Finite Group Databasebeta

← #4419101

Subgroups — #4419101

Order 512.

2671 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2498C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic, complemented
3C2×#2661C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#17039C212yesmaximal, complemented
6#16811C212yesmaximal, complemented
7#16517C212yesmaximal, complemented
8#16516C212yesmaximal, complemented
9#15567C214yesmaximal, complemented
10#15473C214yesmaximal, complemented
11#15174C212yesmaximal, complemented
12#15112C212yesmaximal, complemented
13#14898C212yesmaximal, complemented
14#14894C212yesmaximal, complemented
15#14658C211yesmaximal, characteristic, complemented
16#14271C211yesmaximal, characteristic, complemented
17#13966C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2661—28no—
22#2624—28no—
23#2498—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2212yes—
26C2×#399C2211yescharacteristic
27C2×#398C2211yescharacteristic
28C2×#398C2211yescharacteristic
29C2×#398C2214yes—
30C2×#397C2211yescharacteristic
31C2×#397C2212yes—
32C2×#397C2211yescharacteristic
33C2×#394C2211yescharacteristic
34C2×#394C2212yes—
35C2×#394C2214yes—
36C2×#394C2214yes—
37C2×#392C2211yescharacteristic
38C2×#392C2212yes—
39#2346—28no—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#388C2214yes—
47C2×#387C2212yes—
48C2×#386C2212yes—
49C2×#386C2214yes—
50C2×#382C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2212yes—
57C22×#95C2211yescharacteristic
58#1445C2212yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2212yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1438C2214yes—
78#1438C2212yes—
79#1438C2212yes—
80#1437C2214yes—
81#1437C2214yes—
82#1437C2214yes—
83#1436C2214yes—
84#1434C2212yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C23×D4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117D418yescomplemented
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136#400—24no—
137#399—28no—
138#399—24no—
139#398—24no—
140#398—24no—
141#398—48no—
142#397—28no—
143#397—24no—
144#397—24no—
145#394—216no—
146#394—28no—
147#394—24no—
148#394—48no—
149#392—28no—
150#392—24no—
151#390—28no—
152#390—28no—
153#389—28no—
154#389—28no—
155#388—216no—
156#388—28no—
157#388—28no—
158#387—28no—
159#386—28no—
160#386—48no—
161#382—28no—
162#380—24no—
163#380—24no—
164#380—28no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4D418yes—
167#376—28no—
168C2×#95—24no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2411yescharacteristic
195C25C2412yes—
196C25C2411yescharacteristic
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—44no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#123—416no—
239#121—416no—
240#117—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
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⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C2×D418yes—
331C2×C42C2×D418yes—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95C2×D414yes—
356#95C2×D414yes—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C2×#4112yes—
384C24C2511yescharacteristic
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4#121116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)14yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—216no—
517C42—48no—
518C42—416no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23C2×#12114yes—
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C2×H(5,𝔽2)12yes—
578C23#58514yes—
579C23#58512yes—
580C23#58512yes—
581C23#58512yes—
582D4—88no—
583D4—816no—
584D4—88no—
585D4—816no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
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—44no—
658C2×C4—816no—
659C2×C4C2×#12114yes—
660C2×C4C2×#12114yes—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—81no—
670C22—82no—
671C22—82no—
672C22—81no—
673C22—82no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—84no—
683C22—82no—
684C22—81no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22C22×#12111yescharacteristic
689C22#344912yes—
690C22#344912yes—
691C22#348512yes—
692C22#348812yes—
693C22#349112yes—
694C22#349811yescharacteristic
695C22#350011yescharacteristic
696C22#350011yescharacteristic
697C22#350011yescharacteristic
698C22#350612yes—
699C22#351212yes—
700C22#354514yes—
701C22#355114yes—
702C22#356614yes—
703C22#356714yes—
704C4—24no—
705C4—24no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—84no—
715C4—84no—
716C4—81no—
717C4—82no—
718C2—44no—
719C2—161no—
720C2—162no—
721C2#3508911yescharacteristic
722C2#3532811yescharacteristic
723C2#3553311yescharacteristic
724C2#4138112yes—
725C2#4215112yes—
726C2#4215512yes—
727C2#4239012yes—
728C2#4887514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2661C2
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×#2498C2semidirect
2. C2×#2624C2semidirect
4. C2×#2346C2semidirect
5. #17039C2semidirect
6. #16811C2semidirect
7. #16517C2semidirect
8. #16516C2semidirect
9. #15567C2semidirect
10. #15473C2semidirect
11. #15174C2semidirect
12. #15112C2semidirect
13. #14898C2semidirect
14. #14894C2semidirect
15. #14658C2semidirect
16. #14271C2semidirect
17. #13966C2semidirect
101. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24982
C2×#26242
C2×#26612
C2×#23462
#170392
#168112
#165172
#165162
#155672
#154732
#151742
#151122
#148982
#148942
#146582
#142712
#139662