Finite Group Databasebeta

← #1409923

Subgroups — #1409923

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2491C211yesmaximal, characteristic
3#17066C212yesmaximal, complemented
4#16558C212yesmaximal, complemented
5#16378C214yesmaximal, complemented
6#16371C214yesmaximal, complemented
7#15229C212yesmaximal, complemented
8#15216C212yesmaximal, complemented
9#15195C211yesmaximal, characteristic, complemented
10#15147C211yesmaximal, characteristic, complemented
11#15136C211yesmaximal, characteristic, complemented
12#14894C212yesmaximal, complemented
13#14273C212yesmaximal, complemented
14#14053C212yesmaximal, complemented
15#13998C212yesmaximal
16C4×#397C211yesmaximal, characteristic, complemented
17#13810C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×#124C2212yes—
21C22×#121C2211yescharacteristic
22#2552—28no—
23#2491—28no—
24C2×#398C2214yes—
25C2×#397C2211yescharacteristic
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30C2×#391C2214yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2214yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49#1446C2214yes—
50#1445C2211yescharacteristic
51#1443C2212yes—
52#1442C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1440C2212yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2211yescharacteristic
63#1439C2212yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—28no—
91C2×#125—28no—
92C2×#125—44no—
93C2×#125—44no—
94C2×#124—28no—
95C2×#124—28no—
96C2×#124—44no—
97C2×#121—24no—
98C2×#121—28no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#398—216no—
134#397—24no—
135#397—48no—
136#397—44no—
137#394—28no—
138#393—28no—
139#391—48no—
140#390—24no—
141#390—24no—
142#390—48no—
143#390—44no—
144#390—44no—
145#389—28no—
146#389—28no—
147#388—28no—
148#384—28no—
149#384—24no—
150#380—24no—
151#380—44no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—44no—
157#376—44no—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—44no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—416no—
223#125—432no—
224#124—432no—
225#121—416no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
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—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4118yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4#4114yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C2×D418yes—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24C2×#4112yes—
367C24C2511yescharacteristic
368C2×Q8—48no—
369C2×Q8—416no—
370C2×Q8—48no—
371C2×Q8—84no—
372C2×Q8—88no—
373C2×D4—48no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—84no—
382C2×D4—88no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+412yes—
476C22×C42−1+414yes—
477C4⋊C4—28no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—88no—
488C4⋊C4—84no—
489C4⋊C4#124116yes—
490C22⋊C4—28no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—84no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—28no—
506C42—216no—
507C42—416no—
508C42—44no—
509C42—48no—
510C42—48no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23C23×D411yescharacteristic
558C23C22×#4112yes—
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+412yes—
565C23C2×2−1+411yescharacteristic
566C23#58512yes—
567C23#58511yescharacteristic
568Q8—816no—
569D4—816no—
570D4—88no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—88no—
654C2×C4C2×#12418yes—
655C2×C4C2×#12414yes—
656C2×C4#53514yes—
657C2×C4#53614yes—
658C2×C4#54218yes—
659C2×C4#54218yes—
660C2×C4#54614yes—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—81no—
667C22—81no—
668C22—82no—
669C22—82no—
670C22—81no—
671C22—81no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—82no—
679C22—84no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22C22×#12411yescharacteristic
684C22C2×#53511yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#54212yes—
687C22C2×#54212yes—
688C22C2×#54611yescharacteristic
689C22#348112yes—
690C22#348711yescharacteristic
691C22#348911yescharacteristic
692C22#349811yescharacteristic
693C22#350611yescharacteristic
694C22#351111yescharacteristic
695C22#351512yes—
696C22#351612yes—
697C22#351912yes—
698C22#352412yes—
699C22#354212yes—
700C22#354412yes—
701C22#354611yescharacteristic
702C22#354612yes—
703C22#354712yes—
704C22#354811yescharacteristic
705C22#355912yes—
706C4—28no—
707C4—24no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—42no—
712C4—42no—
713C4—82no—
714C4—81no—
715C4—84no—
716C4—82no—
717C4—84no—
718C4—82no—
719C4—82no—
720C4#264018yes—
721C2—82no—
722C2—162no—
723C2C2×#264011yescharacteristic
724C2#3585411yescharacteristic
725C2#3628911yescharacteristic
726C2#3890711yescharacteristic
727C2#3947111yescharacteristic
728C2#4014712yes—
729C2#4015412yes—
730C2#4257912yes—
731C2#4258812yes—
732C2#4287611yescharacteristic
733C2#4296811yescharacteristic
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×#2552C2semidirect
3. #17066C2semidirect
4. #16558C2semidirect
5. #16378C2semidirect
6. #16371C2semidirect
7. #15229C2semidirect
8. #15216C2semidirect
9. #15195C2semidirect
10. #15147C2semidirect
11. #15136C2semidirect
12. #14894C2semidirect
13. #14273C2semidirect
14. #14053C2semidirect
16. C4×#397C2semidirect
17. #13810C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24912
#170662
#165582
#163782
#163712
#152292
#152162
#151952
#151472
#151362
#148942
#142732
#140532
#139982
C4×#3972
#138102