Finite Group Databasebeta

← #676652

Subgroups — #676652

Order 512.

4042 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3C2×#2567C211yesmaximal, characteristic, complemented
4C2×C4×#124C211yesmaximal, characteristic
5#15279C218yesmaximal, complemented
6#15238C212yesmaximal, complemented
7#15228C212yesmaximal, complemented
8#15179C214yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14130C214yesmaximal, complemented
11#14027C212yesmaximal
12#13977C212yesmaximal
13#13906C211yesmaximal, characteristic
14C4×#391C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28#2582—28no—
29#2567—28no—
30C2×#398C2214yes—
31C2×#395C2214yes—
32C4×#124—28no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2214yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#389C2212yes—
44C2×#388C2214yes—
45C2×#385C2218yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×#382C2211yescharacteristic
49C2×#380C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2214yes—
52C2×C43C2211yescharacteristic
53#1444C2218yes—
54#1442C2218yes—
55#1442C2218yes—
56#1441C2218yes—
57#1440C2218yes—
58#1439C2218yes—
59#1439C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C22⋊#95C2218yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75#554—2128no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8D418yes—
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—28no—
82C2×#124—24no—
83C2×#124—24no—
84C2×#124Q818yescomplemented
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D418yes—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D418yes—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#398—216no—
135#395—216no—
136#391—28no—
137#391—28no—
138#391—24no—
139#391—28no—
140#391—28no—
141#391—48no—
142#390—24no—
143#390—28no—
144#389—216no—
145#389—28no—
146#389—28no—
147#388—216no—
148#385—416no—
149#384—28no—
150#384—24no—
151#382—24no—
152#380—216no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—216no—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C43—24no—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8C2×D414yes—
186C22×Q8C2×D418yes—
187C22×Q8C2×Q814yes—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4C2×D414yes—
193C22×D4C2×Q814yes—
194C22×D4#4114yes—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4C2×D418yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219#125—264no—
220#124—232no—
221C22⋊Q8—264no—
222C22⋊Q8—2128no—
223C4×Q8—232no—
224C4×Q8—416no—
225C4×D4—264no—
226C4×D4—232no—
227C2×C4⋊C4—264no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4#4118yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4C2×Q8116yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—42no—
304C2×C42C2×D418yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×Q814yes—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—216no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—216no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—216no—
321#95—44no—
322#95—48no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24C22×D414yes—
332C24C22×Q814yes—
333C24C2×#4114yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—216no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—416no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4C22⋊Q8116yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—232no—
402C22×C4—216no—
403C22×C4—232no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—816no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—816no—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D418yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q812yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C42−1+414yes—
442C22×C42−1+412yes—
443C22×C42−1+412yes—
444C4⋊C4—232no—
445C4⋊C4—232no—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—84no—
457C4⋊C4—816no—
458C4⋊C4—88no—
459C22⋊C4—232no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—432no—
463C22⋊C4—416no—
464C42—216no—
465C42—28no—
466C42—48no—
467C42—48no—
468C42—44no—
469C42—48no—
470C42—48no—
471C42#120116yes—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23C2×C22⋊Q818yes—
495C23C23×D411yescharacteristic
496C23C23×D411yescharacteristic
497C23C23×D411yescharacteristic
498C23C23×Q811yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23#58512yes—
506Q8—416no—
507Q8—416no—
508Q8—816no—
509D4—232no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—216no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—432no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—48no—
573C2×C4—832no—
574C2×C4C2×#12014yes—
575C2×C4C2×#12014yes—
576C2×C4C2×#12014yes—
577C2×C4C2×C22⋊Q814yes—
578C2×C4#53214yes—
579C2×C4#53614yes—
580C2×C4#53614yes—
581C2×C4#54718yes—
582C2×C4#54814yes—
583C2×C4#54814yes—
584C2×C4#54818yes—
585C2×C4D4×Q814yes—
586C2×C4D4×Q814yes—
587C22—28no—
588C22—216no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22C22×#12011yescharacteristic
599C22C22×C22⋊Q811yescharacteristic
600C22C2×#53211yescharacteristic
601C22C2×#53611yescharacteristic
602C22C2×#53611yescharacteristic
603C22C2×#54712yes—
604C22C2×#54811yescharacteristic
605C22C2×#54812yes—
606C22C2×#54811yescharacteristic
607C22C2×D4×Q811yescharacteristic
608C22C2×D4×Q811yescharacteristic
609C22Q8×#4112yes—
610C22#348112yes—
611C22#348812yes—
612C22#349612yes—
613C22#349811yescharacteristic
614C22#350011yescharacteristic
615C22#350312yes—
616C22#351512yes—
617C22#351512yes—
618C22#352212yes—
619C22#354412yes—
620C22#355512yes—
621C4—24no—
622C4—44no—
623C4—48no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—84no—
631C4—82no—
632C4#240918yes—
633C4#243218yes—
634C2—48no—
635C2C2×#240911yescharacteristic
636C2C2×#243211yescharacteristic
637C2#3551311yescharacteristic
638C2#3627512yes—
639C2#3649611yescharacteristic
640C2#3656211yescharacteristic
641C2#3892112yes—
642C2#3917311yescharacteristic
643C2#3961711yescharacteristic
644C2#5965512yes—
645C2#5966012yes—
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
2. C2×#2582C2semidirect
3. C2×#2567C2semidirect
5. #15279C2semidirect
6. #15238C2semidirect
7. #15228C2semidirect
8. #15179C2semidirect
9. #14714C2semidirect
10. #14130C2semidirect
14. C4×#391C2semidirect
84. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25822
C2×#25672
C2×C4×#1242
#152792
#152382
#152282
#151792
#147142
#141302
#140272
#139772
#139062
C4×#3912