Finite Group Databasebeta

← #6446014

Subgroups — #6446014

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#25065C214yesmaximal, complemented
4#25064C214yesmaximal, complemented
5#25059C212yesmaximal, complemented
6#25058C212yesmaximal, complemented
7#25057C212yesmaximal, complemented
8#25056C212yesmaximal, complemented
9#23735C212yesmaximal
10#23662C212yesmaximal, complemented
11#21300C211yesmaximal, characteristic, complemented
12#21170C211yesmaximal, characteristic, complemented
13#21065C212yesmaximal
14#21039C212yesmaximal, complemented
15#20797C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C2×C4×#102C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#427C2212yes—
23C2×#423C2211yescharacteristic
24C2×#418C2212yes—
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2848—22no—
29#2626—22no—
30#2580C2214yes—
31#2566—22no—
32#2563—21no—
33#2542—21no—
34#2538C2212yes—
35#2538C2212yes—
36#2370—22no—
37#2369—21no—
38#2365—21no—
39#2355C2212yescomplemented
40#2355C2214yescomplemented
41Q8×C4⋊C4C2212yescomplemented
42#2329—22no—
43#2317C2212yes—
44#2317C2212yes—
45#2309C2212yes—
46C4×C22⋊Q8—22no—
47#2285C2211yescharacteristic
48#2285C2211yescharacteristic
49#2274C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2094C2214yes—
52#2093C2214yes—
53#2092C2214yes—
54#2092C2214yes—
55#2085C2214yescomplemented
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2083C2214yescomplemented
59#1984C2212yescomplemented
60#1984C2212yescomplemented
61#1984C2212yescomplemented
62#1984C2212yescomplemented
63#1949C2214yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1926C2214yes—
67#1926C2214yes—
68#1925C2214yes—
69#1925C2214yes—
70#1921C2214yescomplemented
71#1920C2214yescomplemented
72#1919C2214yescomplemented
73#1918C2214yescomplemented
74#1846C2212yescomplemented
75#1846C2212yescomplemented
76#1845C2211yescharacteristic, complemented
77#1845C2211yescharacteristic, complemented
78#1844C2211yescharacteristic, complemented
79#1844C2211yescharacteristic, complemented
80#1804C2214yes—
81#1802C2212yes—
82#1802C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89C2×#340C2211yescharacteristic
90C2×#340C2211yescharacteristic
91#1729C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#533—24no—
99#533—24no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8D411yescharacteristic
102C2×#125D412yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8D412yes—
106C2×C22⋊Q8D412yes—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125#428—44no—
126#427—28no—
127#427—44no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#423—24no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141#418—44no—
142C2×#102—24no—
143C2×#102Q818yescomplemented
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C4×C8—24no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#399—24no—
156#396—24no—
157#396—24no—
158#393—24no—
159#391—24no—
160#391—22no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#387—24no—
172#387—24no—
173#387—24no—
174#385—24no—
175#385—24no—
176#385—24no—
177#385—24no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2314yes—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382C2312yes—
186#382C2314yes—
187#382C2312yes—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—22no—
193#380—22no—
194#379—24no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—24no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—24no—
210C2×#95—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C22×Q8C2×D412yes—
225C22×Q8C2×D412yes—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4C2×Q814yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—44no—
240C23×C4—82no—
241C23×C4—82no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×M16—24no—
246C2×M16—24no—
247C2×M16—24no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8C2×Q814yes—
253C22×C8#4114yes—
254C22×C8C2414yes—
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2411yescharacteristic
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261#125—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—84no—
265C22⋊Q8—84no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—84no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—44no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
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—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×Q814yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D414yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—82no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2414yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—24no—
343#107—42no—
344#107—42no—
345#107—44no—
346#106—24no—
347#106—24no—
348#106—42no—
349#106—42no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#102—24no—
354#102—24no—
355#102—44no—
356#102—44no—
357#102C2×Q818yes—
358#102C2×Q818yes—
359C4×C8—22no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375C24—22no—
376C24—24no—
377C24—42no—
378C24—42no—
379C24—81no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4C22×Q814yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—816no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4#12314yes—
431C22×C4#12314yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q812yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C22×C4C2511yescharacteristic
446M16—48no—
447C2×C8—28no—
448C2×C8—28no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8C22×Q812yes—
461C2×C8C22×Q812yes—
462C2×C8C2×#4112yes—
463C2×C8C2×#4112yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C22×Q814yes—
499C22⋊C4—28no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—84no—
514C42—82no—
515C42—84no—
516C42#12314yes—
517C42#12314yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—24no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23C2×#12312yes—
541C23D4×Q812yes—
542C23#55012yes—
543C23C23×D411yescharacteristic
544Q8—48no—
545Q8—84no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—42no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—88no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4#53314yes—
597C2×C4#54214yes—
598C2×C4D4×Q812yes—
599C2×C4#55012yes—
600C2×C4#55214yes—
601C2×C4#55414yes—
602C2×C4#57612yes—
603C2×C4#57712yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C8—22no—
611C8—22no—
612C8—22no—
613C8—22no—
614C8—44no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—816no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#242314yes—
630C22C22×#12311yescharacteristic
631C22C2×D4×Q811yescharacteristic
632C22C2×#55011yescharacteristic
633C22C2×#57611yescharacteristic
634C22C2×#57711yescharacteristic
635C22#358411yescharacteristic
636C22#358411yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#242314yes—
650C4#309614yes—
651C4C2×#53311yescharacteristic
652C4C2×#54211yescharacteristic
653C4C2×#55211yescharacteristic
654C4C2×#55411yescharacteristic
655C2—24no—
656C2—84no—
657C2C2×#242311yescharacteristic
658C2C2×#309611yescharacteristic
659C2#6051611yescharacteristic
660C2#6137011yescharacteristic
661C2#6137111yescharacteristic
662C2#6140611yescharacteristic
663C2#6140711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
3. #25065C2semidirect
4. #25064C2semidirect
5. #25059C2semidirect
6. #25058C2semidirect
7. #25057C2semidirect
8. #25056C2semidirect
10. #23662C2semidirect
11. #21300C2semidirect
12. #21170C2semidirect
14. #21039C2semidirect
15. #20797C2semidirect
39. #2355C22semidirect
40. #2355C22semidirect
41. Q8×C4⋊C4C22semidirect
55. #2085C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2083C22semidirect
59. #1984C22semidirect
60. #1984C22semidirect
61. #1984C22semidirect
62. #1984C22semidirect
63. #1949C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
70. #1921C22semidirect
71. #1920C22semidirect
72. #1919C22semidirect
73. #1918C22semidirect
74. #1846C22semidirect
75. #1846C22semidirect
76. #1845C22semidirect
77. #1845C22semidirect
78. #1844C22semidirect
79. #1844C22semidirect
143. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
#351722
#342132
#250652
#250642
#250592
#250582
#250572
#250562
#237352
#236622
#213002
#211702
#210652
#210392
#207972
#206462
C2×C4×#1022