Finite Group Databasebeta

← #6534722

Subgroups — #6534722

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30080C212yesmaximal, complemented
4#29688C211yesmaximal, characteristic
5#29395C212yesmaximal, complemented
6#29315C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic
8#27647C211yesmaximal, characteristic
9#27035C214yesmaximal
10#26558C214yesmaximal
11#26093C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#23679C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#22964C212yesmaximal, complemented
16C4⋊C4×Q16C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#483C2212yes—
20C2×#475C2212yes—
21C2×C4×Q16C2212yes—
22C2×C4×Q16C2214yes—
23#2864C2212yes—
24#2857C2211yescharacteristic
25#2571C2211yescharacteristic
26#2568C2211yescharacteristic
27#2533—22no—
28#2530—22no—
29#2511—21no—
30#2499—21no—
31#2495C2212yes—
32#2421C2212yes—
33#2365C2211yescharacteristic
34#2362—21no—
35#2360—22no—
36#2360C2214yes—
37#2356—21no—
38#2355—21no—
39#2355C2212yes—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2211yescharacteristic
42Q8×C4⋊C4C2212yes—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48#2295C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2097C2214yes—
53#2084C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2074C2214yes—
57#2072C2212yes—
58#2068C2214yes—
59#2058C2212yes—
60#2057C2211yescharacteristic
61#2056C2214yes—
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2036C2212yes—
65#2035C2214yes—
66#2024C2211yescharacteristic
67#2016C2211yescharacteristic
68#2001C2212yes—
69#1990C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1942C2214yes—
74#1941C2214yes—
75#1939C2212yes—
76#1927C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1852C2214yes—
84#1845C2212yes—
85#1839C2212yes—
86#1823C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×Q16C2312yes—
96#533—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×#117—21no—
113C2×#117D411yescharacteristic
114C2×#117D411yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#483—44no—
120#475—28no—
121C4×Q16—28no—
122C4×Q16—48no—
123C2×#107C2312yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#400—22no—
145#397—22no—
146#397—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—21no—
152#391C2314yes—
153#391C2312yes—
154#391C2311yescharacteristic
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—24no—
161#389—24no—
162#389—22no—
163#389C2314yes—
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—21no—
178#387—24no—
179#387—24no—
180#387—22no—
181#387—22no—
182#385—21no—
183#385—24no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#383—22no—
192#382—21no—
193#382—22no—
194#382C2312yes—
195#382C2312yes—
196#380—21no—
197#380C2311yescharacteristic
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—24no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×Q16—24no—
242C2×Q16—24no—
243C2×Q16—24no—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250#125—28no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—22no—
255C22⋊Q8—41no—
256C22⋊Q8—44no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—42no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—84no—
266C4×Q8#4118yes—
267#117—28no—
268#117—24no—
269#117—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—44no—
334#107—44no—
335#106—28no—
336#106—44no—
337C4⋊C8—28no—
338C4⋊C8—24no—
339#103—24no—
340#103—28no—
341#103—28no—
342#103—24no—
343#103—24no—
344#103—42no—
345#103—44no—
346C22⋊C8—24no—
347C22⋊C8—42no—
348C4×C8—24no—
349C4×C8—44no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—21no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C2×#4114yes—
391C2×Q8H(5,𝔽2)12yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4#13514yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2511yescharacteristic
424Q16—216no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C82−1+414yes—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4#13714yes—
471C4⋊C4C2×#4114yes—
472C22⋊C4—28no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—81no—
478C42—22no—
479C42—22no—
480C42—28no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—22no—
488C42—42no—
489C42—44no—
490C42—42no—
491C42—42no—
492C42—44no—
493C42—84no—
494C42—84no—
495C42C2×#4114yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C2×#13512yes—
503C23C23×D411yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—24no—
507Q8—48no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—84no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—164no—
564C2×C4#52512yes—
565C2×C4#52512yes—
566C2×C4C2×#13512yes—
567C2×C4C2×#13512yes—
568C2×C4C2×#13714yes—
569C2×C4C2×#13712yes—
570C2×C4#57812yes—
571C2×C4C22×#4112yes—
572C2×C4C2×H(5,𝔽2)11yescharacteristic
573C2×C4C2×2−1+412yes—
574C2×C4#58511yescharacteristic
575C8—24no—
576C8—44no—
577C8—42no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×#52511yescharacteristic
586C22#348811yescharacteristic
587C22#349911yescharacteristic
588C22C22×#13511yescharacteristic
589C22C22×#13711yescharacteristic
590C22C2×#57811yescharacteristic
591C22#358511yescharacteristic
592C4—24no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—44no—
601C4—42no—
602C4—44no—
603C4—82no—
604C4—84no—
605C4—81no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4#310814yes—
610C4#349212yes—
611C4#350712yes—
612C2—42no—
613C2#3568611yescharacteristic
614C2C2×#310811yescharacteristic
615C2#6058911yescharacteristic
616C2#6195311yescharacteristic
617C2#6198411yescharacteristic
618C2#6247311yescharacteristic
619C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #30080C2semidirect
5. #29395C2semidirect
6. #29315C2semidirect
11. #26093C2semidirect
12. #24683C2semidirect
13. #23679C2semidirect
15. #22964C2semidirect
16. C4⋊C4×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#342142
#300802
#296882
#293952
#293152
#279502
#276472
#270352
#265582
#260932
#246832
#236792
#235712
#229642
C4⋊C4×Q162
#226392