Finite Group Databasebeta

← #6651593

Subgroups — #6651593

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#33100C212yesmaximal
4#32694C212yesmaximal, complemented
5#32683C212yesmaximal, complemented
6#32645C212yesmaximal, complemented
7#32631C212yesmaximal, complemented
8#32582C214yesmaximal, complemented
9#32575C214yesmaximal, complemented
10#29641C212yesmaximal
11#27616C212yesmaximal
12#24609C212yesmaximal
13#24601C212yesmaximal
14#24063C212yesmaximal
15#21820C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2855C2212yes—
24#2855C2212yes—
25#2634—21no—
26#2634—21no—
27#2581C2212yes—
28#2581C2212yes—
29#2568C2211yescharacteristic
30#2568C2211yescharacteristic
31#2567—21no—
32#2567—21no—
33#2567—21no—
34#2567—21no—
35#2567C2212yes—
36#2567C2212yes—
37#2567C2212yes—
38#2567C2212yes—
39#2546C2211yescharacteristic
40#2546C2211yescharacteristic
41#2503—21no—
42#2484—21no—
43#2484—21no—
44#2434—21no—
45C2×#395—21no—
46C2×#395—21no—
47#2327—21no—
48#2327—21no—
49C2×#391—21no—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×#384—21no—
53#2283C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yes—
56#2088C2214yes—
57#2088C2214yes—
58#2085C2214yes—
59#2068C2214yes—
60#2066C2214yes—
61#2058C2214yes—
62#2058C2214yes—
63#2058C2214yes—
64#2043C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#1985C2212yes—
68#1985C2212yes—
69#1984C2212yes—
70#1984C2212yes—
71#1943C2214yes—
72#1941C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1872C2212yes—
82#1872C2212yes—
83#1865C2214yes—
84#1865C2214yes—
85#1853C2212yes—
86#1852C2212yes—
87#1823C2212yes—
88#1823C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91C23×Q8—21no—
92C23×Q8D411yescharacteristic
93#533—42no—
94#533—42no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—42no—
110C2×C4×Q8—42no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×#117—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#483—44no—
134#483—44no—
135#432—42no—
136#431—42no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151#400—21no—
152#400—21no—
153#400—21no—
154#400—21no—
155#400—22no—
156#400—22no—
157#398—21no—
158#398—21no—
159#398—21no—
160#398—21no—
161#395—22no—
162#395—22no—
163#395—22no—
164#395—22no—
165#395—22no—
166#395—22no—
167#395—22no—
168#395—22no—
169#395—22no—
170#395—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#389C2314yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#385—22no—
204#385—22no—
205#384—22no—
206#384—22no—
207#384—22no—
208#384—22no—
209#383—21no—
210#382—21no—
211#382—21no—
212#382C2312yes—
213#381—22no—
214#381—22no—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224#376—21no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8—44no—
247C22×Q8—84no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2412yes—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—41no—
269C4⋊Q8—41no—
270C4⋊Q8—41no—
271C4⋊Q8—41no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—44no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—41no—
285C4×Q8—41no—
286C4×Q8—41no—
287C4×Q8—41no—
288C4×Q8—41no—
289C4×Q8—41no—
290C4×Q8—41no—
291C4×Q8—41no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295#117—44no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
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—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#106—28no—
367#106—44no—
368#106—44no—
369C4⋊C8—42no—
370C4⋊C8—42no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#97—42no—
378C4×C8—42no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
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#95—22no—
402C24C22×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—416no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—84no—
425C2×Q8—84no—
426C2×Q8—816no—
427C2×Q8#13714yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4C2511yescharacteristic
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C42−1+418yes—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—44no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23#53412yes—
544C23#57714yes—
545C23C23×D411yescharacteristic
546Q8—44no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—44no—
551Q8—44no—
552Q8—84no—
553Q8—88no—
554Q8—88no—
555C2×C4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—88no—
610C2×C4#53412yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4#57512yes—
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+414yes—
618C8—44no—
619C8—44no—
620C22—22no—
621C22—22no—
622C22—24no—
623C22—24no—
624C22#320014yes—
625C22C2×#53411yescharacteristic
626C22#348612yes—
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57712yes—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—48no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#354412yes—
650C4#354412yes—
651C2—24no—
652C2#3725811yescharacteristic
653C2C2×#320011yescharacteristic
654C2#6075111yescharacteristic
655C2#6183812yes—
656C2#6184112yes—
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. #32694C2semidirect
5. #32683C2semidirect
6. #32645C2semidirect
7. #32631C2semidirect
8. #32582C2semidirect
9. #32575C2semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#25672
#331002
#326942
#326832
#326452
#326312
#325822
#325752
#296412
#276162
#246092
#246012
#240632
#218202