Finite Group Databasebeta

← #483228

Subgroups — #483228

Order 512.

754 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60714C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic, complemented
3#17799C211yesmaximal, characteristic, complemented
4#11986C211yesmaximal, characteristic, complemented
5#11985C211yesmaximal, characteristic, complemented
6#11982C211yesmaximal, characteristic, complemented
7#11981C211yesmaximal, characteristic, complemented
8#11976C211yesmaximal, characteristic, complemented
9#11973C211yesmaximal, characteristic, complemented
10#11964C211yesmaximal, characteristic, complemented
11#11961C211yesmaximal, characteristic, complemented
12#11952C211yesmaximal, characteristic, complemented
13#11951C211yesmaximal, characteristic, complemented
14#11948C211yesmaximal, characteristic, complemented
15#11947C211yesmaximal, characteristic, complemented
16#3508—21no—
17#3184—21no—
18#3184—21no—
19#3176—21no—
20#3176—21no—
21#3176C412yes—
22#3176C412yes—
23#3169C2211yescharacteristic
24#3169C2211yescharacteristic
25#3169C2211yescharacteristic
26#3169C2211yescharacteristic
27#3160C2211yescharacteristic
28C2×#493C2211yescharacteristic
29#3138—21no—
30#3058—21no—
31#3058C412yes—
32#2989—21no—
33#2948—21no—
34#2948C412yes—
35C4×#131C2211yescharacteristic
36#2164—21no—
37#2164—21no—
38#2163C2211yescharacteristic
39#2163C2211yescharacteristic, complemented
40#2162C2211yescharacteristic
41#2162C2211yescharacteristic, complemented
42#2150C2211yescharacteristic, complemented
43C2×C4⋊C16C2211yescharacteristic, complemented
44#2113—21no—
45#2108—21no—
46#1569—21no—
47#1569—21no—
48#1568C2211yescharacteristic
49#1568C2211yescharacteristic, complemented
50#1567C2211yescharacteristic
51#1567C2211yescharacteristic, complemented
52#1565—21no—
53#1564C2211yescharacteristic, complemented
54C2×#335C2211yescharacteristic, complemented
55#1365C2211yescharacteristic, complemented
56#1365C2211yescharacteristic, complemented
57#1365C2211yescharacteristic, complemented
58#1365C2211yescharacteristic, complemented
59#1364C2211yescharacteristic, complemented
60#1364C2211yescharacteristic, complemented
61#1364C2211yescharacteristic, complemented
62#1364C2211yescharacteristic, complemented
63#1361C2211yescharacteristic
64#1361C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67#1361C2211yescharacteristic
68#1361C2211yescharacteristic
69#1361C2211yescharacteristic
70#1361C2211yescharacteristic
71C22×D8—21no—
72C22×D8C2×C412yes—
73C2×#131C2×C411yescharacteristic
74C2×#131D411yescharacteristic
75#559—41no—
76#559—41no—
77#550—41no—
78#550—41no—
79#546—41no—
80#546—41no—
81#546—41no—
82#546—41no—
83#540—22no—
84#540—22no—
85#535—22no—
86#533—21no—
87#532—21no—
88#532—21no—
89#532—21no—
90#532—21no—
91C2×#127—21no—
92#518—22no—
93C4×#41D411yescharacteristic
94C2×M32—21no—
95C22×C16—21no—
96#500C2311yescharacteristic
97#500C2311yescharacteristic
98#499—21no—
99#499C2×C412yes—
100#498—21no—
101#493—21no—
102#493—21no—
103#493C2×C412yes—
104#493C2311yescharacteristic
105#493C2311yescharacteristic
106#490—22no—
107#490—22no—
108#486—21no—
109#486—21no—
110#486—21no—
111#486—21no—
112#469—22no—
113#469—22no—
114#469—22no—
115#469—22no—
116#466—21no—
117#466—21no—
118#466—21no—
119#466—21no—
120#466C2×C412yes—
121#466C2×C412yes—
122#466C2×C412yes—
123#466C2×C412yes—
124#446—41no—
125#442—22no—
126#442—22no—
127#442—22no—
128#442—22no—
129C4×D8—21no—
130C4×D8—21no—
131C4×D8—21no—
132C4×D8—21no—
133C4×D8C2×C412yes—
134C4×D8C2×C412yes—
135C4×D8C2×C412yes—
136C4×D8C2×C412yes—
137#436—41no—
138#435—41no—
139#435—41no—
140#432—21no—
141#432—21no—
142#431—21no—
143#427—21no—
144#427C2×C412yes—
145#423—21no—
146#423—21no—
147C2×C4⋊C8—21no—
148#418—22no—
149#418—22no—
150#405C2×C411yescharacteristic
151#405C2×C411yescharacteristic
152#405C2×C411yescharacteristic
153#405D411yescharacteristic
154#405D411yescharacteristic
155#405D411yescharacteristic
156C4×M16D411yescharacteristic
157C4×M16D411yescharacteristic
158C4×M16D411yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C4⋊C16—21no—
163C4⋊C16—21no—
164C4⋊C16C2311yescharacteristic
165C4⋊C16C2311yescharacteristic
166C4⋊C16C2311yescharacteristic
167C4⋊C16C2311yescharacteristic
168C22⋊C16—21no—
169C22⋊C16—21no—
170C22⋊C16—21no—
171C22⋊C16—21no—
172#346—21no—
173#346—21no—
174C4×C16—21no—
175C4×C16—21no—
176#335—21no—
177#335—21no—
178#335—21no—
179#335C2311yescharacteristic
180#335C2311yescharacteristic
181#335C2311yescharacteristic
182#335C2311yescharacteristic
183#334—21no—
184#334—21no—
185#334—21no—
186C2×#41—41no—
187C2×#41—41no—
188C2×#41—41no—
189C2×#41—41no—
190C2×#41C22⋊C411yescharacteristic
191C22×D4—22no—
192C22×D4—41no—
193C22×D4—41no—
194C2×D8—22no—
195C2×D8—22no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—42no—
199C2×D8—42no—
200C2×D8—44no—
201C2×D8C22×C412yes—
202C2×D8C22×C412yes—
203#131—41no—
204#131—41no—
205#131—42no—
206C2×M16—41no—
207C2×M16—41no—
208C2×M16C22⋊C411yescharacteristic
209C2×M16C22⋊C411yescharacteristic
210C2×M16C22⋊C411yescharacteristic
211C2×M16C22⋊C411yescharacteristic
212C2×M16C22⋊C411yescharacteristic
213C2×M16C22⋊C411yescharacteristic
214C22×C8—41no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223#127—22no—
224#127—21no—
225#127—21no—
226#127—42no—
227#127—41no—
228#127—41no—
229#125—22no—
230#124—42no—
231#124—41no—
232#124—41no—
233#124—41no—
234#124—41no—
235#123—42no—
236#123—42no—
237#123—42no—
238#123—42no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241#121—22no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121—41no—
246#121—41no—
247#121—41no—
248#121—41no—
249#121—42no—
250#121—42no—
251#121—42no—
252#121—42no—
253#120—42no—
254#120—42no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8D811yescharacteristic
258C4×Q8SD1611yescharacteristic
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4D811yescharacteristic
270C4×D4D811yescharacteristic
271C4×D4D811yescharacteristic
272C4×D4SD1611yescharacteristic
273C4×D4SD1611yescharacteristic
274C4×D4SD1611yescharacteristic
275#117—22no—
276#117C22⋊C411yescharacteristic
277#117C22⋊C411yescharacteristic
278#117C22⋊C411yescharacteristic
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284M32—42no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—42no—
290#107—21no—
291#107—21no—
292#107C22×C412yes—
293#107C22×C412yes—
294#106—22no—
295#106—22no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303#102—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314#97—41no—
315#97—41no—
316#97C22⋊C411yescharacteristic
317#97C22⋊C411yescharacteristic
318#97C22⋊C411yescharacteristic
319#97C22⋊C411yescharacteristic
320#97C22⋊C411yescharacteristic
321#97C22⋊C411yescharacteristic
322C4×C8—41no—
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2411yescharacteristic
333C24—42no—
334#41—41no—
335#41—41no—
336#41—161no—
337#41—161no—
338#41—161no—
339#41—161no—
340#41—162no—
341#41—162no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8#10211yescharacteristic
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—161no—
361C2×D4—161no—
362C2×D4—162no—
363C2×D4#10211yescharacteristic
364C2×D4#10211yescharacteristic
365C2×D4#10211yescharacteristic
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×D811yescharacteristic
374C22×C4C2×D811yescharacteristic
375C22×C4C2×D811yescharacteristic
376C22×C4C2×SD1611yescharacteristic
377C22×C4C2×SD1611yescharacteristic
378C22×C4C2×SD1611yescharacteristic
379D8—42no—
380D8—42no—
381D8—44no—
382D8—41no—
383D8—41no—
384D8—41no—
385D8—41no—
386D8—44no—
387M16—22no—
388M16—22no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—161no—
394M16—161no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—161no—
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C2×C22⋊C411yescharacteristic
411C2×C8C2×C22⋊C411yescharacteristic
412C2×C8C2×C22⋊C411yescharacteristic
413C2×C8C2×C22⋊C411yescharacteristic
414C2×C8C2×C22⋊C411yescharacteristic
415C2×C8C23×C411yescharacteristic
416C2×C8C22×D411yescharacteristic
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4#10211yescharacteristic
424C4⋊C4#10211yescharacteristic
425C4⋊C4#10211yescharacteristic
426C4⋊C4#10211yescharacteristic
427C4⋊C4#10211yescharacteristic
428C4⋊C4#10211yescharacteristic
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4#10211yescharacteristic
439C22⋊C4#10211yescharacteristic
440C22⋊C4#10211yescharacteristic
441C22⋊C4#10211yescharacteristic
442C22⋊C4#10211yescharacteristic
443C22⋊C4#10211yescharacteristic
444C42—42no—
445C42C2×C22⋊C411yescharacteristic
446C42C2×C22⋊C411yescharacteristic
447C42C2×C22⋊C411yescharacteristic
448C42C2×D811yescharacteristic
449C42C2×D811yescharacteristic
450C42C2×D811yescharacteristic
451C42C2×SD1611yescharacteristic
452C42C2×SD1611yescharacteristic
453C42C2×SD1611yescharacteristic
454C42C22×D411yescharacteristic
455C16—41no—
456C16—41no—
457C16—41no—
458C16—41no—
459C23—42no—
460C23—42no—
461C23—84no—
462C23—162no—
463C23—162no—
464C23C2×#10211yescharacteristic
465C23C2×#10211yescharacteristic
466C23C2×#10211yescharacteristic
467Q8—41no—
468Q8—82no—
469Q8—81no—
470Q8—81no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—84no—
479D4—82no—
480D4—82no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485D4—162no—
486D4—162no—
487D4—162no—
488D4—162no—
489D4—162no—
490D4—162no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—162no—
509C2×C4—162no—
510C2×C4—162no—
511C2×C4—162no—
512C2×C4C2×#10211yescharacteristic
513C2×C4C2×#10211yescharacteristic
514C2×C4C2×#10211yescharacteristic
515C2×C4C2×#10211yescharacteristic
516C2×C4C2×#10211yescharacteristic
517C2×C4C2×#10211yescharacteristic
518C2×C4C2×#10211yescharacteristic
519C2×C4C2×#10211yescharacteristic
520C2×C4C2×#10211yescharacteristic
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4C22×D811yescharacteristic
523C2×C4C22×SD1611yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—81no—
531C8—81no—
532C8—81no—
533C8—81no—
534C8#41312yes—
535C8#57511yescharacteristic
536C8#57511yescharacteristic
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—324no—
553C22C22×#10211yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—82no—
565C4—82no—
566C4C2×#41311yescharacteristic
567C4#289611yescharacteristic
568C4#341511yescharacteristic
569C4#341611yescharacteristic
570C2—41no—
571C2—41no—
572C2—41no—
573C2—162no—
574C2—162no—
575C2#1770411yescharacteristic
576C2#3338111yescharacteristic
577C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#60714C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33423C2semidirect
3. #17799C2semidirect
4. #11986C2semidirect
5. #11985C2semidirect
6. #11982C2semidirect
7. #11981C2semidirect
8. #11976C2semidirect
9. #11973C2semidirect
10. #11964C2semidirect
11. #11961C2semidirect
12. #11952C2semidirect
13. #11951C2semidirect
14. #11948C2semidirect
15. #11947C2semidirect
39. #2163C22semidirect
41. #2162C22semidirect
42. #2150C22semidirect
43. C2×C4⋊C16C22semidirect
49. #1568C22semidirect
51. #1567C22semidirect
53. #1564C22semidirect
54. C2×#335C22semidirect
55. #1365C22semidirect
56. #1365C22semidirect
57. #1365C22semidirect
58. #1365C22semidirect
59. #1364C22semidirect
60. #1364C22semidirect
61. #1364C22semidirect
62. #1364C22semidirect

Maximal subgroups

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

subgroupindex
#607142
#334232
#177992
#119862
#119852
#119822
#119812
#119762
#119732
#119642
#119612
#119522
#119512
#119482
#119472