Finite Group Databasebeta

← #647578

Subgroups — #647578

Order 512.

2820 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3#17042C212yesmaximal, complemented
4#17040C212yesmaximal, complemented
5#17009C212yesmaximal, complemented
6#16991C212yesmaximal, complemented
7#15108C212yesmaximal, complemented
8#15092C211yesmaximal, characteristic, complemented
9#15091C211yesmaximal, characteristic
10#14969C212yesmaximal, complemented
11#14919C212yesmaximal, complemented
12#14912C212yesmaximal
13#14786C211yesmaximal, characteristic, complemented
14#14776C211yesmaximal, characteristic
15#14684C211yesmaximal, characteristic, complemented
16#14314C212yesmaximal, complemented
17#14313C212yesmaximal, complemented
18#14039C212yesmaximal, complemented
19#14015C212yesmaximal
20C22×#123C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C42C2211yescharacteristic
26C2×#398C2212yes—
27C2×#397C2212yes—
28C4×#123—28no—
29C2×#389—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#387C2211yescharacteristic
37C2×#387C2211yescharacteristic
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2211yescharacteristic
44C2×#385C2212yes—
45C2×#385C2214yes—
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2214yes—
60#1444C2212yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2212yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2212yes—
80#1436C2214yes—
81#1436C2214yes—
82#1434C2212yes—
83#1434C2212yes—
84#1434C2212yes—
85#1434C2212yes—
86C22⋊#95C2212yes—
87C22⋊#95C2212yes—
88C22⋊#95C2212yes—
89C22⋊#95C2212yes—
90C22⋊#95C2211yescharacteristic
91C22⋊#95C2212yes—
92C22⋊#95C2211yescharacteristic
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2312yes—
99C24×C4C2311yescharacteristic
100C2×#123—24no—
101C2×#123—24no—
102C2×#123—24no—
103C2×C4×D4—24no—
104C2×C4×D4—44no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—48no—
110C22×C4⋊C4—44no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2314yes—
136C22×C42—44no—
137C22×C42D418yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#398—28no—
146#397—28no—
147#389—28no—
148#389—28no—
149#389—28no—
150#389—28no—
151#388—24no—
152#388—24no—
153#387—24no—
154#387—24no—
155#387—48no—
156#386—28no—
157#386—24no—
158#386—44no—
159#385—24no—
160#385—28no—
161#385—48no—
162#385—44no—
163#381—24no—
164#381—44no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—24no—
170C2×#95—28no—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C25C2411yescharacteristic
204C25C2411yescharacteristic
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4#4114yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—44no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254#123—232no—
255C4×D4—416no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—416no—
287C2×C4⋊C4—432no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4#4118yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4#4114yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—416no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338#95—28no—
339#95—28no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—28no—
357#95—24no—
358#95—24no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C22×D412yes—
383C24C2×#4112yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24H(5,𝔽2)14yes—
387C242−1+414yes—
388C24C2511yescharacteristic
389C2×D4—28no—
390C2×D4—216no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C22×C4—232no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—416no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4118yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4114yes—
504C22×C4H(5,𝔽2)14yes—
505C22×C42−1+414yes—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—416no—
517C4⋊C4—44no—
518C4⋊C4—88no—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—416no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—88no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—44no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23#52514yes—
568C23#53414yes—
569C23#53614yes—
570C23#53918yes—
571C23#54218yes—
572C23C23×D411yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+412yes—
579C23C2×2−1+412yes—
580C23#58512yes—
581D4—416no—
582C2×C4—232no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—416no—
611C2×C4—432no—
612C2×C4—416no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
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—44no—
659C2×C4—44no—
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—88no—
669C2×C4—816no—
670C2×C4#52514yes—
671C2×C4#53514yes—
672C2×C4#53614yes—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—216no—
677C22—24no—
678C22—216no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—48no—
685C22—81no—
686C22—81no—
687C22—84no—
688C22—81no—
689C22—81no—
690C22#263818yes—
691C22C2×#52511yescharacteristic
692C22C2×#52511yescharacteristic
693C22C2×#53411yescharacteristic
694C22C2×#53511yescharacteristic
695C22C2×#53611yescharacteristic
696C22C2×#53611yescharacteristic
697C22#344911yescharacteristic
698C22C2×#53912yes—
699C22C2×#54212yes—
700C22#348112yes—
701C22#348112yes—
702C22#349214yes—
703C22#351512yes—
704C22#351512yes—
705C22#351614yes—
706C22#353414yes—
707C22#355214yes—
708C4—24no—
709C4—24no—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—48no—
718C4—81no—
719C4—84no—
720C4—84no—
721C4—81no—
722C4—84no—
723C4—81no—
724C4—81no—
725C2—28no—
726C2—82no—
727C2C2×#263811yescharacteristic
728C2#3496311yescharacteristic
729C2#3505711yescharacteristic
730C2#3513811yescharacteristic
731C2#3524611yescharacteristic
732C2#3744011yescharacteristic
733C2#3761511yescharacteristic
734C2#3857314yes—
735C2#3857514yes—
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
2. C22×#389C2semidirect
3. #17042C2semidirect
4. #17040C2semidirect
5. #17009C2semidirect
6. #16991C2semidirect
7. #15108C2semidirect
8. #15092C2semidirect
10. #14969C2semidirect
11. #14919C2semidirect
13. #14786C2semidirect
15. #14684C2semidirect
16. #14314C2semidirect
17. #14313C2semidirect
18. #14039C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C22×#3892
#170422
#170402
#170092
#169912
#151082
#150922
#150912
#149692
#149192
#149122
#147862
#147762
#146842
#143142
#143132
#140392
#140152