Finite Group Databasebeta

← #2737597

Subgroups — #2737597

Order 512.

3118 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C212yesmaximal, complemented
2#17224C212yesmaximal, complemented
3#17218C212yesmaximal, complemented
4#17217C214yesmaximal, complemented
5#16808C212yesmaximal
6#16798C214yesmaximal, complemented
7#15186C212yesmaximal, complemented
8#15182C212yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#15177C212yesmaximal, complemented
11#15150C211yesmaximal, characteristic, complemented
12#15129C212yesmaximal, complemented
13#14627C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic, complemented
15C4×#392C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2409—216no—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#395C2214yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#392C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42#1446C2212yes—
43#1446C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1445C2214yes—
47#1444C2211yescharacteristic
48#1444C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2211yescharacteristic
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2211yescharacteristic
68#1438C2212yes—
69#1438C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×Q8—44no—
77C23×Q8—44no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8D418yescomplemented
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8—44no—
94C2×C4×D4—28no—
95C2×C4×D4—44no—
96C2×C4×D4—42no—
97C2×C4×D4—44no—
98C2×C4×D4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#397—28no—
126#397—44no—
127#395—216no—
128#394—48no—
129#393—216no—
130#392—24no—
131#391—28no—
132#391—28no—
133#391—44no—
134#391—44no—
135#389—44no—
136#386—216no—
137#385—216no—
138#384—42no—
139#380—28no—
140#380—44no—
141#380—44no—
142#380—42no—
143C4×C22⋊C4—42no—
144C4×C22⋊C4—44no—
145C2×#95C2311yescharacteristic
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—432no—
184C22×Q8—416no—
185C22×Q8—416no—
186C22×Q8—432no—
187C22×Q8—416no—
188C22×Q8C2×D418yes—
189C22×Q8C2×D418yes—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—41no—
200C22×D4—42no—
201C22×D4—41no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D418yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C22⋊Q8—432no—
233C22⋊Q8—432no—
234C4×D4—432no—
235C4×D4—416no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
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—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4#4114yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—41no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—41no—
325#95—41no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24H(5,𝔽2)12yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—416no—
368C2×Q8—48no—
369C2×Q8—4128no—
370C2×Q8—48no—
371C2×Q8—416no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—216no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—48no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4C22×D412yes—
475C22×C4C22×D418yes—
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+414yes—
484C22×C42−1+412yes—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—88no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—88no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C42—48no—
509C42—44no—
510C42—44no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23#55014yes—
569C23C23×D411yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)12yes—
574C23C2×H(5,𝔽2)12yes—
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×2−1+412yes—
577C23C2×2−1+412yes—
578C23C2×2−1+411yescharacteristic
579C23#58511yescharacteristic
580Q8—432no—
581Q8—432no—
582D4—432no—
583D4—416no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—432no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—416no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—816no—
661C2×C4#53418yes—
662C2×C4#53414yes—
663C2×C4#56014yes—
664C2×C4#56018yes—
665C2×C4H(3,𝔽4)14yes—
666C22—24no—
667C22—28no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22C2×#53412yes—
693C22C2×#53411yescharacteristic
694C22C2×#55011yescharacteristic
695C22#348612yes—
696C22#349711yescharacteristic
697C22#350511yescharacteristic
698C22#351111yescharacteristic
699C22#352012yes—
700C22#352012yes—
701C22C2×#56011yescharacteristic
702C22C2×#56012yes—
703C22C2×H(3,𝔽4)11yescharacteristic
704C22#353611yescharacteristic
705C22#353612yes—
706C22#353612yes—
707C22#353812yes—
708C22#353811yescharacteristic
709C22#354112yes—
710C22#354712yes—
711C22#355011yescharacteristic
712C22#355012yes—
713C22#355212yes—
714C22#355811yescharacteristic
715C4—42no—
716C4—42no—
717C4—42no—
718C4—48no—
719C4—42no—
720C4—44no—
721C4—82no—
722C4—82no—
723C4—84no—
724C4—84no—
725C4—81no—
726C4—84no—
727C4#281218yes—
728C2—44no—
729C2—84no—
730C2C2×#281211yescharacteristic
731C2#3913312yes—
732C2#3974311yescharacteristic
733C2#4278612yes—
734C2#4280711yescharacteristic
735C2#4566711yescharacteristic
736C2#4567712yes—
737C2#4634211yescharacteristic
738C2#5295811yescharacteristic
739C2#5296212yes—
740C2#5299311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2409C2semidirect
2. #17224C2semidirect
3. #17218C2semidirect
4. #17217C2semidirect
6. #16798C2semidirect
7. #15186C2semidirect
8. #15182C2semidirect
9. #15179C2semidirect
10. #15177C2semidirect
11. #15150C2semidirect
12. #15129C2semidirect
13. #14627C2semidirect
14. #13922C2semidirect
80. C23×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#172242
#172182
#172172
#168082
#167982
#151862
#151822
#151792
#151772
#151502
#151292
#146272
#139222
C4×#3922