Finite Group Databasebeta

← #6389564

Subgroups — #6389564

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60688C211yesmaximal, characteristic
2#60268C212yesmaximal, complemented
3#23019C214yesmaximal, complemented
4#23018C214yesmaximal, complemented
5#22804C214yesmaximal, complemented
6#22687C214yesmaximal, complemented
7#22235C212yesmaximal, complemented
8#22235C212yesmaximal, complemented
9#21857C212yesmaximal, complemented
10#21737C211yesmaximal, characteristic, complemented
11#21735C211yesmaximal, characteristic, complemented
12#21709C211yesmaximal, characteristic, complemented
13#21708C211yesmaximal, characteristic, complemented
14C2×#1849C211yesmaximal, characteristic, complemented
15#20481C211yesmaximal, characteristic, complemented
16#3444C2212yes—
17#3181C414yescomplemented
18#3181C414yescomplemented
19#3180C414yescomplemented
20#3180C414yescomplemented
21#3162C2211yescharacteristic
22#3150C2211yescharacteristic
23#3135C2212yes—
24#3050C2214yes—
25#2994—22no—
26#2994—22no—
27#2993—22no—
28#2993—22no—
29#2991—22no—
30#2990—22no—
31#2989—22no—
32#2988—22no—
33#2960C2212yes—
34#2960C2212yes—
35C2×#444C2214yes—
36C2×#443C2214yes—
37#2943C2214yes—
38#2933C2214yes—
39#2933C2214yes—
40#2931C2212yes—
41#2931C2212yes—
42#2925C2214yes—
43C2×#431C2211yescharacteristic
44C22×#108C2214yes—
45#2906C2214yes—
46C2×#423C2212yes—
47C22×#97C2211yescharacteristic
48#2283C2211yescharacteristic
49#1995C2212yes—
50#1995C2212yes—
51#1975C2214yes—
52#1974C2214yes—
53#1963C2212yes—
54#1963C2212yes—
55#1946C2214yes—
56#1945C2214yes—
57#1929C2218yes—
58#1929C2218yes—
59#1871C2218yes—
60#1866C2218yes—
61#1866C2218yes—
62#1849C2214yes—
63#1849C2211yescharacteristic
64#1849C2211yescharacteristic
65#1849C2212yes—
66#1806C2218yes—
67#1773C2214yes—
68#1772C2214yes—
69#1771C2212yes—
70#1771C2212yes—
71#1752C2212yes—
72C8×C22⋊C4C2212yes—
73#1746C2211yescharacteristic
74#1746C2211yescharacteristic
75#1726C2211yescharacteristic
76#1726C2211yescharacteristic
77C2×#135C2314yes—
78C2×#131C2314yes—
79C2×#131C2314yes—
80C22×M16C2314yes—
81C23×C8C2312yes—
82#533C2×C412yes—
83#533C2×C412yes—
84#533C2312yes—
85#532C2×C412yes—
86#532C2×C412yes—
87#532C2312yes—
88#529—22no—
89#529—22no—
90#529—22no—
91#529—22no—
92#529C2×C414yes—
93#529C2×C414yes—
94#525—22no—
95#525—22no—
96#525C2314yes—
97C4×#41—22no—
98C4×#41—22no—
99C4×#41C2314yes—
100C22×C42C2311yescharacteristic
101#501C2×C414yes—
102#501C2×C414yes—
103#497C2×C414yes—
104#496C2×C414yes—
105#491C2×C414yes—
106#490C2×C414yes—
107#489C2×C414yes—
108#489C2×C414yes—
109#467C2×C414yes—
110#467C2×C414yes—
111#466C2×C414yes—
112#466C2×C414yes—
113#465C2×C418yes—
114#465C2×C418yes—
115#446—22no—
116#446—22no—
117#446—22no—
118#446—22no—
119C8×Q8—22no—
120C8×Q8—22no—
121C8×Q8—22no—
122C8×Q8—22no—
123#444—48no—
124#443—48no—
125#442C2×C418yes—
126#441C2×C418yes—
127#440C2×C418yes—
128#440C2×C418yes—
129#436—24no—
130#436—24no—
131#436—22no—
132#436—22no—
133#436—22no—
134#436—22no—
135#435—24no—
136#435—24no—
137#435—24no—
138#435—24no—
139C8×D4—24no—
140C8×D4—24no—
141C8×D4—22no—
142C8×D4—22no—
143C8×D4—22no—
144C8×D4—22no—
145#433—24no—
146#433—24no—
147#432—24no—
148#431—24no—
149#431D412yes—
150#431D412yes—
151#431D412yes—
152#431D412yes—
153C2×#108—48no—
154C2×#108C2314yes—
155C2×#108C2314yes—
156C2×#108C2318yes—
157#427C2314yes—
158#424—24no—
159#424—24no—
160#424C2314yes—
161#424C2314yes—
162#423—22no—
163#423—22no—
164#423—24no—
165#423C2314yes—
166#423C2312yes—
167#423C2312yes—
168C2×C4⋊C8—22no—
169C2×C4⋊C8—22no—
170C2×C4⋊C8—24no—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×#104C2318yes—
174C2×#104C2318yes—
175#416C2318yes—
176#408C2314yes—
177#408C2314yes—
178C2×C22⋊C8C2311yescharacteristic
179C2×C22⋊C8C2311yescharacteristic
180#405C2314yes—
181#405C2314yes—
182C2×#97C2×C414yes—
183C2×#97C2×C414yes—
184C2×#97C2312yes—
185C2×#97C2312yes—
186C2×#97C2311yescharacteristic
187C2×#97C2311yescharacteristic
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190C2×C4×C8C2312yes—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193#383—21no—
194#382—21no—
195#381—21no—
196#380—21no—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4C2312yes—
199#376C2311yescharacteristic
200#376C2311yescharacteristic
201#339C2318yes—
202#339C2318yes—
203#336C2312yes—
204#336C2312yes—
205#336C2312yes—
206#336C2312yes—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41C2414yes—
210C23×C4#4111yescharacteristic
211C23×C4#4111yescharacteristic
212C23×C4C2411yescharacteristic
213#135—28no—
214#135—28no—
215C2×Q16C22×C414yes—
216C2×SD16C22×C418yes—
217C2×D8C22×C414yes—
218#131—48no—
219#131—48no—
220C2×M16—24no—
221C2×M16—24no—
222C2×M16—24no—
223C2×M16—48no—
224C2×M16C2414yes—
225C2×M16C2414yes—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C22×C414yes—
232C22×C8C22×C414yes—
233C22×C8C2412yes—
234C22×C8C2411yescharacteristic
235C22×C8C2411yescharacteristic
236C22×C8C2411yescharacteristic
237C22×C8C2411yescharacteristic
238C22×C8C2412yes—
239C4⋊Q8C22×C412yes—
240C4⋊Q8C22×C412yes—
241C4⋊Q8C22×C412yes—
242#127C22×C412yes—
243#126—24no—
244#126—24no—
245#125—22no—
246#125—22no—
247#125C22×C414yes—
248#124—22no—
249#124—22no—
250#124C22×C414yes—
251#123—24no—
252#123—24no—
253#123—24no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8C22×C414yes—
259C22⋊Q8C22×C414yes—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—22no—
264#121C22×C414yes—
265#121C22×C414yes—
266#120—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8C22×C414yes—
272C4×Q8C22×C414yes—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4C22×C414yes—
284C4×D4C22×C414yes—
285#117—22no—
286#117—22no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117C2414yes—
291C2×C4⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C42—24no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42C2412yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#108—28no—
305#108—28no—
306#108—48no—
307#107C22×C414yes—
308#107C22×C414yes—
309#106C22×C418yes—
310C4⋊C8—24no—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8C2×D412yes—
321C4⋊C8C2×D412yes—
322C4⋊C8C2×D412yes—
323C4⋊C8C2×D412yes—
324#104—48no—
325#104—48no—
326#103C22×C418yes—
327#103C22×C418yes—
328#102C22×C418yes—
329#102C22×C418yes—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8C2×D411yescharacteristic
335C22⋊C8C2×D412yes—
336C22⋊C8C2×D411yescharacteristic
337C22⋊C8C2×D411yescharacteristic
338C22⋊C8C2×D412yes—
339C22⋊C8C2×D411yescharacteristic
340#97—24no—
341#97—24no—
342#97—24no—
343#97—24no—
344#97C22×C418yes—
345#97C22×C414yes—
346#97C22×C414yes—
347C4×C8—24no—
348C4×C8—24no—
349C4×C8—24no—
350C4×C8—24no—
351C4×C8C2×D412yes—
352C4×C8C2×D412yes—
353C4×C8C2×D412yes—
354C4×C8C2×D412yes—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361C24C2×#4111yescharacteristic
362#41—28no—
363#41—28no—
364#41—48no—
365#41—84no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8C23×C414yes—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4C23×C414yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2511yescharacteristic
398Q16—28no—
399SD16—216no—
400D8—28no—
401M16—28no—
402M16—28no—
403M16—84no—
404M16—84no—
405M16—84no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—28no—
413C2×C8—28no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—84no—
417C2×C8—84no—
418C2×C8—84no—
419C2×C8C23×C414yes—
420C2×C8C23×C414yes—
421C2×C8C22×D411yescharacteristic
422C2×C8C22×D412yes—
423C2×C8C22×D411yescharacteristic
424C2×C8C22×D411yescharacteristic
425C2×C8C22×D412yes—
426C2×C8C22×D411yescharacteristic
427C2×C8H(5,𝔽2)14yes—
428C2×C82−1+414yes—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—82no—
440C4⋊C4C23×C414yes—
441C4⋊C4C23×C414yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42C4×D412yes—
460C42C4×D412yes—
461C42C4×D412yes—
462C42C4×D412yes—
463C42C23×C412yes—
464C42C23×C412yes—
465C42C22×D412yes—
466C42C22×D412yes—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—41no—
471C23C2×C4×D411yescharacteristic
472C23C2×C4×D411yescharacteristic
473C23C22×#4111yescharacteristic
474Q8—28no—
475Q8—44no—
476Q8—44no—
477D4—28no—
478D4—44no—
479D4—44no—
480D4—48no—
481D4—48no—
482D4—84no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4C2×C4×D412yes—
512C2×C4C2×C4×D412yes—
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×D412yes—
516C2×C4C2×C4×D412yes—
517C2×C4C24×C411yescharacteristic
518C2×C4C23×D411yescharacteristic
519C2×C4C2×H(5,𝔽2)12yes—
520C2×C4C2×2−1+412yes—
521C8—24no—
522C8—21no—
523C8—22no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—22no—
528C8—44no—
529C8—44no—
530C8#52018yes—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22#295512yes—
543C22C22×C4×D411yescharacteristic
544C22#351112yes—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—82no—
556C4C2×#52012yes—
557C4#349811yescharacteristic
558C4#349911yescharacteristic
559C2—22no—
560C2—22no—
561C2—41no—
562C2—84no—
563C2#3420711yescharacteristic
564C2C2×#295511yescharacteristic
565C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#60688C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60268C2semidirect
3. #23019C2semidirect
4. #23018C2semidirect
5. #22804C2semidirect
6. #22687C2semidirect
7. #22235C2semidirect
8. #22235C2semidirect
9. #21857C2semidirect
10. #21737C2semidirect
11. #21735C2semidirect
12. #21709C2semidirect
13. #21708C2semidirect
14. C2×#1849C2semidirect
15. #20481C2semidirect
17. #3181C4semidirect
18. #3181C4semidirect
19. #3180C4semidirect
20. #3180C4semidirect

Maximal subgroups

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

subgroupindex
#606882
#602682
#230192
#230182
#228042
#226872
#222352
#218572
#217372
#217352
#217092
#217082
C2×#18492
#204812