Finite Group Databasebeta

← #6856423

Subgroups — #6856423

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36577C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32625C214yesmaximal, complemented
5#32563C214yesmaximal, complemented
6#29715C212yesmaximal
7#29335C212yesmaximal
8#29244C212yesmaximal
9#28196C212yesmaximal, complemented
10#27972C211yesmaximal, characteristic
11#27680C211yesmaximal, characteristic
12#25302C212yesmaximal, complemented
13#24994C212yesmaximal, complemented
14#24191C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#462C2212yes—
22C2×#451C2211yescharacteristic
23C2×#451C2212yes—
24C2×#446C2212yes—
25C2×C4×Q16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2582—21no—
29#2582—21no—
30#2582—22no—
31#2582C2211yescharacteristic
32#2582C2212yes—
33#2582C2212yes—
34#2582C2214yes—
35#2582C2212yes—
36#2568C2212yes—
37#2568C2211yescharacteristic
38#2453—21no—
39#2453—22no—
40#2442—22no—
41#2442—21no—
42C2×#398—22no—
43#2351—21no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389C2212yes—
52C2×#384C2211yescharacteristic
53#2099C2212yes—
54#2099C2214yes—
55#2096C2212yes—
56#2086C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2043C2214yes—
63#2016C2214yes—
64#1985C2214yes—
65#1984C2212yes—
66#1971C2214yes—
67#1956C2212yes—
68#1948C2212yes—
69#1935C2214yes—
70#1935C2214yes—
71#1935C2214yes—
72#1927C2214yes—
73#1920C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1863C2214yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1834C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99#557—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×#125—22no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8—44no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D412yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#462—44no—
134#451—24no—
135#451—28no—
136#446—44no—
137C4×Q16—44no—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#398—22no—
164#398—22no—
165#398—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391—24no—
170#391—22no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#391C2312yes—
179#391C2311yescharacteristic
180#391C2312yes—
181#391C2314yes—
182#389—22no—
183#389—22no—
184#389—24no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389—22no—
193#389—21no—
194#389—22no—
195#389C2314yes—
196#389C2312yes—
197#389C2311yescharacteristic
198#389C2314yes—
199#387—21no—
200#387—22no—
201#385—21no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#382—21no—
210#382—22no—
211#380—22no—
212#380—21no—
213#380—21no—
214#380—22no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8#4114yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×Q16—24no—
259C2×Q16—28no—
260C2×Q16—42no—
261C2×Q16—42no—
262C2×Q16—44no—
263C2×Q16—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269#125—48no—
270#125—44no—
271C22⋊Q8—22no—
272C22⋊Q8—24no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#107—42no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—24no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—42no—
365#103—44no—
366#103—42no—
367#103—42no—
368#103—44no—
369C22⋊C8—22no—
370C22⋊C8—24no—
371C22⋊C8—22no—
372#97—44no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—42no—
418C2×Q8—48no—
419C2×Q8—416no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—48no—
423C2×Q8#13714yes—
424C2×Q8C2×#4114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4#13714yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469Q16—48no—
470Q16—88no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23#54812yes—
543C23C2×#13712yes—
544C23#57812yes—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—24no—
549Q8—48no—
550Q8—48no—
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—42no—
557Q8—44no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4#53412yes—
615C2×C4#53412yes—
616C2×C4#54812yes—
617C2×C4C2×#13714yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58512yes—
625C8—42no—
626C8—44no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#337414yes—
635C22C2×#53411yescharacteristic
636C22C2×#54811yescharacteristic
637C22#348711yescharacteristic
638C22C22×#13711yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358511yescharacteristic
642C4—22no—
643C4—22no—
644C4—24no—
645C4—22no—
646C4—44no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—84no—
658C4—81no—
659C4#319514yes—
660C4#350512yes—
661C4#354112yes—
662C2—24no—
663C2#3972311yescharacteristic
664C2C2×#319511yescharacteristic
665C2#6076411yescharacteristic
666C2C2×#337411yescharacteristic
667C2#6131911yescharacteristic
668C2#6189211yescharacteristic
669C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32625C2semidirect
5. #32563C2semidirect
9. #28196C2semidirect
12. #25302C2semidirect
13. #24994C2semidirect
14. #24191C2semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25822
#329062
#326252
#325632
#297152
#293352
#292442
#281962
#279722
#276802
#253022
#249942
#241912
#225102
#220502
C4×#4512