Finite Group Databasebeta

← #6557818

Subgroups — #6557818

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#30359C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29213C211yesmaximal, characteristic, complemented
6#29165C212yesmaximal, complemented
7#28993C214yesmaximal, complemented
8#28385C211yesmaximal, characteristic, complemented
9#25504C212yesmaximal, complemented
10#25394C212yesmaximal
11#25348C212yesmaximal
12#25039C214yesmaximal, complemented
13#23820C211yesmaximal, characteristic, complemented
14#23571C212yesmaximal, complemented
15#22998C212yesmaximal, complemented
16#22964C212yesmaximal, complemented
17#20567C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#475C2212yes—
20C2×#470C2214yes—
21C2×#462C2212yes—
22C2×C4×Q16C2214yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2855C2212yes—
28#2575—22no—
29#2552—21no—
30#2547C2211yescharacteristic
31#2542—21no—
32#2542C2211yescharacteristic
33#2538C2212yes—
34#2517—22no—
35#2511—21no—
36#2489C2212yes—
37#2447—21no—
38#2441—22no—
39#2429—22no—
40#2423C2214yes—
41#2421C2212yes—
42#2367—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2365C2211yescharacteristic
46#2355—21no—
47#2354—21no—
48Q8×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#125C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2096C2212yes—
56#2077C2214yes—
57#2074C2212yes—
58#2072C2214yes—
59#2058C2212yes—
60#2057C2212yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2036C2214yes—
64#2035C2214yes—
65#2026C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2016C2212yes—
69#2001C2214yes—
70#1992C2212yes—
71#1990C2212yes—
72#1971C2214yes—
73#1956C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1941C2214yes—
77#1933C2214yes—
78#1920C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1878C2214yes—
82#1865C2214yes—
83#1863C2214yes—
84#1847C2212yes—
85#1809C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#103C2214yes—
91#1753C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×Q16C2314yes—
96C22×M16C2314yes—
97#533—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#125C2311yescharacteristic
106C2×#124—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42C2312yes—
122#475—28no—
123#470—48no—
124#462—28no—
125C4×Q16—48no—
126#436—28no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—22no—
145#399—22no—
146#399—24no—
147#398—24no—
148#397—22no—
149#397—22no—
150#397—22no—
151#396—24no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#390—22no—
162#390—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—24no—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—24no—
176#387—21no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#382—21no—
196#382—22no—
197#382C2311yescharacteristic
198#381—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×D4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—44no—
245C2×M16—28no—
246C2×M16—44no—
247C2×M16—44no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#125—22no—
258#125—41no—
259#125—48no—
260#124—44no—
261C22⋊Q8—28no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—44no—
267C22⋊Q8—84no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—42no—
278#117—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—44no—
340#106—28no—
341#106—44no—
342C4⋊C8#4118yes—
343#103—24no—
344#103—24no—
345#103—28no—
346#103—28no—
347#103—24no—
348#103—42no—
349#103—44no—
350#103—44no—
351C22⋊C8—28no—
352#97—42no—
353C4×C8—24no—
354C4×C8—44no—
355#95—21no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24C22×D412yes—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8H(5,𝔽2)14yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427Q16—416no—
428M16—88no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8C2×#4118yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4#13714yes—
475C4⋊C42−1+412yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—84no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—42no—
495C42—48no—
496C42—44no—
497C42—42no—
498C42—44no—
499C42—42no—
500C42—82no—
501C42C2×#4114yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—42no—
513Q8—42no—
514Q8—44no—
515Q8—44no—
516Q8—88no—
517Q8—84no—
518D4—44no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4#53212yes—
571C2×C4#53212yes—
572C2×C4C2×#13714yes—
573C2×C4C2×#13712yes—
574C2×C4#57512yes—
575C2×C4#57712yes—
576C2×C4#57812yes—
577C2×C4C22×#4112yes—
578C2×C4C2×H(5,𝔽2)12yes—
579C2×C4C2×2−1+411yescharacteristic
580C2×C4#58511yescharacteristic
581C8—28no—
582C8—42no—
583C8—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C2×#53211yescharacteristic
592C22#348411yescharacteristic
593C22#348911yescharacteristic
594C22C22×#13711yescharacteristic
595C22C2×#57511yescharacteristic
596C22C2×#57711yescharacteristic
597C22C2×#57811yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—24no—
602C4—42no—
603C4—44no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—81no—
612C4—84no—
613C4—81no—
614C4—81no—
615C4#317214yes—
616C4#351312yes—
617C4#352212yes—
618C2—82no—
619C2#3591211yescharacteristic
620C2C2×#317211yescharacteristic
621C2#6072311yescharacteristic
622C2#6159611yescharacteristic
623C2#6160211yescharacteristic
624C2#6214611yescharacteristic
625C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30359C2semidirect
4. #30205C2semidirect
5. #29213C2semidirect
6. #29165C2semidirect
7. #28993C2semidirect
8. #28385C2semidirect
9. #25504C2semidirect
12. #25039C2semidirect
13. #23820C2semidirect
14. #23571C2semidirect
15. #22998C2semidirect
16. #22964C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#361342
#303592
#302052
#292132
#291652
#289932
#283852
#255042
#253942
#253482
#250392
#238202
#235712
#229982
#229642
#205672