Finite Group Databasebeta

← #628173

Subgroups — #628173

Order 512.

3224 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic, complemented
3#15697C214yesmaximal, complemented
4#15690C214yesmaximal, complemented
5#15636C214yesmaximal, complemented
6#15631C214yesmaximal, complemented
7#14945C212yesmaximal, complemented
8#14873C212yesmaximal, complemented
9#14702C211yesmaximal, characteristic, complemented
10#14698C211yesmaximal, characteristic
11#14281C211yesmaximal, characteristic, complemented
12#14271C211yesmaximal, characteristic
13#14020C212yesmaximal, complemented
14#14019C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2639—28no—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#394C2212yes—
25C2×#393C2212yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#391—28no—
29C2×#391C2214yes—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42—24no—
120C22×C42—44no—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124#400—28no—
125#397—28no—
126#394—28no—
127#393—28no—
128#393—48no—
129#393—48no—
130#391—216no—
131#391—216no—
132#391—44no—
133#390—44no—
134#388—48no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—44no—
140#386—44no—
141#380—24no—
142#380—24no—
143C4×C22⋊C4—44no—
144C2×#95—24no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95D418yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×Q8—232no—
178C22×Q8—28no—
179C22×Q8—22no—
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—28no—
185C22×Q8—28no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8—432no—
193C22×Q8—432no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4—42no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4#4114yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C22⋊Q8—416no—
255#121—416no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
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—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
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×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—416no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C2×D418yes—
343#95C2×D418yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2×#4112yes—
362C24C2×#4112yes—
363C24C2×#4112yes—
364C24H(5,𝔽2)12yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×Q8—232no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—4128no—
373C2×Q8—416no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—416no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—84no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—84no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4#124116yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C42−1+414yes—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—48no—
521C42—48no—
522C42—88no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23#52514yes—
563C23C2×#12414yes—
564C23#53714yes—
565C23#55614yes—
566C23#55614yes—
567C23#55914yes—
568C23#55914yes—
569C23C23×D411yescharacteristic
570C23C22×#4111yescharacteristic
571C23C22×#4111yescharacteristic
572C23C22×#4111yescharacteristic
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+411yescharacteristic
578C23C2×2−1+412yes—
579C23#58512yes—
580Q8—464no—
581Q8—88no—
582D4—88no—
583D4—816no—
584C2×C4—232no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
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—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
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—88no—
662C2×C4—88no—
663C2×C4C2×#12414yes—
664C2×C4C2×#12414yes—
665C2×C4#53818yes—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—28no—
670C22—28no—
671C22—24no—
672C22—24no—
673C22—28no—
674C22—28no—
675C22—24no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—84no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22—84no—
686C22—82no—
687C22#259218yes—
688C22C2×#52511yescharacteristic
689C22C22×#12411yescharacteristic
690C22C2×#53711yescharacteristic
691C22C2×#53812yes—
692C22#346112yes—
693C22C2×#55611yescharacteristic
694C22C2×#55611yescharacteristic
695C22C2×#55911yescharacteristic
696C22C2×#55911yescharacteristic
697C22#348411yescharacteristic
698C22#348411yescharacteristic
699C22#348511yescharacteristic
700C22#348611yescharacteristic
701C22#349012yes—
702C22#350512yes—
703C22#350512yes—
704C22#350712yes—
705C22#351112yes—
706C22#351112yes—
707C22#353612yes—
708C22#353812yes—
709C22#355812yes—
710C22#356112yes—
711C4—24no—
712C4—24no—
713C4—28no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—84no—
722C4—82no—
723C4—84no—
724C2—28no—
725C2—162no—
726C2C2×#259211yescharacteristic
727C2#3498811yescharacteristic
728C2#3509211yescharacteristic
729C2#3537711yescharacteristic
730C2#3559211yescharacteristic
731C2#3671911yescharacteristic
732C2#3695711yescharacteristic
733C2#3976012yes—
734C2#3976812yes—
735C2#3979612yes—
736C2#3980412yes—
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
2. C22×#391C2semidirect
3. #15697C2semidirect
4. #15690C2semidirect
5. #15636C2semidirect
6. #15631C2semidirect
7. #14945C2semidirect
8. #14873C2semidirect
9. #14702C2semidirect
11. #14281C2semidirect
13. #14020C2semidirect
14. #14019C2semidirect
15. #13778C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C22×#3912
#156972
#156902
#156362
#156312
#149452
#148732
#147022
#146982
#142812
#142712
#140202
#140192
#137782