Finite Group Databasebeta

← #6557141

Subgroups — #6557141

Order 512.

1754 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#36416C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30202C212yesmaximal, complemented
5#29338C212yesmaximal, complemented
6#29123C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27777C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27283C214yesmaximal, complemented
11#27277C214yesmaximal, complemented
12#25861C212yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23507C212yesmaximal, complemented
15#22430C212yesmaximal
16#22430C212yesmaximal
17#20568C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#491C2214yes—
21C2×#487C2212yes—
22C2×#436C2212yes—
23C2×#419C2214yes—
24C2×#419C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2589—22no—
30#2587—22no—
31#2580—21no—
32#2571C2211yescharacteristic
33#2566—21no—
34#2552C2211yescharacteristic
35#2551—21no—
36#2551—21no—
37#2546C2212yes—
38#2441C2212yes—
39#2440—22no—
40#2440—22no—
41#2439C2212yes—
42#2434C2214yes—
43#2370—21no—
44#2355—21no—
45#2354C2211yescharacteristic
46#2354C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51#2296C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2100C2212yes—
56#2074C2214yes—
57#2072C2212yes—
58#2043C2211yescharacteristic
59#2043C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2043C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2214yes—
67#2043C2214yes—
68#2026C2214yes—
69#2024C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1960C2214yes—
73#1956C2212yes—
74#1956C2212yes—
75#1956C2212yes—
76#1935C2214yes—
77#1935C2214yes—
78#1933C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1873C2214yes—
82#1873C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1787C2214yes—
86#1787C2214yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×#103C2212yes—
90C4×#103C2212yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C23×Q8C2314yes—
96C22×M16C2314yes—
97D4×Q8—28no—
98D4×Q8—28no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—21no—
110C2×#125C2312yes—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C42C2312yes—
124#491—48no—
125#487—44no—
126#436—28no—
127C2×C4⋊C8C2312yes—
128#419—48no—
129#419—48no—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—24no—
152#396—22no—
153#396—22no—
154#395—24no—
155#395—24no—
156#393—22no—
157#391—21no—
158#391—22no—
159#391C2312yes—
160#391C2314yes—
161#390—21no—
162#390—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—21no—
203#382—21no—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2311yescharacteristic
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—28no—
239C22×Q8—28no—
240C22×Q8—28no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2414yes—
245C22×Q8C2412yes—
246C22×D4C2×D411yescharacteristic
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—44no—
253C2×M16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—41no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270#125—44no—
271#125—44no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—416no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—48no—
281C4×D4#4118yes—
282#117—24no—
283#117—42no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—44no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C4⋊C8—24no—
351#103—24no—
352#103—28no—
353#103—24no—
354#103—24no—
355#103—28no—
356#103—44no—
357#103—42no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365C22⋊C8—28no—
366#97—44no—
367C4×C8—24no—
368C4×C8—42no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D412yes—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—416no—
403C2×Q8—416no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8—84no—
408C2×Q8—88no—
409C2×Q8—88no—
410C2×Q8H(5,𝔽2)14yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4#12414yes—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4118yes—
445C22×C4C2511yescharacteristic
446M16—88no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4#12414yes—
498C4⋊C4H(5,𝔽2)12yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4#12418yes—
507C42—21no—
508C42—24no—
509C42—22no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—41no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—42no—
519C42—82no—
520C42#13714yes—
521C42#13714yes—
522C42C2×#4114yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23C2×#12414yes—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—42no—
534Q8—42no—
535Q8—44no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—816no—
540Q8—84no—
541D4—28no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—816no—
592C2×C4C2×#12412yes—
593C2×C4C2×#12414yes—
594C2×C4C2×#12412yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4#58511yescharacteristic
605C8—42no—
606C8—44no—
607C8—42no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22C22×#12411yescharacteristic
616C22#348411yescharacteristic
617C22#348811yescharacteristic
618C22C22×#13711yescharacteristic
619C22C22×#13711yescharacteristic
620C22#358611yescharacteristic
621C22#358611yescharacteristic
622C4—22no—
623C4—22no—
624C4—24no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—84no—
635C4—81no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4#313914yes—
640C4#351312yes—
641C4#351912yes—
642C2—44no—
643C2#3590311yescharacteristic
644C2C2×#313911yescharacteristic
645C2#6062711yescharacteristic
646C2#6158411yescharacteristic
647C2#6158411yescharacteristic
648C2#6196011yescharacteristic
649C2#6196411yescharacteristic
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. #30220C2semidirect
4. #30202C2semidirect
5. #29338C2semidirect
6. #29123C2semidirect
7. #28748C2semidirect
8. #27777C2semidirect
9. #27757C2semidirect
10. #27283C2semidirect
11. #27277C2semidirect
12. #25861C2semidirect
13. #23816C2semidirect
14. #23507C2semidirect

Maximal subgroups

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

subgroupindex
#364162
D4×C4⋊Q82
#302202
#302022
#293382
#291232
#287482
#277772
#277572
#272832
#272772
#258612
#238162
#235072
#224302
#205682