Finite Group Databasebeta

← #6798658

Subgroups — #6798658

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36175C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#33236C211yesmaximal, characteristic, complemented
4#31316C214yesmaximal, complemented
5#31312C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#28755C211yesmaximal, characteristic, complemented
8#28522C211yesmaximal, characteristic, complemented
9#27896C212yesmaximal, complemented
10#25888C212yesmaximal
11#25884C212yesmaximal
12#25861C212yesmaximal, complemented
13#25278C212yesmaximal, complemented
14#25276C212yesmaximal, complemented
15#21811C212yesmaximal, complemented
16C4×#491C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#491C2211yescharacteristic
21C2×#491C2212yes—
22C2×#487C2212yes—
23C2×#487C2212yes—
24C2×#477C2212yes—
25C2×#475C2212yes—
26C2×#432C2212yes—
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2857C2212yes—
30#2851C2212yes—
31#2563C2211yescharacteristic
32#2551—21no—
33#2551—21no—
34#2547C2211yescharacteristic
35#2546C2212yes—
36#2527—22no—
37#2523—22no—
38#2514—21no—
39#2495C2212yes—
40#2494C2212yes—
41#2447—21no—
42#2440—22no—
43#2440—22no—
44#2434C2214yes—
45#2369C2211yescharacteristic
46#2367—21no—
47#2365C2211yescharacteristic
48#2356—21no—
49D4×C4⋊C4—21no—
50D4×C4⋊C4—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#125C2211yescharacteristic
53#2283C2212yes—
54C42×D4C2211yescharacteristic
55#2097C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2043C2212yes—
62#2036C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2035C2214yes—
66#1992C2211yescharacteristic
67#1981C2212yes—
68#1981C2212yes—
69#1981C2214yes—
70#1981C2214yes—
71#1960C2212yes—
72#1960C2212yes—
73#1960C2211yescharacteristic
74#1960C2212yes—
75#1960C2212yes—
76#1960C2214yes—
77#1954C2212yes—
78#1939C2212yes—
79#1935C2212yes—
80#1935C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1924C2214yes—
84#1847C2212yes—
85#1836C2211yescharacteristic
86#1787C2214yes—
87#1787C2214yes—
88#1787C2214yes—
89#1787C2214yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8C2314yes—
96D4×Q8—44no—
97D4×Q8—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125C2311yescharacteristic
111C2×#124—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2314yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C42C2312yes—
123#491—24no—
124#491—44no—
125#487—44no—
126#487—44no—
127#477—44no—
128#475—44no—
129#432—44no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2311yescharacteristic
151C2×#97C2312yes—
152C2×#97C2311yescharacteristic
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155#401—22no—
156#400—22no—
157#398—24no—
158#397—22no—
159#397—22no—
160#397—22no—
161#395—24no—
162#395—24no—
163#393—22no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391C2312yes—
172#391C2314yes—
173#390—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#387—21no—
188#387—21no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#383—22no—
203#383—22no—
204#382—21no—
205#382C2311yescharacteristic
206#382C2312yes—
207#382C2312yes—
208#381—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231#376—22no—
232C43C2311yescharacteristic
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—28no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2414yes—
249C22×Q8C2412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—24no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8#4114yes—
272#125—22no—
273#125—22no—
274#124—44no—
275C22⋊Q8—416no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—48no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—44no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348#103—24no—
349#103—24no—
350#103—42no—
351#103—42no—
352#103—44no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362C22⋊C8—44no—
363C22⋊C8—44no—
364#97—22no—
365#97—22no—
366#97—24no—
367#97—42no—
368C4×C8—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24C22×D412yes—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—84no—
413C2×Q8—81no—
414C2×Q8—81no—
415C2×Q8—88no—
416C2×Q8—88no—
417C2×Q8—88no—
418C2×Q8—88no—
419C2×Q8C2×#4114yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4#13714yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4#13714yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—44no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—82no—
521C42C2×#4112yes—
522C42H(5,𝔽2)14yes—
523C42H(5,𝔽2)12yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23C2×#13714yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—24no—
534Q8—42no—
535Q8—42no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—816no—
541Q8—84no—
542Q8—162no—
543D4—28no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—816no—
596C2×C4#53512yes—
597C2×C4#53512yes—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4C2×#13714yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4#58512yes—
607C8—42no—
608C8—42no—
609C8—44no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C2×#53511yescharacteristic
618C22C2×#54611yescharacteristic
619C22#350011yescharacteristic
620C22C22×#13711yescharacteristic
621C22C22×#13711yescharacteristic
622C22#358611yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—44no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—84no—
634C4—81no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4#321814yes—
642C4#351412yes—
643C4#354612yes—
644C2—44no—
645C2#3895711yescharacteristic
646C2C2×#321811yescharacteristic
647C2#6082811yescharacteristic
648C2#6115711yescharacteristic
649C2#6115711yescharacteristic
650C2#6257711yescharacteristic
651C2#6258911yescharacteristic
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. #33236C2semidirect
4. #31316C2semidirect
5. #31312C2semidirect
6. #29852C2semidirect
7. #28755C2semidirect
8. #28522C2semidirect
9. #27896C2semidirect
12. #25861C2semidirect
13. #25278C2semidirect
14. #25276C2semidirect
15. #21811C2semidirect
16. C4×#491C2semidirect

Maximal subgroups

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

subgroupindex
#361752
D4×C4⋊Q82
#332362
#313162
#313122
#298522
#287552
#285222
#278962
#258882
#258842
#258612
#252782
#252762
#218112
C4×#4912
#212702