Finite Group Databasebeta

← #6514558

Subgroups — #6514558

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#29287C212yesmaximal, complemented
4#29273C212yesmaximal, complemented
5#29255C212yesmaximal, complemented
6#28283C212yesmaximal, complemented
7#26412C214yesmaximal
8#26340C214yesmaximal
9#26108C212yesmaximal, complemented
10#26088C212yesmaximal, complemented
11#26042C212yesmaximal, complemented
12#26022C212yesmaximal, complemented
13#21981C211yesmaximal, characteristic
14#21972C211yesmaximal, characteristic
15#21885C211yesmaximal, characteristic
16#21385C211yesmaximal, characteristic
17C2×#1948C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#533C2211yescharacteristic
20C2×#475C2214yes—
21C2×#431C2211yescharacteristic
22C22×#106C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24#2552C2211yescharacteristic
25#2552C2211yescharacteristic
26#2551C2211yescharacteristic
27#2551C2211yescharacteristic
28#2550—21no—
29#2550—21no—
30#2549—22no—
31#2440C2212yes—
32#2440C2212yes—
33#2399—22no—
34#2360—22no—
35#2360C2214yes—
36#2355—21no—
37#2355C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40#2330—22no—
41#2317—21no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46#2296—21no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2274—21no—
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2066C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2024C2214yes—
60#1988C2214yes—
61#1987C2211yescharacteristic
62#1987C2212yes—
63#1987C2211yescharacteristic
64#1981C2211yescharacteristic
65#1981C2212yes—
66#1981C2211yescharacteristic
67#1948C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1939C2212yes—
76#1919C2214yes—
77#1919C2214yes—
78#1892C2214yes—
79#1872C2212yes—
80#1872C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1853C2214yes—
84#1850C2214yes—
85#1844C2212yes—
86#1842C2212yes—
87#1834C2212yes—
88#1831C2212yes—
89#1803C2214yes—
90#1803C2212yes—
91#1803C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C23×C8C2312yes—
96#533—24no—
97#533—48no—
98#533Q818yescomplemented
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#125—21no—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2314yes—
118C2×#117—21no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#475—48no—
130#431—24no—
131C2×#106—44no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—22no—
156#395—22no—
157#395—22no—
158#395—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—24no—
173#389—24no—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—24no—
180#387—24no—
181#385—24no—
182#385—22no—
183#385—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#382—22no—
194#382—21no—
195#380—21no—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—24no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C2×#95—21no—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8—24no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8C2×Q814yes—
260C4⋊Q8C2×Q814yes—
261#125—28no—
262#125—44no—
263#125—44no—
264C22⋊Q8—28no—
265C22⋊Q8—24no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8C2×Q818yes—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—48no—
275C4×Q8—84no—
276C4×Q8C2×Q818yes—
277#117—28no—
278#117—24no—
279#117—44no—
280#117SD1618yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
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—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—41no—
312C2×C4⋊C4—41no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×Q814yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#106—28no—
355#106—44no—
356#106—44no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—22no—
362C4⋊C8—44no—
363C4⋊C8—44no—
364#103—24no—
365#103—24no—
366#103—42no—
367#103—42no—
368#103—44no—
369#103—44no—
370C22⋊C8—24no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8C22×Q814yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C22⋊Q814yes—
431C22×C4C22⋊Q814yes—
432C22×C4C2×SD1614yes—
433C22×C4#13714yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×Q812yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—48no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4C2×SD1614yes—
500C4⋊C4C2×SD1614yes—
501C4⋊C4C22×Q814yes—
502C4⋊C4H(5,𝔽2)12yes—
503C4⋊C42−1+412yes—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4C2×SD1618yes—
508C42—28no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—84no—
522C42—84no—
523C42C22⋊Q814yes—
524C42C22⋊Q814yes—
525C42C2×SD1618yes—
526C42C22×Q812yes—
527C42C22×Q812yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23C2×C22⋊Q812yes—
535C23C22×SD1612yes—
536C23C2×#13712yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—88no—
544Q8—88no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—164no—
596C2×C4#475116yescomplemented
597C2×C4C2×C22⋊Q812yes—
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C22×SD1614yes—
603C2×C4C22×SD1614yes—
604C2×C4C22×SD1612yes—
605C2×C4C22×SD1612yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4C23×Q811yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4#58511yescharacteristic
613C2×C4#58511yescharacteristic
614C8—42no—
615C8—44no—
616C8—42no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22C2×#47514yes—
624C22C22×C22⋊Q811yescharacteristic
625C22#348811yescharacteristic
626C22#348911yescharacteristic
627C22C23×SD1611yescharacteristic
628C22C22×#13711yescharacteristic
629C22#358511yescharacteristic
630C22#358611yescharacteristic
631C4—22no—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—24no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4C2×#47514yes—
648C4C2×#47514yes—
649C4#350311yescharacteristic
650C4#350411yescharacteristic
651C4#350611yescharacteristic
652C4#350711yescharacteristic
653C2—24no—
654C2#3551911yescharacteristic
655C2C22×#47511yescharacteristic
656C2#6049211yescharacteristic
657C2#6194811yescharacteristic
658C2#6196511yescharacteristic
659C2#6209611yescharacteristic
660C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #29287C2semidirect
4. #29273C2semidirect
5. #29255C2semidirect
6. #28283C2semidirect
9. #26108C2semidirect
10. #26088C2semidirect
11. #26042C2semidirect
12. #26022C2semidirect
98. #533Q8semidirect
596. C2×C4#475semidirect

Maximal subgroups

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

subgroupindex
#349382
#342142
#292872
#292732
#292552
#282832
#264122
#263402
#261082
#260882
#260422
#260222
#219812
#219722
#218852
#213852
C2×#19482