Finite Group Databasebeta

← #6856200

Subgroups — #6856200

Order 512.

1613 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32432C214yesmaximal, complemented
5#31640C214yesmaximal, complemented
6#29715C212yesmaximal
7#28401C212yesmaximal, complemented
8#27972C212yesmaximal
9#27729C212yesmaximal, complemented
10#27681C212yesmaximal
11#25902C211yesmaximal, characteristic
12#25324C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic
15#22625C211yesmaximal, characteristic
16#22044C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#462C2212yes—
23C2×#451C2211yescharacteristic
24C2×#451C2212yes—
25C2×C4×Q16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2585—22no—
29#2569—22no—
30#2567—22no—
31#2567—21no—
32#2567—21no—
33#2567C2212yes—
34#2567C2212yes—
35#2567C2214yes—
36#2552C2211yescharacteristic
37#2546C2211yescharacteristic
38#2511C2212yes—
39#2509—21no—
40#2446C2212yes—
41#2442—21no—
42C2×#395—22no—
43#2370—21no—
44#2365C2211yescharacteristic
45#2355C2211yescharacteristic
46#2345—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yes—
55#2095C2214yes—
56#2072C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2057C2212yes—
60#2056C2214yes—
61#2056C2214yes—
62#2024C2214yes—
63#2024C2212yes—
64#2016C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1969C2214yes—
68#1956C2214yes—
69#1947C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1928C2214yes—
73#1916C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1834C2214yes—
86#1803C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#487—44no—
128#477—44no—
129#462—44no—
130#451—24no—
131#451—28no—
132C4×Q16—44no—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#400—22no—
153#400—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#395—24no—
158#395—24no—
159#395—24no—
160#395—24no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#389—22no—
171#389—24no—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#387—22no—
187#387—22no—
188#387—21no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#385—21no—
195#385—22no—
196#384—22no—
197#384—22no—
198#384C2314yes—
199#384C2314yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#382—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—24no—
258C2×Q16—28no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273#125—44no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4Q1618yescomplemented
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—24no—
355#103—28no—
356#103—42no—
357#103—42no—
358#103—42no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365C22⋊C8—22no—
366C4×C8—24no—
367C4×C8—44no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—816no—
417C2×Q8C2×#4114yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4C2×Q1614yes—
449C22×C4C2×Q1618yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462Q16—48no—
463Q16—88no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C2×Q1618yes—
510C4⋊C4C2×Q1614yes—
511C4⋊C4C2×Q1614yes—
512C4⋊C4#13714yes—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—82no—
528C42—82no—
529C42H(5,𝔽2)14yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#54812yes—
535C23C22×Q1612yes—
536C23#57712yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—44no—
545Q8—42no—
546Q8—44no—
547Q8—88no—
548Q8—88no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4#54812yes—
606C2×C4C22×Q1614yes—
607C2×C4C22×Q1614yes—
608C2×C4C22×Q1612yes—
609C2×C4C22×Q1612yes—
610C2×C4C2×#13714yes—
611C2×C4C2×#13712yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4C2×2−1+411yescharacteristic
615C2×C4#58512yes—
616C8—42no—
617C8—44no—
618C8—42no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22#336314yes—
626C22C2×#53511yescharacteristic
627C22C2×#54811yescharacteristic
628C22#348511yescharacteristic
629C22C23×Q1611yescharacteristic
630C22C22×#13711yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C4—24no—
634C4—24no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—44no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4#320914yes—
651C4#350512yes—
652C4#354112yes—
653C2—24no—
654C2#3972211yescharacteristic
655C2C2×#320911yescharacteristic
656C2#6081211yescharacteristic
657C2C2×#336311yescharacteristic
658C2#6131911yescharacteristic
659C2#6169811yescharacteristic
660C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #32432C2semidirect
5. #31640C2semidirect
7. #28401C2semidirect
9. #27729C2semidirect
12. #25324C2semidirect
13. #24682C2semidirect
317. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25672
#329062
#324322
#316402
#297152
#284012
#279722
#277292
#276812
#259022
#253242
#246822
#243022
#226252
#220442
C4×#4512