Finite Group Databasebeta

← #6536232

Subgroups — #6536232

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34332C211yesmaximal, characteristic
3#30169C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29296C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28206C211yesmaximal, characteristic, complemented
8#27164C214yesmaximal, complemented
9#26559C214yesmaximal, complemented
10#25016C212yesmaximal
11#24122C212yesmaximal
12#24064C212yesmaximal, complemented
13#23679C211yesmaximal, characteristic, complemented
14#23571C212yesmaximal, complemented
15#22965C212yesmaximal, complemented
16#22835C211yesmaximal, characteristic
17C4⋊C4×Q16C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#533C2212yes—
20C2×#484C2214yes—
21C2×#477C2212yes—
22C2×C4×Q16C2212yes—
23C2×C4×Q16C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542—21no—
28#2538C2212yes—
29#2441—22no—
30#2429—22no—
31#2423C2214yes—
32#2370—21no—
33#2369C2211yescharacteristic
34#2365—21no—
35#2360—22no—
36#2360C2212yes—
37#2354—21no—
38Q8×C4⋊C4C2211yescharacteristic
39Q8×C4⋊C4C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2345C2212yes—
43#2330—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#389C2212yes—
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2095C2212yes—
54#2083C2212yes—
55#2083C2211yescharacteristic
56#2076C2214yes—
57#2072C2211yescharacteristic
58#2068C2212yes—
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2056C2212yes—
63#2043C2211yescharacteristic
64#2043C2212yes—
65#2036C2214yes—
66#2036C2214yes—
67#2024C2212yes—
68#2016C2211yescharacteristic
69#2001C2214yes—
70#1990C2212yes—
71#1954C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1942C2214yes—
75#1942C2214yes—
76#1928C2214yes—
77#1923C2214yes—
78#1921C2214yes—
79#1872C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1845C2212yes—
86#1831C2211yescharacteristic
87#1817C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2212yes—
95C22×Q16C2312yes—
96#554—24no—
97#554—24no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—21no—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2314yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127#484—48no—
128#477—28no—
129C4×Q16—28no—
130C4×Q16—48no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#399—24no—
150#398—24no—
151#396—24no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2311yescharacteristic
158#390—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389C2312yes—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#388—24no—
178#387—21no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221C2×#95—22no—
222C43C2311yescharacteristic
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—24no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#125—28no—
258#125—48no—
259C22⋊Q8—216no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—84no—
266C4×Q8—22no—
267C4×Q8—28no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—82no—
274C4×Q8#4114yes—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4Q1618yescomplemented
281#117—48no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—28no—
349#107—44no—
350#107—44no—
351#107—44no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354#103—24no—
355#103—28no—
356#103—24no—
357#103—24no—
358#103—24no—
359#103—28no—
360#103—44no—
361C22⋊C8—28no—
362C22⋊C8—44no—
363C4×C8—24no—
364C4×C8—44no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—24no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8C2×#4112yes—
401C2×Q8H(5,𝔽2)14yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4C2×Q1614yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4C2×Q1618yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440Q16—216no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C82−1+414yes—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×Q1614yes—
488C4⋊C4C2×#4112yes—
489C4⋊C4C2×#4114yes—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—84no—
496C22⋊C4C2×Q1618yes—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—48no—
507C42—44no—
508C42—42no—
509C42—82no—
510C42—82no—
511C42—82no—
512C42C2×Q1614yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23C22×Q1614yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—24no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—84no—
530Q8—88no—
531D4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4—164no—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4C22×Q1612yes—
587C2×C4C22×Q1612yes—
588C2×C4C22×Q1614yes—
589C2×C4C22×Q1612yes—
590C2×C4#57512yes—
591C2×C4#57612yes—
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)12yes—
595C2×C4C2×2−1+412yes—
596C8—24no—
597C8—42no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22C2×#52511yescharacteristic
607C22#348411yescharacteristic
608C22#349911yescharacteristic
609C22C23×Q1611yescharacteristic
610C22C2×#57511yescharacteristic
611C22C2×#57611yescharacteristic
612C22#358611yescharacteristic
613C4—22no—
614C4—22no—
615C4—24no—
616C4—22no—
617C4—22no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—84no—
629C4—82no—
630C4#310414yes—
631C4#349212yes—
632C4#352212yes—
633C2—44no—
634C2#3569311yescharacteristic
635C2C2×#310411yescharacteristic
636C2#6059611yescharacteristic
637C2#6156311yescharacteristic
638C2#6160511yescharacteristic
639C2#6247211yescharacteristic
640C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30169C2semidirect
4. #29984C2semidirect
5. #29296C2semidirect
6. #29208C2semidirect
7. #28206C2semidirect
8. #27164C2semidirect
9. #26559C2semidirect
12. #24064C2semidirect
13. #23679C2semidirect
14. #23571C2semidirect
15. #22965C2semidirect
17. C4⋊C4×Q16C2semidirect
280. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361342
#343322
#301692
#299842
#292962
#292082
#282062
#271642
#265592
#250162
#241222
#240642
#236792
#235712
#229652
#228352
C4⋊C4×Q162