Finite Group Databasebeta

← #6848225

Subgroups — #6848225

Order 512.

1607 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic
4#32248C214yesmaximal, complemented
5#32150C214yesmaximal, complemented
6#29641C212yesmaximal
7#28799C212yesmaximal, complemented
8#27960C211yesmaximal, characteristic
9#27823C212yesmaximal, complemented
10#27647C211yesmaximal, characteristic
11#27616C212yesmaximal
12#26444C212yesmaximal, complemented
13#26026C212yesmaximal, complemented
14#24609C212yesmaximal
15#22603C211yesmaximal, characteristic
16#21840C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#483C2211yescharacteristic
21C2×#483C2212yes—
22C2×#432C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2571C2211yescharacteristic
26#2568C2211yescharacteristic
27#2551—21no—
28#2551—21no—
29#2551—22no—
30#2551C2212yes—
31#2551C2212yes—
32#2551C2214yes—
33#2533—22no—
34#2514—21no—
35#2514C2212yes—
36#2509C2212yes—
37#2503—21no—
38#2493—22no—
39C2×#395—22no—
40#2365C2211yescharacteristic
41#2360—21no—
42#2356—21no—
43#2355C2211yescharacteristic
44C4×C22⋊Q8C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2099C2214yes—
53#2088C2214yes—
54#2085C2214yes—
55#2084C2212yes—
56#2072C2212yes—
57#2068C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#1993C2214yes—
63#1993C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1987C2214yes—
67#1987C2214yes—
68#1948C2212yes—
69#1947C2212yes—
70#1943C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1927C2212yes—
74#1921C2214yes—
75#1920C2212yes—
76#1920C2214yes—
77#1918C2214yes—
78#1917C2214yes—
79#1872C2211yescharacteristic
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1834C2212yes—
83#1823C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#483—24no—
124#483—44no—
125#432—44no—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2311yescharacteristic
137C2×C22⋊C8C2311yescharacteristic
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#401—22no—
141#400—22no—
142#400—22no—
143#400—22no—
144#400—22no—
145#400—22no—
146#395—24no—
147#395—24no—
148#395—24no—
149#395—24no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391C2312yes—
155#391C2312yes—
156#391C2311yescharacteristic
157#389—21no—
158#389—22no—
159#389—24no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—21no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—21no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2312yes—
185#383—22no—
186#383—22no—
187#382—21no—
188#382—21no—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#381—22no—
193#380—21no—
194#380C2312yes—
195#380C2311yescharacteristic
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213#376—22no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×Q8C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—48no—
259C4⋊Q8—416no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—44no—
265C4×Q8—24no—
266C4×Q8—44no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4#4114yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—28no—
334#106—24no—
335#106—24no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#103—24no—
340#103—24no—
341#103—44no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—24no—
345#97—44no—
346C4×C8—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24C22×D411yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—832no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4AGL(1,ℤ/8)14yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4#13714yes—
478C4⋊C4C2×#4114yes—
479C4⋊C42−1+414yes—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—42no—
493C42—44no—
494C42—44no—
495C42—42no—
496C42—44no—
497C42—82no—
498C422−1+414yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23#54612yes—
504C23C2×AGL(1,ℤ/8)12yes—
505C23#57712yes—
506C23C23×D411yescharacteristic
507Q8—42no—
508Q8—42no—
509Q8—816no—
510Q8—816no—
511Q8—84no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4#53612yes—
569C2×C4#53612yes—
570C2×C4#54612yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×#13714yes—
574C2×C4C2×#13712yes—
575C2×C4C22×#4111yescharacteristic
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4C2×2−1+412yes—
578C2×C4C2×2−1+412yes—
579C8—42no—
580C8—42no—
581C8—44no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#332414yes—
589C22C2×#53611yescharacteristic
590C22C2×#54611yescharacteristic
591C22#348611yescharacteristic
592C22C22×AGL(1,ℤ/8)11yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57711yescharacteristic
595C22#358611yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—24no—
600C4—44no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—42no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4—81no—
611C4—84no—
612C4—81no—
613C4#323814yes—
614C4#350912yes—
615C4#354412yes—
616C2—24no—
617C2#3946511yescharacteristic
618C2C2×#323811yescharacteristic
619C2#6087211yescharacteristic
620C2C2×#332411yescharacteristic
621C2#6115711yescharacteristic
622C2#6182711yescharacteristic
623C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32248C2semidirect
5. #32150C2semidirect
7. #28799C2semidirect
9. #27823C2semidirect
12. #26444C2semidirect
13. #26026C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25512
#331462
#322482
#321502
#296412
#287992
#279602
#278232
#276472
#276162
#264442
#260262
#246092
#226032
#218402
C4×#4832