Finite Group Databasebeta

← #556146

Subgroups — #556146

Order 512.

3538 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3C22×#388C211yesmaximal, characteristic
4#14697C211yesmaximal, characteristic, complemented
5#14697C211yesmaximal, characteristic, complemented
6#14514C214yesmaximal, complemented
7#14408C214yesmaximal, complemented
8#14408C214yesmaximal, complemented
9#14402C214yesmaximal, complemented
10#14275C211yesmaximal, characteristic, complemented
11#14275C211yesmaximal, characteristic, complemented
12#14149C212yesmaximal, complemented
13#14130C212yesmaximal, complemented
14#13877C212yesmaximal
15#13877C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2315C4116yescomplemented
23#2312—28no—
24C2×#389C2214yescomplemented
25C2×#389C2212yescomplemented
26C2×#389C2212yescomplemented
27C2×#388—28no—
28C2×#388C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2214yes—
43C2×#380C2214yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#379C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1444C2214yescomplemented
52#1444C2212yescomplemented
53#1444C2212yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1442C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C4⋊#95C2214yescomplemented
70C4⋊#95C2212yescomplemented
71C4⋊#95C2212yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C26C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4C2×C4116yes—
107C22×C22⋊C4C2×C418yes—
108C22×C22⋊C4D418yes—
109C22×C22⋊C4Q818yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42—24no—
124C22×C42—44no—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#389—44no—
130#389—48no—
131#389—44no—
132#388—28no—
133#388—28no—
134#388—28no—
135#388—28no—
136#388—216no—
137#388—216no—
138#385—44no—
139#385—44no—
140#385—44no—
141#385—44no—
142#385C2×C4132yescomplemented
143#385C2×C4132yescomplemented
144#380—24no—
145#380—24no—
146#380—48no—
147#380—48no—
148#380C2×C4116yes—
149#380C2×C418yes—
150#380C2×C418yes—
151#379—28no—
152C4×C22⋊C4—28no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
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—22no—
178C25—22no—
179C25—22no—
180C25—22no—
181C25—42no—
182C25—42no—
183C25—42no—
184C25—42no—
185C25C22×C414yes—
186C25C2×D414yes—
187C25C2×Q814yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C22×C4116yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×Q814yes—
226C23×C4C2×Q814yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C22×C4116yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C418yes—
273C2×C22⋊C4—24no—
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—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—432no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4—416no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C4⋊C4164yescomplemented
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C22×C418yes—
305C2×C22⋊C4C22×C4116yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4C2×D418yes—
310C2×C22⋊C4C2×Q818yes—
311C2×C22⋊C4C2×Q818yes—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—416no—
321C2×C42—44no—
322C2×C42—44no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C22×C4116yes—
343#95C22×C418yes—
344#95C22×C418yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—28no—
358C24—28no—
359C24—28no—
360C24—28no—
361C24—28no—
362C24—28no—
363C24—44no—
364C24—44no—
365C24—48no—
366C24—48no—
367C24—44no—
368C24—44no—
369C24—48no—
370C24—48no—
371C24—84no—
372C24—84no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—84no—
377C24—84no—
378C24—84no—
379C24C2×C4⋊C4116yes—
380C24C23×C414yes—
381C24C23×C412yes—
382C24C22×D414yes—
383C24C22×D412yes—
384C24C22×Q814yes—
385C24C22×Q812yes—
386C24H(5,𝔽2)12yes—
387C24H(5,𝔽2)12yes—
388C242−1+412yes—
389C242−1+412yes—
390C24C2511yescharacteristic
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
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—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4C2×C4⋊C4116yes—
487C22×C4C2×C4⋊C4116yes—
488C22×C4C23×C418yes—
489C22×C4C23×C418yes—
490C22×C4C23×C412yes—
491C22×C4C23×C412yes—
492C22×C4C23×C414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×Q812yes—
497C22×C4C22×Q812yes—
498C22×C4C22×Q814yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4H(5,𝔽2)14yes—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4C2×C4⋊C4132yes—
525C22⋊C4C2×C4⋊C4132yes—
526C42—48no—
527C42—48no—
528C42—84no—
529C42—84no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—28no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—232no—
561C23—416no—
562C23—416no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—88no—
572C23—88no—
573C23—88no—
574C23—88no—
575C23—816no—
576C23—816no—
577C23—816no—
578C23—816no—
579C23C22×C4⋊C418yes—
580C23C22×C4⋊C414yes—
581C23#51814yes—
582C23#51914yes—
583C23#53714yes—
584C23#53714yes—
585C23#54414yes—
586C23#54414yes—
587C23C24×C411yescharacteristic
588C23C23×D411yescharacteristic
589C23C23×Q811yescharacteristic
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)12yes—
593C23C2×H(5,𝔽2)12yes—
594C23C2×2−1+411yescharacteristic
595C23C2×2−1+411yescharacteristic
596C23#58512yes—
597C23#58512yes—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—832no—
673C2×C4—88no—
674C2×C4—88no—
675C2×C4C22×C4⋊C414yes—
676C2×C4C22×C4⋊C414yes—
677C2×C4C22×C4⋊C418yes—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—28no—
683C22—28no—
684C22—28no—
685C22—28no—
686C22—24no—
687C22—24no—
688C22—28no—
689C22—28no—
690C22—28no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—48no—
696C22—48no—
697C22—48no—
698C22—48no—
699C22—832no—
700C22#231618yes—
701C22C23×C4⋊C411yescharacteristic
702C22C2×#51811yescharacteristic
703C22C2×#51911yescharacteristic
704C22C4×H(5,𝔽2)12yes—
705C22C4×H(5,𝔽2)12yes—
706C22C2×#53711yescharacteristic
707C22C2×#53711yescharacteristic
708C22C2×#54411yescharacteristic
709C22C2×#54411yescharacteristic
710C22#348411yescharacteristic
711C22#348411yescharacteristic
712C22#348511yescharacteristic
713C22#348511yescharacteristic
714C22#350112yes—
715C22#350112yes—
716C22#353412yes—
717C22#353512yes—
718C22#353612yes—
719C22#353712yes—
720C22#355812yes—
721C22#356012yes—
722C22#356112yes—
723C22#356412yes—
724C4—24no—
725C4—24no—
726C4—28no—
727C4—44no—
728C4—44no—
729C4—48no—
730C4—48no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C2—28no—
738C2—48no—
739C2C2×#231611yescharacteristic
740C2#3424811yescharacteristic
741C2#3424811yescharacteristic
742C2#3443312yes—
743C2#3443712yes—
744C2#3670611yescharacteristic
745C2#3670911yescharacteristic
746C2#3674912yes—
747C2#3675012yes—
748C2#3695711yescharacteristic
749C2#3696011yescharacteristic
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. #14697C2semidirect
5. #14697C2semidirect
6. #14514C2semidirect
7. #14408C2semidirect
8. #14408C2semidirect
9. #14402C2semidirect
10. #14275C2semidirect
11. #14275C2semidirect
12. #14149C2semidirect
13. #14130C2semidirect
22. #2315C4semidirect
24. C2×#389C22semidirect
25. C2×#389C22semidirect
26. C2×#389C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
71. C4⋊#95C22semidirect
142. #385C2×C4semidirect
143. #385C2×C4semidirect
302. 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×#23152
C2×#23122
C22×#3882
#146972
#145142
#144082
#144022
#142752
#141492
#141302
#138772