Finite Group Databasebeta

← #6487722

Subgroups — #6487722

Order 512.

1943 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic, complemented
2C2×#3117C211yesmaximal, characteristic
3#37633C211yesmaximal, characteristic
4#34182C211yesmaximal, characteristic, complemented
5#32918C212yesmaximal
6#32915C212yesmaximal, complemented
7#29768C212yesmaximal
8#29764C212yesmaximal, complemented
9#24831C212yesmaximal, complemented
10#24828C212yesmaximal, complemented
11#24826C214yesmaximal, complemented
12#24746C212yesmaximal, complemented
13#24743C212yesmaximal, complemented
14#24741C214yesmaximal, complemented
15#21956C211yesmaximal, characteristic, complemented
16#21870C211yesmaximal, characteristic, complemented
17#21868C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C2×#519C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#3117—28no—
23#3113—28no—
24C2×#484C2211yescharacteristic
25C2×#481C2211yescharacteristic
26C2×#483C2212yes—
27C2×#479C2212yes—
28C2×#475C2212yes—
29C2×#464C2212yes—
30C2×#464C2214yes—
31C2×#462C2212yes—
32C2×#462C2214yes—
33C2×#451C2211yescharacteristic
34C2×#451C2212yes—
35C2×#448C2211yescharacteristic
36C2×#441C2212yes—
37C2×#441C2214yes—
38C2×#440C2212yes—
39C2×#432C2211yescharacteristic
40C2×#432C2211yescharacteristic
41#2814—22no—
42#2637—22no—
43#2629—22no—
44#2621C2212yes—
45#2620C2212yes—
46#2567C2211yescharacteristic
47#2565C2212yes—
48#2563C2211yescharacteristic
49#2547C2211yescharacteristic
50#2546C2211yescharacteristic
51#2545C2212yes—
52#2485—22no—
53#2351—22no—
54#2351C2212yes—
55#2351C2212yes—
56Q8×C4⋊C4—22no—
57Q8×C4⋊C4C2214yes—
58#2316—22no—
59#2316C2212yes—
60#2285—22no—
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63#2085C2214yes—
64#2084C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1993C2212yes—
72#1992C2212yes—
73#1960C2212yes—
74#1959C2212yes—
75#1940C2214yes—
76#1940C2212yes—
77#1940C2212yes—
78#1880C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1875C2212yes—
83#1873C2212yes—
84#1873C2212yes—
85#1873C2212yes—
86#1805C2212yes—
87#1805C2214yes—
88#1804C2212yes—
89#1803C2214yes—
90#1803C2214yes—
91#1798C2212yes—
92#1798C2212yes—
93C23×Q8C2311yescharacteristic
94C22×Q16C2314yes—
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102#529—24no—
103#529—24no—
104C2×#126—21no—
105C2×#125—21no—
106C2×#125C2311yescharacteristic
107C2×#124C2311yescharacteristic
108C2×#124C2312yes—
109C2×#123—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121C2311yescharacteristic
114#519—48no—
115C2×C4×Q8—24no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—24no—
120C2×C4×Q8D418yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2314yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—24no—
131C2×#117—21no—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#484—24no—
139#483—28no—
140#481—24no—
141#479—28no—
142#475—28no—
143#464—28no—
144#464—48no—
145#462—28no—
146#462—48no—
147#451—28no—
148#451—24no—
149#448—24no—
150#441—216no—
151#441—28no—
152#440—28no—
153#432—24no—
154#432—24no—
155C2×#107C2312yes—
156C2×#106C2314yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2311yescharacteristic
174C2×C22⋊C8C2311yescharacteristic
175C2×#97C2312yes—
176C2×#97C2312yes—
177#398—24no—
178#398—24no—
179#398—22no—
180#397—24no—
181#397—22no—
182#396—24no—
183#396—24no—
184#395—22no—
185#394—24no—
186#393—22no—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#390C2312yes—
191#389—22no—
192#389—24no—
193#389—24no—
194#389—24no—
195#389—22no—
196#389—24no—
197#389—24no—
198#389C2314yes—
199#389C2314yes—
200#388—24no—
201#387—24no—
202#384—24no—
203#384D412yes—
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#382—22no—
208#382D412yes—
209#382C2312yes—
210#382C2312yes—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—24no—
216#380D412yes—
217#380D412yes—
218C4×C4⋊C4—24no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D414yes—
238C22×Q8C2411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×Q16—28no—
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—24no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266#126—44no—
267#125—22no—
268#125—22no—
269#125—44no—
270#124—22no—
271#124—22no—
272#124—44no—
273#123—44no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279#121—22no—
280#121—22no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—416no—
294C4×Q8—44no—
295C4×Q8C2×D418yes—
296C4×Q8C2×D418yes—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—44no—
300#117—44no—
301#117—44no—
302#117—416no—
303#117—44no—
304#117—44no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D414yes—
347C2×C42—28no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#106—28no—
365#106—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—24no—
371#103—24no—
372#103—28no—
373#103—28no—
374#103—24no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—24no—
379#102—44no—
380C22⋊C8—22no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383#97—24no—
384#97—24no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D414yes—
396C24—21no—
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—216no—
400C2×Q8—24no—
401C2×Q8—216no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×Q8C22×D414yes—
417C2×Q8H(5,𝔽2)14yes—
418C2×Q82−1+414yes—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—82no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456Q16—416no—
457Q16—48no—
458SD16—48no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C22×D414yes—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—28no—
513C42—28no—
514C42—24no—
515C42—24no—
516C42—48no—
517C42—48no—
518C42—44no—
519C42—42no—
520C42—44no—
521C42#12114yes—
522C42#12114yes—
523C42C22×D412yes—
524C42C22×D412yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×#12112yes—
537C23C23×D411yescharacteristic
538Q8—216no—
539Q8—24no—
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—44no—
544Q8—42no—
545Q8—44no—
546Q8—84no—
547Q8#53718yes—
548D4—42no—
549D4—42no—
550D4—84no—
551C2×C4—216no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C2×#12112yes—
591C2×C4C2×#12112yes—
592C2×C4C2×#12112yes—
593C2×C4C2×#12112yes—
594C2×C4C2×#12112yes—
595C2×C4C23×D411yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)12yes—
598C2×C4C2×2−1+412yes—
599C8—42no—
600C8—42no—
601C8—44no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22C22×#12111yescharacteristic
613C22#348411yescharacteristic
614C22#348611yescharacteristic
615C22#358612yes—
616C22#358612yes—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—28no—
622C4—22no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—81no—
630C4—82no—
631C4—84no—
632C4C2×#53712yes—
633C4#349212yes—
634C2—42no—
635C2—161no—
636C2#3522311yescharacteristic
637C2#6042411yescharacteristic
638C2#6042711yescharacteristic
639C2#6167212yes—
640C2#6187512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3113C2semidirect
4. #34182C2semidirect
6. #32915C2semidirect
8. #29764C2semidirect
9. #24831C2semidirect
10. #24828C2semidirect
11. #24826C2semidirect
12. #24746C2semidirect
13. #24743C2semidirect
14. #24741C2semidirect
15. #21956C2semidirect
16. #21870C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#31172
#376332
#341822
#329182
#329152
#297682
#297642
#248312
#248282
#248262
#247462
#247432
#247412
#219562
#218702
#218682