Finite Group Databasebeta

← #636796

Subgroups — #636796

Order 512.

4670 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2409C212yesmaximal, complemented
3C22×#391C211yesmaximal, characteristic, complemented
4#16656C214yesmaximal, complemented
5#16608C212yesmaximal, complemented
6#16559C212yesmaximal, complemented
7#15201C212yesmaximal, complemented
8#15177C212yesmaximal, complemented
9#14914C212yesmaximal, complemented
10#14913C212yesmaximal
11#14912C214yesmaximal, complemented
12#14685C211yesmaximal, characteristic, complemented
13#14258C214yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C24×Q8C2211yescharacteristic
16C2×#529—28no—
17C2×#529—216no—
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2214yes—
24C22×#121C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×#117C2212yes—
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2409—216no—
32C2×#395C2214yes—
33C2×#393C2214yes—
34C2×#391—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#385C2214yes—
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C22×#95C2212yes—
47C22×#95C2212yes—
48#1446C2214yes—
49#1444C2214yes—
50#1442C2218yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2218yes—
54#1441C2218yes—
55#1440C2218yes—
56#1440C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59#1438C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8—28no—
68C23×Q8—24no—
69C23×Q8—28no—
70C23×Q8—44no—
71C23×Q8C2312yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82#529—2128no—
83C2×#126—28no—
84C2×#126—28no—
85C2×#126—28no—
86C2×#125—28no—
87C2×#125—24no—
88C2×#124—28no—
89C2×#124—24no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—48no—
98C2×#121—28no—
99C2×#121—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—28no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—24no—
106C2×C4×D4D4116yes—
107C2×#117—28no—
108C2×#117—28no—
109C2×#117—28no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2318yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2318yes—
130C22×C42—28no—
131C22×C42D418yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#395—216no—
138#393—216no—
139#391—216no—
140#391—28no—
141#391—28no—
142#391—28no—
143#391—28no—
144#386—216no—
145#386—48no—
146#385—216no—
147#381—44no—
148#380—28no—
149#380—48no—
150C2×#95—28no—
151C2×#95—28no—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×Q8—232no—
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—28no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—42no—
181C22×Q8—464no—
182C22×Q8—416no—
183C22×Q8—416no—
184C22×Q8#4118yes—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4C2×D418yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D418yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4C2418yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#126—264no—
236#125—232no—
237#124—232no—
238#123—264no—
239C22⋊Q8—232no—
240C22⋊Q8—464no—
241#121—232no—
242C4×Q8—232no—
243C4×D4—232no—
244C4×D4—264no—
245#117—264no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
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—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4C2×D418yes—
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—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D418yes—
297C2×C42—232no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42C2×D414yes—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—216no—
315#95—216no—
316#95—28no—
317#95—44no—
318#95—48no—
319#95—48no—
320#95—44no—
321#95—44no—
322#95—48no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24C22×D414yes—
338C24C22×D414yes—
339C24C22×D412yes—
340C24H(5,𝔽2)12yes—
341C242−1+414yes—
342C242−1+412yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—216no—
346C2×Q8—216no—
347C2×Q8—232no—
348C2×Q8—416no—
349C2×Q8—432no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—4128no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—416no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—416no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—44no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—216no—
411C22×C4—28no—
412C22×C4—232no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—432no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4#120116yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+414yes—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—432no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—432no—
472C22⋊C4—416no—
473C22⋊C4—816no—
474C42—28no—
475C42—28no—
476C42—28no—
477C42—28no—
478C42—216no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12014yes—
515C23#53614yes—
516C23#53618yes—
517C23#53718yes—
518C23#53714yes—
519C23C23×D411yescharacteristic
520C23C23×D412yes—
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×2−1+412yes—
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+412yes—
527C23#58512yes—
528Q8—416no—
529Q8—432no—
530Q8—432no—
531D4—416no—
532D4—432no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—232no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—216no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—416no—
559C2×C4—432no—
560C2×C4—416no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—432no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—816no—
593C2×C4C2×#12014yes—
594C2×C4C2×#12014yes—
595C2×C4#52514yes—
596C2×C4#53514yes—
597C2×C4#53614yes—
598C2×C4#546116yes—
599C22—216no—
600C22—216no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—84no—
613C22—84no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22#261618yes—
619C22C22×#12011yescharacteristic
620C22C2×#52511yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#53612yes—
623C22C2×#53611yescharacteristic
624C22C2×#53611yescharacteristic
625C22C2×#53712yes—
626C22C2×#53711yescharacteristic
627C22#344911yescharacteristic
628C22C2×#54614yes—
629C22#348914yes—
630C22#351614yes—
631C22#352212yes—
632C22#352312yes—
633C22#355214yes—
634C22#355812yes—
635C22#356112yes—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—42no—
644C4—44no—
645C4—88no—
646C4—84no—
647C4#240418yes—
648C2—28no—
649C2—84no—
650C2C2×#261611yescharacteristic
651C2C2×#240411yescharacteristic
652C2#3524511yescharacteristic
653C2#3524711yescharacteristic
654C2#3530511yescharacteristic
655C2#3754611yescharacteristic
656C2#3761511yescharacteristic
657C2#3802514yes—
658C2#3802714yes—
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×#2409C2semidirect
3. C22×#391C2semidirect
4. #16656C2semidirect
5. #16608C2semidirect
6. #16559C2semidirect
7. #15201C2semidirect
8. #15177C2semidirect
9. #14914C2semidirect
11. #14912C2semidirect
12. #14685C2semidirect
13. #14258C2semidirect
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×#24092
C22×#3912
#166562
#166082
#165592
#152012
#151772
#149142
#149132
#149122
#146852
#142582
#138722