Finite Group Databasebeta

← #705518

Subgroups — #705518

Order 512.

3198 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2339C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3#16699C212yesmaximal, complemented
4#16699C212yesmaximal, complemented
5#15954C214yesmaximal, complemented
6#15952C214yesmaximal, complemented
7#15192C212yesmaximal, complemented
8#15186C212yesmaximal, complemented
9#15015C212yesmaximal, complemented
10#14936C212yesmaximal, complemented
11#14264C211yesmaximal, characteristic, complemented
12#14018C212yesmaximal, complemented
13#13937C211yesmaximal, characteristic
14#13924C211yesmaximal, characteristic, complemented
15#13916C212yesmaximal, complemented
16#13904C211yesmaximal, characteristic, complemented
17#13844C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2339—28no—
24#2311—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#381C2212yes—
37C2×#381C2214yes—
38C2×#381C2214yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C22×#95C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2212yes—
53#1444C2211yescharacteristic
54#1444C2211yescharacteristic
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2211yescharacteristic
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C22×#41—28no—
86C22×#41—24no—
87C22×#41—24no—
88C22×#41—44no—
89C22×#41D418yes—
90C23×Q8C2311yescharacteristic
91C23×D4C2312yes—
92C23×D4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2312yes—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—28no—
101C2×C4×D4—44no—
102C2×#117—24no—
103C2×#117—44no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#391—28no—
134#391—24no—
135#391—24no—
136#391—44no—
137#390—28no—
138#390—28no—
139#389—44no—
140#386—216no—
141#386—28no—
142#386—28no—
143#385—48no—
144#381—28no—
145#381—48no—
146#381—48no—
147#380—28no—
148#380—28no—
149#380—24no—
150#380—24no—
151C4×C4⋊C4—44no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—24no—
154C2×#95—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C25C2412yes—
188C2×#41—232no—
189C2×#41—232no—
190C2×#41—416no—
191C2×#41—432no—
192C2×#41—432no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8C2×D414yes—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—44no—
205C22×D4C2×D418yes—
206C22×D4C2×D414yes—
207C22×D4#4118yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D418yes—
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C4×D4—416no—
249C4×D4—416no—
250#117—416no—
251C2×C4⋊C4—24no—
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—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24C22×D412yes—
370C24C22×D414yes—
371C24C2×#4118yes—
372C24C2×#4112yes—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375#41—4128no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—48no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4#123116yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—216no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—88no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—88no—
490C22×C4#123116yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—88no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C22⋊C4—84no—
522C22⋊C4—88no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C42—44no—
526C42—48no—
527C42—44no—
528C42—48no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23C2×#12318yes—
574C23C2×#12314yes—
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)12yes—
581C23C2×2−1+411yescharacteristic
582C23C2×2−1+411yescharacteristic
583C23#58511yescharacteristic
584C23#58511yescharacteristic
585C23#58512yes—
586Q8—416no—
587D4—232no—
588D4—432no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—232no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—432no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—816no—
673C2×C4—816no—
674C2×C4—816no—
675C2×C4C2×#12314yes—
676C2×C4C2×#12314yes—
677C2×C4C2×#12314yes—
678C2×C4#53414yes—
679C2×C4#55118yes—
680C2×C4#55118yes—
681C22—28no—
682C22—24no—
683C22—28no—
684C22—24no—
685C22—28no—
686C22—24no—
687C22—44no—
688C22—48no—
689C22—44no—
690C22—48no—
691C22—44no—
692C22—48no—
693C22—44no—
694C22—48no—
695C22—42no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22—44no—
703C22C22×#12311yescharacteristic
704C22C22×#12311yescharacteristic
705C22C2×#53411yescharacteristic
706C22#346111yescharacteristic
707C22#346111yescharacteristic
708C22C2×#55112yes—
709C22C2×#55112yes—
710C22#348911yescharacteristic
711C22#348911yescharacteristic
712C22#349412yes—
713C22#349811yescharacteristic
714C22#350011yescharacteristic
715C22#350612yes—
716C22#350911yescharacteristic
717C22#351011yescharacteristic
718C22#351312yes—
719C22#351512yes—
720C22#352012yes—
721C22#352112yes—
722C22#353612yes—
723C22#353912yes—
724C22#354612yes—
725C22#354712yes—
726C4—24no—
727C4—44no—
728C4—42no—
729C4—44no—
730C4—42no—
731C4—82no—
732C4—84no—
733C4—82no—
734C4—84no—
735C4—81no—
736C4—82no—
737C4—82no—
738C4—81no—
739C4—82no—
740C4#248318yes—
741C2—48no—
742C2—44no—
743C2C2×#248311yescharacteristic
744C2#3530711yescharacteristic
745C2#3553111yescharacteristic
746C2#3598711yescharacteristic
747C2#3606711yescharacteristic
748C2#3820012yes—
749C2#3904012yes—
750C2#4298811yescharacteristic
751C2#4299211yescharacteristic
752C2#4536612yes—
753C2#4536812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16699C2semidirect
4. #16699C2semidirect
5. #15954C2semidirect
6. #15952C2semidirect
7. #15192C2semidirect
8. #15186C2semidirect
9. #15015C2semidirect
10. #14936C2semidirect
11. #14264C2semidirect
12. #14018C2semidirect
14. #13924C2semidirect
15. #13916C2semidirect
16. #13904C2semidirect
17. #13844C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23392
C2×#23112
#166992
#159542
#159522
#151922
#151862
#150152
#149362
#142642
#140182
#139372
#139242
#139162
#139042
#138442