Finite Group Databasebeta

← #6537115

Subgroups — #6537115

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#34337C211yesmaximal, characteristic, complemented
3#30078C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29575C212yesmaximal, complemented
6#29366C212yesmaximal
7#28512C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic
9#26902C214yesmaximal, complemented
10#26373C214yesmaximal, complemented
11#25058C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23799C212yesmaximal
14#23678C211yesmaximal, characteristic
15#22996C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17C4⋊C4×SD16C212yesmaximal, complemented
18C2×#557C2211yescharacteristic
19C2×#554—21no—
20C2×#475C2214yes—
21C2×#461C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2214yes—
24C2×C4×SD16C2214yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2566—22no—
28#2563—21no—
29#2538—21no—
30#2538C2212yes—
31#2452—22no—
32#2452C2214yes—
33#2370—21no—
34#2370—22no—
35#2365C2212yes—
36#2365C2211yescharacteristic
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2211yescharacteristic
39Q8×C4⋊C4C2212yes—
40Q8×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43#2330—21no—
44#2330—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49#2295C2211yescharacteristic
50#2295C2212yes—
51#2276—21no—
52C42×Q8C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2094C2214yes—
58#2084C2214yes—
59#2084C2211yescharacteristic
60#2072C2211yescharacteristic
61#2072C2212yes—
62#2065C2212yes—
63#2058C2212yes—
64#2055C2212yes—
65#2043C2214yes—
66#2041C2211yescharacteristic
67#2024C2212yes—
68#2014C2211yescharacteristic
69#2014C2212yes—
70#2013C2214yes—
71#1984C2212yes—
72#1956C2212yes—
73#1948C2212yes—
74#1941C2214yes—
75#1933C2214yes—
76#1926C2214yes—
77#1919C2214yes—
78#1916C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1865C2214yes—
85#1864C2214yes—
86#1844C2212yes—
87#1839C2212yes—
88#1831C2211yescharacteristic
89#1803C2212yes—
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1744C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#557—24no—
98#557—24no—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125—22no—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2314yes—
127C22×C4⋊C4—22no—
128#475—48no—
129#461—28no—
130C4×SD16—28no—
131C4×SD16—48no—
132C4×SD16—48no—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#396—24no—
151#396—24no—
152#393—24no—
153#391C2312yes—
154#391C2311yescharacteristic
155#391C2311yescharacteristic
156#391C2312yes—
157#390—22no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390C2312yes—
162#390C2314yes—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2314yes—
178#389C2311yescharacteristic
179#389C2314yes—
180#389C2312yes—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—22no—
200#380—22no—
201#380—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220#376—22no—
221#376—21no—
222C43C2311yescharacteristic
223C43C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—24no—
245C2×SD16—24no—
246C2×SD16—24no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—24no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#127—24no—
262#125—216no—
263#125—28no—
264#125—48no—
265C22⋊Q8—48no—
266C22⋊Q8—416no—
267C4×Q8—28no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—42no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8SD1618yescomplemented
279C4×Q8#4114yes—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—84no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
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—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#106—28no—
353#106—24no—
354#106—44no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358#103—24no—
359#103—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—24no—
365#102—24no—
366#102—44no—
367C4×C8—28no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8C2×SD1614yes—
400C2×Q8C2×#4114yes—
401C2×Q8C2×#4112yes—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D42−1+414yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439SD16—216no—
440SD16—416no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8H(5,𝔽2)14yes—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×SD1614yes—
488C4⋊C4C2×SD1618yes—
489C4⋊C4C2×#4112yes—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—28no—
497C42—24no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—41no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—82no—
512C42—82no—
513C42—82no—
514C42C2×SD1618yes—
515C42C2×SD1614yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—24no—
528Q8—28no—
529Q8—44no—
530Q8—42no—
531Q8—164no—
532D4—24no—
533D4—24no—
534D4—48no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—88no—
584C2×C4—168no—
585C2×C4#52512yes—
586C2×C4#52512yes—
587C2×C4C22×SD1612yes—
588C2×C4C22×SD1614yes—
589C2×C4C22×SD1612yes—
590C2×C4C22×SD1614yes—
591C2×C4C22×SD1612yes—
592C2×C4#57512yes—
593C2×C4#57712yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×2−1+412yes—
598C8—24no—
599C8—42no—
600C8—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#52511yescharacteristic
609C22#348611yescharacteristic
610C22#349811yescharacteristic
611C22C23×SD1611yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358611yescharacteristic
615C4—24no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—24no—
620C4—22no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—82no—
628C4—81no—
629C4—84no—
630C4—81no—
631C4—82no—
632C4#310314yes—
633C4#349212yes—
634C4#352012yes—
635C2—84no—
636C2#3570111yescharacteristic
637C2C2×#310311yescharacteristic
638C2#6059911yescharacteristic
639C2#6181811yescharacteristic
640C2#6184111yescharacteristic
641C2#6241311yescharacteristic
642C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34337C2semidirect
3. #30078C2semidirect
4. #29983C2semidirect
5. #29575C2semidirect
7. #28512C2semidirect
9. #26902C2semidirect
10. #26373C2semidirect
11. #25058C2semidirect
12. #24187C2semidirect
15. #22996C2semidirect
16. #22828C2semidirect
17. C4⋊C4×SD16C2semidirect
278. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365592
#343372
#300782
#299832
#295752
#293662
#285122
#281572
#269022
#263732
#250582
#241872
#237992
#236782
#229962
#228282
C4⋊C4×SD162