Finite Group Databasebeta

← #6626456

Subgroups — #6626456

Order 512.

1699 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33151C212yesmaximal, complemented
4#33148C211yesmaximal, characteristic, complemented
5#29303C212yesmaximal, complemented
6#29235C212yesmaximal
7#27014C214yesmaximal, complemented
8#26874C214yesmaximal, complemented
9#26447C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25452C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23163C212yesmaximal, complemented
14#22968C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21578C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#498C2212yes—
21C2×#480C2211yescharacteristic
22C2×#474C2212yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2589—22no—
26#2563C2211yescharacteristic
27#2552C2211yescharacteristic
28#2509C2212yes—
29#2497—21no—
30#2487—21no—
31#2454C2212yes—
32#2450—22no—
33#2369C2211yescharacteristic
34#2365—22no—
35#2365—21no—
36#2365—21no—
37#2365C2212yes—
38#2365C2212yes—
39#2365C2214yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2345—21no—
43C4×#121C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391—21no—
46C2×#389C2212yes—
47C2×#387—22no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2088C2214yes—
52#2085C2214yes—
53#2065C2214yes—
54#2059C2214yes—
55#2058C2214yes—
56#2057C2212yes—
57#2023C2212yes—
58#1990C2212yes—
59#1989C2214yes—
60#1988C2212yes—
61#1987C2212yes—
62#1949C2214yes—
63#1948C2212yes—
64#1944C2214yes—
65#1943C2214yes—
66#1939C2212yes—
67#1926C2214yes—
68#1925C2212yes—
69#1921C2214yes—
70#1921C2214yes—
71#1920C2214yes—
72#1920C2214yes—
73#1919C2214yes—
74#1918C2212yes—
75#1917C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1864C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2211yescharacteristic
83#1852C2212yes—
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
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×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—48no—
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—22no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#498—28no—
126#480Q818yescomplemented
127#474—28no—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2311yescharacteristic
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#399—22no—
147#398—22no—
148#396—22no—
149#393—22no—
150#391—22no—
151#391—22no—
152#391—24no—
153#391—21no—
154#391—22no—
155#390C2312yes—
156#390C2311yescharacteristic
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389C2311yescharacteristic
164#389C2312yes—
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#387—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#386—21no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384C2311yescharacteristic
187#382C2312yes—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—22no—
193#380C2311yescharacteristic
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8#4118yes—
260#125—44no—
261C22⋊Q8—82no—
262#121—22no—
263#121C2×Q814yes—
264C4×Q8—416no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—84no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×Q814yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42—48no—
329C2×C42—44no—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—28no—
339#107—42no—
340#107C2×Q818yescomplemented
341#106—28no—
342#106—28no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—44no—
346#102—28no—
347#102—42no—
348#102C2×Q818yescomplemented
349C22⋊C8—24no—
350C22⋊C8—22no—
351C22⋊C8C2×Q814yes—
352#97—44no—
353C4×C8—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—21no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D411yescharacteristic
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—432no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C22×Q812yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4AGL(1,ℤ/8)14yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×Q812yes—
441C22×C42−1+412yes—
442C22×C4C2511yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8C22×Q814yes—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C22×Q814yes—
485C4⋊C4C22×Q812yes—
486C4⋊C4C2×#4118yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—48no—
504C42—48no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—82no—
509C42C2×#4114yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23D4×Q812yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23#57712yes—
524C23C23×D411yescharacteristic
525Q8—416no—
526Q8—416no—
527Q8—162no—
528D4—24no—
529D4—48no—
530D4—42no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—164no—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4D4×Q812yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4#57512yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4112yes—
595C2×C4C2×2−1+411yescharacteristic
596C2×C4#58512yes—
597C8—24no—
598C8—42no—
599C8—44no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#338414yes—
614C22C2×#52511yescharacteristic
615C22C2×D4×Q811yescharacteristic
616C22#348511yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57711yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—48no—
629C4—41no—
630C4—44no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4#311414yes—
638C4Q8×#4112yes—
639C4#351612yes—
640C2—24no—
641C2—82no—
642C2#3654711yescharacteristic
643C2C2×#311411yescharacteristic
644C2#6058211yescharacteristic
645C2C2×#338411yescharacteristic
646C2#6136911yescharacteristic
647C2#6171711yescharacteristic
648C2#6173911yescharacteristic
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×#2365C2semidirect
3. #33151C2semidirect
4. #33148C2semidirect
5. #29303C2semidirect
7. #27014C2semidirect
8. #26874C2semidirect
9. #26447C2semidirect
10. #25811C2semidirect
11. #25452C2semidirect
13. #23163C2semidirect
14. #22968C2semidirect
15. #22629C2semidirect
126. #480Q8semidirect
340. #107C2×Q8semidirect
348. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#23652
#331512
#331482
#293032
#292352
#270142
#268742
#264472
#258112
#254522
#242982
#231632
#229682
#226292
#215782
C4×#4802