Finite Group Databasebeta

← #6593797

Subgroups — #6593797

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#33008C212yesmaximal
5#27583C212yesmaximal
6#27085C214yesmaximal, complemented
7#26775C214yesmaximal, complemented
8#26528C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic
10#25324C212yesmaximal, complemented
11#24508C212yesmaximal
12#24212C211yesmaximal, characteristic
13#23221C212yesmaximal, complemented
14#23102C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic
17#20740C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#483C2211yescharacteristic
22C2×#462C2212yes—
23C2×#451C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2567C2211yescharacteristic
28#2552C2211yescharacteristic
29#2520—22no—
30#2511C2212yes—
31#2508—21no—
32#2507C2212yes—
33#2442—21no—
34#2438—22no—
35#2370—21no—
36#2369C2211yescharacteristic
37#2365—21no—
38#2365—21no—
39#2365—22no—
40#2365C2212yes—
41#2365C2214yes—
42#2365C2212yes—
43#2355C2211yescharacteristic
44#2344—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#391C2212yes—
47C2×#389—21no—
48C2×#387—22no—
49#2285C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2096C2214yes—
54#2072C2214yes—
55#2066C2214yes—
56#2056C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2016C2212yes—
60#1987C2212yes—
61#1981C2212yes—
62#1971C2214yes—
63#1956C2212yes—
64#1948C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1927C2214yes—
68#1927C2212yes—
69#1927C2214yes—
70#1920C2212yes—
71#1920C2214yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1872C2211yescharacteristic
76#1865C2214yes—
77#1865C2214yes—
78#1853C2211yescharacteristic
79#1853C2212yes—
80#1823C2211yescharacteristic
81#1823C2212yes—
82#1817C2212yes—
83#1803C2214yes—
84#1803C2214yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—28no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×#117—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—22no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#487—28no—
126#483—24no—
127#462—28no—
128#451—44no—
129#432—44no—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#395—22no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2314yes—
161#391C2312yes—
162#391C2314yes—
163#391C2311yescharacteristic
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—24no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2311yescharacteristic
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—21no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—22no—
188#385—22no—
189#385—21no—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#382—21no—
194#382—22no—
195#381—22no—
196#380—21no—
197#380—22no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8C2411yescharacteristic
243C22×Q8C2414yes—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8#4118yes—
264#125—44no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—82no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—48no—
275#117—84no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—48no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#106—24no—
343#106—42no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—28no—
348#103—28no—
349#103—28no—
350#103—24no—
351#103—44no—
352#103—44no—
353#103—44no—
354#103—42no—
355#103—44no—
356C22⋊C8—22no—
357C22⋊C8—22no—
358C22⋊C8—24no—
359#97—44no—
360C4×C8—24no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
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#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24C22×D411yescharacteristic
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—416no—
398C2×Q8—416no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8C2×#4118yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4#13714yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4C2511yescharacteristic
449Q16—88no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4#13714yes—
495C4⋊C4C2×#4114yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—48no—
509C42—48no—
510C42—42no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—82no—
515C42—82no—
516C42C2×#4114yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#54612yes—
522C23C2×#13712yes—
523C23#57712yes—
524C23C23×D411yescharacteristic
525Q8—28no—
526Q8—28no—
527Q8—416no—
528Q8—44no—
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—162no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4#52512yes—
590C2×C4#52512yes—
591C2×C4#54612yes—
592C2×C4C2×#13712yes—
593C2×C4C2×#13714yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4#58512yes—
600C8—42no—
601C8—42no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22#333114yes—
610C22C2×#52511yescharacteristic
611C22C2×#54611yescharacteristic
612C22#348711yescharacteristic
613C22C22×#13711yescharacteristic
614C22C22×#13711yescharacteristic
615C22C2×#57711yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—48no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4#311314yes—
635C4#347512yes—
636C4#351312yes—
637C2—24no—
638C2#3616211yescharacteristic
639C2C2×#311311yescharacteristic
640C2#6058211yescharacteristic
641C2C2×#333111yescharacteristic
642C2#6115711yescharacteristic
643C2#6188911yescharacteristic
644C2#6190611yescharacteristic
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
6. #27085C2semidirect
7. #26775C2semidirect
8. #26528C2semidirect
10. #25324C2semidirect
13. #23221C2semidirect
14. #23102C2semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#23652
#331482
#330082
#275832
#270852
#267752
#265282
#258562
#253242
#245082
#242122
#232212
#231022
#226032
C4×#4832
#207402