Finite Group Databasebeta

← #6447014

Subgroups — #6447014

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#24984C214yesmaximal, complemented
4#24983C214yesmaximal, complemented
5#24980C214yesmaximal, complemented
6#24979C214yesmaximal, complemented
7#23643C212yesmaximal, complemented
8#23484C212yesmaximal, complemented
9#21188C211yesmaximal, characteristic
10#21170C211yesmaximal, characteristic, complemented
11#20913C212yesmaximal, complemented
12#20762C211yesmaximal, characteristic, complemented
13#20703C212yesmaximal, complemented
14#20647C211yesmaximal, characteristic, complemented
15C2×#1763C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#419C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C2×#408C2212yes—
26C2×#408C2212yes—
27C22×#97C2211yescharacteristic
28#2854—22no—
29#2618—22no—
30#2548—21no—
31#2546—21no—
32#2545—22no—
33#2538C2212yes—
34#2538C2212yes—
35#2536C2214yes—
36#2367—22no—
37#2366—21no—
38#2365—21no—
39#2354C2212yes—
40#2352C2214yes—
41D4×C4⋊C4C2212yes—
42#2325—22no—
43#2311C2212yes—
44#2311C2212yes—
45C4×#121—22no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2077C2214yes—
50#2077C2214yes—
51#2076C2214yes—
52#2075C2214yes—
53#2036C2214yes—
54#2036C2214yes—
55#2035C2214yes—
56#2035C2214yes—
57#1955C2214yes—
58#1955C2214yes—
59#1949C2214yes—
60#1948C2212yes—
61#1947C2212yes—
62#1873C2214yes—
63#1873C2214yes—
64#1872C2214yes—
65#1872C2214yes—
66#1836C2212yes—
67#1836C2212yes—
68#1836C2212yes—
69#1836C2212yes—
70#1825C2214yes—
71#1825C2214yes—
72#1824C2214yes—
73#1823C2214yes—
74#1805C2214yes—
75#1803C2212yes—
76#1803C2212yes—
77#1763C2214yes—
78#1763C2214yes—
79#1736C2212yes—
80#1736C2212yes—
81#1735C2212yes—
82#1735C2212yes—
83C2×#340C2211yescharacteristic
84C2×#340C2211yescharacteristic
85C22×#41C2312yes—
86C23×Q8C2311yescharacteristic
87C22×M16C2312yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90#536—28no—
91#536—44no—
92#532—24no—
93#532—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8D411yescharacteristic
96C2×#127D411yescharacteristic
97C2×#124—22no—
98C2×#124D412yes—
99C2×#123—24no—
100C2×C22⋊Q8C2314yes—
101C2×C22⋊Q8C2314yes—
102C2×#121D412yes—
103C2×#121D412yes—
104#520—28no—
105#520—44no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2314yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#428—44no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132#423—24no—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135#419—44no—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142#408—44no—
143#408—44no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97—24no—
149C2×#97C2312yes—
150C2×#97C2312yes—
151#398—24no—
152#396—24no—
153#396—24no—
154#394—24no—
155#391—22no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#388—24no—
166#388—24no—
167#388—24no—
168#388—24no—
169#387—24no—
170#386—24no—
171#386—24no—
172#386—24no—
173#385—24no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382C2312yes—
179#382C2312yes—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380C2314yes—
185#380C2314yes—
186#380C2314yes—
187#380C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—24no—
198C2×#95—21no—
199#340C2314yes—
200#340C2314yes—
201#340C2312yes—
202#340C2314yes—
203#340C2312yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—24no—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8C2×D412yes—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×D4C2×D414yes—
229C22×D4C2×D412yes—
230C22×D4C2×D412yes—
231C22×D4C2×D412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—21no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—24no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—48no—
255#127—24no—
256#124—28no—
257#124—48no—
258#123—416no—
259C22⋊Q8—28no—
260C22⋊Q8—28no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263#121—28no—
264#121—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—48no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—48no—
277C4×D4—416no—
278C4×D4—84no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—28no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42—82no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4114yes—
342C2×C42#4114yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—44no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349#103—28no—
350#103—28no—
351#103—44no—
352#103—44no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—24no—
358#97—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369C24C22×D414yes—
370C24C22×D411yescharacteristic
371#41—48no—
372#41—48no—
373#41—48no—
374#41—48no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8C2×#4114yes—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4#12314yes—
437C22×C4#12314yes—
438C22×C4C22×D414yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4C2511yescharacteristic
451M16—48no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—44no—
462C2×C8—44no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
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—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4C2×#4114yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—48no—
507C42—44no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—82no—
513C42#12314yes—
514C42#12314yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23C2×#12312yes—
531C23#54612yes—
532C23#54612yes—
533C23C23×D411yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—48no—
537Q8—48no—
538Q8—42no—
539Q8—42no—
540D4—48no—
541D4—416no—
542D4—48no—
543D4—84no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
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—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—168no—
590C2×C4C2×#12312yes—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4#52914yes—
596C2×C4#54014yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#55314yes—
600C2×C4#55514yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C8—44no—
608C8—42no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22#242014yes—
624C22C22×#12311yescharacteristic
625C22C2×#54611yescharacteristic
626C22C2×#54611yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C22#358611yescharacteristic
630C22#358611yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#242014yes—
645C4C2×#52911yescharacteristic
646C4C2×#54011yescharacteristic
647C4C2×#55311yescharacteristic
648C4C2×#55511yescharacteristic
649C2—24no—
650C2—84no—
651C2C2×#242011yescharacteristic
652C2#6051711yescharacteristic
653C2#6051911yescharacteristic
654C2#6121911yescharacteristic
655C2#6122011yescharacteristic
656C2#6122111yescharacteristic
657C2#6122111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #24984C2semidirect
4. #24983C2semidirect
5. #24980C2semidirect
6. #24979C2semidirect
7. #23643C2semidirect
8. #23484C2semidirect
10. #21170C2semidirect
11. #20913C2semidirect
12. #20762C2semidirect
13. #20703C2semidirect
14. #20647C2semidirect
15. C2×#1763C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#342062
#249842
#249832
#249802
#249792
#236432
#234842
#211882
#211702
#209132
#207622
#207032
#206472
C2×#17632