Finite Group Databasebeta

← #2737613

Subgroups — #2737613

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2426C211yesmaximal, characteristic
3#17119C214yesmaximal, complemented
4#17092C214yesmaximal, complemented
5#16814C212yesmaximal, complemented
6#16811C212yesmaximal, complemented
7#16803C212yesmaximal, complemented
8#15226C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#15157C211yesmaximal, characteristic, complemented
11#15155C211yesmaximal, characteristic, complemented
12#15137C212yesmaximal, complemented
13#15135C211yesmaximal, characteristic, complemented
14#15126C212yesmaximal
15#14628C212yesmaximal, complemented
16#13931C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2582—28no—
24#2426—28no—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#399C2214yes—
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#396C2212yes—
32C2×#394C2212yes—
33C2×#394C2214yes—
34C2×#393C2211yescharacteristic
35C2×#393C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2212yes—
43C2×#389C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#388C2212yes—
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#380C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1446C2211yescharacteristic
54#1445C2212yes—
55#1445C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2211yescharacteristic
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2211yescharacteristic
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#124—24no—
91C2×#124—44no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—44no—
94C2×#121—24no—
95C2×#121—44no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C2×C4×Q8D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#399—28no—
136#399—44no—
137#399—44no—
138#399—48no—
139#398—216no—
140#398—48no—
141#396—28no—
142#394—28no—
143#394—48no—
144#393—24no—
145#393—44no—
146#391—28no—
147#391—28no—
148#391—44no—
149#390—24no—
150#390—28no—
151#390—44no—
152#390—44no—
153#389—28no—
154#389—24no—
155#389—28no—
156#389—44no—
157#388—28no—
158#384—24no—
159#384—44no—
160#380—44no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—44no—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C25C2412yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—42no—
199C22×Q8C2×D414yes—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—44no—
208C22×D4—42no—
209C22×D4—44no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#124—416no—
231C22⋊Q8—416no—
232#121—416no—
233C4×Q8—416no—
234C4×Q8—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
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×D418yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
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×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D414yes—
324C2×C42C2×D418yes—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—24no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95#4114yes—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C242−1+412yes—
372C24C2511yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+414yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—28no—
516C42—48no—
517C42—48no—
518C42—44no—
519C42—48no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+411yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23C2×2−1+412yes—
575C23C2×2−1+411yescharacteristic
576C23#58511yescharacteristic
577Q8—416no—
578Q8—416no—
579D4—816no—
580D4—88no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
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—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—88no—
663C2×C4#53414yes—
664C2×C4#53514yes—
665C2×C4#53614yes—
666C2×C4#55014yes—
667C2×C4#56218yes—
668C2×C4#56218yes—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—82no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—84no—
690C22—82no—
691C22C2×#53411yescharacteristic
692C22C2×#53511yescharacteristic
693C22C2×#53611yescharacteristic
694C22C2×#55011yescharacteristic
695C22#348611yescharacteristic
696C22#348911yescharacteristic
697C22#349811yescharacteristic
698C22#350812yes—
699C22#351511yescharacteristic
700C22#352012yes—
701C22#352311yescharacteristic
702C22C2×#56212yes—
703C22C2×#56212yes—
704C22#353612yes—
705C22#353612yes—
706C22#354312yes—
707C22#354412yes—
708C22#354612yes—
709C22#355012yes—
710C22#355211yescharacteristic
711C22#355212yes—
712C22#355412yes—
713C22#355411yescharacteristic
714C4—24no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—42no—
720C4—44no—
721C4—81no—
722C4—81no—
723C4—81no—
724C4—82no—
725C4—84no—
726C4—84no—
727C4—82no—
728C4#281618yes—
729C2—82no—
730C2—162no—
731C2C2×#281611yescharacteristic
732C2#3913311yescharacteristic
733C2#3913911yescharacteristic
734C2#3974511yescharacteristic
735C2#4278911yescharacteristic
736C2#4281511yescharacteristic
737C2#4298811yescharacteristic
738C2#4567512yes—
739C2#4610412yes—
740C2#5296412yes—
741C2#5299212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2582C2semidirect
3. #17119C2semidirect
4. #17092C2semidirect
5. #16814C2semidirect
6. #16811C2semidirect
7. #16803C2semidirect
8. #15226C2semidirect
9. #15182C2semidirect
10. #15157C2semidirect
11. #15155C2semidirect
12. #15137C2semidirect
13. #15135C2semidirect
15. #14628C2semidirect
16. #13931C2semidirect
17. C4×#393C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#24262
#171192
#170922
#168142
#168112
#168032
#152262
#151822
#151572
#151552
#151372
#151352
#151262
#146282
#139312
C4×#3932