Finite Group Databasebeta

← #6795852

Subgroups — #6795852

Order 512.

1475 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32200C214yesmaximal, complemented
5#30672C214yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#29208C211yesmaximal, characteristic, complemented
8#28284C211yesmaximal, characteristic, complemented
9#28228C212yesmaximal
10#27629C212yesmaximal, complemented
11#26092C212yesmaximal, complemented
12#25340C212yesmaximal
13#24752C212yesmaximal, complemented
14#24682C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21961C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#477C2212yes—
19C2×#477C2212yes—
20C2×#467C2214yes—
21C2×#462C2211yescharacteristic
22C2×C4×Q16C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2574—22no—
28#2571—21no—
29#2567C2211yescharacteristic
30#2563—21no—
31#2563C2211yescharacteristic
32#2552C2211yescharacteristic
33#2538C2211yescharacteristic
34#2538C2212yes—
35#2508—21no—
36#2497—22no—
37#2497C2212yes—
38#2492C2214yes—
39#2454—21no—
40#2439—22no—
41#2423C2212yes—
42#2396—22no—
43#2369—21no—
44#2355C2212yes—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125—21no—
49C4×C22⋊Q8C2212yes—
50#2293—21no—
51#2285C2212yes—
52#2285C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2095C2214yes—
56#2076C2214yes—
57#2072C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2057C2214yes—
61#2056C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2036C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1981C2212yes—
68#1981C2211yescharacteristic
69#1947C2212yes—
70#1947C2212yes—
71#1947C2214yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1947C2214yes—
77#1939C2212yes—
78#1928C2214yes—
79#1928C2212yes—
80#1921C2214yes—
81#1892C2214yes—
82#1878C2212yes—
83#1850C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1808C2214yes—
87#1808C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C23×C8C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C2×#117—22no—
115C22×C42C2312yes—
116#477—44no—
117#477—44no—
118#467—48no—
119#462—24no—
120C4×Q16—44no—
121C8×D4—44no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2312yes—
127C2×C4⋊C8—28no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#400—22no—
145#399—22no—
146#398—22no—
147#398—22no—
148#396—22no—
149#396—22no—
150#396—24no—
151#395—22no—
152#393—24no—
153#393—22no—
154#391—21no—
155#391—24no—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#382—21no—
191#382—22no—
192#382—22no—
193#382—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C22×D4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—42no—
246C22×C8—28no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#125—44no—
261C22⋊Q8—24no—
262C22⋊Q8—42no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—41no—
273C4×Q8—82no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—42no—
277C4×D4Q1618yescomplemented
278#117—84no—
279#117—84no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
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—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
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#107—42no—
346#107—44no—
347#107—44no—
348#107—44no—
349C4⋊C8—28no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—42no—
354C4⋊C8#4114yes—
355#103—28no—
356#103—24no—
357#103—42no—
358#103—42no—
359#103—42no—
360#103—44no—
361#103—44no—
362C22⋊C8—44no—
363C4×C8—24no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24C22×D412yes—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8H(5,𝔽2)12yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4C2×Q1614yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4C2×Q1618yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437Q16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—88no—
447C2×C8C2×#4114yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4C2×Q1614yes—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4C2×Q1618yes—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—22no—
505C42—21no—
506C42—41no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—81no—
515C42—84no—
516C42C2×Q1614yes—
517C42C2×#4112yes—
518C42H(5,𝔽2)12yes—
519C422−1+412yes—
520C422−1+412yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23C22×Q1614yes—
528C23C23×D411yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—84no—
538Q8—164no—
539D4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—216no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—84no—
590C2×C4#53712yes—
591C2×C4#53712yes—
592C2×C4#54712yes—
593C2×C4#54712yes—
594C2×C4C22×Q1612yes—
595C2×C4C22×Q1614yes—
596C2×C4C22×Q1612yes—
597C2×C4C22×Q1612yes—
598C2×C4#57512yes—
599C2×C4#57612yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C2×C4#58511yescharacteristic
606C8—24no—
607C8—44no—
608C8—42no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22C2×#53711yescharacteristic
617C22C2×#54711yescharacteristic
618C22#348811yescharacteristic
619C22C23×Q1611yescharacteristic
620C22C2×#57511yescharacteristic
621C22C2×#57611yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—22no—
625C4—24no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4#324914yes—
640C4#333814yes—
641C4#351811yescharacteristic
642C4#351911yescharacteristic
643C4#354411yescharacteristic
644C4#354711yescharacteristic
645C2—44no—
646C2#3892211yescharacteristic
647C2C2×#324911yescharacteristic
648C2#6092811yescharacteristic
649C2C2×#333811yescharacteristic
650C2#6125711yescharacteristic
651C2#6195211yescharacteristic
652C2#6198811yescharacteristic
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. #33198C2semidirect
4. #32200C2semidirect
5. #30672C2semidirect
6. #29923C2semidirect
7. #29208C2semidirect
8. #28284C2semidirect
10. #27629C2semidirect
11. #26092C2semidirect
13. #24752C2semidirect
14. #24682C2semidirect
16. #21961C2semidirect
17. C4×#462C2semidirect
277. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361942
#349412
#331982
#322002
#306722
#299232
#292082
#282842
#282282
#276292
#260922
#253402
#247522
#246822
D4×C4⋊C82
#219612
C4×#4622