Finite Group Databasebeta

← #6556862

Subgroups — #6556862

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#36538C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic
5#29352C212yesmaximal, complemented
6#29346C211yesmaximal, characteristic
7#29148C212yesmaximal, complemented
8#29141C211yesmaximal, characteristic, complemented
9#27765C214yesmaximal, complemented
10#27758C212yesmaximal
11#27739C214yesmaximal, complemented
12#27737C212yesmaximal, complemented
13#23821C212yesmaximal, complemented
14#23816C211yesmaximal, characteristic, complemented
15#22546C212yesmaximal
16#22524C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#557—21no—
20C2×#557C2211yescharacteristic
21C2×#446C2212yes—
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2861C2212yes—
25#2858C2211yescharacteristic
26#2568C2211yescharacteristic
27#2566C2212yes—
28#2558—22no—
29#2558—22no—
30#2552—21no—
31#2552—22no—
32#2552C2212yes—
33#2552C2214yes—
34#2548C2211yescharacteristic
35#2546—21no—
36#2546—22no—
37#2545—21no—
38#2545—21no—
39#2545C2212yes—
40#2367—21no—
41#2367—21no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2212yes—
48C42×Q8C2211yescharacteristic
49#2101C2214yes—
50#2100C2212yes—
51#2072C2211yescharacteristic
52#2072C2211yescharacteristic
53#2072C2214yes—
54#2071C2214yes—
55#2071C2212yes—
56#2071C2214yes—
57#2043C2212yes—
58#2042C2214yes—
59#2024C2211yescharacteristic
60#2024C2211yescharacteristic
61#2024C2212yes—
62#2024C2212yes—
63#2023C2212yes—
64#2023C2212yes—
65#2023C2212yes—
66#1992C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1983C2214yes—
71#1982C2212yes—
72#1959C2214yes—
73#1959C2214yes—
74#1957C2212yes—
75#1956C2212yes—
76#1936C2214yes—
77#1935C2214yes—
78#1934C2214yes—
79#1836C2214yes—
80#1834C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2214yes—
84#1802C2214yes—
85#1771C2214yes—
86#1770C2212yes—
87#1767C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89#1763C2214yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95#559—28no—
96#557—28no—
97#557—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#125—22no—
109C2×#125—24no—
110C2×#125—22no—
111C2×#125C2312yes—
112C2×#124—21no—
113C2×#124—21no—
114C2×#124—21no—
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130#446—28no—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#97C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#398—24no—
149#398—24no—
150#398—24no—
151#396—22no—
152#396—22no—
153#396—22no—
154#396—22no—
155#394—22no—
156#391—21no—
157#391—21no—
158#391—22no—
159#391—21no—
160#391C2311yescharacteristic
161#391C2311yescharacteristic
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—24no—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2314yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—22no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2314yes—
197#384C2312yes—
198#384C2312yes—
199#384C2314yes—
200#382C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C43C2311yescharacteristic
216C43C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—41no—
250C4⋊Q8—48no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—82no—
254C4⋊Q8—84no—
255#127—24no—
256#125—28no—
257#125—416no—
258#125—48no—
259#125—48no—
260#124—24no—
261#124—28no—
262#124—42no—
263#124—44no—
264#124—44no—
265#124—44no—
266#123—48no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8#4118yes—
277C4×D4—28no—
278C4×D4—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C4⋊C8—24no—
345C4⋊C8—28no—
346#103—24no—
347#103—42no—
348#103—44no—
349#103—44no—
350#103—42no—
351#103—42no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#97—28no—
357#97—44no—
358C4×C8—24no—
359C4×C8—42no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8#12414yes—
395C2×Q82−1+412yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4#12414yes—
475C4⋊C4#12418yes—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—42no—
496C42—41no—
497C42—42no—
498C42—48no—
499C42—42no—
500C42—44no—
501C42—42no—
502C42—44no—
503C42AGL(1,ℤ/8)14yes—
504C42#13714yes—
505C42C2×#4114yes—
506C42C2×#4118yes—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C23×D411yescharacteristic
515Q8—28no—
516Q8—24no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—88no—
521Q8—84no—
522Q8—84no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—162no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—164no—
572C2×C4C2×#12414yes—
573C2×C4C2×#12414yes—
574C2×C4C2×#12412yes—
575C2×C4C2×#12412yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4C22×#4112yes—
583C2×C4C2×2−1+411yescharacteristic
584C2×C4#58512yes—
585C2×C4#58511yescharacteristic
586C8—42no—
587C8—42no—
588C8—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C22×#12411yescharacteristic
597C22#348711yescharacteristic
598C22#348911yescharacteristic
599C22C22×AGL(1,ℤ/8)11yescharacteristic
600C22C22×#13711yescharacteristic
601C22#358511yescharacteristic
602C22#358611yescharacteristic
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—24no—
608C4—24no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—81no—
614C4—82no—
615C4—82no—
616C4—84no—
617C4—82no—
618C4—81no—
619C4—81no—
620C4#314114yes—
621C4#351612yes—
622C4#352412yes—
623C2—162no—
624C2#3585411yescharacteristic
625C2C2×#314111yescharacteristic
626C2#6062511yescharacteristic
627C2#6189011yescharacteristic
628C2#6189311yescharacteristic
629C2#6211111yescharacteristic
630C2#6211211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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. #36593C2semidirect
3. #30216C2semidirect
5. #29352C2semidirect
7. #29148C2semidirect
8. #29141C2semidirect
9. #27765C2semidirect
11. #27739C2semidirect
12. #27737C2semidirect
13. #23821C2semidirect
14. #23816C2semidirect
16. #22524C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
#365932
#365382
#302162
#302032
#293522
#293462
#291482
#291412
#277652
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