Finite Group Databasebeta

← #6558639

Subgroups — #6558639

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#30367C212yesmaximal, complemented
4#30305C211yesmaximal, characteristic, complemented
5#29166C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#29123C211yesmaximal, characteristic, complemented
8#28871C214yesmaximal, complemented
9#28232C212yesmaximal
10#28151C212yesmaximal
11#26988C214yesmaximal, complemented
12#25504C212yesmaximal, complemented
13#23801C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#22763C212yesmaximal, complemented
16#22674C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#470C2214yes—
20C2×#441C2214yes—
21C2×C4×Q16C2212yes—
22C2×C4×Q16C2212yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2212yes—
27#2855C2212yes—
28#2608—22no—
29#2587—22no—
30#2570—21no—
31#2566—21no—
32#2552—21no—
33#2542—21no—
34#2538C2212yes—
35#2538C2211yescharacteristic
36#2538C2211yescharacteristic
37#2497C2212yes—
38#2441—22no—
39#2429—22no—
40#2429C2212yes—
41#2423C2214yes—
42#2370—21no—
43#2365—21no—
44#2357—21no—
45#2355C2211yescharacteristic
46#2354—21no—
47Q8×C4⋊C4C2212yes—
48Q8×C4⋊C4C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2212yes—
54#2095C2211yescharacteristic
55#2095C2212yes—
56#2095C2211yescharacteristic
57#2084C2214yes—
58#2075C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2043C2212yes—
62#2035C2214yes—
63#2035C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2016C2211yescharacteristic
67#2016C2212yes—
68#2016C2211yescharacteristic
69#2001C2214yes—
70#1956C2212yes—
71#1943C2214yes—
72#1935C2214yes—
73#1933C2214yes—
74#1918C2214yes—
75#1894C2214yes—
76#1894C2214yes—
77#1878C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1844C2212yes—
81#1834C2212yes—
82#1834C2212yes—
83#1809C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
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⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42C2312yes—
122#470—48no—
123#441—48no—
124C4×Q16—28no—
125C4×Q16—28no—
126#436—28no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#399—24no—
146#398—22no—
147#398—24no—
148#396—22no—
149#396—22no—
150#396—22no—
151#396—22no—
152#396—24no—
153#391—22no—
154#391C2311yescharacteristic
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2312yes—
158#390—21no—
159#390—21no—
160#390—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389—24no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—24no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#382—21no—
200#382—21no—
201#381—22no—
202#380—22no—
203#380C2312yes—
204#380C2314yes—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×Q8C2412yes—
237C22×Q8C2411yescharacteristic
238C22×D4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C2×Q16—24no—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C2×M16—28no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—41no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#125—48no—
261#125—44no—
262C22⋊Q8—28no—
263C22⋊Q8—24no—
264C22⋊Q8—42no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—84no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280#117—48no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
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—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—24no—
347#106—44no—
348#106—44no—
349C4⋊C8—24no—
350#103—28no—
351#103—24no—
352#103—24no—
353#103—24no—
354#103—24no—
355#103—44no—
356#103—44no—
357#103—44no—
358C22⋊C8—28no—
359#97—44no—
360C4×C8—42no—
361C4×C8#4118yes—
362#95—21no—
363#95—22no—
364#95—21no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24C22×D412yes—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8H(5,𝔽2)14yes—
398C2×Q82−1+412yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431Q16—216no—
432Q16—48no—
433M16—88no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8C2×#4114yes—
442C2×C8C2×#4114yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—84no—
493C42—24no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—41no—
501C42—48no—
502C42—44no—
503C42—42no—
504C42—42no—
505C42—82no—
506C42—82no—
507C42#13714yes—
508C42#13714yes—
509C42C2×#4114yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—88no—
527D4—44no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—84no—
578C2×C4#53212yes—
579C2×C4#53212yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4C22×#4112yes—
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4C2×2−1+411yescharacteristic
589C2×C4#58511yescharacteristic
590C8—24no—
591C8—24no—
592C8—44no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22C2×#53211yescharacteristic
601C22#348411yescharacteristic
602C22#348911yescharacteristic
603C22C22×#13711yescharacteristic
604C22C22×#13711yescharacteristic
605C22#358511yescharacteristic
606C22#358611yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4#318514yes—
625C4#349212yes—
626C4#349412yes—
627C2—82no—
628C2#3592311yescharacteristic
629C2C2×#318511yescharacteristic
630C2#6072111yescharacteristic
631C2#6158411yescharacteristic
632C2#6158811yescharacteristic
633C2#6211211yescharacteristic
634C2#6211311yescharacteristic
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. #30367C2semidirect
4. #30305C2semidirect
5. #29166C2semidirect
6. #29141C2semidirect
7. #29123C2semidirect
8. #28871C2semidirect
11. #26988C2semidirect
12. #25504C2semidirect
13. #23801C2semidirect
14. #23577C2semidirect
15. #22763C2semidirect
16. #22674C2semidirect

Maximal subgroups

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

subgroupindex
#364772
#361342
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