Finite Group Databasebeta

← #920270

Subgroups — #920270

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 787 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic, complemented
3#17168C212yesmaximal, complemented
4#17088C212yesmaximal, complemented
5#16824C212yesmaximal, complemented
6#16338C212yesmaximal, complemented
7#16277C214yesmaximal, complemented
8#15334C214yesmaximal, complemented
9#14912C212yesmaximal, complemented
10#14719C212yesmaximal, complemented
11#14718C212yesmaximal, complemented
12#14256C212yesmaximal, complemented
13#14134C212yesmaximal, complemented
14#14039C212yesmaximal, complemented
15#13869C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2539—28no—
25C2×#399—28no—
26C2×#399C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#399C2212yes—
31C2×#399C2212yes—
32C2×#394C2214yes—
33C2×#394C2214yescomplemented
34C2×#392C2211yescharacteristic
35C2×#392C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#387C2214yescomplemented
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#381C2212yes—
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2212yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2212yes—
60#1440C2214yescomplemented
61#1439C2214yes—
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1438C2214yes—
68#1438C2214yes—
69#1436C2214yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—44no—
90C2×#121—24no—
91C2×#121—42no—
92C2×#121—42no—
93C2×#120—28no—
94C2×#120—28no—
95C2×#120—44no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—28no—
110C22×C22⋊C4—48no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4—42no—
113C22×C22⋊C4—42no—
114C22×C22⋊C4—42no—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C42C2312yes—
138#399—28no—
139#399—28no—
140#399—28no—
141#399—28no—
142#399—24no—
143#399—24no—
144#394—48no—
145#394—48no—
146#392—24no—
147#392—24no—
148#390—28no—
149#390—28no—
150#387—48no—
151#386—48no—
152#386—44no—
153#386—44no—
154#386—44no—
155#386—44no—
156#385—28no—
157#381—44no—
158#379—42no—
159C4×C22⋊C4—24no—
160C2×#95—24no—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C25—24no—
181C25—24no—
182C25—22no—
183C25—24no—
184C25—22no—
185C25—44no—
186C25—42no—
187C25C2×D414yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248#123—432no—
249#121—416no—
250#120—432no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
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—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—432no—
276C2×C4⋊C4—416no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
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—24no—
288C2×C22⋊C4—24no—
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—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—432no—
321C2×C22⋊C4—416no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4#4118yes—
341C2×C42—24no—
342C2×C42—44no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—28no—
372C24—22no—
373C24—24no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—28no—
380C24—216no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—48no—
393C24—44no—
394C24—416no—
395C24—48no—
396C24—48no—
397C24—416no—
398C24—48no—
399C24—44no—
400C24—44no—
401C24—88no—
402C24—88no—
403C24—84no—
404C24—84no—
405C24C22×D412yes—
406C24C22×D412yes—
407C24C22×D412yes—
408C24C2×#4114yes—
409C24H(5,𝔽2)12yes—
410C24H(5,𝔽2)12yes—
411C24H(5,𝔽2)14yes—
412C242−1+414yes—
413C24C2511yescharacteristic
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—416no—
417C2×D4—44no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—84no—
428C2×D4—88no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—84no—
434C2×D4—84no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—28no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—48no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—88no—
511C22×C4—88no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4—84no—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4114yes—
521C22×C4H(5,𝔽2)18yes—
522C4⋊C4—416no—
523C4⋊C4—416no—
524C4⋊C4—416no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C22⋊C4—216no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C42—84no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—28no—
568C23—24no—
569C23—24no—
570C23—28no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—416no—
577C23—432no—
578C23—42no—
579C23—41no—
580C23—44no—
581C23—41no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—416no—
587C23—41no—
588C23—44no—
589C23—44no—
590C23—41no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—88no—
621C23—816no—
622C23—816no—
623C23—816no—
624C23—832no—
625C23#53414yes—
626C23#53614yes—
627C23#54018yes—
628C23#54618yes—
629C23#55914yes—
630C23#56014yes—
631C23C23×D411yescharacteristic
632C23C22×#4112yes—
633C23C2×H(5,𝔽2)11yescharacteristic
634C23C2×H(5,𝔽2)11yescharacteristic
635C23C2×H(5,𝔽2)12yes—
636C23C2×H(5,𝔽2)14yes—
637C23C2×2−1+412yes—
638C23#58512yes—
639D4—816no—
640D4—816no—
641D4—88no—
642D4—816no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—432no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—416no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—88no—
709C2×C4—816no—
710C2×C4—816no—
711C22—28no—
712C22—28no—
713C22—24no—
714C22—24no—
715C22—28no—
716C22—216no—
717C22—28no—
718C22—24no—
719C22—28no—
720C22—44no—
721C22—44no—
722C22—42no—
723C22—48no—
724C22—42no—
725C22—44no—
726C22—42no—
727C22—42no—
728C22—44no—
729C22—42no—
730C22—44no—
731C22—832no—
732C22—82no—
733C22—82no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—84no—
742C22—84no—
743C22—82no—
744C22—82no—
745C22—82no—
746C22#270918yes—
747C22C2×#53411yescharacteristic
748C22C2×#53611yescharacteristic
749C22#344911yescharacteristic
750C22C2×#54012yes—
751C22C2×#54612yes—
752C22C2×#55911yescharacteristic
753C22#348412yes—
754C22#351314yes—
755C22#352014yes—
756C22C2×#56011yescharacteristic
757C22#353414yes—
758C22#353512yes—
759C22#353512yes—
760C22#353614yes—
761C22#354312yes—
762C22#354312yes—
763C4—42no—
764C4—42no—
765C4—44no—
766C4—42no—
767C4—42no—
768C4—48no—
769C4—44no—
770C4—84no—
771C4—84no—
772C4—84no—
773C4—82no—
774C2—28no—
775C2—44no—
776C2—82no—
777C2—162no—
778C2#3723211yescharacteristic
779C2#3744711yescharacteristic
780C2#3802112yes—
781C2#3890012yes—
782C2#4084614yes—
783C2#4328812yes—
784C2C2×#270911yescharacteristic
785C2#5254812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#399C2semidirect
3. #17168C2semidirect
4. #17088C2semidirect
5. #16824C2semidirect
6. #16338C2semidirect
7. #16277C2semidirect
8. #15334C2semidirect
9. #14912C2semidirect
10. #14719C2semidirect
11. #14718C2semidirect
12. #14256C2semidirect
13. #14134C2semidirect
14. #14039C2semidirect
15. #13869C2semidirect
33. C2×#394C22semidirect
38. C2×#387C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
60. #1440C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
69. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
C22×#3992
#171682
#170882
#168242
#163382
#162772
#153342
#149122
#147192
#147182
#142562
#141342
#140392
#138692