Finite Group Databasebeta

← #6798431

Subgroups — #6798431

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36165C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31372C214yesmaximal, complemented
5#31370C214yesmaximal, complemented
6#29883C212yesmaximal, complemented
7#29149C211yesmaximal, characteristic, complemented
8#28694C211yesmaximal, characteristic, complemented
9#27969C212yesmaximal, complemented
10#26111C212yesmaximal, complemented
11#26107C212yesmaximal, complemented
12#25676C212yesmaximal, complemented
13#24967C212yesmaximal, complemented
14#24965C212yesmaximal, complemented
15#21865C212yesmaximal, complemented
16C4×#490C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#501C2212yes—
21C2×#490C2211yescharacteristic
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2571C2211yescharacteristic
28#2571C2211yescharacteristic
29#2547—21no—
30#2537—21no—
31#2537C2212yes—
32#2508—21no—
33#2508—21no—
34#2505—22no—
35#2505—22no—
36#2442C2212yes—
37#2442C2212yes—
38#2432—22no—
39#2422—22no—
40#2422C2214yes—
41#2349—21no—
42#2349—21no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#125C2211yescharacteristic
49C2×#382C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2088C2214yes—
54#2087C2214yes—
55#2084C2212yes—
56#2083C2212yes—
57#2077C2214yes—
58#2075C2214yes—
59#2034C2214yes—
60#2033C2214yes—
61#2025C2214yes—
62#1993C2212yes—
63#1993C2212yes—
64#1993C2212yes—
65#1992C2214yes—
66#1992C2211yescharacteristic
67#1961C2212yes—
68#1961C2211yescharacteristic
69#1954C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1939C2212yes—
73#1936C2212yes—
74#1936C2212yes—
75#1924C2214yes—
76#1874C2214yes—
77#1874C2214yes—
78#1864C2214yes—
79#1862C2214yes—
80#1836C2211yescharacteristic
81#1836C2212yes—
82#1834C2212yes—
83#1834C2212yes—
84#1824C2214yes—
85#1823C2214yes—
86#1804C2214yes—
87#1804C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1767C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C22×#41—22no—
96#547—48no—
97#532—28no—
98#532—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125C2311yescharacteristic
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#123—24no—
107C2×C22⋊Q8—22no—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121C2314yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120C2×C4×D4C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C42C2312yes—
125#501—44no—
126#490—24no—
127#432—44no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#400—22no—
147#400—22no—
148#400—22no—
149#398—22no—
150#398—22no—
151#397—24no—
152#394—24no—
153#394—24no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#387—21no—
173#387—21no—
174#387—22no—
175#387—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—24no—
183#385—24no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#382—21no—
193#382—22no—
194#382—22no—
195#382C2311yescharacteristic
196#382C2312yes—
197#382C2312yes—
198#382C2312yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C2×#95—22no—
218C2×#95—22no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234C2×#41—48no—
235C2×#41—48no—
236C2×#41—84no—
237C22×Q8—22no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#127—22no—
258#127#4114yes—
259#125—22no—
260#125—22no—
261#125—44no—
262#124—48no—
263#123—416no—
264C22⋊Q8—48no—
265#121—28no—
266#121—44no—
267#121—48no—
268#121—48no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—42no—
275C4×D4—44no—
276C4×D4—48no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—44no—
346#102—24no—
347#102—24no—
348#102—42no—
349#102—42no—
350#102—44no—
351#102—44no—
352C22⋊C8—44no—
353C22⋊C8—44no—
354#97—22no—
355#97—22no—
356#97—24no—
357#97—42no—
358C4×C8—24no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D412yes—
382#41—832no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—84no—
402C2×D4AGL(1,ℤ/8)14yes—
403C2×D4C2×#4114yes—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—42no—
511C42—42no—
512C42C2×#4112yes—
513C42H(5,𝔽2)12yes—
514C422−1+414yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C2×AGL(1,ℤ/8)14yes—
529C23C23×D411yescharacteristic
530Q8—88no—
531D4—28no—
532D4—24no—
533D4—24no—
534D4—48no—
535D4—48no—
536D4—42no—
537D4—42no—
538D4—816no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—816no—
589C2×C4—88no—
590C2×C4—164no—
591C2×C4—164no—
592C2×C4#53612yes—
593C2×C4#53612yes—
594C2×C4#54612yes—
595C2×C4#54612yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4#57512yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×2−1+412yes—
601C2×C4#58512yes—
602C8—42no—
603C8—42no—
604C8—44no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#53611yescharacteristic
620C22C2×#54611yescharacteristic
621C22#350011yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C2×#57511yescharacteristic
624C22#358511yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4—81no—
642C4#323414yes—
643C4#351512yes—
644C4#354712yes—
645C2—44no—
646C2—84no—
647C2#3895511yescharacteristic
648C2C2×#323411yescharacteristic
649C2#6088611yescharacteristic
650C2#6115511yescharacteristic
651C2#6115911yescharacteristic
652C2#6258311yescharacteristic
653C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36165C2semidirect
3. #33297C2semidirect
4. #31372C2semidirect
5. #31370C2semidirect
6. #29883C2semidirect
7. #29149C2semidirect
8. #28694C2semidirect
9. #27969C2semidirect
10. #26111C2semidirect
11. #26107C2semidirect
12. #25676C2semidirect
13. #24967C2semidirect
14. #24965C2semidirect
15. #21865C2semidirect
16. C4×#490C2semidirect
17. #21270C2semidirect

Maximal subgroups

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

subgroupindex
#361652
#361272
#332972
#313722
#313702
#298832
#291492
#286942
#279692
#261112
#261072
#256762
#249672
#249652
#218652
C4×#4902
#212702