Finite Group Databasebeta

← #6795030

Subgroups — #6795030

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32444C214yesmaximal, complemented
5#32358C214yesmaximal, complemented
6#29908C212yesmaximal
7#28542C212yesmaximal, complemented
8#27738C212yesmaximal, complemented
9#27589C212yesmaximal
10#25811C211yesmaximal, characteristic
11#25453C212yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#24508C212yesmaximal
14#24302C211yesmaximal, characteristic
15#22603C211yesmaximal, characteristic
16#21987C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#451C2211yescharacteristic
24C2×C4×Q16C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2567—21no—
31#2567—21no—
32#2567—22no—
33#2567C2212yes—
34#2567C2214yes—
35#2567C2212yes—
36#2533—22no—
37#2514—21no—
38#2514C2212yes—
39#2509C2212yes—
40#2503—21no—
41#2493—22no—
42C2×#395—22no—
43#2365C2211yescharacteristic
44#2360—21no—
45#2356—21no—
46#2355C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2212yes—
52#2283C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2214yes—
55#2097C2214yes—
56#2096C2214yes—
57#2084C2214yes—
58#2074C2214yes—
59#2072C2214yes—
60#2066C2214yes—
61#2058C2214yes—
62#2057C2212yes—
63#2056C2214yes—
64#2024C2212yes—
65#1990C2212yes—
66#1987C2212yes—
67#1971C2214yes—
68#1948C2214yes—
69#1947C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1917C2214yes—
75#1878C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1865C2214yes—
79#1863C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1847C2214yes—
86#1836C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#498—44no—
125#477—44no—
126#475—44no—
127#451—24no—
128C4×Q16—44no—
129#432—28no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#401—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#395—24no—
156#395—24no—
157#395—24no—
158#395—24no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391—21no—
165#391C2314yes—
166#391C2314yes—
167#391C2312yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#389—22no—
171#389—22no—
172#389—24no—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—21no—
192#384—22no—
193#384—22no—
194#384C2314yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#383—22no—
198#383—22no—
199#382—21no—
200#382—21no—
201#382C2312yes—
202#382C2312yes—
203#381—22no—
204#381—22no—
205#380—21no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—82no—
246C22×Q8—82no—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—84no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—42no—
342#107—44no—
343#106—44no—
344#106—44no—
345#106—44no—
346C4⋊C8—28no—
347C4⋊C8—24no—
348#103—28no—
349#103—24no—
350#103—24no—
351#103—42no—
352#103—42no—
353#103—44no—
354#103—44no—
355C22⋊C8—24no—
356C22⋊C8—22no—
357C22⋊C8#4114yes—
358#97—28no—
359C4×C8—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×Q82−1+414yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4#13714yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C42−1+412yes—
456C22×C4C2511yescharacteristic
457Q16—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8C2×#4114yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4#13714yes—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—82no—
520C42H(5,𝔽2)14yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#54712yes—
526C23C2×#13712yes—
527C23#57712yes—
528C23C23×D411yescharacteristic
529Q8—24no—
530Q8—24no—
531Q8—48no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—88no—
537Q8—88no—
538Q8—84no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4#53512yes—
596C2×C4#53512yes—
597C2×C4#54712yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×2−1+412yes—
605C2×C4C2×2−1+411yescharacteristic
606C8—24no—
607C8—44no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22#335614yes—
616C22C2×#53511yescharacteristic
617C22C2×#54711yescharacteristic
618C22#348611yescharacteristic
619C22C22×#13711yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358511yescharacteristic
623C4—22no—
624C4—24no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4#321814yes—
641C4#352012yes—
642C4#353812yes—
643C2—24no—
644C2#3891711yescharacteristic
645C2C2×#321811yescharacteristic
646C2#6081211yescharacteristic
647C2C2×#335611yescharacteristic
648C2#6125411yescharacteristic
649C2#6183011yescharacteristic
650C2#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. #32444C2semidirect
5. #32358C2semidirect
7. #28542C2semidirect
8. #27738C2semidirect
11. #25453C2semidirect
12. #24682C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25672
#329542
#324442
#323582
#299082
#285422
#277382
#275892
#258112
#254532
#246822
#245082
#243022
#226032
#219872
C4×#4512