Finite Group Databasebeta

← #6497417

Subgroups — #6497417

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic
2#35132C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#33293C211yesmaximal, characteristic, complemented
5#33281C211yesmaximal, characteristic, complemented
6#33281C211yesmaximal, characteristic, complemented
7#30234C211yesmaximal, characteristic, complemented
8#30233C211yesmaximal, characteristic, complemented
9#30215C211yesmaximal, characteristic, complemented
10#30214C211yesmaximal, characteristic, complemented
11#27818C214yesmaximal, complemented
12#27734C214yesmaximal, complemented
13#27724C214yesmaximal
14#27722C214yesmaximal
15#21866C211yesmaximal, characteristic
16#21842C211yesmaximal, characteristic
17#21404C211yesmaximal, characteristic
18#21323C211yesmaximal, characteristic
19#20480C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868C2211yescharacteristic
25#2864C2211yescharacteristic
26#2864C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2666—22no—
33#2621—22no—
34#2612—22no—
35#2570C2212yes—
36#2565—22no—
37#2565C2212yes—
38#2565C2212yes—
39#2550C2212yes—
40#2499—22no—
41#2499C2212yes—
42#2499C2212yes—
43#2442C2212yes—
44#2440C2212yes—
45#2438—22no—
46#2329—22no—
47#2327—22no—
48#2283C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
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#1991C2214yes—
72#1991C2214yes—
73#1990C2214yes—
74#1989C2214yes—
75#1960C2211yescharacteristic
76#1960C2211yescharacteristic
77#1960C2211yescharacteristic
78#1960C2211yescharacteristic
79#1935C2214yes—
80#1935C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1894C2214yes—
84#1872C2214yes—
85#1836C2214yes—
86#1836C2214yes—
87#1825C2214yes—
88#1805C2214yes—
89#1770C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92#1753C2212yes—
93#1753C2212yes—
94#1745C2211yescharacteristic
95#1745C2211yescharacteristic
96#1736C2214yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C22×M16C2312yes—
100#533—24no—
101#533—24no—
102#533—24no—
103#533—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×#117—21no—
123C2×#117D412yes—
124C2×#117C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127#432—24no—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×C22⋊C8C2312yes—
134C4×M16—24no—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×#97C2311yescharacteristic
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×#97C2311yescharacteristic
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#398—24no—
147#398—24no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#397—24no—
153#397—24no—
154#395—22no—
155#395—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#384—22no—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#383—21no—
191#382—22no—
192#382—22no—
193#382—21no—
194#382—21no—
195#382C2312yes—
196#382C2312yes—
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×Q8C2412yes—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×M16—24no—
246C2×M16—24no—
247C2×M16—42no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—48no—
262#125—24no—
263#125—24no—
264#125—44no—
265#125—44no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277#117—24no—
278#117—24no—
279#117—44no—
280#117—44no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42#4114yes—
327C2×C42#4114yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C4⋊C8—24no—
335#103—28no—
336#103—44no—
337#103—44no—
338#103—44no—
339C22⋊C8—24no—
340#97—24no—
341#97—24no—
342#97—22no—
343#97—22no—
344#97—22no—
345#97—22no—
346C4×C8—24no—
347C4×C8—42no—
348C4×C8—42no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D411yescharacteristic
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—82no—
390C2×Q8—82no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4#12414yes—
407C22×C4#12414yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2511yescharacteristic
420M16—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C82−1+414yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—44no—
481C42—82no—
482C42#12414yes—
483C42#12414yes—
484C42C2×#4112yes—
485C42C2×#4112yes—
486C42C2×#4112yes—
487C42C2×#4112yes—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23C2×#12412yes—
493C23C23×D411yescharacteristic
494Q8—44no—
495Q8—44no—
496Q8—84no—
497Q8—84no—
498Q8—84no—
499Q8—164no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—164no—
543C2×C4C2×#12412yes—
544C2×C4C2×#12412yes—
545C2×C4C2×#12412yes—
546C2×C4C2×#12412yes—
547C2×C4C2×#12412yes—
548C2×C4#57514yes—
549C2×C4C22×#4111yescharacteristic
550C2×C4C22×#4111yescharacteristic
551C2×C4C2×2−1+412yes—
552C2×C4#58512yes—
553C8—24no—
554C8—42no—
555C8—42no—
556C8—42no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22C22×#12411yescharacteristic
563C22#349911yescharacteristic
564C22#350011yescharacteristic
565C22C2×#57512yes—
566C22#358612yes—
567C4—24no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4#346112yes—
585C4#349512yes—
586C2—42no—
587C2#3531211yescharacteristic
588C2#6061811yescharacteristic
589C2#6062011yescharacteristic
590C2#6251112yes—
591C2#6258512yes—
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. #33296C2semidirect
4. #33293C2semidirect
5. #33281C2semidirect
6. #33281C2semidirect
7. #30234C2semidirect
8. #30233C2semidirect
9. #30215C2semidirect
10. #30214C2semidirect
11. #27818C2semidirect
12. #27734C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#351322
#332962
#332932
#332812
#302342
#302332
#302152
#302142
#278182
#277342
#277242
#277222
#218662
#218422
#214042
#213232
#204802