Finite Group Databasebeta

← #6497397

Subgroups — #6497397

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic
2#35202C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33281C211yesmaximal, characteristic, complemented
6#33281C211yesmaximal, characteristic, complemented
7#30234C211yesmaximal, characteristic, complemented
8#30233C211yesmaximal, characteristic, complemented
9#30214C211yesmaximal, characteristic, complemented
10#30214C211yesmaximal, characteristic, complemented
11#27831C214yesmaximal, complemented
12#27741C214yesmaximal, complemented
13#27716C214yesmaximal
14#27700C214yesmaximal
15#21870C211yesmaximal, characteristic
16#21842C211yesmaximal, characteristic
17#21407C211yesmaximal, characteristic
18#21321C211yesmaximal, characteristic
19#20479C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2868C2211yescharacteristic
25#2864C2211yescharacteristic
26#2864C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2667—22no—
33#2621—22no—
34#2606—22no—
35#2565—22no—
36#2565C2212yes—
37#2565C2212yes—
38#2552C2212yes—
39#2552C2212yes—
40#2499—22no—
41#2499C2212yes—
42#2499C2212yes—
43#2441—22no—
44#2441C2212yes—
45#2441C2212yes—
46#2330—22no—
47#2327—22no—
48#2285C2211yescharacteristic
49#2283C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2099C2214yes—
55#2072C2214yes—
56#2072C2214yes—
57#2026C2214yes—
58#2026C2214yes—
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#1992C2214yes—
72#1992C2214yes—
73#1991C2214yes—
74#1989C2214yes—
75#1960C2211yescharacteristic
76#1960C2211yescharacteristic
77#1960C2211yescharacteristic
78#1960C2211yescharacteristic
79#1935C2214yes—
80#1935C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1892C2214yes—
84#1873C2214yes—
85#1836C2214yes—
86#1834C2214yes—
87#1825C2214yes—
88#1803C2214yes—
89#1771C2212yes—
90#1771C2212yes—
91#1763C2214yes—
92#1752C2212yes—
93C8×C22⋊C4C2212yes—
94#1745C2211yescharacteristic
95#1745C2211yescharacteristic
96#1735C2214yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C23×C8C2312yes—
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×#125—22no—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×#125C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
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—
125C2×#117C2312yes—
126C22×C42C2311yescharacteristic
127#432—24no—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×C22⋊C8C2312yes—
134C2×#97—24no—
135C2×#97C2312yes—
136C2×#97C2311yescharacteristic
137C2×#97C2311yescharacteristic
138C2×#97C2312yes—
139C2×#97C2311yescharacteristic
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#399—22no—
145#399—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#396—22no—
155#396—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—22no—
185#387—22no—
186#384—22no—
187#384—22no—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#383—21no—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#382C2312yes—
198#382C2311yescharacteristic
199#382C2311yescharacteristic
200#382C2311yescharacteristic
201#382C2311yescharacteristic
202#381—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2311yescharacteristic
215C4×C22⋊C4C2311yescharacteristic
216#376—21no—
217#376D412yes—
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×Q8C2412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—42no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—48no—
259#125—24no—
260#125—24no—
261#125—44no—
262#125—44no—
263#125—48no—
264#125—44no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275#117—24no—
276#117—24no—
277#117—44no—
278#117—44no—
279#117—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
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—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—22no—
321C2×C42—22no—
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—22no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
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—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4#12414yes—
401C22×C4#12414yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2511yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—48no—
426C2×C82−1+414yes—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—44no—
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—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
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—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
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×C4C2×#12412yes—
542C2×C4C2×#12412yes—
543C2×C4C2×#12412yes—
544C2×C4C2×#12412yes—
545C2×C4C2×#12412yes—
546C2×C4#57514yes—
547C2×C4C22×#4111yescharacteristic
548C2×C4C22×#4111yescharacteristic
549C2×C4C2×2−1+412yes—
550C2×C4#58512yes—
551C8—24no—
552C8—42no—
553C8—42no—
554C8—42no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—42no—
559C22—42no—
560C22C22×#12411yescharacteristic
561C22#349911yescharacteristic
562C22#350011yescharacteristic
563C22C2×#57512yes—
564C22#358612yes—
565C4—24no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#346112yes—
583C4#349512yes—
584C2—42no—
585C2#3531211yescharacteristic
586C2#6061811yescharacteristic
587C2#6062011yescharacteristic
588C2#6250512yes—
589C2#6258712yes—
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. #33296C2semidirect
5. #33281C2semidirect
6. #33281C2semidirect
7. #30234C2semidirect
8. #30233C2semidirect
9. #30214C2semidirect
10. #30214C2semidirect
11. #27831C2semidirect
12. #27741C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#352022
#332972
#332962
#332812
#302342
#302332
#302142
#278312
#277412
#277162
#277002
#218702
#218422
#214072
#213212
#204792