Finite Group Databasebeta

← #6626455

Subgroups — #6626455

Order 512.

1591 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33150C212yesmaximal, complemented
4#33149C211yesmaximal, characteristic, complemented
5#29303C212yesmaximal, complemented
6#29234C212yesmaximal
7#27018C214yesmaximal, complemented
8#26878C214yesmaximal, complemented
9#26445C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25450C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic
13#23162C212yesmaximal, complemented
14#22969C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#500C2212yes—
20C2×#481C2211yescharacteristic
21C2×#476C2212yes—
22#2857C2211yescharacteristic
23#2851C2212yes—
24#2589—22no—
25#2563C2211yescharacteristic
26#2552C2211yescharacteristic
27#2509C2212yes—
28#2497—21no—
29#2487—21no—
30#2454C2212yes—
31#2450—22no—
32#2369—22no—
33#2369—21no—
34#2369—21no—
35#2369C2212yes—
36#2369C2212yes—
37#2369C2214yes—
38#2369C2211yescharacteristic
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2345—21no—
42C4×#121C2212yes—
43C4×#121C2211yescharacteristic
44C2×#391—22no—
45C2×#389—21no—
46C2×#389C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2086C2214yes—
52#2085C2214yes—
53#2067C2214yes—
54#2059C2214yes—
55#2058C2212yes—
56#2057C2214yes—
57#2023C2212yes—
58#1990C2212yes—
59#1989C2214yes—
60#1987C2212yes—
61#1987C2212yes—
62#1949C2214yes—
63#1947C2212yes—
64#1944C2214yes—
65#1943C2214yes—
66#1939C2212yes—
67#1926C2212yes—
68#1925C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1920C2214yes—
72#1919C2212yes—
73#1918C2214yes—
74#1918C2214yes—
75#1917C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1862C2214yes—
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×#121C2311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4—22no—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42—22no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#500—28no—
122#481Q818yescomplemented
123#476—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×C22⋊C8C2311yescharacteristic
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#400—22no—
142#399—22no—
143#398—22no—
144#396—22no—
145#393—22no—
146#391—21no—
147#391—22no—
148#391—24no—
149#391—24no—
150#391—24no—
151#391—24no—
152#390C2312yes—
153#390C2311yescharacteristic
154#389—22no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—24no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2311yescharacteristic
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#388—21no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#386—21no—
176#386—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384C2311yescharacteristic
183#382C2312yes—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380C2311yescharacteristic
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C2×#95—22no—
214C2×#95—22no—
215C2×#95—21no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—82no—
237C22×Q8—82no—
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8#4118yes—
259#125—44no—
260C22⋊Q8—82no—
261#121—22no—
262#121C2×Q814yes—
263C4×D4—24no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—84no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—84no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×Q814yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2414yes—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C42—24no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—84no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—28no—
336#107—44no—
337#106—28no—
338#106—42no—
339#106C2×Q818yescomplemented
340C4⋊C8—24no—
341C4⋊C8—44no—
342#102—28no—
343#102—42no—
344#102C2×Q818yescomplemented
345C22⋊C8—22no—
346C22⋊C8—24no—
347C22⋊C8C2×Q814yes—
348#97—44no—
349C4×C8—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—21no—
372C24C22×D411yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—42no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×Q8—81no—
381C2×Q8—81no—
382C2×Q8—816no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C22×Q812yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4#13714yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×Q812yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8C22×Q814yes—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4C22×Q814yes—
481C4⋊C4C22×Q812yes—
482C4⋊C4C2×#4118yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—82no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—42no—
501C42—44no—
502C42—84no—
503C42—84no—
504C42—82no—
505C42C2×#4114yes—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23D4×Q812yes—
518C23C2×#13712yes—
519C23#57612yes—
520C23C23×D411yescharacteristic
521Q8—88no—
522Q8—88no—
523Q8—162no—
524D4—24no—
525D4—48no—
526D4—42no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—164no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4D4×Q812yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4#57512yes—
589C2×C4C23×Q811yescharacteristic
590C2×C4C22×#4112yes—
591C2×C4C2×2−1+411yescharacteristic
592C2×C4#58512yes—
593C8—24no—
594C8—42no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22#338514yes—
610C22C2×#52511yescharacteristic
611C22C2×D4×Q811yescharacteristic
612C22#348511yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57611yescharacteristic
616C22#358511yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—48no—
626C4—42no—
627C4—44no—
628C4—81no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4#311714yes—
634C4Q8×#4112yes—
635C4#351612yes—
636C2—24no—
637C2—82no—
638C2#3654711yescharacteristic
639C2C2×#311711yescharacteristic
640C2#6057811yescharacteristic
641C2C2×#338511yescharacteristic
642C2#6136711yescharacteristic
643C2#6171911yescharacteristic
644C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33150C2semidirect
4. #33149C2semidirect
5. #29303C2semidirect
7. #27018C2semidirect
8. #26878C2semidirect
9. #26445C2semidirect
10. #25813C2semidirect
11. #25450C2semidirect
13. #23162C2semidirect
14. #22969C2semidirect
15. #22629C2semidirect
122. #481Q8semidirect
339. #106C2×Q8semidirect
344. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#23692
#331502
#331492
#293032
#292342
#270182
#268782
#264452
#258132
#254502
#242992
#231622
#229692
#226292
#215802
C4×#4812