Finite Group Databasebeta

← #560177

Subgroups — #560177

Order 512.

3710 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic
3C2×C4×#124C211yesmaximal, characteristic
4#14709C212yesmaximal, complemented
5#14664C212yesmaximal, complemented
6#14505C212yesmaximal
7#14475C214yesmaximal, complemented
8#14450C212yesmaximal, complemented
9#14364C214yesmaximal, complemented
10#14102C214yesmaximal, complemented
11#14052C214yesmaximal, complemented
12#14024C212yesmaximal, complemented
13#13968C212yesmaximal
14C2×#563—28no—
15C2×#563—28no—
16C2×#563C4116yescomplemented
17C22×C4⋊Q8C2212yes—
18C22×#126C2214yes—
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#2379—28no—
26C4×#124—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#382C2212yes—
37C2×#382C2211yescharacteristic
38C2×#381C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C22×#95C2212yes—
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C4×#95C2212yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78#563—2128no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8C2×C4116yescomplemented
82C2×#126—216no—
83C2×#126—216no—
84C2×#126C2×C4132yescomplemented
85C2×#125—28no—
86C2×#125—28no—
87C2×#125C2×C4116yescomplemented
88C2×#124C2×C418yes—
89C2×#124D418yes—
90C2×#124D418yes—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123C2×C4116yes—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8C2×C4132yescomplemented
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×D4—24no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#391—28no—
130#391—24no—
131#391—44no—
132#391—44no—
133#390—24no—
134#388—216no—
135#388—216no—
136#386—28no—
137#385—216no—
138#382—28no—
139#382—24no—
140#381—44no—
141C4×C22⋊C4—216no—
142C2×#95—44no—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C43—24no—
165C25C2412yes—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8C22×C418yes—
177C22×Q8C22×C414yes—
178C22×Q8C2×D414yes—
179C22×Q8C2×D414yes—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4C22×C414yes—
185C22×D4C2×D414yes—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4C22×C418yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—264no—
217#126—2128no—
218#125—264no—
219#124C22⋊C4164yes—
220#123—264no—
221C22⋊Q8—2128no—
222C4×Q8—416no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C4116yes—
254C2×C4⋊C4C22×C4116yes—
255C2×C4⋊C4C22×C4116yes—
256C2×C4⋊C4C22×C418yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4C22×C4116yes—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C4116yes—
276C2×C22⋊C4C2×D4116yes—
277C2×C22⋊C4C2×D4116yes—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C22×C418yes—
299C2×C42C22×C414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24C23×C414yes—
332C24C22×D414yes—
333C24C22×D414yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8C2×C22⋊C4116yes—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4C2×C22⋊C4116yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—84no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—88no—
434C22×C4C23×C418yes—
435C22×C4C23×C418yes—
436C22×C4C23×C414yes—
437C22×C4C23×C414yes—
438C22×C4C23×C412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—232no—
451C4⋊C4—232no—
452C4⋊C4—232no—
453C4⋊C4—216no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C22⋊C4C2×C22⋊C4164yes—
467C42—28no—
468C42—216no—
469C42—48no—
470C42—48no—
471C42—48no—
472C42—44no—
473C42—84no—
474C42C2×C22⋊C4116yes—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23C22×C22⋊C418yes—
498C23C24×C411yescharacteristic
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+412yes—
504C23C2×2−1+412yes—
505C23#58511yescharacteristic
506C23#58511yescharacteristic
507C23#58512yes—
508C23#58512yes—
509Q8—416no—
510Q8—816no—
511D4—416no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4—816no—
586C2×C4—816no—
587C2×C4C22×C22⋊C418yes—
588C2×C4C22×C22⋊C418yes—
589C2×C4C22×C22⋊C414yes—
590C2×C4#56314yescomplemented
591C22—28no—
592C22—216no—
593C22—44no—
594C22—48no—
595C22—44no—
596C22—44no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—48no—
602C22C23×C22⋊C411yescharacteristic
603C22C4×H(5,𝔽2)11yescharacteristic
604C22C4×H(5,𝔽2)11yescharacteristic
605C22C4×2−1+412yes—
606C22C4×2−1+412yes—
607C22#348411yescharacteristic
608C22#348511yescharacteristic
609C22#348912yes—
610C22#348912yes—
611C22#349611yescharacteristic
612C22#349611yescharacteristic
613C22#349912yes—
614C22#350012yes—
615C22C2×#56311yescharacteristic
616C22#352911yescharacteristic
617C22#353112yes—
618C22#354511yescharacteristic
619C22#355011yescharacteristic
620C22#355312yes—
621C22#355912yes—
622C22#356112yes—
623C22#356312yes—
624C22#356812yes—
625C4—24no—
626C4—24no—
627C4—28no—
628C4—28no—
629C4—44no—
630C4—42no—
631C4—48no—
632C4—48no—
633C4—82no—
634C4—84no—
635C4—82no—
636C4—84no—
637C2—48no—
638C2#3425611yescharacteristic
639C2#3425912yes—
640C2C4×#56311yescharacteristic
641C2#3452311yescharacteristic
642C2#3452812yes—
643C2#4613212yes—
644C2#4613611yescharacteristic
645C2#4614711yescharacteristic
646C2#4614912yes—
647C2#4622811yescharacteristic
648C2#4623711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14709C2semidirect
5. #14664C2semidirect
7. #14475C2semidirect
8. #14450C2semidirect
9. #14364C2semidirect
10. #14102C2semidirect
11. #14052C2semidirect
12. #14024C2semidirect
16. C2×#563C4semidirect
81. C2×C4⋊Q8C2×C4semidirect
84. C2×#126C2×C4semidirect
87. C2×#125C2×C4semidirect
96. C2×C22⋊Q8C2×C4semidirect
590. C2×C4#563semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#23792
C2×C4×#1242
#147092
#146642
#145052
#144752
#144502
#143642
#141022
#140522
#140242
#139682