Finite Group Databasebeta

← #6497398

Subgroups — #6497398

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#35202C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33280C211yesmaximal, characteristic, complemented
6#33280C211yesmaximal, characteristic, complemented
7#30234C211yesmaximal, characteristic, complemented
8#30233C211yesmaximal, characteristic, complemented
9#30212C211yesmaximal, characteristic, complemented
10#30212C211yesmaximal, characteristic, complemented
11#27829C214yesmaximal, complemented
12#27739C214yesmaximal, complemented
13#27718C214yesmaximal, complemented
14#27702C214yesmaximal, complemented
15#21869C211yesmaximal, characteristic
16#21842C211yesmaximal, characteristic, complemented
17#21408C211yesmaximal, characteristic
18#21321C211yesmaximal, characteristic, complemented
19#20479C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2868C2211yescharacteristic
25#2864C2211yescharacteristic
26#2864C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2667—22no—
33#2620—22no—
34#2606—22no—
35#2566C2212yes—
36#2566C2212yes—
37#2552C2212yes—
38#2552C2212yes—
39#2545—22no—
40#2500C2212yes—
41#2500C2212yes—
42#2441—22no—
43#2441C2212yes—
44#2441C2212yes—
45#2432—22no—
46#2330—22no—
47#2326—22no—
48#2285C2211yescharacteristic
49#2283C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2099C2214yes—
55#2071C2214yes—
56#2071C2214yes—
57#2025C2214yes—
58#2025C2214yes—
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#1961C2211yescharacteristic
76#1961C2211yescharacteristic
77#1961C2211yescharacteristic
78#1961C2211yescharacteristic
79#1936C2214yes—
80#1936C2214yes—
81#1934C2214yes—
82#1934C2214yes—
83#1893C2214yes—
84#1875C2214yes—
85#1836C2214yes—
86#1834C2214yes—
87#1825C2214yes—
88#1802C2214yes—
89#1771C2212yes—
90#1771C2212yes—
91#1762C2214yes—
92#1752C2212yes—
93C8×C22⋊C4C2212yes—
94#1745C2211yescharacteristic
95#1745C2211yescharacteristic
96#1735C2214yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C22×#41—21no—
100C23×C8C2312yes—
101#533—24no—
102#533—24no—
103#532—24no—
104#532—24no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—22no—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×#125C2312yes—
114C2×#124—22no—
115C2×#124—21no—
116C2×#124—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×#121—21no—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×#117D412yes—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C42C2311yescharacteristic
129#432—24no—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×#97—24no—
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×#97C2312yes—
141C2×#97C2311yescharacteristic
142C2×#97C2311yescharacteristic
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146#399—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#396—24no—
154#396—22no—
155#396—22no—
156#394—24no—
157#394—24no—
158#391—22no—
159#391—22no—
160#390—22no—
161#390—22no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—24no—
179#388—24no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—22no—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#383—21no—
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—
236C2×#41—44no—
237C2×#41—84no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
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—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—84no—
263#127—22no—
264#127—22no—
265#125—24no—
266#125—24no—
267#125—44no—
268#125—44no—
269#125—48no—
270#124—48no—
271#124—44no—
272#124—84no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275#121—24no—
276#121—24no—
277#121—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283#117—24no—
284#117—24no—
285#117—44no—
286#117—44no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
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—22no—
300C2×C4⋊C4—22no—
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—22no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42#4114yes—
338C2×C42#4114yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C4⋊C8—24no—
346#102—28no—
347#102—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351#97—24no—
352#97—24no—
353#97—22no—
354#97—22no—
355#97—22no—
356#97—22no—
357#97—42no—
358C4×C8—24no—
359C4×C8—42no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D411yescharacteristic
382#41—816no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—84no—
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—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—84no—
412C22×C4#12414yes—
413C22×C4#12414yes—
414C22×C4C22×D412yes—
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×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2511yescharacteristic
426C2×C8—22no—
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—48no—
438C2×C82−1+414yes—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—44no—
492C42—44no—
493C42—82no—
494C42#12414yes—
495C42#12414yes—
496C42C2×#4112yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C42C2×#4112yes—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23C2×#12412yes—
510C23C23×D411yescharacteristic
511Q8—84no—
512Q8—84no—
513Q8—84no—
514D4—44no—
515D4—44no—
516D4—84no—
517D4—84no—
518D4—88no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—88no—
559C2×C4C2×#12412yes—
560C2×C4C2×#12412yes—
561C2×C4C2×#12412yes—
562C2×C4C2×#12412yes—
563C2×C4C2×#12412yes—
564C2×C4#57514yes—
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4111yescharacteristic
567C2×C4C2×2−1+412yes—
568C2×C4#58512yes—
569C8—24no—
570C8—42no—
571C8—42no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—84no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22C22×#12411yescharacteristic
584C22#349911yescharacteristic
585C22#350011yescharacteristic
586C22C2×#57512yes—
587C22#358512yes—
588C4—24no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4#346112yes—
605C4#349512yes—
606C2—42no—
607C2—162no—
608C2#3531211yescharacteristic
609C2#6061811yescharacteristic
610C2#6061911yescharacteristic
611C2#6250612yes—
612C2#6258612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. #35202C2semidirect
3. #33297C2semidirect
4. #33296C2semidirect
5. #33280C2semidirect
6. #33280C2semidirect
7. #30234C2semidirect
8. #30233C2semidirect
9. #30212C2semidirect
10. #30212C2semidirect
11. #27829C2semidirect
12. #27739C2semidirect
13. #27718C2semidirect
14. #27702C2semidirect
16. #21842C2semidirect
18. #21321C2semidirect
19. #20479C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#352022
#332972
#332962
#332802
#302342
#302332
#302122
#278292
#277392
#277182
#277022
#218692
#218422
#214082
#213212
#204792