Finite Group Databasebeta

← #6564146

Subgroups — #6564146

Order 512.

1623 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#36445C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29923C211yesmaximal, characteristic, complemented
4#29922C212yesmaximal, complemented
5#27916C212yesmaximal, complemented
6#27905C211yesmaximal, characteristic, complemented
7#27685C212yesmaximal, complemented
8#27680C211yesmaximal, characteristic, complemented
9#26247C214yesmaximal, complemented
10#26231C212yesmaximal, complemented
11#25029C214yesmaximal, complemented
12#25015C212yesmaximal, complemented
13#23763C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#22729C212yesmaximal, complemented
16#22547C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#485C2214yes—
22C2×#483C2214yes—
23C2×#446C2212yes—
24#2857C2211yescharacteristic
25#2857C2212yes—
26#2542C2211yescharacteristic
27#2542C2212yes—
28#2538C2211yescharacteristic
29#2538C2212yes—
30#2454—22no—
31#2454—21no—
32#2441—22no—
33#2441—21no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2211yescharacteristic
42#2341—21no—
43#2314—22no—
44#2311—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51#2274—21no—
52#2100C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2095C2211yescharacteristic
58#2095C2212yes—
59#2083C2211yescharacteristic
60#2077C2214yes—
61#2076C2212yes—
62#2072C2211yescharacteristic
63#2072C2212yes—
64#2036C2214yes—
65#2036C2212yes—
66#2036C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#1981C2212yes—
70#1981C2214yes—
71#1960C2214yes—
72#1956C2212yes—
73#1955C2214yes—
74#1954C2212yes—
75#1949C2214yes—
76#1948C2212yes—
77#1948C2214yes—
78#1947C2212yes—
79#1873C2214yes—
80#1872C2212yes—
81#1854C2214yes—
82#1853C2212yes—
83#1847C2214yes—
84#1844C2212yes—
85#1805C2214yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#557—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—21no—
100C2×#125—22no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C4×#41—28no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—21no—
118C2×#117—22no—
119C2×#117D412yes—
120C2×#117D411yescharacteristic
121C2×#117C2314yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#485—48no—
128#485—48no—
129#483—48no—
130#446—28no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#399—22no—
153#399—22no—
154#399—22no—
155#396—22no—
156#396—22no—
157#396—22no—
158#391—22no—
159#391C2311yescharacteristic
160#391C2312yes—
161#390—22no—
162#390—21no—
163#390—22no—
164#390—21no—
165#390—22no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389C2311yescharacteristic
175#388—24no—
176#388—24no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—21no—
181#386—24no—
182#385—21no—
183#385—22no—
184#384—21no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#382—21no—
195#382—22no—
196#382C2314yes—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380C2311yescharacteristic
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—24no—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×Q8C2412yes—
240C22×D4C2×D411yescharacteristic
241C22×D4C2×D412yes—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—28no—
254#125—48no—
255#125—44no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—41no—
259C22⋊Q8—42no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—28no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8#4118yes—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—28no—
273#117—24no—
274#117—28no—
275#117—44no—
276#117—44no—
277#117—44no—
278#117—84no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
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—21no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—24no—
349#107—44no—
350#106—28no—
351#106—44no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—28no—
355#103—28no—
356#103—24no—
357#103—42no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362C22⋊C8—24no—
363C22⋊C8—28no—
364C22⋊C8—42no—
365C22⋊C8—44no—
366#97—28no—
367C4×C8—24no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—416no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8#12314yes—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4#13714yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8H(5,𝔽2)14yes—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4#12318yes—
491C4⋊C4#12314yes—
492C4⋊C42−1+412yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—81no—
500C22⋊C4—82no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42—42no—
516C42—84no—
517C42—84no—
518C42C2×#4114yes—
519C42C2×#4118yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23C2×#13712yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—44no—
537Q8—42no—
538Q8—44no—
539Q8—42no—
540D4—48no—
541D4—44no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—164no—
594C2×C4C2×#12314yes—
595C2×C4C2×#12312yes—
596C2×C4C2×#12314yes—
597C2×C4C2×#12312yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4#57512yes—
601C2×C4#57712yes—
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×2−1+411yescharacteristic
605C2×C4#58511yescharacteristic
606C8—24no—
607C8—42no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C22×#12311yescharacteristic
624C22#348911yescharacteristic
625C22#349811yescharacteristic
626C22C22×#13711yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—24no—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4#309514yes—
647C4#351712yes—
648C4#352212yes—
649C2—42no—
650C2—82no—
651C2#3600511yescharacteristic
652C2C2×#309511yescharacteristic
653C2#6052211yescharacteristic
654C2#6211311yescharacteristic
655C2#6213411yescharacteristic
656C2#6242211yescharacteristic
657C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29923C2semidirect
4. #29922C2semidirect
5. #27916C2semidirect
6. #27905C2semidirect
7. #27685C2semidirect
8. #27680C2semidirect
9. #26247C2semidirect
10. #26231C2semidirect
11. #25029C2semidirect
12. #25015C2semidirect
13. #23763C2semidirect
14. #23757C2semidirect
15. #22729C2semidirect
16. #22547C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#341882
#299232
#299222
#279162
#279052
#276852
#276802
#262472
#262312
#250292
#250152
#237632
#237572
#227292
#225472
#225322