Finite Group Databasebeta

← #4452779

Subgroups — #4452779

Order 512.

2893 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2413C211yesmaximal, characteristic
2C2×#2623C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic, complemented
4C2×#2316C211yesmaximal, characteristic, complemented
5#17070C212yesmaximal, complemented
6#17040C212yesmaximal, complemented
7#16992C212yesmaximal, complemented
8#16821C212yesmaximal, complemented
9#16594C214yesmaximal, complemented
10#16550C214yesmaximal, complemented
11#14935C212yesmaximal, complemented
12#14927C212yesmaximal
13#14729C212yesmaximal
14#14728C212yesmaximal, complemented
15#14135C211yesmaximal, characteristic
16#13962C211yesmaximal, characteristic, complemented
17#13870C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25#2623—28no—
26#2413—28no—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#398—28no—
30C2×#398C2211yescharacteristic
31C2×#398C2211yescharacteristic
32C2×#398C2212yes—
33C2×#398C2212yes—
34C2×#398C2214yes—
35C2×#398C2212yes—
36C2×#398C2214yes—
37C2×#397C2211yescharacteristic
38C2×#397C2211yescharacteristic
39C2×#396C2212yes—
40C2×#396C2214yes—
41#2316—28no—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#388C2212yes—
47C2×#386C2212yes—
48C2×#386C2211yescharacteristic
49C2×#386C2212yes—
50C2×#385C2212yes—
51C2×#385C2211yescharacteristic
52C2×#385C2212yes—
53C2×#381C2211yescharacteristic
54C2×#381C2211yescharacteristic
55C2×#381C2212yes—
56C2×#380C2212yes—
57C2×#380C2212yes—
58C22×#95C2211yescharacteristic
59C22×#95C2212yes—
60#1444C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1440C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2214yes—
79#1437C2212yes—
80#1436C2214yes—
81#1436C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×#123—44no—
98C2×#117—24no—
99C2×#117—44no—
100C2×#117—44no—
101C2×#117D418yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—48no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142#400—28no—
143#399—28no—
144#398—24no—
145#398—24no—
146#398—28no—
147#398—28no—
148#398—216no—
149#398—28no—
150#398—48no—
151#397—24no—
152#397—24no—
153#396—28no—
154#396—48no—
155#389—216no—
156#389—28no—
157#389—28no—
158#389—28no—
159#388—28no—
160#386—28no—
161#386—24no—
162#386—44no—
163#385—28no—
164#385—24no—
165#385—44no—
166#381—24no—
167#381—24no—
168#381—44no—
169#380—28no—
170#380—28no—
171C2×#95—24no—
172C2×#95—28no—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C25C2411yescharacteristic
201C25C2411yescharacteristic
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D418yes—
231C23×C4#4114yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247#123—416no—
248#123—416no—
249#117—416no—
250#117—416no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—432no—
290C2×C4⋊C4—416no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D418yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
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⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D418yes—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4112yes—
379C24H(5,𝔽2)14yes—
380C242−1+414yes—
381C24C2511yescharacteristic
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—48no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4#121116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+418yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C42—48no—
523C42—48no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×#12114yes—
552C23#53414yes—
553C23#53614yes—
554C23#53718yes—
555C23#53918yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4111yescharacteristic
559C23C2×H(5,𝔽2)12yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23C2×2−1+414yes—
563C23#58512yes—
564D4—88no—
565C2×C4—232no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—432no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—816no—
647C2×C4—88no—
648C2×C4C2×#12114yes—
649C2×C4C2×#12114yes—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—216no—
657C22—81no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—81no—
665C22#261818yes—
666C22C22×#12111yescharacteristic
667C22C2×#53411yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#53712yes—
670C22C2×#53912yes—
671C22#348411yescharacteristic
672C22#348511yescharacteristic
673C22#348612yes—
674C22#348812yes—
675C22#349412yes—
676C22#349912yes—
677C22#350912yes—
678C22#353614yes—
679C22#353814yes—
680C22#355914yes—
681C22#356314yes—
682C4—24no—
683C4—24no—
684C4—48no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—81no—
694C4—84no—
695C4—82no—
696C4—81no—
697C2—28no—
698C2—161no—
699C2C2×#261811yescharacteristic
700C2#3551411yescharacteristic
701C2#3566711yescharacteristic
702C2#3671912yes—
703C2#3696812yes—
704C2#4193712yes—
705C2#4193912yes—
706C2#4196514yes—
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. C22×#398C2semidirect
4. C2×#2316C2semidirect
5. #17070C2semidirect
6. #17040C2semidirect
7. #16992C2semidirect
8. #16821C2semidirect
9. #16594C2semidirect
10. #16550C2semidirect
11. #14935C2semidirect
14. #14728C2semidirect
16. #13962C2semidirect
17. #13870C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
C2×#26232
C22×#3982
C2×#23162
#170702
#170402
#169922
#168212
#165942
#165502
#149352
#149272
#147292
#147282
#141352
#139622
#138702