Finite Group Databasebeta

← #6611551

Subgroups — #6611551

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36182C211yesmaximal, characteristic
2#36131C211yesmaximal, characteristic, complemented
3#33295C211yesmaximal, characteristic, complemented
4#33052C211yesmaximal, characteristic, complemented
5#33052C211yesmaximal, characteristic, complemented
6#29227C211yesmaximal, characteristic, complemented
7#28722C211yesmaximal, characteristic, complemented
8#27967C211yesmaximal, characteristic, complemented
9#27965C211yesmaximal, characteristic, complemented
10#27819C214yesmaximal, complemented
11#27718C214yesmaximal, complemented
12#27711C214yesmaximal, complemented
13#25913C211yesmaximal, characteristic, complemented
14#25911C211yesmaximal, characteristic, complemented
15#22488C214yesmaximal, complemented
16C4×#490C211yesmaximal, characteristic, complemented
17#21321C211yesmaximal, characteristic, complemented
18C2×#1992C211yesmaximal, characteristic, complemented
19D4×#97C211yesmaximal, characteristic, complemented
20C2×#553C2212yes—
21C2×#533C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#490C2211yescharacteristic
24C2×#490C2211yescharacteristic
25C2×#490C2211yescharacteristic
26C2×C8×D4C2212yes—
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2856C2211yescharacteristic
32#2856C2211yescharacteristic
33#2851C2211yescharacteristic
34#2851C2211yescharacteristic
35#2566C2212yes—
36#2556—22no—
37#2550C2212yes—
38#2512—22no—
39#2504C2212yes—
40#2500C2212yes—
41#2462—22no—
42#2449—22no—
43#2441C2212yes—
44#2440C2212yes—
45#2422—22no—
46#2416—22no—
47#2361—22no—
48#2360—22no—
49#2352C2212yes—
50C4×#127C2211yescharacteristic
51C4×#125C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2099C2214yes—
58#2071C2214yes—
59#2025C2214yes—
60#2023C2214yes—
61#1993C2214yes—
62#1992C2211yescharacteristic
63#1992C2211yescharacteristic
64#1992C2211yescharacteristic
65#1992C2211yescharacteristic
66#1992C2211yescharacteristic
67#1992C2211yescharacteristic
68#1992C2211yescharacteristic
69#1992C2211yescharacteristic
70#1992C2214yes—
71#1992C2211yescharacteristic
72#1991C2214yes—
73#1989C2214yes—
74#1961C2211yescharacteristic
75#1936C2214yes—
76#1936C2214yes—
77#1936C2214yes—
78#1893C2214yes—
79#1893C2214yes—
80#1875C2214yes—
81#1874C2214yes—
82#1836C2214yes—
83#1836C2211yescharacteristic
84#1836C2211yescharacteristic
85#1836C2211yescharacteristic
86#1825C2214yes—
87#1824C2214yes—
88#1804C2214yes—
89#1770C2212yes—
90#1767C2211yescharacteristic
91C4×#102C2214yes—
92#1752C2212yes—
93#1752C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1735C2214yes—
97C2×#340C2214yes—
98C4×#97C2211yescharacteristic
99C23×C8C2312yes—
100C23×C8C2312yes—
101#553—216no—
102#533—24no—
103#533—42no—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—22no—
107C2×#125—24no—
108C2×#125—22no—
109C2×#125—42no—
110C2×#125C2312yes—
111C2×#125C2311yescharacteristic
112C2×#125C2311yescharacteristic
113C2×#125C2312yes—
114C2×#125C2311yescharacteristic
115C2×#123—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2312yes—
122C2×#121C2312yes—
123C2×C4×Q8—22no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4D412yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#490—24no—
138#490—42no—
139#490—42no—
140C8×D4—28no—
141C2×C4⋊C8C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97—42no—
149C2×#97—42no—
150C2×#97C2312yes—
151C2×#97C2311yescharacteristic
152C2×#97C2311yescharacteristic
153C2×#97C2311yescharacteristic
154C2×#97C2311yescharacteristic
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#399—22no—
159#398—22no—
160#398—22no—
161#397—24no—
162#396—24no—
163#396—22no—
164#395—22no—
165#395—22no—
166#394—24no—
167#391—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—24no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#386—22no—
191#386—24no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—24no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382C2311yescharacteristic
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2311yescharacteristic
218C4×C22⋊C4C2311yescharacteristic
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—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—42no—
260#127—22no—
261#127—22no—
262#126—48no—
263#125—24no—
264#125—22no—
265#125—22no—
266#125—28no—
267#125—22no—
268#125—22no—
269#125—22no—
270#125—42no—
271#125—48no—
272#125#4114yes—
273#123—48no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276#121—28no—
277#121—28no—
278#121—48no—
279#121—48no—
280#121—84no—
281C4×Q8—48no—
282C4×D4—28no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—84no—
286C4×D4#4118yes—
287#117—28no—
288#117—24no—
289#117—42no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
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—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D411yescharacteristic
336C2×C22⋊C4C2×D411yescharacteristic
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C4⋊C8—24no—
354#102—28no—
355#102—28no—
356#102—44no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360#97—24no—
361#97—22no—
362#97—22no—
363#97—22no—
364#97—22no—
365#97—42no—
366#97—42no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D411yescharacteristic
384C24C22×D411yescharacteristic
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—84no—
400C2×D4#12414yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—84no—
425C22×C4—82no—
426C22×C4—84no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2511yescharacteristic
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—48no—
450C2×C8—48no—
451C2×C82−1+414yes—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4#12414yes—
482C4⋊C4C2×#4114yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—82no—
495C22⋊C4#12414yes—
496C22⋊C4#12414yes—
497C42—24no—
498C42—24no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—44no—
507C42—44no—
508C42—82no—
509C42C2×#4114yes—
510C42C2×#4112yes—
511C42H(5,𝔽2)12yes—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23C2×#12412yes—
525C23C2×#12412yes—
526C23#57514yes—
527C23C23×D411yescharacteristic
528Q8—88no—
529D4—28no—
530D4—44no—
531D4—44no—
532D4—88no—
533D4—88no—
534D4—164no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—88no—
577C2×C4—164no—
578C2×C4—164no—
579C2×C4C2×#12412yes—
580C2×C4C2×#12412yes—
581C2×C4C2×#12412yes—
582C2×C4C2×#12412yes—
583C2×C4#54614yes—
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4112yes—
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×2−1+412yes—
588C8—24no—
589C8—42no—
590C8—42no—
591C8—42no—
592C22—24no—
593C22—22no—
594C22—22no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22#313514yes—
607C22C22×#12411yescharacteristic
608C22C2×#54611yescharacteristic
609C22#349911yescharacteristic
610C22C2×#57512yes—
611C22#358512yes—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—24no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#348212yes—
628C4#352012yes—
629C2—24no—
630C2—42no—
631C2—162no—
632C2#3630311yescharacteristic
633C2C2×#313511yescharacteristic
634C2#6062611yescharacteristic
635C2#6116412yes—
636C2#6250612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36182C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36131C2semidirect
3. #33295C2semidirect
4. #33052C2semidirect
5. #33052C2semidirect
6. #29227C2semidirect
7. #28722C2semidirect
8. #27967C2semidirect
9. #27965C2semidirect
10. #27819C2semidirect
11. #27718C2semidirect
12. #27711C2semidirect
13. #25913C2semidirect
14. #25911C2semidirect
15. #22488C2semidirect
16. C4×#490C2semidirect
17. #21321C2semidirect
18. C2×#1992C2semidirect
19. D4×#97C2semidirect

Maximal subgroups

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

subgroupindex
#361822
#361312
#332952
#330522
#292272
#287222
#279672
#279652
#278192
#277182
#277112
#259132
#259112
#224882
C4×#4902
#213212
C2×#19922
D4×#972