Finite Group Databasebeta

← #6567811

Subgroups — #6567811

Order 512.

1629 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#29981C211yesmaximal, characteristic, complemented
4#29970C212yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29162C212yesmaximal, complemented
8#27678C212yesmaximal, complemented
9#26508C212yesmaximal, complemented
10#26427C214yesmaximal, complemented
11#26106C212yesmaximal, complemented
12#26007C214yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23745C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22503C212yesmaximal
18C2×#555C2211yescharacteristic
19C2×#554—21no—
20C2×#436C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2567—21no—
24#2552—21no—
25#2552C2212yes—
26#2502—22no—
27#2440—22no—
28#2440C2214yes—
29#2369C2211yescharacteristic
30#2367—22no—
31#2363C2212yes—
32#2361—21no—
33#2354C2211yescharacteristic
34D4×C4⋊C4—21no—
35D4×C4⋊C4C2211yescharacteristic
36D4×C4⋊C4C2212yes—
37#2347—21no—
38#2347C2212yes—
39#2337—22no—
40#2324—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#121C2212yes—
44#2296C2212yes—
45#2293C2211yescharacteristic
46C2×#384C2212yes—
47#2274—21no—
48C42×D4C2211yescharacteristic
49#2100C2212yes—
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2100C2211yescharacteristic
54#2099C2214yes—
55#2099C2214yes—
56#2094C2212yes—
57#2094C2211yescharacteristic
58#2088C2214yes—
59#2088C2212yes—
60#2086C2214yes—
61#2065C2212yes—
62#2058C2211yescharacteristic
63#2058C2212yes—
64#2057C2212yes—
65#2023C2212yes—
66#2023C2211yescharacteristic
67#1990C2212yes—
68#1987C2214yes—
69#1987C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1947C2212yes—
73#1939C2212yes—
74#1921C2214yes—
75#1919C2214yes—
76#1895C2214yes—
77#1874C2214yes—
78#1862C2214yes—
79#1854C2214yes—
80#1853C2214yes—
81#1851C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1825C2214yes—
85#1809C2214yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×M16C2314yes—
96#555—24no—
97#555Q818yescomplemented
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2314yes—
127C22×C42C2312yes—
128#436—28no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141#398—24no—
142#398—24no—
143#395—24no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—22no—
148#390—21no—
149#390—22no—
150#390C2311yescharacteristic
151#390C2311yescharacteristic
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—21no—
169#388—22no—
170#388—21no—
171#388—22no—
172#386—22no—
173#386—22no—
174#385—22no—
175#385—22no—
176#385—24no—
177#385—24no—
178#385—22no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2314yes—
187#384C2312yes—
188#383—22no—
189#380—22no—
190#380—21no—
191#380C2312yes—
192#380C2314yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2311yescharacteristic
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C2×M16—28no—
245C2×M16—44no—
246C2×M16—44no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—22no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8C2×Q814yes—
256#125—48no—
257#124—216no—
258C22⋊Q8—416no—
259C22⋊Q8—48no—
260#121—28no—
261#121—24no—
262#121C2×Q818yes—
263C4×Q8—44no—
264C4×Q8—82no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—84no—
271C4×D4C2×Q814yes—
272C4×D4C2×Q814yes—
273C4×D4#4114yes—
274#117—24no—
275#117—28no—
276#117—44no—
277#117—44no—
278#117C2×Q818yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—84no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—44no—
348#106—42no—
349#106—42no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353C4⋊C8—42no—
354#102—42no—
355#102—42no—
356#102—44no—
357C22⋊C8—28no—
358C4×C8—24no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—24no—
371#95—24no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D412yes—
377C2×Q8—24no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4C22⋊Q814yes—
395C2×D4C22×Q812yes—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q814yes—
431C22×C4C2×#4114yes—
432C22×C4C2511yescharacteristic
433M16—88no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C22⋊Q814yes—
477C4⋊C4AGL(1,ℤ/8)14yes—
478C4⋊C4C22×Q812yes—
479C4⋊C4C22×Q814yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—48no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4C22⋊Q818yes—
494C42—28no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—42no—
502C42—41no—
503C42—44no—
504C42—42no—
505C42—82no—
506C42—82no—
507C42—82no—
508C42#13714yes—
509C42C22×Q812yes—
510C42C2×#4112yes—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×C22⋊Q814yes—
525C23C23×D411yescharacteristic
526Q8—84no—
527Q8—84no—
528Q8—164no—
529D4—28no—
530D4—24no—
531D4—42no—
532D4—88no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—164no—
581C2×C4—168no—
582C2×C4C2×C22⋊Q814yes—
583C2×C4C2×C22⋊Q812yes—
584C2×C4C2×C22⋊Q812yes—
585C2×C4C2×AGL(1,ℤ/8)14yes—
586C2×C4C2×AGL(1,ℤ/8)12yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4#57712yes—
591C2×C4C23×Q811yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4#58512yes—
594C2×C4#58512yes—
595C8—42no—
596C8—44no—
597C8—42no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C22×C22⋊Q811yescharacteristic
613C22#348511yescharacteristic
614C22#350011yescharacteristic
615C22C22×AGL(1,ℤ/8)11yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358511yescharacteristic
619C4—22no—
620C4—22no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—41no—
629C4—81no—
630C4—82no—
631C4—82no—
632C4—81no—
633C4—82no—
634C4—84no—
635C4#307914yes—
636C4#350412yes—
637C4#352112yes—
638C2—44no—
639C2—82no—
640C2#3603111yescharacteristic
641C2C2×#307911yescharacteristic
642C2#6049511yescharacteristic
643C2#6171911yescharacteristic
644C2#6172811yescharacteristic
645C2#6254011yescharacteristic
646C2#6258811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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
1. #36180C2semidirect
3. #29981C2semidirect
4. #29970C2semidirect
5. #29277C2semidirect
6. #29218C2semidirect
7. #29162C2semidirect
8. #27678C2semidirect
9. #26508C2semidirect
10. #26427C2semidirect
11. #26106C2semidirect
12. #26007C2semidirect
13. #23775C2semidirect
14. #23745C2semidirect
15. #22715C2semidirect
16. #22700C2semidirect
97. #555Q8semidirect

Maximal subgroups

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

subgroupindex
#361802
#343212
#299812
#299702
#292772
#292182
#291622
#276782
#265082
#264272
#261062
#260072
#237752
#237452
#227152
#227002
#225032