Finite Group Databasebeta

← #2736550

Subgroups — #2736550

Order 512.

2550 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C212yesmaximal, complemented
2C2×#2538C211yesmaximal, characteristic
3C2×#2452C211yesmaximal, characteristic
4#17124C214yesmaximal, complemented
5#17122C214yesmaximal, complemented
6#16601C212yesmaximal, complemented
7#16394C212yesmaximal, complemented
8#16366C212yesmaximal, complemented
9#15158C211yesmaximal, characteristic, complemented
10#15153C211yesmaximal, characteristic, complemented
11#15152C211yesmaximal, characteristic, complemented
12#15147C212yesmaximal
13#15139C212yesmaximal, complemented
14#15138C212yesmaximal
15#15128C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2817—216no—
24#2538—28no—
25#2452—28no—
26C2×#400C2212yes—
27C2×#400C2214yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#396C2214yes—
33C2×#396C2212yes—
34C2×#395C2214yes—
35C2×#395C2212yes—
36C2×#395C2211yescharacteristic
37C2×#393C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#385C2212yes—
47C2×#384C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52#1446C2214yes—
53#1446C2211yescharacteristic
54#1445C2212yes—
55#1445C2214yes—
56#1444C2211yescharacteristic
57#1444C2214yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2211yescharacteristic
67#1441C2212yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×C4×Q8—24no—
99C2×C4×Q8—44no—
100C2×C4×D4—44no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—28no—
135#400—216no—
136#400—28no—
137#400—44no—
138#397—216no—
139#397—216no—
140#396—28no—
141#396—216no—
142#395—28no—
143#395—216no—
144#395D418yescomplemented
145#393—28no—
146#390—24no—
147#390—24no—
148#390—28no—
149#390—44no—
150#389—24no—
151#389—28no—
152#389—24no—
153#389—44no—
154#385—28no—
155#384—24no—
156#381—28no—
157#380—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—44no—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C25C2412yes—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C2×D414yes—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4C2×D414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D418yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227C22⋊Q8—432no—
228C22⋊Q8—432no—
229C4×Q8—416no—
230C4×D4—416no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
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—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
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—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D418yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42C2×D414yes—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—22no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D414yes—
346#95C2×D418yes—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24C22×D414yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—84no—
367C2×Q8—88no—
368C2×Q8—88no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—48no—
373C2×D4—48no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—216no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
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—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C42−1+414yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—88no—
499C22⋊C4—84no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—28no—
503C42—48no—
504C42—44no—
505C42—44no—
506C42—48no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+412yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58512yes—
539C23#58511yescharacteristic
540C23#58512yes—
541Q8—416no—
542Q8—816no—
543D4—416no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
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—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—816no—
635C2×C4#53414yes—
636C2×C4#53614yes—
637C2×C4#53614yes—
638C2×C4D4214yes—
639C2×C4#56318yes—
640C2×C4#56318yes—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22C2×#53411yescharacteristic
653C22C2×#53611yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×D4211yescharacteristic
656C22#348511yescharacteristic
657C22#348911yescharacteristic
658C22#349612yes—
659C22#349712yes—
660C22#349711yescharacteristic
661C22#349911yescharacteristic
662C22#350011yescharacteristic
663C22C2×#56312yes—
664C22C2×#56312yes—
665C22#354112yes—
666C22#354212yes—
667C22#354711yescharacteristic
668C22#355011yescharacteristic
669C22#355112yes—
670C22#355412yes—
671C22#355512yes—
672C22#355712yes—
673C22#355712yes—
674C22#356512yes—
675C4—24no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4#281718yescomplemented
691C2—84no—
692C2C2×#281711yescharacteristic
693C2#3933811yescharacteristic
694C2#3934311yescharacteristic
695C2#3936311yescharacteristic
696C2#4264711yescharacteristic
697C2#4267611yescharacteristic
698C2#4290811yescharacteristic
699C2#4548512yes—
700C2#4632912yes—
701C2#5311112yes—
702C2#5311312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2817C2semidirect
4. #17124C2semidirect
5. #17122C2semidirect
6. #16601C2semidirect
7. #16394C2semidirect
8. #16366C2semidirect
9. #15158C2semidirect
10. #15153C2semidirect
11. #15152C2semidirect
13. #15139C2semidirect
15. #15128C2semidirect
17. C4×#395C2semidirect
144. #395D4semidirect
690. C4#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#25382
C2×#24522
#171242
#171222
#166012
#163942
#163662
#151582
#151532
#151522
#151472
#151392
#151382
#151282
#139162
C4×#3952