Finite Group Databasebeta

← #6495190

Subgroups — #6495190

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35335C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33234C211yesmaximal, characteristic, complemented
6#33234C211yesmaximal, characteristic, complemented
7#30211C214yesmaximal, complemented
8#29151C214yesmaximal, complemented
9#28725C211yesmaximal, characteristic, complemented
10#28724C211yesmaximal, characteristic, complemented
11#28723C211yesmaximal, characteristic, complemented
12#28722C211yesmaximal, characteristic, complemented
13#27984C214yesmaximal, complemented
14#25913C211yesmaximal, characteristic
15#25912C211yesmaximal, characteristic, complemented
16#25911C211yesmaximal, characteristic
17#25910C211yesmaximal, characteristic, complemented
18#21871C214yesmaximal, complemented
19#20465C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#533C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#491C2211yescharacteristic
24C2×#491C2211yescharacteristic
25C2×#491C2211yescharacteristic
26C2×#491C2211yescharacteristic
27C2×#490C2211yescharacteristic
28C2×#490C2211yescharacteristic
29C2×#490C2211yescharacteristic
30C2×#490C2211yescharacteristic
31C2×#433C2212yes—
32#2868C2211yescharacteristic
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2859C2211yescharacteristic
36#2859C2211yescharacteristic
37#2858C2211yescharacteristic
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2660C2212yes—
41#2660C2212yes—
42#2640C2212yes—
43#2620C2212yes—
44#2584—22no—
45#2584—22no—
46#2583—22no—
47#2583—22no—
48#2512—22no—
49#2491—22no—
50#2452—22no—
51#2422—22no—
52#2326C2212yes—
53#2317C2212yes—
54#2317C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#2074C2214yes—
59#2073C2214yes—
60#2072C2214yes—
61#2071C2214yes—
62#2026C2214yes—
63#2025C2214yes—
64#2024C2214yes—
65#2023C2214yes—
66#1993C2214yes—
67#1992C2211yescharacteristic
68#1992C2211yescharacteristic
69#1992C2211yescharacteristic
70#1992C2211yescharacteristic
71#1992C2211yescharacteristic
72#1992C2211yescharacteristic
73#1992C2211yescharacteristic
74#1992C2211yescharacteristic
75#1990C2214yes—
76#1959C2214yes—
77#1959C2214yes—
78#1958C2214yes—
79#1958C2214yes—
80#1875C2214yes—
81#1874C2214yes—
82#1873C2214yes—
83#1872C2214yes—
84#1836C2214yes—
85#1834C2214yes—
86#1805C2214yes—
87#1804C2214yes—
88#1803C2214yes—
89#1802C2214yes—
90#1768C2212yes—
91#1767C2211yescharacteristic
92#1767C2211yescharacteristic
93#1750C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1746C2212yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99#542—216no—
100#533—42no—
101#533—42no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#125—22no—
109C2×#125—42no—
110C2×#125—42no—
111C2×#125C2311yescharacteristic
112C2×#125C2311yescharacteristic
113C2×#125C2311yescharacteristic
114C2×#125C2311yescharacteristic
115C2×#125C2312yes—
116C2×#124—22no—
117C2×#124C2312yes—
118C2×#124C2312yes—
119C2×#123—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8C2312yes—
123C2×#121—22no—
124C2×#121C2312yes—
125C2×C4×Q8—22no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4C2312yes—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#491—42no—
136#491—42no—
137#491—42no—
138#491—42no—
139#490—42no—
140#490—42no—
141#490—42no—
142#490—42no—
143#433—28no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×#97C2312yes—
157C2×#97C2311yescharacteristic
158C2×#97C2311yescharacteristic
159C2×#97C2311yescharacteristic
160C2×#97C2311yescharacteristic
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#397—24no—
169#396—24no—
170#395—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#394—24no—
175#393—24no—
176#391C2312yes—
177#391C2312yes—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#388—24no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—24no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#384C2312yes—
210#384C2312yes—
211#384C2312yes—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#382C2312yes—
216#380—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222#376C2311yescharacteristic
223#376C2311yescharacteristic
224C43C2311yescharacteristic
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2411yescharacteristic
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—84no—
258#127—22no—
259#127—22no—
260#126—48no—
261#125—22no—
262#125—22no—
263#125—22no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125—22no—
269#125—42no—
270#125—42no—
271#125—48no—
272#124—28no—
273#124—28no—
274#124—48no—
275#123—48no—
276C22⋊Q8—28no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279#121—28no—
280#121—84no—
281C4×Q8—28no—
282C4×Q8—48no—
283C4×D4—28no—
284C4×D4—84no—
285#117—24no—
286#117—42no—
287#117—42no—
288#117#4118yes—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
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—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D411yescharacteristic
335C2×C22⋊C4C2×D411yescharacteristic
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C4⋊C8—24no—
352C4⋊C8—24no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#97—24no—
364#97—22no—
365#97—22no—
366#97—22no—
367#97—22no—
368#97—24no—
369C4×C8—24no—
370C4×C8—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
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—22no—
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—84no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—84no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—84no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4114yes—
433C22×C4C2511yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
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—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4#12414yes—
481C4⋊C4#12414yes—
482C22⋊C4—24no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4#12414yes—
493C22⋊C4#12414yes—
494C42—24no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—44no—
504C42C2×#4114yes—
505C42C2×#4114yes—
506C42H(5,𝔽2)12yes—
507C42H(5,𝔽2)12yes—
508C422−1+414yes—
509C23—22no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×#12412yes—
519C23C23×D411yescharacteristic
520Q8—44no—
521Q8—44no—
522Q8—88no—
523Q8—164no—
524D4—44no—
525D4—44no—
526D4—88no—
527D4—164no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—84no—
566C2×C4—88no—
567C2×C4—164no—
568C2×C4C2×#12412yes—
569C2×C4C2×#12412yes—
570C2×C4C2×#12412yes—
571C2×C4C2×#12412yes—
572C2×C4C2×#12412yes—
573C2×C4#53514yes—
574C2×C4#53614yes—
575C2×C4C22×#4112yes—
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4C2×2−1+412yes—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—82no—
593C22C22×#12411yescharacteristic
594C22C2×#53511yescharacteristic
595C22C2×#53611yescharacteristic
596C22#358512yes—
597C22#358612yes—
598C4—24no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—24no—
604C4—22no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#351112yes—
615C4#351112yes—
616C2—42no—
617C2—162no—
618C2#3528911yescharacteristic
619C2#6062611yescharacteristic
620C2#6062711yescharacteristic
621C2#6081012yes—
622C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. #35335C2semidirect
3. #33297C2semidirect
4. #33294C2semidirect
5. #33234C2semidirect
6. #33234C2semidirect
7. #30211C2semidirect
8. #29151C2semidirect
9. #28725C2semidirect
10. #28724C2semidirect
11. #28723C2semidirect
12. #28722C2semidirect
13. #27984C2semidirect
15. #25912C2semidirect
17. #25910C2semidirect
18. #21871C2semidirect
19. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353352
#352772
#332972
#332942
#332342
#302112
#291512
#287252
#287242
#287232
#287222
#279842
#259132
#259122
#259112
#259102
#218712
#204652