Finite Group Databasebeta

← #951606

Subgroups — #951606

Order 512.

3598 conjugacy classes of subgroups.

The subgroup lattice has 794 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2420C212yesmaximal, complemented
2C2×#2428C212yesmaximal, complemented
3#17118C212yesmaximal, complemented
4#17083C212yesmaximal, complemented
5#16272C214yesmaximal, complemented
6#16259C214yesmaximal, complemented
7#15952C212yesmaximal, complemented
8#15278C212yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#15130C212yesmaximal, complemented
11#15123C212yesmaximal, complemented
12#14277C211yesmaximal, characteristic, complemented
13#14272C211yesmaximal, characteristic, complemented
14#14018C212yesmaximal, complemented
15#13847C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2214yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2428—216no—
23#2420—216no—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#392C2214yes—
30C2×#392C2214yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#381C2214yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1445C2211yescharacteristic
50#1445C2211yescharacteristic
51#1444C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1438C2211yescharacteristic
69#1438C2211yescharacteristic
70#1438C2212yes—
71#1438C2212yes—
72#1438C2212yes—
73#1438C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C23×D4C2314yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2312yes—
84C24×C4C2314yes—
85C2×#127—42no—
86C2×#127—42no—
87C2×#127D418yescomplemented
88C2×#121—216no—
89C2×#121—28no—
90C2×#121—28no—
91C2×#121—28no—
92C2×#121—48no—
93C2×#121—48no—
94C2×#121—44no—
95C2×#121—44no—
96C2×#121—44no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#397—216no—
133#394—216no—
134#394—216no—
135#394—48no—
136#394—48no—
137#392—216no—
138#392—48no—
139#390—28no—
140#390—28no—
141#390—28no—
142#389—28no—
143#389—28no—
144#386—216no—
145#386—216no—
146#385—48no—
147#384—42no—
148#382—42no—
149#382D418yescomplemented
150#381—216no—
151#380—28no—
152#380—44no—
153C4×C4⋊C4—42no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156#376—42no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2414yes—
180C25C2412yes—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4C2×D418yes—
215C22×D4C2×D418yes—
216C22×D4C2×D418yes—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244#127—416no—
245#121—464no—
246#121—432no—
247#121—432no—
248C4×D4—416no—
249C4×D4—432no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D418yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—41no—
344#95—41no—
345#95—41no—
346#95—41no—
347#95—41no—
348#95—41no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95C2×D418yes—
356#95C2×D418yes—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—22no—
384C24—22no—
385C24C22×D418yes—
386C24H(5,𝔽2)14yes—
387C24C2511yescharacteristic
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—416no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—416no—
403C2×D4—416no—
404C2×D4—416no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—88no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—88no—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D418yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4H(5,𝔽2)14yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C4⋊C4—216no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—416no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—88no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C22⋊C4—216no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C42—28no—
543C42—28no—
544C42—28no—
545C42—48no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—44no—
584C23—42no—
585C23—44no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—44no—
597C23—42no—
598C23—44no—
599C23—42no—
600C23—44no—
601C23—42no—
602C23—44no—
603C23—42no—
604C23—44no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—44no—
613C23—44no—
614C23—42no—
615C23—42no—
616C23C23×D411yescharacteristic
617C23C23×D411yescharacteristic
618C23C22×#4111yescharacteristic
619C23C2×H(5,𝔽2)11yescharacteristic
620C23C2×H(5,𝔽2)11yescharacteristic
621C23C2×H(5,𝔽2)11yescharacteristic
622C23C2×H(5,𝔽2)12yes—
623C23C2×H(5,𝔽2)12yes—
624C23C2×2−1+411yescharacteristic
625C23#58512yes—
626C23#58512yes—
627D4—416no—
628D4—432no—
629D4—832no—
630D4—816no—
631D4—816no—
632D4—832no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—832no—
706C2×C4—816no—
707C2×C4#53414yes—
708C2×C4#53714yes—
709C2×C4#53814yes—
710C2×C4D4214yes—
711C2×C4#54714yes—
712C2×C4#55014yes—
713C2×C4#56214yes—
714C22—44no—
715C22—48no—
716C22—48no—
717C22—44no—
718C22—48no—
719C22—44no—
720C22—48no—
721C22—48no—
722C22—44no—
723C22—44no—
724C22—48no—
725C22—44no—
726C22—48no—
727C22—44no—
728C22—82no—
729C22—84no—
730C22—84no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—84no—
736C22—84no—
737C22—82no—
738C22—84no—
739C22—84no—
740C22—82no—
741C22—84no—
742C22—82no—
743C22—84no—
744C22C2×#53411yescharacteristic
745C22C2×#53711yescharacteristic
746C22C2×#53811yescharacteristic
747C22#346012yes—
748C22C2×D4211yescharacteristic
749C22C2×#54711yescharacteristic
750C22C2×#55011yescharacteristic
751C22#348412yes—
752C22#348512yes—
753C22#348812yes—
754C22#348812yes—
755C22#349412yes—
756C22C2×#56211yescharacteristic
757C22#353711yescharacteristic
758C22#354011yescharacteristic
759C22#354212yes—
760C22#354212yes—
761C22#354312yes—
762C22#354712yes—
763C22#354911yescharacteristic
764C22#355111yescharacteristic
765C22#355912yes—
766C22#356212yes—
767C4—44no—
768C4—48no—
769C4—42no—
770C4—42no—
771C4—42no—
772C4—42no—
773C4—42no—
774C4—42no—
775C4—82no—
776C4—82no—
777C4—84no—
778C4—84no—
779C4#268218yes—
780C2—84no—
781C2—88no—
782C2#3728411yescharacteristic
783C2#3751311yescharacteristic
784C2#3776412yes—
785C2#3777512yes—
786C2#4247511yescharacteristic
787C2#4267311yescharacteristic
788C2#4295211yescharacteristic
789C2#4297111yescharacteristic
790C2#4493212yes—
791C2C2×#268211yescharacteristic
792C2#5311512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2420C2semidirect
2. C2×#2428C2semidirect
3. #17118C2semidirect
4. #17083C2semidirect
5. #16272C2semidirect
6. #16259C2semidirect
7. #15952C2semidirect
8. #15278C2semidirect
9. #15220C2semidirect
10. #15130C2semidirect
11. #15123C2semidirect
12. #14277C2semidirect
13. #14272C2semidirect
14. #14018C2semidirect
15. #13847C2semidirect
87. C2×#127D4semidirect
149. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24202
C2×#24282
#171182
#170832
#162722
#162592
#159522
#152782
#152202
#151302
#151232
#142772
#142722
#140182
#138472