Finite Group Databasebeta

← #6492056

Subgroups — #6492056

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35292C211yesmaximal, characteristic
2#34129C211yesmaximal, characteristic, complemented
3#33114C212yesmaximal, complemented
4#33113C212yesmaximal, complemented
5#29922C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#26328C214yesmaximal, complemented
8#26245C214yesmaximal, complemented
9#24876C214yesmaximal, complemented
10#24870C214yesmaximal, complemented
11#21959C211yesmaximal, characteristic, complemented
12#21851C211yesmaximal, characteristic, complemented
13#21809C211yesmaximal, characteristic, complemented
14#21364C211yesmaximal, characteristic, complemented
15#21257C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#485C2214yes—
19C2×#482C2214yes—
20C2×#432C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#416C2214yes—
23#2857C2212yes—
24#2856C2212yes—
25#2624—21no—
26#2615—21no—
27#2592C2212yes—
28#2580—21no—
29#2540C2212yes—
30#2535—21no—
31#2414—22no—
32#2407C2214yes—
33#2354C2212yes—
34#2354C2212yes—
35#2354C2212yes—
36Q8×C4⋊C4C2212yes—
37#2346—22no—
38#2346—21no—
39#2341—21no—
40#2324—22no—
41#2293—22no—
42#2293—21no—
43#2285C2211yescharacteristic
44#2285C2212yes—
45#2285C2211yescharacteristic
46C42×Q8—21no—
47C4×#117C2211yescharacteristic
48C4×#117C2212yes—
49#2096C2214yes—
50#2094C2214yes—
51#2072C2214yes—
52#2071C2214yes—
53#2035C2214yes—
54#2035C2214yes—
55#2034C2214yes—
56#2034C2214yes—
57#1985C2212yes—
58#1983C2214yes—
59#1982C2211yescharacteristic
60#1982C2211yescharacteristic
61#1982C2212yes—
62#1981C2211yescharacteristic
63#1981C2211yescharacteristic
64#1981C2212yes—
65#1959C2214yes—
66#1955C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2214yes—
71#1947C2212yes—
72#1928C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1875C2214yes—
77#1873C2214yes—
78#1869C2214yes—
79#1854C2214yes—
80#1807C2212yes—
81#1807C2212yes—
82#1797C2212yes—
83#1797C2212yes—
84#1788C2212yes—
85#1788C2212yes—
86#1776C2214yes—
87C22×#41D411yescharacteristic
88C22×#41C2312yes—
89C23×C8C2312yes—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124C2312yes—
95C2×C22⋊Q8C2314yes—
96C2×#121—22no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—22no—
100C2×#121C2314yes—
101C4×#41—24no—
102C4×#41—24no—
103C2×C4×Q8—22no—
104C2×C4×Q8D411yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4D412yes—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2312yes—
111C2×C4×D4C2311yescharacteristic
112C2×#117—21no—
113C2×#117—22no—
114C2×#117—21no—
115C2×#117D412yes—
116C2×#117D411yescharacteristic
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#485—48no—
123#482—48no—
124#432—24no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134#416—48no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2312yes—
147#399—22no—
148#397—22no—
149#394—24no—
150#393—22no—
151#392—24no—
152#392—22no—
153#391C2314yes—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—24no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#388—24no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#386—24no—
174#385—24no—
175#384C2312yes—
176#384C2312yes—
177#384C2312yes—
178#384C2312yes—
179#382—21no—
180#382—22no—
181#382—21no—
182#380—22no—
183#380—22no—
184#380—21no—
185#380—22no—
186#380—21no—
187#380C2311yescharacteristic
188#380C2311yescharacteristic
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C22⋊C4—22no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209#376—22no—
210#376—21no—
211C43—22no—
212C43—21no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—44no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8C2×D411yescharacteristic
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×D412yes—
232C22×D4C2×D411yescharacteristic
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4C2×D412yes—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C22×C8—24no—
242C22×C8—44no—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248#127—22no—
249#127—22no—
250#124—24no—
251#124—24no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254#121—48no—
255#121—44no—
256#121—44no—
257#121—44no—
258#121—44no—
259#121—84no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—48no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—44no—
271C4×D4—44no—
272#117—24no—
273#117—28no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117—42no—
278#117—42no—
279#117—42no—
280#117—82no—
281#117—84no—
282#117—82no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—28no—
342#106—44no—
343#106—44no—
344#106—44no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349#103—28no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—44no—
359C22⋊C8—44no—
360#97—24no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375C24—22no—
376C24C22×D412yes—
377C24C22×D411yescharacteristic
378#41—416no—
379#41—416no—
380#41—84no—
381#41—84no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—84no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—84no—
432C22×C4#12314yes—
433C22×C4#12314yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—84no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4H(5,𝔽2)14yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—44no—
503C42—44no—
504C42—44no—
505C42—42no—
506C42—44no—
507C42—42no—
508C42—82no—
509C42#12314yes—
510C42#12314yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23C2×#12312yes—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—44no—
535Q8—42no—
536Q8—42no—
537D4—48no—
538D4—44no—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—84no—
543D4—88no—
544D4—84no—
545D4—168no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4C2×#12312yes—
589C2×C4C2×#12312yes—
590C2×C4C2×#12312yes—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4#57514yes—
594C2×C4#57712yes—
595C2×C4#57712yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4#58512yes—
600C8—44no—
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C22×#12311yescharacteristic
621C22#348812yes—
622C22C2×#57512yes—
623C22C2×#57711yescharacteristic
624C22C2×#57711yescharacteristic
625C4—22no—
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#309914yes—
639C4#346012yes—
640C4#350811yescharacteristic
641C4#351011yescharacteristic
642C2—42no—
643C2—82no—
644C2—162no—
645C2#3525411yescharacteristic
646C2#6051111yescharacteristic
647C2C2×#309911yescharacteristic
648C2#6198612yes—
649C2#6198712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35292C2
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. #34129C2semidirect
3. #33114C2semidirect
4. #33113C2semidirect
5. #29922C2semidirect
6. #29921C2semidirect
7. #26328C2semidirect
8. #26245C2semidirect
9. #24876C2semidirect
10. #24870C2semidirect
11. #21959C2semidirect
12. #21851C2semidirect
13. #21809C2semidirect
14. #21364C2semidirect
15. #21257C2semidirect

Maximal subgroups

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

subgroupindex
#352922
#341292
#331142
#331132
#299222
#299212
#263282
#262452
#248762
#248702
#219592
#218512
#218092
#213642
#212572