Finite Group Databasebeta

← #6490787

Subgroups — #6490787

Order 512.

1586 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#35331C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#33119C212yesmaximal, complemented
4#33118C212yesmaximal
5#30074C212yesmaximal, complemented
6#30072C212yesmaximal, complemented
7#29266C212yesmaximal, complemented
8#29265C212yesmaximal, complemented
9#29211C212yesmaximal, complemented
10#28754C212yesmaximal, complemented
11#27914C211yesmaximal, characteristic, complemented
12#27913C211yesmaximal, characteristic, complemented
13#25909C211yesmaximal, characteristic, complemented
14#25909C211yesmaximal, characteristic, complemented
15#24074C212yesmaximal, complemented
16#24073C212yesmaximal, complemented
17#22224C212yesmaximal, complemented
18#22223C212yesmaximal, complemented
19#20783C211yesmaximal, characteristic, complemented
20C2×#542C2211yescharacteristic
21C2×#542C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#484C2212yes—
26C2×#481C2212yes—
27C2×#483C2212yes—
28C2×#480C2212yes—
29#2862C2211yescharacteristic
30#2858C2211yescharacteristic
31#2850C2211yescharacteristic
32#2850C2211yescharacteristic
33#2640C2211yescharacteristic
34#2640C2211yescharacteristic
35#2621C2211yescharacteristic
36#2619C2211yescharacteristic
37#2560—22no—
38#2560—22no—
39#2557—22no—
40#2557—22no—
41#2360—22no—
42#2360—22no—
43#2355—22no—
44Q8×C4⋊C4—22no—
45#2330C2212yes—
46#2330C2212yes—
47#2328C2211yescharacteristic
48#2327C2211yescharacteristic
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#2058C2214yes—
62#2057C2214yes—
63#2042C2214yes—
64#2042C2214yes—
65#2026C2212yes—
66#2026C2212yes—
67#2025C2212yes—
68#2025C2212yes—
69#1993C2212yes—
70#1993C2212yes—
71#1982C2214yes—
72#1981C2214yes—
73#1959C2212yes—
74#1959C2212yes—
75#1949C2214yes—
76#1949C2214yes—
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#542—44no—
100#542—44no—
101#533—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—21no—
108C2×#126—21no—
109C2×#125C2312yes—
110C2×#124—21no—
111C2×#124—21no—
112C2×#124C2311yescharacteristic
113C2×#124C2311yescharacteristic
114C2×#124C2311yescharacteristic
115C2×#124C2311yescharacteristic
116C2×#123—21no—
117C2×#123—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2311yescharacteristic
122C2×C22⋊Q8C2311yescharacteristic
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#398—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#394—22no—
166#393—22no—
167#393—22no—
168#392—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#391C2311yescharacteristic
176#391C2311yescharacteristic
177#390—22no—
178#390—22no—
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#390C2311yescharacteristic
182#390C2311yescharacteristic
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#387—22no—
194#387—22no—
195#387—24no—
196#387—24no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384—22no—
205#384C2312yes—
206#384C2312yes—
207#382—22no—
208#382C2311yescharacteristic
209#382C2311yescharacteristic
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×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4C2411yescharacteristic
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8#4118yes—
259#127—24no—
260#126—44no—
261#126—44no—
262#125—28no—
263#124—24no—
264#124—24no—
265#124—42no—
266#124—42no—
267#124—44no—
268#124—44no—
269#123—44no—
270#123—44no—
271C22⋊Q8—216no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—24no—
277#121—24no—
278C4×Q8—216no—
279C4×Q8—42no—
280C4×Q8—42no—
281C4×Q8—84no—
282C4×Q8—84no—
283C4×D4—42no—
284C4×D4—42no—
285#117—24no—
286#117—28no—
287#117—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
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—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—84no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2414yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
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⋊C4C2×D411yescharacteristic
333C2×C22⋊C4C2×D411yescharacteristic
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#107—44no—
354#106—44no—
355#106—44no—
356C4⋊C8—28no—
357#103—28no—
358#103—28no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367#97—24no—
368#97—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
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×Q8—42no—
398C2×Q8—42no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—82no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4C2511yescharacteristic
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4C2×#4118yes—
476C22⋊C4—28no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
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—
531Q8—162no—
532Q8—162no—
533D4—44no—
534D4—44no—
535D4—164no—
536D4—162no—
537D4—162no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
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—22no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—162no—
581C2×C4—162no—
582C2×C4—164no—
583C2×C4#52514yes—
584C2×C4#53512yes—
585C2×C4#53614yes—
586C2×C4C22×#4112yes—
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4C2×2−1+412yes—
589C8—44no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—82no—
605C22—82no—
606C22C2×#52511yescharacteristic
607C22C2×#53511yescharacteristic
608C22C2×#53611yescharacteristic
609C22#358512yes—
610C22#358612yes—
611C4—24no—
612C4—22no—
613C4—24no—
614C4—42no—
615C4—48no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—84no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4#352214yes—
629C2—42no—
630C2—161no—
631C2—161no—
632C2#3524911yescharacteristic
633C2#6056512yes—
634C2#6082511yescharacteristic
635C2#6082611yescharacteristic
636C2#6088512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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. #33119C2semidirect
5. #30074C2semidirect
6. #30072C2semidirect
7. #29266C2semidirect
8. #29265C2semidirect
9. #29211C2semidirect
10. #28754C2semidirect
11. #27914C2semidirect
12. #27913C2semidirect
13. #25909C2semidirect
14. #25909C2semidirect
15. #24074C2semidirect
16. #24073C2semidirect
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
#353312
#342142
#331192
#331182
#300742
#300722
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