Finite Group Databasebeta

← #6559613

Subgroups — #6559613

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36476C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#30326C212yesmaximal, complemented
4#30262C211yesmaximal, characteristic, complemented
5#29347C211yesmaximal, characteristic, complemented
6#28925C214yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28608C212yesmaximal
9#28581C212yesmaximal, complemented
10#28489C212yesmaximal
11#26988C214yesmaximal, complemented
12#25504C212yesmaximal, complemented
13#23837C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23187C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#494C2212yes—
20C2×#487C2212yes—
21C2×#487C2212yes—
22C2×#470C2214yes—
23C2×#470C2214yes—
24C2×#441C2214yes—
25C2×#436C2212yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2857C2212yes—
29#2855C2212yes—
30#2607—22no—
31#2589—22no—
32#2580—21no—
33#2570—21no—
34#2563C2211yescharacteristic
35#2552—21no—
36#2542C2212yes—
37#2542C2211yescharacteristic
38#2538—21no—
39#2492C2212yes—
40#2441—22no—
41#2429C2214yes—
42#2423—22no—
43#2423C2212yes—
44#2369C2211yescharacteristic
45#2365C2212yes—
46#2365C2211yescharacteristic
47#2357—21no—
48#2355—21no—
49#2354—21no—
50Q8×C4⋊C4—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C22⋊Q8C2212yes—
54C42×D4C2211yescharacteristic
55#2084C2212yes—
56#2084C2211yescharacteristic
57#2084C2211yescharacteristic
58#2072C2212yes—
59#2058C2211yescharacteristic
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2043C2212yes—
63#2043C2212yes—
64#2035C2214yes—
65#2035C2214yes—
66#2024C2212yes—
67#2016C2212yes—
68#2016C2214yes—
69#2016C2212yes—
70#2001C2214yes—
71#2001C2214yes—
72#1987C2212yes—
73#1981C2212yes—
74#1981C2212yes—
75#1981C2212yes—
76#1971C2214yes—
77#1956C2212yes—
78#1935C2214yes—
79#1933C2214yes—
80#1927C2214yes—
81#1894C2214yes—
82#1894C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1865C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×M16C2314yes—
98#533—48no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42C2312yes—
122#494—28no—
123#487—28no—
124#487—44no—
125#470—48no—
126#470—48no—
127#441—48no—
128#436—28no—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#399—22no—
148#399—22no—
149#399—24no—
150#398—22no—
151#398—24no—
152#396—24no—
153#393—22no—
154#393—22no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391C2312yes—
162#391C2312yes—
163#391C2314yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—24no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#382—21no—
201#382—21no—
202#381—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2414yes—
240C22×Q8C2412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×M16—28no—
252C2×M16—44no—
253C2×M16—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—41no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267#125—48no—
268C22⋊Q8—28no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C4×Q8—28no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285#117—48no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—24no—
348#106—24no—
349#106—24no—
350#106—42no—
351C4⋊C8—24no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—28no—
361#97—44no—
362C4×C8—42no—
363C4×C8#4118yes—
364#95—21no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437Q16—416no—
438Q16—88no—
439M16—88no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8C2×#4114yes—
448C2×C8C2×#4114yes—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4H(5,𝔽2)12yes—
487C4⋊C4H(5,𝔽2)14yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C42—21no—
498C42—24no—
499C42—22no—
500C42—21no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—41no—
505C42—48no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—82no—
510C42—82no—
511C42#13714yes—
512C42#13714yes—
513C42C2×#4114yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23C23×D411yescharacteristic
522Q8—48no—
523Q8—44no—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—44no—
529Q8—84no—
530Q8—162no—
531D4—44no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4#53212yes—
583C2×C4#53212yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4C22×#4112yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4#58511yescharacteristic
594C8—24no—
595C8—24no—
596C8—44no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22C2×#53211yescharacteristic
605C22#348411yescharacteristic
606C22#348811yescharacteristic
607C22C22×#13711yescharacteristic
608C22C22×#13711yescharacteristic
609C22#358611yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—81no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4#318514yes—
629C4#350612yes—
630C4#350812yes—
631C2—82no—
632C2#3593011yescharacteristic
633C2C2×#318511yescharacteristic
634C2#6072011yescharacteristic
635C2#6158411yescharacteristic
636C2#6158611yescharacteristic
637C2#6196011yescharacteristic
638C2#6196611yescharacteristic
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. #30326C2semidirect
4. #30262C2semidirect
5. #29347C2semidirect
6. #28925C2semidirect
7. #28748C2semidirect
9. #28581C2semidirect
11. #26988C2semidirect
12. #25504C2semidirect
13. #23837C2semidirect
14. #23577C2semidirect
15. #23221C2semidirect
16. #23187C2semidirect

Maximal subgroups

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

subgroupindex
#364762
#361342
#303262
#302622
#293472
#289252
#287482
#286082
#285812
#284892
#269882
#255042
#238372
#235772
#232212
#231872
#205682