Finite Group Databasebeta

← #6993718

Subgroups — #6993718

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32808C212yesmaximal, complemented
4#32052C214yesmaximal
5#31840C214yesmaximal
6#30205C212yesmaximal, complemented
7#29984C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic
9#29553C212yesmaximal, complemented
10#28401C212yesmaximal, complemented
11#27815C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic
13#25324C212yesmaximal, complemented
14#24682C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#462C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2212yes—
25#2862C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2660—22no—
29#2612—22no—
30#2574—22no—
31#2567C2212yes—
32#2567C2212yes—
33#2551C2211yescharacteristic
34#2546—21no—
35#2546—21no—
36#2546C2211yescharacteristic
37#2508—21no—
38#2502C2214yes—
39#2492C2212yes—
40#2439—21no—
41#2434—22no—
42#2429C2212yes—
43#2369—21no—
44#2365C2211yescharacteristic
45#2355C2211yescharacteristic
46#2346—21no—
47#2317—21no—
48#2317—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2072C2214yes—
55#2072C2214yes—
56#2057C2211yescharacteristic
57#2057C2212yes—
58#2056C2214yes—
59#2056C2214yes—
60#2043C2211yescharacteristic
61#2036C2212yes—
62#2024C2211yescharacteristic
63#2024C2212yes—
64#2024C2212yes—
65#2024C2214yes—
66#2024C2212yes—
67#2024C2212yes—
68#2016C2211yescharacteristic
69#2016C2214yes—
70#2016C2212yes—
71#2001C2212yes—
72#2001C2214yes—
73#1990C2212yes—
74#1990C2214yes—
75#1987C2212yes—
76#1969C2214yes—
77#1956C2214yes—
78#1947C2214yes—
79#1939C2212yes—
80#1933C2214yes—
81#1928C2214yes—
82#1923C2214yes—
83#1878C2214yes—
84#1872C2214yes—
85#1872C2212yes—
86#1852C2212yes—
87#1831C2212yes—
88#1803C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127#487—44no—
128#477—44no—
129#462—44no—
130C4×Q16—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#398—22no—
150#398—22no—
151#398—24no—
152#398—24no—
153#396—22no—
154#395—24no—
155#395—24no—
156#395—22no—
157#393—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#391C2312yes—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389C2311yescharacteristic
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—21no—
191#385—24no—
192#385—24no—
193#384—22no—
194#384—22no—
195#384C2314yes—
196#384C2314yes—
197#384C2311yescharacteristic
198#382—21no—
199#382—22no—
200#382—22no—
201#382—22no—
202#382—22no—
203#380—21no—
204#380—21no—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×Q16—42no—
249C2×Q16—42no—
250C2×Q16—42no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×Q16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—84no—
269C4⋊Q8—84no—
270#125—44no—
271C22⋊Q8—22no—
272C22⋊Q8—41no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—44no—
283#117—24no—
284#117—44no—
285#117Q1618yescomplemented
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#107—42no—
346#107—44no—
347C4⋊C8—24no—
348C4⋊C8—44no—
349#103—24no—
350#103—44no—
351#103—42no—
352#103—42no—
353#103—42no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C4×C8—24no—
361C4×C8—44no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—816no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C2×Q1614yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—48no—
446Q16—88no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4C2×Q1614yes—
491C4⋊C4C2×Q1614yes—
492C4⋊C4#13714yes—
493C4⋊C42−1+412yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—81no—
498C22⋊C4—82no—
499C22⋊C4C2×Q1618yes—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—22no—
508C42—42no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—42no—
513C42—82no—
514C42C2×Q1618yes—
515C42H(5,𝔽2)14yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23C22×Q1612yes—
523C23C23×D411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—44no—
528Q8—42no—
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—84no—
533Q8—88no—
534Q8—88no—
535Q8—164no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—88no—
586C2×C4#53512yes—
587C2×C4#53512yes—
588C2×C4C22×Q1612yes—
589C2×C4C22×Q1614yes—
590C2×C4C22×Q1614yes—
591C2×C4C22×Q1612yes—
592C2×C4C2×#13714yes—
593C2×C4C2×#13712yes—
594C2×C4#57712yes—
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4C2×2−1+411yescharacteristic
597C2×C4#58511yescharacteristic
598C2×C4#58512yes—
599C8—42no—
600C8—44no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22C2×#53511yescharacteristic
610C22#348911yescharacteristic
611C22#350011yescharacteristic
612C22C23×Q1611yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57711yescharacteristic
615C22#358511yescharacteristic
616C4—24no—
617C4—22no—
618C4—24no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4#320914yes—
634C4#354012yes—
635C4#354712yes—
636C2—42no—
637C2#4299511yescharacteristic
638C2C2×#320911yescharacteristic
639C2#6081211yescharacteristic
640C2#6210011yescharacteristic
641C2#6214711yescharacteristic
642C2#6253011yescharacteristic
643C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32808C2semidirect
6. #30205C2semidirect
7. #29984C2semidirect
9. #29553C2semidirect
10. #28401C2semidirect
11. #27815C2semidirect
13. #25324C2semidirect
14. #24682C2semidirect
285. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#362882
#352812
#328082
#320522
#318402
#302052
#299842
#296642
#295532
#284012
#278152
#259022
#253242
#246822
#243022
#235712
#226322