Finite Group Databasebeta

← #636928

Subgroups — #636928

Order 512.

4350 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×#529C211yesmaximal, characteristic
2C2×#2486C212yesmaximal, complemented
3C22×#389C211yesmaximal, characteristic, complemented
4#16673C214yesmaximal, complemented
5#16607C212yesmaximal, complemented
6#16558C212yesmaximal, complemented
7#15202C212yesmaximal, complemented
8#15181C212yesmaximal, complemented
9#15069C214yesmaximal, complemented
10#14934C212yesmaximal, complemented
11#14926C212yesmaximal
12#14685C211yesmaximal, characteristic, complemented
13#14264C214yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C2×#529—28no—
16C2×#529—216no—
17C22×#126C2212yes—
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#123C2212yes—
21C22×#123C2214yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2212yes—
31C23×C42C2211yescharacteristic
32#2486—216no—
33C2×#398C2214yes—
34C2×#397C2214yes—
35C2×#389—28no—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#386C2214yes—
42C2×#381C2214yes—
43C2×#381C2212yes—
44C2×#380C2214yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C22×#95C2212yes—
48#1446C2214yes—
49#1444C2218yes—
50#1444C2214yes—
51#1444C2218yes—
52#1442C2218yes—
53#1442C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76#529—2128no—
77C2×#126—28no—
78C2×#126—28no—
79C2×#126—28no—
80C2×#125—28no—
81C2×#125—24no—
82C2×#124—28no—
83C2×#124—24no—
84C2×#123—28no—
85C2×#123—28no—
86C2×#123—216no—
87C2×#123—28no—
88C2×#123—48no—
89C2×#123—48no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C2×#121—28no—
93C2×#121—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4D4116yes—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117—28no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—48no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2318yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2318yes—
132C22×C42—28no—
133C22×C42D418yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#398—216no—
140#397—216no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#386—48no—
147#381—216no—
148#381—44no—
149#380—216no—
150#380—28no—
151#380—216no—
152C2×#95—28no—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×Q8—24no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—42no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4C2×D418yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2×D418yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D418yes—
209C23×C4C2418yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#126—264no—
222#125—232no—
223#124—232no—
224#123—264no—
225#123—464no—
226C22⋊Q8—232no—
227#121—232no—
228C4×Q8—232no—
229C4×D4—232no—
230C4×D4—264no—
231#117—264no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4#4118yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C42—232no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—28no—
310#95—216no—
311#95—216no—
312#95—28no—
313#95—44no—
314#95—48no—
315#95—44no—
316#95—44no—
317#95—48no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24C22×D414yes—
333C24C22×D414yes—
334C24C22×D412yes—
335C24H(5,𝔽2)12yes—
336C24H(5,𝔽2)14yes—
337C242−1+412yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—416no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×D4—28no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—416no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—44no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—232no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—232no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—44no—
411C22×C4—416no—
412C22×C4—432no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4#120116yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4114yes—
439C22×C42−1+414yes—
440C22×C42−1+414yes—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—432no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—432no—
461C22⋊C4—416no—
462C22⋊C4—816no—
463C42—28no—
464C42—28no—
465C42—28no—
466C42—28no—
467C42—216no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#12014yes—
504C23#53518yes—
505C23#53514yes—
506C23#53714yes—
507C23#53718yes—
508C23C23×D411yescharacteristic
509C23C23×D412yes—
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×2−1+412yes—
514C23C2×2−1+412yes—
515C23C2×2−1+411yescharacteristic
516C23#58512yes—
517Q8—416no—
518D4—416no—
519D4—432no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—232no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—216no—
545C2×C4—416no—
546C2×C4—432no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—432no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—816no—
580C2×C4C2×#12014yes—
581C2×C4C2×#12014yes—
582C2×C4#52514yes—
583C2×C4#53614yes—
584C2×C4#53614yes—
585C2×C4#547116yes—
586C22—24no—
587C22—216no—
588C22—216no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—44no—
595C22—48no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22#261518yes—
606C22C22×#12011yescharacteristic
607C22C2×#52511yescharacteristic
608C22C2×#53511yescharacteristic
609C22C2×#53512yes—
610C22C2×#53611yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#53712yes—
613C22C2×#53711yescharacteristic
614C22#344911yescharacteristic
615C22C2×#54714yes—
616C22#348914yes—
617C22#351314yes—
618C22#352212yes—
619C22#352312yes—
620C22#354414yes—
621C22#355912yes—
622C22#356312yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—48no—
629C4—44no—
630C4—42no—
631C4—44no—
632C4—84no—
633C4—88no—
634C4#240618yes—
635C2—28no—
636C2—84no—
637C2C2×#261511yescharacteristic
638C2C2×#240611yescharacteristic
639C2#3523211yescharacteristic
640C2#3524711yescharacteristic
641C2#3530511yescharacteristic
642C2#3754911yescharacteristic
643C2#3763211yescharacteristic
644C2#3804114yes—
645C2#3804314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2486C2semidirect
3. C22×#389C2semidirect
4. #16673C2semidirect
5. #16607C2semidirect
6. #16558C2semidirect
7. #15202C2semidirect
8. #15181C2semidirect
9. #15069C2semidirect
10. #14934C2semidirect
12. #14685C2semidirect
13. #14264C2semidirect
14. #13872C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#24862
C22×#3892
#166732
#166072
#165582
#152022
#151812
#150692
#149342
#149262
#146852
#142642
#138722