Finite Group Databasebeta

← #711902

Subgroups — #711902

Order 512.

2672 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2C2×#2395C211yesmaximal, characteristic
3C2×#2346C211yesmaximal, characteristic, complemented
4C2×#2316C211yesmaximal, characteristic, complemented
5#16564C212yesmaximal, complemented
6#16517C212yesmaximal, complemented
7#15486C214yesmaximal, complemented
8#15473C214yesmaximal
9#15220C212yesmaximal, complemented
10#15149C212yesmaximal, complemented
11#14908C212yesmaximal, complemented
12#14894C212yesmaximal, complemented
13#14343C211yesmaximal, characteristic, complemented
14#14150C212yesmaximal, complemented
15#13993C212yesmaximal, complemented
16#13881C211yesmaximal, characteristic, complemented
17#13872C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2485—28no—
23#2395—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2211yescharacteristic
26C2×#398C2211yescharacteristic
27C2×#398C2214yes—
28C2×#397C2211yescharacteristic
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33#2346—28no—
34#2316—28no—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#382C2212yes—
47C2×#382C2212yes—
48C2×#380C2214yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×#376C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1444C2214yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1440C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D418yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136#400—24no—
137#399—24no—
138#398—24no—
139#398—48no—
140#397—28no—
141#397—28no—
142#397—24no—
143#394—28no—
144#394—28no—
145#389—28no—
146#389—28no—
147#389—24no—
148#389—24no—
149#389—28no—
150#389—28no—
151#389—28no—
152#388—216no—
153#388—28no—
154#387—28no—
155#386—44no—
156#382—28no—
157#382—28no—
158#380—24no—
159#380—24no—
160#380—24no—
161#380—24no—
162#380—28no—
163#380—216no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4D418yes—
168#376—28no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—41no—
204C22×D4—41no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4×D4—416no—
236#117—232no—
237#117—416no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
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—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4#4118yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×D418yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D418yes—
334C2×C42C2×D418yes—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—21no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C2×#4112yes—
383C242−1+412yes—
384C24C2511yescharacteristic
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—216no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4#121116yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4118yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4114yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+414yes—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—216no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C4⋊C4—84no—
514C4⋊C4—88no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—216no—
528C42—216no—
529C42—416no—
530C42—48no—
531C42—44no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C2×#12114yes—
578C23C23×D411yescharacteristic
579C23C23×D411yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4112yes—
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×2−1+412yes—
584C23C2×2−1+411yescharacteristic
585C23#58512yes—
586C23#58512yes—
587C23#58511yescharacteristic
588C23#58511yescharacteristic
589D4—416no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—232no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
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—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—816no—
677C2×C4—88no—
678C2×C4C2×#12114yes—
679C2×C4C2×#12114yes—
680C2×C4#52514yes—
681C2×C4#53614yes—
682C2×C4#54718yes—
683C2×C4#55218yes—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—81no—
701C22—82no—
702C22—82no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22C22×#12111yescharacteristic
707C22C2×#52511yescharacteristic
708C22C2×#53611yescharacteristic
709C22#344912yes—
710C22C2×#54712yes—
711C22C2×#55212yes—
712C22#348811yescharacteristic
713C22#348811yescharacteristic
714C22#348911yescharacteristic
715C22#348911yescharacteristic
716C22#349211yescharacteristic
717C22#349311yescharacteristic
718C22#350011yescharacteristic
719C22#350011yescharacteristic
720C22#351212yes—
721C22#351412yes—
722C22#351612yes—
723C22#352112yes—
724C22#352412yes—
725C22#354712yes—
726C22#356312yes—
727C22#356512yes—
728C22#356512yes—
729C4—24no—
730C4—44no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—48no—
736C4—81no—
737C4—82no—
738C4—82no—
739C4—84no—
740C4—81no—
741C4—82no—
742C4—82no—
743C4#241318yes—
744C2—44no—
745C2—82no—
746C2C2×#241311yescharacteristic
747C2#3532811yescharacteristic
748C2#3551811yescharacteristic
749C2#3598511yescharacteristic
750C2#3607511yescharacteristic
751C2#3805012yes—
752C2#3891512yes—
753C2#4296511yescharacteristic
754C2#4297811yescharacteristic
755C2#4491912yes—
756C2#4518612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2346C2semidirect
4. C2×#2316C2semidirect
5. #16564C2semidirect
6. #16517C2semidirect
7. #15486C2semidirect
9. #15220C2semidirect
10. #15149C2semidirect
11. #14908C2semidirect
12. #14894C2semidirect
13. #14343C2semidirect
14. #14150C2semidirect
15. #13993C2semidirect
16. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#23952
C2×#23462
C2×#23162
#165642
#165172
#154862
#154732
#152202
#151492
#149082
#148942
#143432
#141502
#139932
#138812
#138722