Finite Group Databasebeta

← #6529808

Subgroups — #6529808

Order 512.

1812 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic
3#35179C211yesmaximal, characteristic
4C2×#2581C211yesmaximal, characteristic
5#28278C214yesmaximal, complemented
6#28198C214yesmaximal, complemented
7#25487C214yesmaximal, complemented
8#25306C214yesmaximal, complemented
9#24609C212yesmaximal
10#24601C212yesmaximal
11#24508C212yesmaximal
12#24502C212yesmaximal
13#21986C211yesmaximal, characteristic
14#21866C211yesmaximal, characteristic
15#20740C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#3113—28no—
20C2×#483C2212yes—
21C2×#475C2212yes—
22C2×#462C2212yes—
23C2×#451C2212yes—
24#2983—28no—
25C2×#446C2214yes—
26C2×#446C2214yes—
27C2×#441C2214yes—
28C2×#432C2211yescharacteristic
29C2×#432C2212yes—
30C2×#432C2211yescharacteristic
31C2×#431C2212yes—
32#2855C2212yes—
33#2855C2212yes—
34#2627—21no—
35#2627—21no—
36#2582C2212yes—
37#2582C2212yes—
38#2581—22no—
39#2581—22no—
40#2581—22no—
41#2581C2214yes—
42#2581C2214yes—
43#2568C2212yes—
44#2567C2212yes—
45#2565—22no—
46C2×#400—21no—
47C2×#395—21no—
48#2317—21no—
49#2310—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#387—21no—
55C2×#387—21no—
56#2285C2211yescharacteristic
57C2×#382C2211yescharacteristic
58#2088C2214yes—
59#2088C2214yes—
60#2068C2214yes—
61#2068C2214yes—
62#2066C2214yes—
63#2066C2214yes—
64#2056C2214yes—
65#2056C2214yes—
66#1993C2212yes—
67#1971C2214yes—
68#1969C2214yes—
69#1960C2212yes—
70#1949C2214yes—
71#1943C2214yes—
72#1941C2214yes—
73#1935C2214yes—
74#1935C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1847C2212yes—
82#1836C2212yes—
83#1823C2212yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1787C2212yes—
88#1763C2214yes—
89C23×Q8D411yescharacteristic
90C23×Q8C2311yescharacteristic
91C22×Q16C2314yes—
92#533—24no—
93#533—24no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—24no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8D412yes—
107C2×C4×Q8D412yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×#117—21no—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—28no—
129#475—28no—
130#462—28no—
131#451—28no—
132#446—216no—
133#446—216no—
134#441—216no—
135#432—24no—
136#432—28no—
137#432D418yes—
138#431—28no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2314yes—
158#400—22no—
159#400—22no—
160#400—24no—
161#400—24no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#395—24no—
167#395—24no—
168#395—22no—
169#395—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#391C2314yes—
179#391C2314yes—
180#391C2312yes—
181#391C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2314yes—
191#389C2314yes—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#385—22no—
201#385—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382—22no—
206#382—21no—
207#382—21no—
208#382—21no—
209#382C2312yes—
210#382C2312yes—
211#381—22no—
212#381—22no—
213#380—21no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C2×D412yes—
241C22×Q8C2×D412yes—
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—44no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264#125—44no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—44no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×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—44no—
283#117—44no—
284#117—44no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D414yes—
343C2×C42#4114yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—44no—
349#106—28no—
350#106—44no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353C4⋊C8—24no—
354C4⋊C8C2×D418yes—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—44no—
359#103—44no—
360#103—44no—
361C22⋊C8—24no—
362C22⋊C8C2×D418yes—
363#97—28no—
364#97—24no—
365#97C2×D418yes—
366C4×C8—28no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—416no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—81no—
415C2×Q8—81no—
416C2×Q82−1+414yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4#12114yes—
455C22×C4#12114yes—
456C22×C4#13714yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469Q16—416no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8C22×D418yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—44no—
519C42—44no—
520C42—44no—
521C42—44no—
522C42#12114yes—
523C42#12114yes—
524C42C22×D412yes—
525C42C22×D412yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23C2×#12112yes—
533C23C2×#13712yes—
534C23#57512yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—24no—
538Q8—48no—
539Q8—48no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—44no—
544Q8—44no—
545Q8—84no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12112yes—
606C2×C4C2×#12112yes—
607C2×C4C2×#12112yes—
608C2×C4C2×#13714yes—
609C2×C4C23×D411yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×2−1+412yes—
612C2×C4#58512yes—
613C8—28no—
614C8—44no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22#305614yes—
622C22C22×#12111yescharacteristic
623C22#348611yescharacteristic
624C22#348711yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22#358611yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4#348912yes—
647C4#349512yes—
648C2—24no—
649C2#3566111yescharacteristic
650C2C2×#305611yescharacteristic
651C2#6045511yescharacteristic
652C2#6182911yescharacteristic
653C2#6183211yescharacteristic
654C2#6188911yescharacteristic
655C2#6189411yescharacteristic
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
5. #28278C2semidirect
6. #28198C2semidirect
7. #25487C2semidirect
8. #25306C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#29832
#351792
C2×#25812
#282782
#281982
#254872
#253062
#246092
#246012
#245082
#245022
#219862
#218662
#207402