Finite Group Databasebeta

← #6435200

Subgroups — #6435200

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic, complemented
2#34056C211yesmaximal, characteristic
3#29993C214yesmaximal, complemented
4#29992C214yesmaximal, complemented
5#29991C214yesmaximal, complemented
6#29990C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23745C211yesmaximal, characteristic, complemented
9#23734C211yesmaximal, characteristic
10#23733C211yesmaximal, characteristic
11#23553C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#23453C211yesmaximal, characteristic
14#23452C211yesmaximal, characteristic
15#21172C211yesmaximal, characteristic, complemented
16#21171C211yesmaximal, characteristic, complemented
17#20753C211yesmaximal, characteristic, complemented
18#20649C211yesmaximal, characteristic
19#20648C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#417C2211yescharacteristic
33#2660—22no—
34#2660—22no—
35#2660C2214yes—
36#2660C2214yes—
37C2×#398—22no—
38C2×#395—22no—
39#2320C2212yes—
40#2320C2212yes—
41#2319C2212yes—
42#2319C2212yes—
43#2318—22no—
44#2318—22no—
45#2298—22no—
46#2297C2211yescharacteristic
47#2297C2211yescharacteristic
48#2296C2211yescharacteristic
49#2296C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2283—22no—
53C2×#382C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C4×#117C2211yescharacteristic
56C2×#376C2211yescharacteristic
57#2101C2214yes—
58#2101C2214yes—
59#2100C2214yes—
60#2099C2214yes—
61#2097C2212yes—
62#2097C2212yes—
63#2096C2212yes—
64#2095C2212yes—
65#2094C2212yes—
66#2093C2212yes—
67#2092C2212yes—
68#2092C2212yes—
69#2059C2214yes—
70#2059C2214yes—
71#2058C2214yes—
72#2057C2214yes—
73#2026C2214yes—
74#2025C2214yes—
75#2024C2214yes—
76#2023C2214yes—
77#1949C2214yes—
78#1949C2214yes—
79#1948C2214yes—
80#1947C2214yes—
81#1832C2214yes—
82#1832C2214yes—
83#1805C2214yes—
84#1804C2214yes—
85#1803C2214yes—
86#1802C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2211yescharacteristic
96#1728C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C23×Q8—21no—
101C22×M16C2311yescharacteristic
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—42no—
107C2×C4⋊Q8—42no—
108C2×C4⋊Q8C2314yes—
109C2×#125—22no—
110C2×#125—42no—
111C2×#125—42no—
112C2×#125C2314yes—
113C4×#41—24no—
114C4×#41Q818yescomplemented
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#428—42no—
136#428—42no—
137#428—42no—
138#428—42no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147#423—24no—
148#423—24no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151#417—42no—
152#417—42no—
153#417—42no—
154#417—42no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163#398—24no—
164#398—24no—
165#398—24no—
166#398—24no—
167#395—24no—
168#395—24no—
169#395—24no—
170#395—24no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#391C2312yes—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#390C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—21no—
190#386—21no—
191#386—22no—
192#385—21no—
193#385—21no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2314yes—
199#384C2312yes—
200#384C2312yes—
201#383—22no—
202#383—22no—
203#382—22no—
204#382—21no—
205#382—21no—
206#382—22no—
207#382—22no—
208#382C2314yes—
209#382C2312yes—
210#382C2312yes—
211#381—22no—
212#381—22no—
213#380—22no—
214#380—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#376—21no—
223#376D412yes—
224#376D412yes—
225#376C2312yes—
226#376C2312yes—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95D412yes—
230C2×#95D412yes—
231C43—21no—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2314yes—
241#336C2314yes—
242#336C2312yes—
243#336C2312yes—
244#336C2312yes—
245#336C2312yes—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C2×Q814yes—
252C22×Q8—24no—
253C22×Q8—44no—
254C22×Q8—84no—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—48no—
288#125—44no—
289#125—44no—
290#125—48no—
291C4×Q8—24no—
292C4×Q8C2×Q818yescomplemented
293C4×D4—216no—
294C4×D4—24no—
295C4×D4C2×Q818yescomplemented
296#117—22no—
297#117—28no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117C2×Q814yes—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—28no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×Q814yes—
364C2×C42#4114yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369#107—42no—
370#107—42no—
371#107—42no—
372#107—42no—
373#106—42no—
374#106—42no—
375#106—42no—
376#106—42no—
377C4⋊C8—28no—
378C4⋊C8—28no—
379#103—42no—
380#103—42no—
381#103—42no—
382#103—42no—
383#102—42no—
384#102—42no—
385#102—42no—
386#102—42no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—21no—
410C24C22×D411yescharacteristic
411#41—28no—
412#41—28no—
413#41—44no—
414#41—44no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—48no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8—816no—
424C2×Q8C22×Q812yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C22×Q812yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4C22⋊Q818yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481M16—44no—
482M16—44no—
483M16—44no—
484M16—44no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×Q814yes—
529C22⋊C4—24no—
530C22⋊C4—28no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—24no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—42no—
549C42—82no—
550C42—82no—
551C42C22⋊Q814yes—
552C42C22⋊Q814yes—
553C42C22×Q814yes—
554C42C2×#4114yes—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23C2×C22⋊Q812yes—
563C23#53512yes—
564C23#53612yes—
565C23C23×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—41no—
569Q8—41no—
570Q8—816no—
571D4—22no—
572D4—22no—
573D4—48no—
574D4—41no—
575D4—41no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4C2×C22⋊Q814yes—
624C2×C4C2×C22⋊Q814yes—
625C2×C4C2×C22⋊Q812yes—
626C2×C4#53512yes—
627C2×C4#53612yes—
628C2×C4#54214yes—
629C2×C4#54214yes—
630C2×C4#54414yes—
631C2×C4#54414yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—42no—
647C22—44no—
648C22—81no—
649C22—81no—
650C22—82no—
651C22#262814yes—
652C22C22×C22⋊Q811yescharacteristic
653C22C2×#53511yescharacteristic
654C22C2×#53611yescharacteristic
655C22#358512yes—
656C22#358612yes—
657C4—24no—
658C4—24no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4#262814yes—
673C4C2×#54211yescharacteristic
674C4C2×#54211yescharacteristic
675C4C2×#54411yescharacteristic
676C4C2×#54411yescharacteristic
677C2—24no—
678C2—82no—
679C2C2×#262811yescharacteristic
680C2#6049312yes—
681C2#6080611yescharacteristic
682C2#6080711yescharacteristic
683C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2660C2semidirect
3. #29993C2semidirect
4. #29992C2semidirect
5. #29991C2semidirect
6. #29990C2semidirect
7. #23746C2semidirect
8. #23745C2semidirect
11. #23553C2semidirect
12. #23552C2semidirect
15. #21172C2semidirect
16. #21171C2semidirect
17. #20753C2semidirect
114. C4×#41Q8semidirect
292. C4×Q8C2×Q8semidirect
295. C4×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
#340562
#299932
#299922
#299912
#299902
#237462
#237452
#237342
#237332
#235532
#235522
#234532
#234522
#211722
#211712
#207532
#206492
#206482