Finite Group Databasebeta

← #6490793

Subgroups — #6490793

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#33121C212yesmaximal, complemented
4#33120C212yesmaximal, complemented
5#30073C212yesmaximal, complemented
6#30071C212yesmaximal, complemented
7#29269C212yesmaximal, complemented
8#29268C212yesmaximal, complemented
9#29212C212yesmaximal, complemented
10#28753C212yesmaximal, complemented
11#27914C211yesmaximal, characteristic, complemented
12#27913C211yesmaximal, characteristic, complemented
13#25909C211yesmaximal, characteristic, complemented
14#25909C211yesmaximal, characteristic, complemented
15#24071C212yesmaximal, complemented
16#24069C212yesmaximal
17#22224C212yesmaximal, complemented
18#22223C212yesmaximal, complemented
19#20783C211yesmaximal, characteristic, complemented
20C2×#541C2211yescharacteristic
21C2×#540C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#484C2212yes—
26C2×#481C2212yes—
27C2×#483C2212yes—
28C2×#480C2212yes—
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2850C2211yescharacteristic
32#2850C2211yescharacteristic
33#2642C2211yescharacteristic
34#2641C2211yescharacteristic
35#2620C2211yescharacteristic
36#2620C2211yescharacteristic
37#2559—22no—
38#2559—22no—
39#2558—22no—
40#2558—22no—
41#2360—22no—
42#2360—22no—
43#2355—22no—
44Q8×C4⋊C4—22no—
45#2330C2212yes—
46#2330C2212yes—
47#2326C2211yescharacteristic
48#2326C2211yescharacteristic
49#2317C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×#124C2211yescharacteristic
53C4×#124C2211yescharacteristic
54C2×#384C2211yescharacteristic
55#2274C2212yes—
56C4×#117C2211yescharacteristic
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2059C2214yes—
62#2059C2214yes—
63#2043C2214yes—
64#2041C2214yes—
65#2026C2212yes—
66#2026C2212yes—
67#2025C2212yes—
68#2025C2212yes—
69#1993C2212yes—
70#1993C2212yes—
71#1983C2214yes—
72#1983C2214yes—
73#1959C2212yes—
74#1959C2212yes—
75#1948C2214yes—
76#1947C2214yes—
77#1928C2214yes—
78#1927C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1875C2212yes—
82#1875C2212yes—
83#1873C2212yes—
84#1873C2212yes—
85#1865C2214yes—
86#1864C2214yes—
87#1863C2214yes—
88#1862C2214yes—
89#1847C2212yes—
90#1847C2212yes—
91#1840C2212yes—
92#1832C2212yes—
93#1818C2212yes—
94#1798C2212yes—
95#1767C2212yes—
96#1749C2211yescharacteristic
97#1749C2211yescharacteristic
98#1745C2212yes—
99#541—44no—
100#540—44no—
101#533—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—21no—
106C2×#126—21no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#125C2312yes—
110C2×#125C2311yescharacteristic
111C2×#125C2311yescharacteristic
112C2×#124C2311yescharacteristic
113C2×#124C2311yescharacteristic
114C2×#124C2311yescharacteristic
115C2×#124C2311yescharacteristic
116C2×#123—21no—
117C2×#123—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C2311yescharacteristic
120C2×C22⋊Q8C2311yescharacteristic
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2311yescharacteristic
124C2×#121C2311yescharacteristic
125C2×C4×Q8—22no—
126C2×C4×Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138#489—28no—
139#489—28no—
140#484—44no—
141#483—44no—
142#481—44no—
143#480—44no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161#400—22no—
162#400—22no—
163#396—22no—
164#396—22no—
165#394—22no—
166#394—22no—
167#394—22no—
168#394—22no—
169#391—22no—
170#391—22no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#390—22no—
176#390—22no—
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—24no—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—24no—
196#387—24no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384—22no—
205#384C2312yes—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#382—22no—
210#382C2311yescharacteristic
211#382C2311yescharacteristic
212C4×C4⋊C4—24no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2311yescharacteristic
225#376C2311yescharacteristic
226C43C2311yescharacteristic
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255C4⋊Q8#4118yes—
256#127—42no—
257#126—44no—
258#126—44no—
259#125—28no—
260#125—42no—
261#125—42no—
262#125—44no—
263#125—44no—
264#124—24no—
265#124—24no—
266#124—24no—
267#124—24no—
268#123—44no—
269#123—44no—
270C22⋊Q8—216no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273#121—24no—
274#121—24no—
275#121—44no—
276#121—44no—
277C4×Q8—216no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—84no—
281C4×Q8—84no—
282C4×D4—42no—
283C4×D4—42no—
284#117—24no—
285#117—28no—
286#117—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D411yescharacteristic
335C2×C22⋊C4C2×D411yescharacteristic
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#107—44no—
356#106—44no—
357#106—44no—
358C4⋊C8—28no—
359#103—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#102—28no—
364#102—28no—
365#102—44no—
366#102—44no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370#97—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4C2511yescharacteristic
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4C2×#4118yes—
478C22⋊C4—28no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—28no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—84no—
509C42—84no—
510C42C2×#4114yes—
511C422−1+414yes—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23#53512yes—
527C23C23×D411yescharacteristic
528Q8—416no—
529Q8—44no—
530Q8—44no—
531D4—44no—
532D4—44no—
533D4—162no—
534D4—162no—
535D4—162no—
536D4—162no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—162no—
580C2×C4—162no—
581C2×C4—164no—
582C2×C4#52514yes—
583C2×C4#53512yes—
584C2×C4#53614yes—
585C2×C4C22×#4112yes—
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4C2×2−1+412yes—
588C8—44no—
589C8—44no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—82no—
605C22C2×#52511yescharacteristic
606C22C2×#53511yescharacteristic
607C22C2×#53611yescharacteristic
608C22#358512yes—
609C22#358612yes—
610C4—24no—
611C4—22no—
612C4—24no—
613C4—42no—
614C4—48no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—84no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4#352214yes—
628C2—42no—
629C2—161no—
630C2—161no—
631C2#3524911yescharacteristic
632C2#6056512yes—
633C2#6082511yescharacteristic
634C2#6082611yescharacteristic
635C2#6088712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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. #34214C2semidirect
3. #33121C2semidirect
4. #33120C2semidirect
5. #30073C2semidirect
6. #30071C2semidirect
7. #29269C2semidirect
8. #29268C2semidirect
9. #29212C2semidirect
10. #28753C2semidirect
11. #27914C2semidirect
12. #27913C2semidirect
13. #25909C2semidirect
14. #25909C2semidirect
15. #24071C2semidirect
17. #22224C2semidirect
18. #22223C2semidirect
19. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#342142
#331212
#331202
#300732
#300712
#292692
#292682
#292122
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