Finite Group Databasebeta

← #454232

Subgroups — #454232

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C2×#2423C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2035C212yesmaximal
5#15125C211yesmaximal, characteristic
6#11323C212yesmaximal, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11190C212yesmaximal, complemented
9#11095C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#475—28no—
16C2×#475—24no—
17C22×#106C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2212yes—
23#2423—24no—
24#2423—24no—
25C2×#399—21no—
26C2×#398—21no—
27C2×#393—21no—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388—21no—
31C2×#387—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2058—24no—
37#2035—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—42no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#475—232no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—22no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103D418yes—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—42no—
103#399—28no—
104#398—28no—
105#393—44no—
106#391—22no—
107#391—22no—
108#389—22no—
109#389D414yes—
110#388—28no—
111#387—28no—
112#384—22no—
113#384—22no—
114#380—22no—
115#380—22no—
116#380—24no—
117#380—42no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—42no—
134#340—42no—
135#336—24no—
136#336—22no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—44no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—82no—
151C22×Q8C2×D414yes—
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D414yes—
188C22×C8#4114yes—
189C4⋊Q8—28no—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C22⋊Q8—48no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C4×Q8—28no—
196C4×Q8—44no—
197C4×Q8—48no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4SD1614yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260#106—216no—
261#106—48no—
262C4⋊C8—28no—
263#103—216no—
264#103—48no—
265C22⋊C8—48no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—22no—
271#95—22no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—82no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24C2×SD1612yes—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—48no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8#12014yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4#12012yes—
384C22×C4#12012yes—
385C22×C4C2×SD1612yes—
386C22×C4C2×SD1612yes—
387C22×C4#13712yes—
388C22×C4#13712yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+412yes—
398C22×C42−1+411yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—28no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—84no—
441C22⋊C4—88no—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—48no—
446C42—41no—
447C42—41no—
448C42—42no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C2×#12011yescharacteristic
473C23#53611yescharacteristic
474C23#53711yescharacteristic
475C23D4212yes—
476C23#54712yes—
477C23C22×SD1611yescharacteristic
478C23C2×#13711yescharacteristic
479C23C2×#13711yescharacteristic
480C23#57511yescharacteristic
481C23#57712yes—
482C23#57712yes—
483Q8—24no—
484Q8—24no—
485Q8—42no—
486Q8—44no—
487Q8—44no—
488Q8—42no—
489Q8—816no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#44814yes—
555C2×C4C2×#12011yescharacteristic
556C2×C4#53211yescharacteristic
557C2×C4#53411yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53612yes—
560C2×C4#53612yes—
561C2×C4#53611yescharacteristic
562C2×C4#53611yescharacteristic
563C2×C4#53711yescharacteristic
564C2×C4#53711yescharacteristic
565C2×C4#53912yes—
566C2×C4#54212yes—
567C2×C4D4212yes—
568C2×C4#54612yes—
569C2×C4#54712yes—
570C2×C4#54812yes—
571C2×C4#56212yes—
572C2×C4#56312yes—
573C8—44no—
574C8—44no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—82no—
585C22—82no—
586C22#240311yescharacteristic
587C22#243211yescharacteristic
588C22#261611yescharacteristic
589C22#267212yes—
590C22#268012yes—
591C22C2×#44811yescharacteristic
592C22#300711yescharacteristic
593C22#300911yescharacteristic
594C22#301011yescharacteristic
595C22#322911yescharacteristic
596C22#323711yescharacteristic
597C22#324412yes—
598C22#324611yescharacteristic
599C22#325511yescharacteristic
600C22#326312yes—
601C22#326612yes—
602C22#327112yes—
603C22#327812yes—
604C22D4×SD1612yes—
605C22#333512yes—
606C22#334412yes—
607C22#334812yes—
608C22#335612yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—44no—
614C4—82no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—162no—
620C4#243011yescharacteristic
621C4#261411yescharacteristic
622C4#266311yescharacteristic
623C4#267412yes—
624C4#268112yes—
625C4#281411yescharacteristic
626C2—82no—
627C2#1625111yescharacteristic
628C2#2385511yescharacteristic
629C2#2387011yescharacteristic
630C2#2570211yescharacteristic
631C2#2571511yescharacteristic
632C2#2968811yescharacteristic
633C2#2969711yescharacteristic
634C2#3066412yes—
635C2#3068412yes—
636C2#3108912yes—
637C2#3110712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#475C2semidirect
3. C2×#2058C2semidirect
6. #11323C2semidirect
7. #11228C2semidirect
8. #11190C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#24232
C2×#20582
C2×#20352
#151252
#113232
#112282
#111902
#110952
#110422
#107892