Finite Group Databasebeta

← #460513

Subgroups — #460513

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11202C211yesmaximal, characteristic, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2452—24no—
19#2452—24no—
20C2×#400—21no—
21C2×#396—21no—
22C2×#393—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2099—24no—
31#1802—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—42no—
62C2×C4×D4D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—42no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91#400—44no—
92#396—28no—
93#393—28no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389D414yes—
102#385—28no—
103#384—22no—
104#384—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—22no—
124#336—42no—
125#336—42no—
126C25—21no—
127C22×Q8—24no—
128C22×Q8—44no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4#4114yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—82no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C4⋊Q8—48no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176C22⋊Q8—432no—
177C4×D4—28no—
178C4×D4—44no—
179C4×D4—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
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—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4D814yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C4⋊C8—44no—
253#102—48no—
254#102—48no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—82no—
264#95—82no—
265C24—22no—
266C24—24no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24C22×D411yescharacteristic
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—88no—
280C2×Q8—84no—
281C2×Q8—88no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4#12414yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—48no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4C2×D812yes—
364C22×C4C2×D812yes—
365C22×C4C2×D812yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4#13712yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+412yes—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4#12314yes—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—42no—
432C23—24no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—81no—
447C23—82no—
448C23—81no—
449C23—81no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23C2×#12011yescharacteristic
456C23#53611yescharacteristic
457C23#53711yescharacteristic
458C23#54612yes—
459C23#54812yes—
460C23C22×D811yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57511yescharacteristic
464C23#57712yes—
465C23#57712yes—
466Q8—816no—
467D4—24no—
468D4—24no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—44no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—816no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#44714yes—
537C2×C4C2×#12311yescharacteristic
538C2×C4C2×#12411yescharacteristic
539C2×C4#53411yescharacteristic
540C2×C4#53611yescharacteristic
541C2×C4#53611yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53711yescharacteristic
544C2×C4#53711yescharacteristic
545C2×C4#53812yes—
546C2×C4#53912yes—
547C2×C4#54112yes—
548C2×C4#54212yes—
549C2×C4#54612yes—
550C2×C4#54812yes—
551C2×C4#55212yes—
552C2×C4#55212yes—
553C2×C4#56312yes—
554C2×C4#56312yes—
555C8—82no—
556C8—82no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22#240511yescharacteristic
569C22#243611yescharacteristic
570C22#261611yescharacteristic
571C22#269012yes—
572C22#270212yes—
573C22C2×#44711yescharacteristic
574C22#300711yescharacteristic
575C22#300911yescharacteristic
576C22#301011yescharacteristic
577C22#322611yescharacteristic
578C22#323511yescharacteristic
579C22#324412yes—
580C22#324511yescharacteristic
581C22#325411yescharacteristic
582C22#326312yes—
583C22#329412yes—
584C22#331012yes—
585C22#331612yes—
586C22#332612yes—
587C22#335812yes—
588C22#337212yes—
589C22#337312yes—
590C22#337612yes—
591C4—24no—
592C4—42no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—82no—
599C4—82no—
600C4—162no—
601C4—162no—
602C4#250311yescharacteristic
603C4#262511yescharacteristic
604C4#263911yescharacteristic
605C4#269012yes—
606C4#270212yes—
607C4#281711yescharacteristic
608C2—84no—
609C2#1636611yescharacteristic
610C2#2403211yescharacteristic
611C2#2405211yescharacteristic
612C2#2593311yescharacteristic
613C2#2594511yescharacteristic
614C2#2968711yescharacteristic
615C2#2969711yescharacteristic
616C2#3145412yes—
617C2#3148212yes—
618C2#3180212yes—
619C2#3182212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2099C2semidirect
4. #15152C2semidirect
5. #11384C2semidirect
6. #11292C2semidirect
7. #11227C2semidirect
8. #11202C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#20992
C2×#18022
#151522
#113842
#112922
#112272
#112022
#111892
#107882