Finite Group Databasebeta

← #640078

Subgroups — #640078

Order 512.

3180 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
1C2×#2623C212yesmaximal
2C2×C4×#123C211yesmaximal, characteristic
3C22×#384C211yesmaximal, characteristic, complemented
4#17063C212yesmaximal, complemented
5#17062C212yesmaximal, complemented
6#17061C212yesmaximal, complemented
7#17060C212yesmaximal, complemented
8#15152C212yesmaximal, complemented
9#15151C212yesmaximal, complemented
10#15150C212yesmaximal, complemented
11#15135C212yesmaximal, complemented
12#14786C212yesmaximal, complemented
13#14776C212yesmaximal
14#14685C211yesmaximal, characteristic, complemented
15#14342C212yesmaximal, complemented
16#14130C212yesmaximal, complemented
17#13872C212yesmaximal
18C22×#123C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#2623—216no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#397C2214yes—
30C2×#397C2214yes—
31C4×#123—28no—
32C2×#389C2214yes—
33C2×#388C2211yescharacteristic
34C2×#388C2211yescharacteristic
35C2×#387C2211yescharacteristic
36C2×#387C2211yescharacteristic
37C2×#386C2214yes—
38C2×#386C2211yescharacteristic
39C2×#385C2214yes—
40C2×#385C2211yescharacteristic
41C2×#384—28no—
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×#384C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#380C2212yes—
49C2×#380C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1445C2214yes—
59#1445C2214yes—
60#1445C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C2×#123—24no—
85C2×#123—24no—
86C2×#123D418yes—
87C2×C4×D4—24no—
88C2×C4×D4—28no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4—48no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2318yes—
119C22×C42—28no—
120C22×C42D418yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#400—216no—
129#399—216no—
130#398—216no—
131#398—48no—
132#397—216no—
133#397—48no—
134#389—48no—
135#388—24no—
136#388—24no—
137#387—24no—
138#387—24no—
139#386—216no—
140#386—24no—
141#385—24no—
142#385—216no—
143#384—28no—
144#384—28no—
145#384—28no—
146#384—28no—
147#381—28no—
148#381—24no—
149#380—28no—
150#380—48no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2318yes—
174C2×#95C2318yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4C2×D414yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—48no—
199C23×C4—48no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4#4118yes—
206C23×C4C2418yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2418yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#123—232no—
222C4×D4—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—432no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—432no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D414yes—
285C2×C42—232no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—28no—
311#95—28no—
312#95—216no—
313#95—216no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—48no—
318#95—48no—
319#95—48no—
320#95—48no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24C22×D412yes—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2×#4114yes—
338C242−1+414yes—
339C24C2511yescharacteristic
340C2×D4—28no—
341C2×D4—216no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C22×C4—232no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—232no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—416no—
415C22×C4—816no—
416C22×C4—816no—
417C22×C4#121116yes—
418C22×C4#123116yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+414yes—
437C22×C42−1+414yes—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—416no—
450C4⋊C4—816no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—416no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—816no—
461C42—28no—
462C42—28no—
463C42—28no—
464C42—28no—
465C42—44no—
466C42—44no—
467C42—44no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23C2×#12114yes—
498C23C2×#12314yes—
499C23#53614yes—
500C23#54718yes—
501C23#55218yes—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+412yes—
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511D4—416no—
512C2×C4—232no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—232no—
531C2×C4—232no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—832no—
578C2×C4—88no—
579C2×C4C2×#12114yes—
580C2×C4C2×#12114yes—
581C2×C4C2×#12314yes—
582C2×C4C2×#12314yes—
583C2×C4#53514yes—
584C2×C4#53514yes—
585C2×C4#53614yes—
586C2×C4#53614yes—
587C2×C4#539116yes—
588C22—24no—
589C22—216no—
590C22—24no—
591C22—216no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—48no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—84no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22#248618yes—
606C22C22×#12111yescharacteristic
607C22C22×#12311yescharacteristic
608C22C2×#53511yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#53914yes—
614C22C2×#54712yes—
615C22C2×#55212yes—
616C22#349214yes—
617C22#349812yes—
618C22#349912yes—
619C22#350812yes—
620C22#350912yes—
621C22#355214yes—
622C22#355214yes—
623C4—24no—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—88no—
634C4—88no—
635C4—81no—
636C4—81no—
637C4#262318yescomplemented
638C2—28no—
639C2—82no—
640C2C2×#262311yescharacteristic
641C2C2×#248611yescharacteristic
642C2#3522711yescharacteristic
643C2#3527211yescharacteristic
644C2#3531611yescharacteristic
645C2#3761511yescharacteristic
646C2#3761711yescharacteristic
647C2#3934814yes—
648C2#3947614yes—
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
3. C22×#384C2semidirect
4. #17063C2semidirect
5. #17062C2semidirect
6. #17061C2semidirect
7. #17060C2semidirect
8. #15152C2semidirect
9. #15151C2semidirect
10. #15150C2semidirect
11. #15135C2semidirect
12. #14786C2semidirect
14. #14685C2semidirect
15. #14342C2semidirect
16. #14130C2semidirect
637. C4#2623semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C2×C4×#1232
C22×#3842
#170632
#170622
#170612
#170602
#151522
#151512
#151502
#151352
#147862
#147762
#146852
#143422
#141302
#138722