Finite Group Databasebeta

← #482081

Subgroups — #482081

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60459C211yesmaximal, characteristic
2#33334C211yesmaximal, characteristic, complemented
3#20664C211yesmaximal, characteristic, complemented
4#11890C212yesmaximal, complemented
5#11889C212yesmaximal, complemented
6#11887C212yesmaximal, complemented
7#11885C212yesmaximal, complemented
8#11706C211yesmaximal, characteristic, complemented
9#11704C211yesmaximal, characteristic, complemented
10#11703C211yesmaximal, characteristic, complemented
11C2×#1348C211yesmaximal, characteristic, complemented
12C2×#578C411yescharacteristic, complemented
13C22×#131C2211yescharacteristic
14#3489—21no—
15C2×#504—21no—
16#3063—21no—
17#3063—22no—
18#3061C411yescharacteristic, complemented
19#3061C412yescomplemented
20#3059C411yescharacteristic, complemented
21#3059C412yescomplemented
22#3056—21no—
23#3056—22no—
24#3050C2211yescharacteristic
25#3050C2212yes—
26C2×#467C2211yescharacteristic
27C2×#467C2212yes—
28#2913C411yescharacteristic, complemented
29C2×#428—21no—
30#2898—21no—
31#2311—21no—
32#2113—21no—
33#2113—22no—
34#2112C2212yes—
35#2112C2211yescharacteristic
36C2×C22⋊C16C2212yes—
37C2×C22⋊C16C2211yescharacteristic
38#1809—22no—
39#1808C2212yes—
40#1807C2212yes—
41#1735C2211yescharacteristic
42C2×#340C2211yescharacteristic
43#1348C2212yes—
44#1348C2212yes—
45#1348C2211yescharacteristic
46#1348C2212yes—
47#1348C2212yes—
48#1348C2211yescharacteristic
49#1348C2212yes—
50#1347C2211yescharacteristic
51#1347C2212yes—
52#1347C2211yescharacteristic
53C2×2−1+4—21no—
54C22×#41D411yescharacteristic
55#578—44no—
56C2×#137—22no—
57C2×#137—21no—
58C2×#135C2×C411yescharacteristic
59C2×#135C2×C412yescomplemented
60C22×Q16C2×C411yescharacteristic
61C22×Q16C2×C412yescomplemented
62C2×#131—44no—
63C2×#131C2×C411yescharacteristic
64C2×#131C2×C411yescharacteristic
65C2×#131C2×C412yes—
66C2×#131D411yescharacteristic
67C2×#131D411yescharacteristic
68C2×#131D412yes—
69C22×M16D411yescharacteristic
70C22×M16D412yes—
71C23×C8C2312yes—
72C23×C8C2311yescharacteristic
73D4×Q8—41no—
74#548—41no—
75#547—42no—
76#547—41no—
77#546—42no—
78#546—41no—
79#537—21no—
80#537—22no—
81#536—21no—
82#536—22no—
83#525—21no—
84#525—22no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—22no—
87#520—21no—
88C2×#117—21no—
89C22×C4⋊C4—21no—
90#504—44no—
91C2×M32—21no—
92C2×M32—22no—
93C22×C16—21no—
94C22×C16—22no—
95#470—22no—
96#470—22no—
97#470—22no—
98#468—22no—
99#468—22no—
100#468—22no—
101#467—21no—
102#467—22no—
103#467—21no—
104#467—22no—
105#467C2×C412yes—
106#467C2×C411yescharacteristic
107#467C2×C412yescomplemented
108#467C2×C412yescomplemented
109#467C2×C412yescomplemented
110#467C2312yes—
111#467C2311yescharacteristic
112#466C2×C411yescharacteristic
113#466C2×C412yescomplemented
114#466C2311yescharacteristic
115#465—21no—
116#465—22no—
117#465—21no—
118#465—22no—
119#428—22no—
120#428—21no—
121#428—44no—
122#427C2×C411yescharacteristic
123#427C2×C412yescomplemented
124C2×#107—22no—
125C2×#107C2×C411yescharacteristic
126C2×#107C2×C412yescomplemented
127C2×#106—22no—
128#419—22no—
129#419—21no—
130#417—22no—
131#417—21no—
132#416—21no—
133#416—22no—
134C2×#103—21no—
135C2×#103—22no—
136#391—41no—
137#390—41no—
138#386—42no—
139#380—22no—
140#380—22no—
141C22⋊C16—22no—
142C22⋊C16—21no—
143C22⋊C16—22no—
144C22⋊C16—21no—
145C22⋊C16C2311yescharacteristic
146C22⋊C16C2312yes—
147C22⋊C16C2311yescharacteristic
148#340—21no—
149#340—21no—
150#340C2311yescharacteristic
151#340C2312yes—
152#340C2311yescharacteristic
1532−1+4—44no—
1542−1+4—44no—
155C2×#41—21no—
156C2×#41—22no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—41no—
160C2×#41—42no—
161C2×#41C22⋊C411yescharacteristic
162C2×#41C22⋊C411yescharacteristic
163C2×#41C22⋊C412yes—
164C2×#41SD1612yes—
165C2×#41SD1611yescharacteristic
166C2×#41SD1611yescharacteristic
167C22×Q8—21no—
168C22×Q8—22no—
169C22×Q8—41no—
170C22×Q8D811yescharacteristic
171C22×D4D812yes—
172C22×D4D811yescharacteristic
173C23×C4—41no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176#137—48no—
177#137—44no—
178#135—42no—
179#135—44no—
180C2×Q16—21no—
181C2×Q16—22no—
182C2×Q16—42no—
183C2×Q16—44no—
184C2×Q16C22×C412yes—
185C2×Q16C22×C411yescharacteristic
186C2×SD16—21no—
187C2×SD16—22no—
188C2×D8C22×C411yescharacteristic
189#131—24no—
190#131—24no—
191#131—44no—
192#131—44no—
193#131—42no—
194#131—42no—
195C2×M16—24no—
196C2×M16—44no—
197C2×M16C22⋊C412yes—
198C2×M16C22⋊C411yescharacteristic
199C2×M16C22⋊C412yes—
200C2×M16C22⋊C412yes—
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C412yes—
203C2×M16C22⋊C411yescharacteristic
204C22×C8—44no—
205C22×C8—42no—
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C412yes—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D412yes—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D412yes—
215C22×C8C2×D411yescharacteristic
216C22×C8C2411yescharacteristic
217C4⋊Q8—41no—
218C4⋊Q8—42no—
219#124—41no—
220#124—42no—
221#123—42no—
222#123—42no—
223#123—42no—
224#123—41no—
225#123—42no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—21no—
230C22⋊Q8—22no—
231C22⋊Q8—21no—
232C22⋊Q8—22no—
233C22⋊Q8—41no—
234C22⋊Q8—42no—
235C22⋊Q8—41no—
236C22⋊Q8—42no—
237C22⋊Q8—41no—
238C22⋊Q8—42no—
239#121—21no—
240#121—22no—
241#121—21no—
242#121—42no—
243#121—42no—
244#121—42no—
245#121—41no—
246#121—42no—
247#120—41no—
248#120—42no—
249C4×Q8—41no—
250C4×D4—42no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—42no—
254C4×D4—41no—
255C4×D4—42no—
256#117—21no—
257#117—22no—
258#117—42no—
259#117—42no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C42—41no—
276M32—42no—
277M32—44no—
278C2×C16—21no—
279C2×C16—22no—
280C2×C16—21no—
281C2×C16—42no—
282C2×C16—44no—
283#107—21no—
284#107—22no—
285#107—44no—
286#107C22×C412yes—
287#107C22×C411yescharacteristic
288#107C22×C411yescharacteristic
289#106—21no—
290#106—22no—
291#106—44no—
292#103—22no—
293#103—22no—
294#103—22no—
295#103—21no—
296#103—22no—
297#103—21no—
298#103—22no—
299#102—21no—
300#102—22no—
301#102—21no—
302#95—42no—
303#95—42no—
304C24C2×D811yescharacteristic
305C24C2×D812yes—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—42no—
311#41—42no—
312#41—44no—
313#41—42no—
314#41—42no—
315#41—44no—
316#41—44no—
317#41—42no—
318#41—42no—
319#41—88no—
320#41—84no—
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—84no—
331C2×Q8#10211yescharacteristic
332C2×Q8#10211yescharacteristic
333C2×Q8#10212yes—
334C2×D4—21no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4#10212yes—
343C2×D4#10211yescharacteristic
344C2×D4#10212yes—
345C2×D4#10211yescharacteristic
346C2×D4#10212yes—
347C2×D4#10212yes—
348C2×D4#10212yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—82no—
363C22×C4—161no—
364C22×C4—161no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×D811yescharacteristic
370C22×C4C2×D812yes—
371C22×C4C2×SD1611yescharacteristic
372C22×C4C2×SD1611yescharacteristic
373C22×C4C2×SD1612yes—
374C22×C4C2×SD1612yes—
375C22×C4C22×D411yescharacteristic
376Q16—42no—
377Q16—41no—
378Q16—42no—
379Q16—44no—
380SD16—42no—
381SD16—44no—
382D8—41no—
383M16—22no—
384M16—24no—
385M16—24no—
386M16—24no—
387M16—42no—
388M16—42no—
389M16#9914yes—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8C2×C22⋊C411yescharacteristic
402C2×C8C2×C22⋊C412yes—
403C2×C8C2×C22⋊C411yescharacteristic
404C2×C8C2×C22⋊C412yes—
405C2×C8C23×C411yescharacteristic
406C2×C8C22×D411yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—82no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—82no—
439C22⋊C4—162no—
440C22⋊C4—162no—
441C42—42no—
442C42—41no—
443C42—82no—
444C16—41no—
445C16—41no—
446C16—42no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23C2×#10211yescharacteristic
457C23C2×#10212yes—
458C23C2×#10212yes—
459C23C2×#10211yescharacteristic
460C23C22×D811yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—42no—
464Q8—41no—
465Q8—42no—
466Q8—41no—
467Q8—42no—
468Q8—84no—
469Q8—82no—
470D4—24no—
471D4—24no—
472D4—22no—
473D4—22no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—41no—
478D4—41no—
479D4—84no—
480D4—81no—
481D4—81no—
482D4—82no—
483D4—161no—
484D4—162no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—161no—
518C2×C4—162no—
519C2×C4—164no—
520C2×C4C2×#9912yes—
521C2×C4C2×#10212yes—
522C2×C4C2×#10211yescharacteristic
523C2×C4C2×#10212yes—
524C2×C4C2×#10211yescharacteristic
525C2×C4#42112yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C22×SD1611yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—22no—
532C8—21no—
533C8C2×#9914yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22#151912yes—
555C22C22×#10211yescharacteristic
556C22#341611yescharacteristic
557C22#341711yescharacteristic
558C4—21no—
559C4—21no—
560C4—22no—
561C4—22no—
562C4—81no—
563C4—82no—
564C4—81no—
565C4—82no—
566C4—81no—
567C4—161no—
568C4—161no—
569C4—162no—
570C4#151912yes—
571C4C22×#9911yescharacteristic
572C4C2×#42111yescharacteristic
573C2—21no—
574C2—21no—
575C2—22no—
576C2—41no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C2×#151911yescharacteristic
581C2#3338211yescharacteristic
582C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#60459C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33334C2semidirect
3. #20664C2semidirect
4. #11890C2semidirect
5. #11889C2semidirect
6. #11887C2semidirect
7. #11885C2semidirect
8. #11706C2semidirect
9. #11704C2semidirect
10. #11703C2semidirect
11. C2×#1348C2semidirect
12. C2×#578C4semidirect
18. #3061C4semidirect
19. #3061C4semidirect
20. #3059C4semidirect
21. #3059C4semidirect
28. #2913C4semidirect
59. C2×#135C2×C4semidirect
61. C22×Q16C2×C4semidirect
107. #467C2×C4semidirect
108. #467C2×C4semidirect
109. #467C2×C4semidirect
113. #466C2×C4semidirect
123. #427C2×C4semidirect
126. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
#604592
#333342
#206642
#118902
#118892
#118872
#118852
#117062
#117042
#117032
C2×#13482