Finite Group Databasebeta

← #6989339

Subgroups — #6989339

Order 512.

1633 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36472C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#32834C212yesmaximal, complemented
4#31350C214yesmaximal, complemented
5#30629C214yesmaximal, complemented
6#30376C212yesmaximal, complemented
7#30075C212yesmaximal, complemented
8#29920C211yesmaximal, characteristic, complemented
9#28789C212yesmaximal, complemented
10#28716C212yesmaximal, complemented
11#27973C211yesmaximal, characteristic, complemented
12#27767C212yesmaximal, complemented
13#27644C211yesmaximal, characteristic, complemented
14#26526C212yesmaximal, complemented
15#26106C212yesmaximal, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C2×#544—21no—
20C2×#486C2214yes—
21C2×#480C2212yes—
22#2868C2212yes—
23#2860C2212yes—
24#2859C2212yes—
25#2856C2211yescharacteristic
26#2660—22no—
27#2603—22no—
28#2552—21no—
29#2552C2211yescharacteristic
30#2552C2212yes—
31#2551—21no—
32#2551C2212yes—
33#2541C2211yescharacteristic
34#2490—21no—
35#2464—22no—
36#2440—22no—
37#2440C2214yes—
38#2428C2212yes—
39#2422C2212yes—
40#2414—21no—
41#2366C2211yescharacteristic
42#2361—21no—
43D4×C4⋊C4C2211yescharacteristic
44#2317—21no—
45C4×#127C2212yes—
46C4×#121C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2214yes—
51#2099C2211yescharacteristic
52#2093C2214yes—
53#2093C2211yescharacteristic
54#2093C2212yes—
55#2086C2214yes—
56#2083C2211yescharacteristic
57#2083C2212yes—
58#2083C2214yes—
59#2075C2212yes—
60#2071C2211yescharacteristic
61#2071C2212yes—
62#2071C2212yes—
63#2071C2212yes—
64#2071C2212yes—
65#2067C2214yes—
66#2057C2214yes—
67#2041C2214yes—
68#2033C2214yes—
69#2033C2212yes—
70#2023C2214yes—
71#1990C2214yes—
72#1982C2212yes—
73#1957C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1942C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1923C2214yes—
80#1874C2214yes—
81#1862C2214yes—
82#1862C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1823C2212yes—
86#1802C2214yes—
87#1802C2214yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#553—48no—
96#544—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C2314yes—
101C2×#127C2311yescharacteristic
102C2×#127C2312yes—
103C2×#126—22no—
104C2×#125—22no—
105C2×#125—21no—
106C2×#125—22no—
107C2×#123—22no—
108C2×C22⋊Q8—24no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121—22no—
114C2×#121C2311yescharacteristic
115C2×#121C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131#486—48no—
132#480—44no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#398—22no—
151#398—24no—
152#398—24no—
153#397—22no—
154#395—24no—
155#395—24no—
156#394—22no—
157#394—22no—
158#392—22no—
159#392—22no—
160#390—21no—
161#390—22no—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2311yescharacteristic
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#389C2311yescharacteristic
177#389C2314yes—
178#389C2312yes—
179#388—21no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—21no—
184#386—22no—
185#386—22no—
186#385—24no—
187#385—22no—
188#385—22no—
189#384—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2314yes—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382—21no—
202#380—21no—
203#380—21no—
204#380C2311yescharacteristic
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376D412yes—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—24no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—48no—
254C4⋊Q8—48no—
255C4⋊Q8—84no—
256#127—24no—
257#127—24no—
258#127—42no—
259#126—48no—
260#125—48no—
261#125—44no—
262#125—48no—
263#123—48no—
264C22⋊Q8—416no—
265#121—22no—
266#121—41no—
267#121—44no—
268#121—44no—
269#121—44no—
270#121—48no—
271#121—44no—
272#121—82no—
273C4×D4—24no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—48no—
279C4×D4—84no—
280#117—24no—
281#117—44no—
282#117D818yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—44no—
352#107—42no—
353#107—44no—
354C4⋊C8—24no—
355C4⋊C8—44no—
356#102—28no—
357#102—24no—
358#102—42no—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363C22⋊C8—42no—
364C22⋊C8—42no—
365C4×C8—24no—
366C4×C8—44no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24—22no—
382C24C22×D411yescharacteristic
383C2×Q8—48no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4H(5,𝔽2)12yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C2×D814yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4C2×D814yes—
494C4⋊C4C2×D814yes—
495C4⋊C4#13714yes—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4C2×D818yes—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42C2×D818yes—
525C422−1+414yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C22×D812yes—
547C23C23×D411yescharacteristic
548Q8—168no—
549D4—24no—
550D4—44no—
551D4—42no—
552D4—44no—
553D4—88no—
554D4—88no—
555D4—84no—
556D4—84no—
557D4—162no—
558D4—164no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—164no—
604C2×C4—164no—
605C2×C4—164no—
606C2×C4#53612yes—
607C2×C4#53612yes—
608C2×C4C22×D812yes—
609C2×C4C22×D814yes—
610C2×C4C22×D814yes—
611C2×C4C22×D812yes—
612C2×C4C2×#13714yes—
613C2×C4C2×#13712yes—
614C2×C4#57712yes—
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+412yes—
617C2×C4#58512yes—
618C2×C4#58511yescharacteristic
619C8—42no—
620C8—44no—
621C8—42no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#53611yescharacteristic
644C22#348811yescharacteristic
645C22#350011yescharacteristic
646C22C23×D811yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358511yescharacteristic
650C4—24no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—24no—
656C4—42no—
657C4—41no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—84no—
663C4—81no—
664C4—82no—
665C4#322614yes—
666C4#355912yes—
667C4#356512yes—
668C2—42no—
669C2—82no—
670C2—162no—
671C2#4294411yescharacteristic
672C2C2×#322611yescharacteristic
673C2#6087311yescharacteristic
674C2#6194711yescharacteristic
675C2#6198011yescharacteristic
676C2#6252911yescharacteristic
677C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36472C2
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
2. #35275C2semidirect
3. #32834C2semidirect
4. #31350C2semidirect
5. #30629C2semidirect
6. #30376C2semidirect
7. #30075C2semidirect
8. #29920C2semidirect
9. #28789C2semidirect
10. #28716C2semidirect
11. #27973C2semidirect
12. #27767C2semidirect
13. #27644C2semidirect
14. #26526C2semidirect
15. #26106C2semidirect
16. #23754C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#364722
#352752
#328342
#313502
#306292
#303762
#300752
#299202
#287892
#287162
#279732
#277672
#276442
#265262
#261062
#237542
#226392