Finite Group Databasebeta

← #6571325

Subgroups — #6571325

Order 512.

1585 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29922C211yesmaximal, characteristic, complemented
4#29922C212yesmaximal, complemented
5#27980C212yesmaximal, complemented
6#27972C211yesmaximal, characteristic, complemented
7#27652C212yesmaximal, complemented
8#27648C211yesmaximal, characteristic, complemented
9#26249C214yesmaximal, complemented
10#26233C212yesmaximal, complemented
11#24983C214yesmaximal, complemented
12#24972C212yesmaximal, complemented
13#23763C212yesmaximal, complemented
14#23756C211yesmaximal, characteristic, complemented
15#22742C212yesmaximal, complemented
16#22547C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2212yes—
21C2×#485C2214yes—
22C2×#446C2212yes—
23#2857C2211yescharacteristic
24#2857C2212yes—
25#2542C2211yescharacteristic
26#2542C2212yes—
27#2538C2211yescharacteristic
28#2538C2212yes—
29#2454—22no—
30#2454—21no—
31#2441—22no—
32#2441—21no—
33#2355—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2212yes—
37#2352—22no—
38#2352C2214yes—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2211yescharacteristic
41#2341—21no—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2099C2211yescharacteristic
52#2096C2211yescharacteristic
53#2096C2212yes—
54#2084C2211yescharacteristic
55#2083C2212yes—
56#2077C2214yes—
57#2077C2212yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2211yescharacteristic
63#2036C2214yes—
64#2035C2214yes—
65#2035C2212yes—
66#2035C2214yes—
67#2035C2214yes—
68#1981C2212yes—
69#1981C2214yes—
70#1960C2214yes—
71#1956C2212yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1948C2214yes—
77#1947C2212yes—
78#1873C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1836C2214yes—
82#1834C2212yes—
83#1825C2214yes—
84#1824C2212yes—
85#1805C2214yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
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⋊Q8C2312yes—
102C2×C22⋊Q8C2311yescharacteristic
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#484—44no—
129#446—28no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#399—22no—
150#399—22no—
151#399—22no—
152#396—22no—
153#396—22no—
154#396—22no—
155#391—22no—
156#391C2311yescharacteristic
157#391C2312yes—
158#390—22no—
159#390—21no—
160#390—22no—
161#390—21no—
162#390—22no—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389C2311yescharacteristic
172#388—24no—
173#388—24no—
174#387—24no—
175#387—24no—
176#387—22no—
177#387—21no—
178#386—24no—
179#385—21no—
180#385—22no—
181#384—21no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#382—21no—
192#382—22no—
193#382C2314yes—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—24no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×D4C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—28no—
253#125—48no—
254#125—44no—
255C22⋊Q8—22no—
256C22⋊Q8—24no—
257C22⋊Q8—41no—
258C22⋊Q8—42no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8#4118yes—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—28no—
272#117—24no—
273#117—28no—
274#117—44no—
275#117—44no—
276#117—44no—
277#117—84no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
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—21no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
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—28no—
348#106—24no—
349#106—42no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—28no—
353#103—42no—
354#103—44no—
355#103—44no—
356#103—42no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—42no—
361C22⋊C8—44no—
362C22⋊C8—42no—
363C22⋊C8—44no—
364#97—28no—
365C4×C8—24no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—21no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383#41—416no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8#12314yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4#12318yes—
489C4⋊C4#12314yes—
490C4⋊C4H(5,𝔽2)12yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—81no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—44no—
513C42—42no—
514C42—84no—
515C42—84no—
516C42C2×#4114yes—
517C42C2×#4118yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23C2×#13712yes—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—42no—
535Q8—44no—
536Q8—42no—
537Q8—44no—
538D4—48no—
539D4—44no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—164no—
592C2×C4C2×#12314yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12314yes—
595C2×C4C2×#12312yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4#57512yes—
599C2×C4#57712yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4#58512yes—
603C2×C4#58511yescharacteristic
604C8—42no—
605C8—44no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C22×#12311yescharacteristic
622C22#348811yescharacteristic
623C22#350011yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—24no—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4#309514yes—
645C4#351712yes—
646C4#352412yes—
647C2—42no—
648C2—82no—
649C2#3607611yescharacteristic
650C2C2×#309511yescharacteristic
651C2#6052211yescharacteristic
652C2#6196611yescharacteristic
653C2#6198611yescharacteristic
654C2#6254011yescharacteristic
655C2#6258711yescharacteristic
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. #29922C2semidirect
4. #29922C2semidirect
5. #27980C2semidirect
6. #27972C2semidirect
7. #27652C2semidirect
8. #27648C2semidirect
9. #26249C2semidirect
10. #26233C2semidirect
11. #24983C2semidirect
12. #24972C2semidirect
13. #23763C2semidirect
14. #23756C2semidirect
15. #22742C2semidirect
16. #22547C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#341882
#299222
#279802
#279722
#276522
#276482
#262492
#262332
#249832
#249722
#237632
#237562
#227422
#225472
#225322