Finite Group Databasebeta

← #6458520

Subgroups — #6458520

Order 512.

2064 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#27984C218yesmaximal, complemented
4#27975C212yesmaximal, complemented
5#27974C212yesmaximal, complemented
6#27973C212yesmaximal, complemented
7#27972C212yesmaximal, complemented
8#23634C214yesmaximal, complemented
9#21200C211yesmaximal, characteristic, complemented
10#21195C211yesmaximal, characteristic, complemented
11#20915C214yesmaximal, complemented
12#20764C211yesmaximal, characteristic, complemented
13C2×#1834C211yesmaximal, characteristic, complemented
14C4×#408C211yesmaximal, characteristic, complemented
15C2×#537C2212yes—
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C2×C4×#41C2211yescharacteristic
19C2×#417C2214yes—
20C2×#416C2214yes—
21C2×#408C2212yes—
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2857C2212yes—
25#2856C2212yes—
26#2620C2214yes—
27#2538—21no—
28#2538—21no—
29#2537—21no—
30#2537—21no—
31#2416—22no—
32#2416—22no—
33Q8×C4⋊C4—21no—
34Q8×C4⋊C4—21no—
35D4×C4⋊C4—22no—
36D4×C4⋊C4—22no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39#2317C2214yes—
40#2317C2212yes—
41#2311C2212yes—
42#2285C2211yescharacteristic
43#2285C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47#2074C2218yes—
48#2073C2218yes—
49#2072C2218yes—
50#2071C2218yes—
51#1959C2214yes—
52#1958C2214yes—
53#1957C2211yescharacteristic
54#1957C2212yes—
55#1957C2211yescharacteristic
56#1956C2211yescharacteristic
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1875C2218yes—
60#1874C2218yes—
61#1873C2218yes—
62#1872C2218yes—
63#1836C2214yes—
64#1834C2212yes—
65#1834C2212yes—
66#1834C2214yes—
67#1834C2212yes—
68#1834C2212yes—
69#1750C2214yes—
70C4×C22⋊C8C2212yes—
71C4×C22⋊C8C2212yes—
72#1736C2214yes—
73C2×#340C2214yes—
74#1729C2212yes—
75C22×#41C2312yes—
76C22×#41C2311yescharacteristic
77C22×M16C2312yes—
78C23×C8C2312yes—
79#537—216no—
80#532—24no—
81#532—24no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#124C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8C2314yes—
87C2×#121—24no—
88C2×#121—22no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94#520—48no—
95C4×#41—24no—
96C4×#41—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8D411yescharacteristic
99C2×C4×Q8D411yescharacteristic
100C2×C4×D4—24no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4D411yescharacteristic
107C2×C4×D4D411yescharacteristic
108C2×#117C2312yes—
109C2×#117C2314yes—
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#417—48no—
124#416—48no—
125C2×#103C2314yes—
126C2×#103C2318yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2318yes—
130C2×#102C2314yes—
131#408—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C4×M16—24no—
135C2×#97C2312yes—
136C2×C4×C8—24no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#396—24no—
142#396—24no—
143#394—24no—
144#394—24no—
145#391C2314yes—
146#390—22no—
147#390—22no—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389—24no—
152#389—24no—
153#389—22no—
154#389—22no—
155#386—24no—
156#385—24no—
157#385—24no—
158#385—24no—
159#384—22no—
160#384—22no—
161#384—21no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384C2314yes—
166#384C2312yes—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#382C2314yes—
173#382C2312yes—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—21no—
179#380—21no—
180#380—22no—
181#380—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186#340C2318yes—
187#340C2318yes—
188#340C2318yes—
189#340C2318yes—
190#336C2314yes—
191#336C2314yes—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—28no—
198C2×#41—28no—
199C2×#41#4118yes—
200C22×Q8C2×D411yescharacteristic
201C22×Q8C2412yes—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4C2×D412yes—
207C22×D4C2×D411yescharacteristic
208C22×D4C2412yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2414yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230#127—22no—
231#127—22no—
232#124—24no—
233#124—24no—
234C22⋊Q8—216no—
235C22⋊Q8—48no—
236#121—216no—
237#121—48no—
238#121—416no—
239#121—44no—
240#121—44no—
241#121—84no—
242#121—84no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—48no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—416no—
251C4×D4—48no—
252C4×D4—44no—
253C4×D4—44no—
254#117—22no—
255#117—22no—
256#117—24no—
257#117—24no—
258#117—28no—
259#117—48no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—84no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—84no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42#4114yes—
317C2×C42#4114yes—
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#103—44no—
322#103—44no—
323#103—48no—
324#102—44no—
325#102—44no—
326#102—48no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329#97—24no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—24no—
333C4×C8—24no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340C24—22no—
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343#41—216no—
344#41—216no—
345#41—416no—
346#41—416no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8#12414yes—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4#12418yes—
375C2×D4#12414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—84no—
406C22×C4#12314yes—
407C22×C4#12314yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4118yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4C2511yescharacteristic
422M16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—48no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—48no—
477C42—44no—
478C42#12314yes—
479C42#12314yes—
480C42C2×#4112yes—
481C42C2×#4112yes—
482C42C2×#4112yes—
483C42C2×#4112yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C2×#12312yes—
502C23C2×#12414yes—
503C23C2×#12412yes—
504C23#53512yes—
505C23C23×D411yescharacteristic
506Q8—28no—
507Q8—44no—
508Q8—44no—
509D4—216no—
510D4—28no—
511D4—44no—
512D4—44no—
513D4—88no—
514D4—84no—
515D4—84no—
516D4—168no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—88no—
551C2×C4—164no—
552C2×C4—164no—
553C2×C4—168no—
554C2×C4C2×#12312yes—
555C2×C4C2×#12312yes—
556C2×C4C2×#12312yes—
557C2×C4C2×#12312yes—
558C2×C4C2×#12312yes—
559C2×C4C2×#12414yes—
560C2×C4C2×#12412yes—
561C2×C4#53512yes—
562C2×C4#55318yes—
563C2×C4#55318yes—
564C2×C4#57612yes—
565C2×C4#57712yes—
566C2×C4#57712yes—
567C2×C4#57812yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4112yes—
571C2×C4C2×H(5,𝔽2)12yes—
572C8—42no—
573C8—42no—
574C8—44no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22#250414yes—
596C22C22×#12311yescharacteristic
597C22C22×#12411yescharacteristic
598C22C2×#53511yescharacteristic
599C22C2×#57611yescharacteristic
600C22C2×#57711yescharacteristic
601C22C2×#57711yescharacteristic
602C22C2×#57811yescharacteristic
603C4—22no—
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4#250414yes—
614C4#309814yes—
615C4#309914yes—
616C4C2×#55312yes—
617C4C2×#55312yes—
618C2—24no—
619C2—44no—
620C2—162no—
621C2C2×#250411yescharacteristic
622C2C2×#309811yescharacteristic
623C2C2×#309911yescharacteristic
624C2#6062211yescharacteristic
625C2#6062311yescharacteristic
626C2#6082211yescharacteristic
627C2#6082411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #27984C2semidirect
4. #27975C2semidirect
5. #27974C2semidirect
6. #27973C2semidirect
7. #27972C2semidirect
8. #23634C2semidirect
9. #21200C2semidirect
10. #21195C2semidirect
11. #20915C2semidirect
12. #20764C2semidirect
13. C2×#1834C2semidirect
14. C4×#408C2semidirect

Maximal subgroups

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

subgroupindex
#352732
#341832
#279842
#279752
#279742
#279732
#279722
#236342
#212002
#211952
#209152
#207642
C2×#18342
C4×#4082