—
184C25C22×C414yes—
185C25C2×D414yes—
186C25C2×Q814yes—
187C25C2411yescharacteristic
188C25C2411yescharacteristic
189C25C2411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×Q814yes—
231C23×C4C2×Q814yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C4116yes—
263C2×C22⋊C4—232no—
264C2×C22⋊C4—232no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
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—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
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⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—416no—
300C2×C22⋊C4—416no—
301C2×C22⋊C4C4⋊C4164yescomplemented
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C22×C418yes—
305C2×C22⋊C4C22×C418yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×Q818yes—
313C2×C22⋊C4C2×Q818yes—
314C2×C22⋊C4#4118yes—
315C2×C22⋊C4#4118yes—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—416no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C22×C4116yes—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C22×C4116yes—
350#95C22×C4116yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—216no—
368C24—28no—
369C24—28no—
370C24—216no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24—48no—
375C24—44no—
376C24—44no—
377C24—48no—
378C24—48no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24C2×C4⋊C4116yes—
384C24C23×C412yes—
385C24C23×C412yes—
386C24C23×C412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C22×Q812yes—
391C24C22×Q812yes—
392C24C22×Q812yes—
393C24C2×#4114yes—
394C24C2×#4114yes—
395C24C2511yescharacteristic
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
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—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4C2×C4⋊C4116yes—
487C22×C4C2×C4⋊C4116yes—
488C22×C4C23×C418yes—
489C22×C4C23×C418yes—
490C22×C4C23×C414yes—
491C22×C4C23×C414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×Q814yes—
495C22×C4C22×Q814yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)14yes—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—416no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4C2×C4⋊C4132yes—
523C22⋊C4C2×C4⋊C4132yes—
524C42—416no—
525C42—416no—
526C42—84no—
527C42—84no—
528C42—84no—
529C42—84no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—232no—
567C23—416no—
568C23—416no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—88no—
578C23—88no—
579C23—832no—
580C23—88no—
581C23—88no—
582C23—832no—
583C23C22×C4⋊C414yes—
584C23C22×C4⋊C414yes—
585C23C22×C4⋊C414yes—
586C23C4×#4118yes—
587C23#52514yes—
588C23#52514yes—
589C23#53318yes—
590C23C24×C411yescharacteristic
591C23C23×D411yescharacteristic
592C23C23×Q811yescharacteristic
593C23C22×#4112yes—
594C23C22×#4112yes—
595C23C2×H(5,𝔽2)12yes—
596C23C2×H(5,𝔽2)12yes—
597C23#58512yes—
598C23#58512yes—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
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—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—88no—
666C2×C4—88no—
667C2×C4—88no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4C22×C4⋊C418yes—
672C2×C4C22×C4⋊C418yes—
673C22—24no—
674C22—24no—
675C22—216no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—216no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—48no—
692C22—48no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—48no—
698C22—48no—
699C22—832no—
700C22#229618yes—
701C22C23×C4⋊C411yescharacteristic
702C22C2×C4×#4112yes—
703C22C4×H(5,𝔽2)12yes—
704C22C4×H(5,𝔽2)12yes—
705C22C2×#52511yescharacteristic
706C22C2×#52511yescharacteristic
707C22C2×#53312yes—
708C22#348411yescharacteristic
709C22#348411yescharacteristic
710C22#348511yescharacteristic
711C22#348511yescharacteristic
712C22#350112yes—
713C22#350112yes—
714C22#351314yes—
715C22#351414yes—
716C22#351914yes—
717C22#352114yes—
718C4—28no—
719C4—28no—
720C4—44no—
721C4—44no—
722C4—44no—
723C4—44no—
724C4—44no—
725C4—44no—
726C4—48no—
727C4—48no—
728C4—82no—
729C4—82no—
730C2—28no—
731C2—44no—
732C2—44no—
733C2C2×#229611yescharacteristic
734C2#3424811yescharacteristic
735C2#3424811yescharacteristic
736C2#3428314yes—
737C2#3535411yescharacteristic
738C2#3535511yescharacteristic
739C2#3540214yes—
740C2#3545111yescharacteristic
741C2#3545211yescharacteristic
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
4. #14149C2semidirect
5. #14130C2semidirect
10. #14118C2semidirect
11. #14116C2semidirect
12. #14114C2semidirect
13. #14112C2semidirect
14. #13782C2semidirect
15. C4×#385C2semidirect
23. #2312C4semidirect
24. C2×#389C22semidirect
42. C2×C4×C4⋊C4C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
77. C4×#95C22semidirect
135. #388C2×C4semidirect
136. #388C2×C4semidirect
145. #380C2×C4semidirect
155. C4×C22⋊C4C2×C4semidirect
301. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
C22×#3882
C2×#22732
#141492
#141302
#141272
#141182
#141162
#141142
#141122
#137822
C4×#3852