Finite Group Databasebeta

← #6560041

Subgroups — #6560041

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#30205C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29141C211yesmaximal, characteristic, complemented
7#29111C212yesmaximal, complemented
8#27824C212yesmaximal, complemented
9#27777C212yesmaximal, complemented
10#27707C214yesmaximal, complemented
11#27283C214yesmaximal, complemented
12#25910C212yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23455C212yesmaximal, complemented
15#22502C212yesmaximal
16#22430C212yesmaximal
17#20568C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#533C2212yes—
21C2×#491C2214yes—
22C2×#436C2212yes—
23C2×#419C2214yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2857C2212yes—
28#2567C2211yescharacteristic
29#2552C2212yes—
30#2550—21no—
31#2550—21no—
32#2549—22no—
33#2549—22no—
34#2546C2211yescharacteristic
35#2542—21no—
36#2538—21no—
37#2502C2212yes—
38#2441C2214yes—
39#2434C2212yes—
40#2429—22no—
41#2423—22no—
42#2365—21no—
43#2354C2212yes—
44Q8×C4⋊C4—21no—
45#2349—21no—
46#2349—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C2×#391C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2099C2214yes—
53#2072C2211yescharacteristic
54#2072C2211yescharacteristic
55#2072C2214yes—
56#2072C2212yes—
57#2043C2212yes—
58#2043C2214yes—
59#2026C2214yes—
60#2026C2214yes—
61#2024C2211yescharacteristic
62#2024C2211yescharacteristic
63#2024C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#1992C2214yes—
69#1990C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1956C2212yes—
73#1956C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1933C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1873C2214yes—
80#1873C2214yes—
81#1834C2212yes—
82#1823C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1787C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C23×Q8C2312yes—
96C22×M16C2314yes—
97#554—28no—
98D4×Q8—28no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—21no—
110C2×#125C2314yes—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2312yes—
131#491—48no—
132#436—28no—
133C2×C4⋊C8C2312yes—
134#419—48no—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#399—24no—
150#398—24no—
151#398—22no—
152#398—22no—
153#398—22no—
154#396—24no—
155#395—22no—
156#395—22no—
157#395—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389C2312yes—
184#389C2314yes—
185#388—24no—
186#387—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#382—21no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2314yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—24no—
239C22×Q8—28no—
240C22×Q8—42no—
241C22×Q8—44no—
242C22×Q8C2411yescharacteristic
243C22×Q8C2412yes—
244C22×Q8C2411yescharacteristic
245C22×D4C2×D411yescharacteristic
246C23×C4—22no—
247C23×C4—22no—
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—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—41no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#125—28no—
268#125—44no—
269#125—44no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—84no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4#4118yes—
287#117—28no—
288#117—44no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
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—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C4⋊C8—24no—
350#103—24no—
351#103—24no—
352#103—28no—
353#103—24no—
354#103—44no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—28no—
361#97—44no—
362C4×C8—24no—
363C4×C8—42no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24C22×D412yes—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—416no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—88no—
409C2×Q82−1+412yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4#12414yes—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4118yes—
447C22×C4C2511yescharacteristic
448M16—88no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4#12414yes—
497C4⋊C4H(5,𝔽2)14yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4#12418yes—
505C42—24no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—42no—
511C42—41no—
512C42—41no—
513C42—48no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42#13714yes—
518C42#13714yes—
519C42C2×#4114yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23C2×#12414yes—
527C23C23×D411yescharacteristic
528Q8—24no—
529Q8—48no—
530Q8—42no—
531Q8—42no—
532Q8—44no—
533Q8—42no—
534Q8—44no—
535Q8—88no—
536Q8—88no—
537Q8—162no—
538D4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—164no—
590C2×C4C2×#12412yes—
591C2×C4C2×#12414yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)12yes—
601C2×C4C2×2−1+411yescharacteristic
602C2×C4#58511yescharacteristic
603C8—44no—
604C8—42no—
605C8—42no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C22×#12411yescharacteristic
614C22#348411yescharacteristic
615C22#348911yescharacteristic
616C22C22×#13711yescharacteristic
617C22C22×#13711yescharacteristic
618C22#358511yescharacteristic
619C22#358611yescharacteristic
620C4—24no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—81no—
632C4—84no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4#313914yes—
638C4#351412yes—
639C4#352012yes—
640C2—44no—
641C2#3593911yescharacteristic
642C2C2×#313911yescharacteristic
643C2#6062511yescharacteristic
644C2#6158611yescharacteristic
645C2#6158811yescharacteristic
646C2#6211311yescharacteristic
647C2#6211411yescharacteristic
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. #30205C2semidirect
4. #30199C2semidirect
5. #29347C2semidirect
6. #29141C2semidirect
7. #29111C2semidirect
8. #27824C2semidirect
9. #27777C2semidirect
10. #27707C2semidirect
11. #27283C2semidirect
12. #25910C2semidirect
13. #23816C2semidirect
14. #23455C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#361342
#302052
#301992
#293472
#291412
#291112
#278242
#277772
#277072
#272832
#259102
#238162
#234552
#225022
#224302
#205682