Finite Group Databasebeta

← #5021365

Subgroups — #5021365

Order 512.

4210 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#562C211yesmaximal, characteristic, complemented
2C2×#2851C211yesmaximal, characteristic, complemented
3C2×#2850C212yesmaximal, complemented
4#15985C214yesmaximal, complemented
5#15951C214yesmaximal, complemented
6#15397C214yesmaximal, complemented
7#15368C214yesmaximal, complemented
8#15338C214yesmaximal, complemented
9#15263C214yesmaximal, complemented
10#14664C211yesmaximal, characteristic, complemented
11#14662C212yesmaximal, complemented
12C2×#562—28no—
13C2×#562—216no—
14C22×#125C2211yescharacteristic
15C22×#124C2212yes—
16C22×#123C2214yes—
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2214yes—
20C22×#121C2212yes—
21C23×C42C2211yescharacteristic
22#2851—28no—
23#2850—216no—
24C2×#394C2214yes—
25C2×#392C2214yes—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40#1446C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C23×Q8C2312yes—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C23×D4C2312yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2314yes—
68#562—2128no—
69C2×#125—28no—
70C2×#125D418yescomplemented
71C2×#124—28no—
72C2×#124—28no—
73C2×#124D4116yescomplemented
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×#121—216no—
84C2×#121—28no—
85C2×#121—216no—
86C2×#121—216no—
87C2×#121—44no—
88C2×#121—44no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42—28no—
110C22×C42—24no—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#394—48no—
115#392—48no—
116#391—28no—
117#390—28no—
118#389—28no—
119#388—216no—
120#388—216no—
121#387—216no—
122#386—216no—
123#386—216no—
124#386—48no—
125#385—216no—
126#385—48no—
127#382—28no—
128#382—24no—
129#380—28no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C25C2414yes—
146C25C2411yescharacteristic
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8C2×D418yes—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4C2×D418yes—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—28no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203#125—232no—
204#124—264no—
205#123—2128no—
206C22⋊Q8—2128no—
207C22⋊Q8—432no—
208#121—2128no—
209#121—432no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D4116yes—
233C2×C4⋊C4C2×D418yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D4116yes—
269C2×C22⋊C4C2×D4116yes—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—416no—
276C2×C42C2×D414yes—
277C2×C42C2×D418yes—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24C22×D418yes—
315C24H(5,𝔽2)14yes—
316C242−1+412yes—
317C24C2511yescharacteristic
318C2×Q8—216no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×D4—232no—
326C2×D4—216no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—216no—
354C22×C4—28no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—232no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4C22×D418yes—
416C22×C4C22×D418yes—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C42−1+412yes—
424C4⋊C4—232no—
425C4⋊C4—232no—
426C4⋊C4—216no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C22⋊C4—232no—
433C22⋊C4—232no—
434C22⋊C4—232no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C42—44no—
445C42—48no—
446C42#120116yes—
447C23—24no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23#56214yescomplemented
503C23C23×D412yes—
504C23C23×D411yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58512yes—
511C23#58512yes—
512Q8—816no—
513D4—816no—
514D4—832no—
515D4—832no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—416no—
538C2×C4—432no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
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—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—832no—
576C2×C4C2×#12014yes—
577C2×C4C2×#12018yes—
578C22—28no—
579C22—24no—
580C22—44no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—48no—
588C22—48no—
589C22—48no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22C22×#12011yescharacteristic
603C22#348412yes—
604C22#348512yes—
605C22#348611yescharacteristic
606C22#348711yescharacteristic
607C22#348812yes—
608C22#348812yes—
609C22#348812yes—
610C22#348812yes—
611C22#348912yes—
612C22#348912yes—
613C22C2×#56211yescharacteristic
614C22#352912yes—
615C22#353111yescharacteristic
616C22#354512yes—
617C22#354811yescharacteristic
618C22#355311yescharacteristic
619C22#355812yes—
620C22#355912yes—
621C22#356112yes—
622C22#356712yes—
623C4—24no—
624C4—28no—
625C4—42no—
626C4—44no—
627C4—48no—
628C4—48no—
629C4—84no—
630C4—84no—
631C4—84no—
632C2—44no—
633C2—88no—
634C2#3551212yes—
635C2#3565911yescharacteristic
636C2#4597212yes—
637C2#4597412yes—
638C2#4597612yes—
639C2#4597811yescharacteristic
640C2#4613312yes—
641C2#4622412yes—
642C2#4622611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#562C2semidirect
2. C2×#2851C2semidirect
3. C2×#2850C2semidirect
4. #15985C2semidirect
5. #15951C2semidirect
6. #15397C2semidirect
7. #15368C2semidirect
8. #15338C2semidirect
9. #15263C2semidirect
10. #14664C2semidirect
11. #14662C2semidirect
70. C2×#125D4semidirect
73. C2×#124D4semidirect
502. C23#562semidirect

Maximal subgroups

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

subgroupindex
C22×#5622
C2×#28512
C2×#28502
#159852
#159512
#153972
#153682
#153382
#152632
#146642
#146622