Finite Group Databasebeta

← #6497392

Subgroups — #6497392

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#35132C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33293C211yesmaximal, characteristic, complemented
5#33281C211yesmaximal, characteristic, complemented
6#33281C211yesmaximal, characteristic, complemented
7#30233C211yesmaximal, characteristic, complemented
8#30233C211yesmaximal, characteristic, complemented
9#30215C211yesmaximal, characteristic, complemented
10#30214C211yesmaximal, characteristic, complemented
11#27832C214yesmaximal, complemented
12#27746C214yesmaximal, complemented
13#27715C214yesmaximal
14#27697C214yesmaximal
15#21866C211yesmaximal, characteristic
16#21841C211yesmaximal, characteristic
17#21407C211yesmaximal, characteristic
18C2×#1993C211yesmaximal, characteristic
19#20479C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2864C2211yescharacteristic
27#2864C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2666—22no—
33#2660—22no—
34#2612—22no—
35#2570C2212yes—
36#2567C2212yes—
37#2567C2212yes—
38#2550C2212yes—
39#2546—22no—
40#2502C2212yes—
41#2502C2212yes—
42#2442C2212yes—
43#2440C2212yes—
44#2438—22no—
45#2434—22no—
46#2329—22no—
47#2317—22no—
48#2283C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2100C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2099C2214yes—
55#2074C2214yes—
56#2072C2214yes—
57#2026C2214yes—
58#2024C2214yes—
59#1993C2211yescharacteristic
60#1993C2211yescharacteristic
61#1993C2211yescharacteristic
62#1993C2211yescharacteristic
63#1993C2211yescharacteristic
64#1993C2211yescharacteristic
65#1993C2211yescharacteristic
66#1993C2211yescharacteristic
67#1993C2211yescharacteristic
68#1993C2211yescharacteristic
69#1993C2211yescharacteristic
70#1993C2211yescharacteristic
71#1993C2214yes—
72#1992C2214yes—
73#1990C2214yes—
74#1988C2214yes—
75#1960C2211yescharacteristic
76#1960C2211yescharacteristic
77#1960C2211yescharacteristic
78#1960C2211yescharacteristic
79#1935C2214yes—
80#1935C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1892C2214yes—
84#1872C2214yes—
85#1836C2214yes—
86#1834C2214yes—
87#1824C2214yes—
88#1805C2214yes—
89#1771C2212yes—
90#1771C2212yes—
91#1763C2214yes—
92#1752C2212yes—
93C8×C22⋊C4C2212yes—
94#1745C2211yescharacteristic
95#1745C2211yescharacteristic
96C2×#340C2214yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C23×Q8—21no—
100C23×C8C2312yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×#117—21no—
123C2×#117D412yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128#432—24no—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2312yes—
135C2×#97—24no—
136C2×#97C2312yes—
137C2×#97C2311yescharacteristic
138C2×#97C2311yescharacteristic
139C2×#97C2312yes—
140C2×#97C2311yescharacteristic
141C2×#97C2311yescharacteristic
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#400—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—24no—
152#398—24no—
153#395—22no—
154#395—24no—
155#395—24no—
156#395—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#389—24no—
167#389—24no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—24no—
182#385—24no—
183#385—22no—
184#385—22no—
185#384—22no—
186#384—22no—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#383—21no—
193#382—22no—
194#382—22no—
195#382—21no—
196#382—21no—
197#382C2312yes—
198#382C2312yes—
199#382C2311yescharacteristic
200#382C2311yescharacteristic
201#382C2311yescharacteristic
202#382C2311yescharacteristic
203#381—21no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2311yescharacteristic
213C4×C22⋊C4C2311yescharacteristic
214#376—21no—
215#376D412yes—
216C2×#95—21no—
217C2×#95—21no—
218C43C2311yescharacteristic
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—84no—
243C22×Q8C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—84no—
270C4⋊Q8—84no—
271#125—44no—
272#125—44no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280#117—28no—
281#117—44no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42#4114yes—
331C2×C42#4114yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C4⋊C8—24no—
339#103—28no—
340#103—44no—
341#103—44no—
342#103—44no—
343C22⋊C8—24no—
344#97—24no—
345#97—24no—
346#97—22no—
347#97—22no—
348#97—22no—
349#97—22no—
350#97—42no—
351C4×C8—24no—
352C4×C8—42no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24C22×D411yescharacteristic
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—816no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—84no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4#12414yes—
414C22×C4#12414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—48no—
439C2×C82−1+414yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C22⋊C4—24no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4#13718yes—
479C42—24no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—82no—
489C42#12414yes—
490C42#12414yes—
491C42C2×#4112yes—
492C42C2×#4112yes—
493C42C2×#4112yes—
494C42C2×#4112yes—
495C23—22no—
496C23—21no—
497C23—21no—
498C23C2×#12412yes—
499C23C2×#13714yes—
500C23C23×D411yescharacteristic
501Q8—44no—
502Q8—44no—
503Q8—84no—
504Q8—84no—
505Q8—84no—
506Q8—88no—
507Q8—164no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—88no—
550C2×C4—164no—
551C2×C4C2×#12412yes—
552C2×C4C2×#12412yes—
553C2×C4C2×#12412yes—
554C2×C4C2×#12412yes—
555C2×C4C2×#12412yes—
556C2×C4C2×#13714yes—
557C2×C4C2×#13714yes—
558C2×C4C22×#4111yescharacteristic
559C2×C4C22×#4111yescharacteristic
560C2×C4C2×2−1+412yes—
561C2×C4#58512yes—
562C8—24no—
563C8—42no—
564C8—42no—
565C8—42no—
566C22—24no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22#313914yes—
571C22C22×#12411yescharacteristic
572C22#349911yescharacteristic
573C22#350011yescharacteristic
574C22C22×#13712yes—
575C22#358512yes—
576C4—24no—
577C4—24no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4#346112yes—
594C4#349512yes—
595C2—24no—
596C2#3531211yescharacteristic
597C2C2×#313911yescharacteristic
598C2#6061911yescharacteristic
599C2#6250412yes—
600C2#6258212yes—
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. #33297C2semidirect
4. #33293C2semidirect
5. #33281C2semidirect
6. #33281C2semidirect
7. #30233C2semidirect
8. #30233C2semidirect
9. #30215C2semidirect
10. #30214C2semidirect
11. #27832C2semidirect
12. #27746C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#351322
#332972
#332932
#332812
#302332
#302152
#302142
#278322
#277462
#277152
#276972
#218662
#218412
#214072
C2×#19932
#204792