Finite Group Databasebeta

← #6805645

Subgroups — #6805645

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36562C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#32454C214yesmaximal, complemented
5#32450C214yesmaximal
6#30214C212yesmaximal, complemented
7#29212C212yesmaximal, complemented
8#28726C211yesmaximal, characteristic
9#28620C212yesmaximal
10#28522C211yesmaximal, characteristic, complemented
11#24682C212yesmaximal, complemented
12#24678C212yesmaximal
13#24332C212yesmaximal, complemented
14#24331C212yesmaximal
15#22223C212yesmaximal
16C4×#497C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#497C2211yescharacteristic
21C2×#489C2212yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×C4×Q16C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2862C2211yescharacteristic
29#2859C2212yes—
30#2582C2212yes—
31#2571—21no—
32#2570—22no—
33#2568—21no—
34#2567C2211yescharacteristic
35#2566C2212yes—
36#2565C2212yes—
37#2556—22no—
38#2556—22no—
39#2542C2211yescharacteristic
40#2511C2214yes—
41#2446—22no—
42#2429—21no—
43#2429—21no—
44#2365—21no—
45#2360—21no—
46#2360—21no—
47#2355—21no—
48#2355C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2212yes—
53C42×Q8C2211yescharacteristic
54#2097C2214yes—
55#2072C2214yes—
56#2072C2214yes—
57#2072C2212yes—
58#2071C2212yes—
59#2059C2214yes—
60#2059C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2026C2214yes—
64#2026C2214yes—
65#1993C2212yes—
66#1993C2212yes—
67#1993C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1984C2212yes—
71#1973C2212yes—
72#1971C2212yes—
73#1960C2211yescharacteristic
74#1960C2212yes—
75#1959C2211yescharacteristic
76#1948C2214yes—
77#1947C2214yes—
78#1939C2212yes—
79#1928C2214yes—
80#1926C2214yes—
81#1917C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1865C2214yes—
85#1862C2214yes—
86#1847C2211yescharacteristic
87#1847C2212yes—
88#1836C2214yes—
89#1803C2212yes—
90#1802C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97#557—44no—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C22×C4⋊C4—21no—
125#497—24no—
126#489—28no—
127#477—44no—
128#475—44no—
129C4×Q16—44no—
130C4×SD16—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×#97C2311yescharacteristic
148C2×#97C2311yescharacteristic
149C2×C4×C8C2312yes—
150#400—24no—
151#400—24no—
152#399—22no—
153#398—22no—
154#398—22no—
155#398—24no—
156#396—22no—
157#396—22no—
158#395—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2311yescharacteristic
169#391C2312yes—
170#391C2314yes—
171#391C2314yes—
172#390C2311yescharacteristic
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—21no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—21no—
188#388—21no—
189#387—24no—
190#387—22no—
191#387—22no—
192#387—24no—
193#387—24no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#382—22no—
205#382—22no—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#125—44no—
264#125—44no—
265#125—48no—
266#124—22no—
267#124#4114yes—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—84no—
281C4×D4—24no—
282C4×D4—44no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—42no—
345#107—44no—
346#106—42no—
347#106—44no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350#103—28no—
351#103—42no—
352#103—42no—
353#103—42no—
354#103—44no—
355#103—44no—
356#102—28no—
357#102—42no—
358#97—24no—
359#97—24no—
360#97—22no—
361#97—22no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×Q8C2×#4112yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4C2×#4112yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436Q16—48no—
437SD16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C82−1+414yes—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4#13714yes—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—41no—
503C42—44no—
504C42—44no—
505C42—42no—
506C42—84no—
507C42C2×#4112yes—
508C42H(5,𝔽2)14yes—
509C422−1+412yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—44no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—88no—
526Q8—84no—
527Q8—162no—
528D4—24no—
529D4—42no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4#53512yes—
580C2×C4#53512yes—
581C2×C4#54812yes—
582C2×C4#54812yes—
583C2×C4C2×#13714yes—
584C2×C4C2×#13712yes—
585C2×C4#57512yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4C2×2−1+412yes—
589C2×C4C2×2−1+411yescharacteristic
590C8—24no—
591C8—44no—
592C8—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×#53511yescharacteristic
601C22C2×#54811yescharacteristic
602C22#349911yescharacteristic
603C22C22×#13711yescharacteristic
604C22C2×#57511yescharacteristic
605C22#358511yescharacteristic
606C22#358611yescharacteristic
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—82no—
619C4—84no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4—82no—
624C4#321614yes—
625C4#352012yes—
626C4#355012yes—
627C2—82no—
628C2#3916711yescharacteristic
629C2C2×#321611yescharacteristic
630C2#6082911yescharacteristic
631C2#6132211yescharacteristic
632C2#6132411yescharacteristic
633C2#6250411yescharacteristic
634C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33293C2semidirect
4. #32454C2semidirect
6. #30214C2semidirect
7. #29212C2semidirect
10. #28522C2semidirect
11. #24682C2semidirect
13. #24332C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365622
#332932
#324542
#324502
#302142
#292122
#287262
#286202
#285222
#246822
#246782
#243322
#243312
#222232
C4×#4972
#213452