Finite Group Databasebeta

← #6540317

Subgroups — #6540317

Order 512.

1685 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic
3#30175C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29111C212yesmaximal, complemented
7#28161C211yesmaximal, characteristic, complemented
8#26679C214yesmaximal, complemented
9#24969C212yesmaximal
10#24918C214yesmaximal, complemented
11#24243C212yesmaximal
12#23934C212yesmaximal, complemented
13#23679C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23004C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic
17#22763C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2212yes—
20C2×#462C2212yes—
21C2×#451C2214yes—
22C2×#441C2214yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542—21no—
28#2538C2212yes—
29#2441—22no—
30#2429—22no—
31#2423C2214yes—
32#2369—21no—
33#2368—22no—
34#2365—21no—
35#2365C2211yescharacteristic
36#2360C2212yes—
37#2355C2211yescharacteristic
38#2354—21no—
39Q8×C4⋊C4C2212yes—
40#2349—21no—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4C2212yes—
43#2331—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#391C2212yes—
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2083C2211yescharacteristic
55#2077C2214yes—
56#2072C2212yes—
57#2072C2211yescharacteristic
58#2068C2212yes—
59#2058C2212yes—
60#2056C2212yes—
61#2043C2211yescharacteristic
62#2036C2214yes—
63#2035C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2016C2212yes—
68#2016C2211yescharacteristic
69#2001C2214yes—
70#1956C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1935C2214yes—
74#1935C2214yes—
75#1924C2214yes—
76#1918C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1872C2214yes—
81#1865C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1823C2214yes—
85#1817C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2214yes—
89#1787C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98D4×Q8—24no—
99D4×Q8—24no—
100#533—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2314yes—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127#462—28no—
128#451—48no—
129#441—48no—
130C4×Q16—28no—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#399—24no—
149#398—24no—
150#396—24no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2311yescharacteristic
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—24no—
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2314yes—
176#388—24no—
177#387—21no—
178#387—22no—
179#387—21no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—21no—
220C2×#95—22no—
221C2×#95—22no—
222C43C2311yescharacteristic
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C22×Q8C2412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×Q16—24no—
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—24no—
254C2×Q16—44no—
255C2×Q16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8#4114yes—
266#125—48no—
267C22⋊Q8—216no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C22⋊Q8—84no—
274C4×Q8—22no—
275C4×Q8—28no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×Q8#4114yes—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—44no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—42no—
353#103—24no—
354#103—28no—
355#103—24no—
356#103—24no—
357#103—42no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362C22⋊C8—44no—
363C22⋊C8—44no—
364#97—44no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D412yes—
389C2×Q8—28no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—88no—
407C2×Q8—88no—
408C2×Q8C2×#4112yes—
409C2×Q8C2×#4114yes—
410C2×Q8H(5,𝔽2)14yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4#13714yes—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449Q16—48no—
450Q16—416no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C2×#4112yes—
496C22⋊C4—28no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—42no—
510C42—42no—
511C42—41no—
512C42—48no—
513C42—44no—
514C42—42no—
515C42—82no—
516C42—82no—
517C42—82no—
518C42#13714yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23C2×#13714yes—
528C23C23×D411yescharacteristic
529Q8—24no—
530Q8—24no—
531Q8—24no—
532Q8—28no—
533Q8—42no—
534Q8—44no—
535Q8—42no—
536Q8—42no—
537Q8—88no—
538Q8—88no—
539D4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—164no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57812yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4#58512yes—
604C8—42no—
605C8—42no—
606C8—44no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22C2×#52511yescharacteristic
615C22#348411yescharacteristic
616C22#350011yescharacteristic
617C22C22×#13711yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57811yescharacteristic
620C22#358511yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—82no—
634C4—81no—
635C4—84no—
636C4—81no—
637C4—82no—
638C4#311314yes—
639C4#349412yes—
640C4#351312yes—
641C2—44no—
642C2#3573011yescharacteristic
643C2C2×#311311yescharacteristic
644C2#6060011yescharacteristic
645C2#6158811yescharacteristic
646C2#6160211yescharacteristic
647C2#6253611yescharacteristic
648C2#6258211yescharacteristic
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. #30175C2semidirect
4. #29983C2semidirect
5. #29141C2semidirect
6. #29111C2semidirect
7. #28161C2semidirect
8. #26679C2semidirect
10. #24918C2semidirect
12. #23934C2semidirect
13. #23679C2semidirect
14. #23577C2semidirect
15. #23004C2semidirect
17. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343342
#301752
#299832
#291412
#291112
#281612
#266792
#249692
#249182
#242432
#239342
#236792
#235772
#230042
#228392
#227632