Finite Group Databasebeta

← #510498

Subgroups — #510498

Order 512.

719 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27660C211yesmaximal, characteristic, complemented
2#24267C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13493C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13189C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13147C211yesmaximal, characteristic, complemented
12#12992C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427C2211yescharacteristic, complemented
21#2485—21no—
22#2408—21no—
23C2×#367C2211yescharacteristic, complemented
24C2×#366C2211yescharacteristic, complemented
25C2×C4⋊C16C2211yescharacteristic, complemented
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic, complemented
37C2×C22⋊C16C2211yescharacteristic, complemented
38C2×C4×C16C2211yescharacteristic, complemented
39#2088—21no—
40#2077—21no—
41#2054—21no—
42#1999—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1942C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1825—21no—
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic, complemented
52#1786—21no—
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59C23×D4—21no—
60C22×D8—22no—
61C22×D8—22no—
62C22×D8—21no—
63C22×D8—41no—
64C22×D8—41no—
65C22×D8—41no—
66C22×D8—41no—
67C22×D8C2311yescharacteristic
68C22×D8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×#121—21no—
72C2×#121—21no—
73C2×#120—41no—
74C2×C4×D4D411yescharacteristic
75C2×#117—21no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#466—22no—
81#466—22no—
82#466—41no—
83#466—41no—
84C8×D4—41no—
85C8×D4—41no—
86#427—41no—
87#427—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#397—41no—
104#394—41no—
105#394—41no—
106#392—41no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#387—41no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#379—41no—
118#367—41no—
119#366—41no—
120C4⋊C16—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128#357—41no—
129C22⋊C16—41no—
130C22⋊C16—41no—
131C22⋊C16—41no—
132C22⋊C16—41no—
133C4×C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C25—41no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—41no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4—81no—
151C22×D4—81no—
152C22×D4D811yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—42no—
166C2×D8—42no—
167C2×D8—42no—
168C2×D8—44no—
169C2×D8—44no—
170C2×D8—44no—
171C2×D8—44no—
172C2×D8#4112yes—
173C2×D8#4112yes—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2411yescharacteristic
190#121—42no—
191#121—42no—
192#121—81no—
193#121—81no—
194#120—84no—
195C4×D4—41no—
196C4×D4—41no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4D811yescharacteristic
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4D811yescharacteristic
220C2×C22⋊C4D811yescharacteristic
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#106—42no—
248C4⋊C8—41no—
249#102—42no—
250#102—42no—
251#102—81no—
252#102—81no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255C4×C8—21no—
256C4×C8—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24—41no—
264C24—41no—
265C24—82no—
266C24—82no—
267C24—82no—
268C24—82no—
269C24—82no—
270C24—161no—
271C24—161no—
272C24C2×D811yescharacteristic
273C24C2×D811yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—48no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—84no—
294C2×D4—84no—
295C2×D4—84no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—161no—
301C2×D4—161no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C2×D811yescharacteristic
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C22×D411yescharacteristic
325D8—22no—
326D8—22no—
327D8—22no—
328D8—22no—
329D8—48no—
330D8—41no—
331D8—41no—
332D8—41no—
333D8—41no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8C2×#4111yescharacteristic
352C2×C8C2×#4111yescharacteristic
353C2×C8H(5,𝔽2)11yescharacteristic
354C2×C8H(5,𝔽2)11yescharacteristic
355C2×C8H(5,𝔽2)11yescharacteristic
356C2×C8H(5,𝔽2)11yescharacteristic
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—82no—
371C4⋊C4—161no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—161no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C22⋊C4—161no—
385C42—21no—
386C42—21no—
387C42—82no—
388C16—41no—
389C16—41no—
390C16—41no—
391C16—41no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—42no—
406C23—42no—
407C23—88no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—164no—
421C23—164no—
422C23—164no—
423C23—164no—
424C23—164no—
425C23—164no—
426C23#50812yes—
427C23C22×D811yescharacteristic
428D4—44no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—88no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—164no—
439D4—164no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4#50912yes—
478C2×C4#53411yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#57511yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57711yescharacteristic
484C8—41no—
485C8—41no—
486C8—41no—
487C8#53811yescharacteristic
488C8#53811yescharacteristic
489C8#55111yescharacteristic
490C8#56011yescharacteristic
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—1616no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22#318611yescharacteristic
521C22#329411yescharacteristic
522C22#329411yescharacteristic
523C22C2×#50811yescharacteristic
524C22C2×#50911yescharacteristic
525C22#341511yescharacteristic
526C22#341611yescharacteristic
527C4—22no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4—161no—
536C4—161no—
537C4#269711yescharacteristic
538C4#320011yescharacteristic
539C4#332311yescharacteristic
540C4#332311yescharacteristic
541C2—42no—
542C2—81no—
543C2—162no—
544C2—162no—
545C2#3166611yescharacteristic
546C2#3375411yescharacteristic
547C2#3376411yescharacteristic
548C2#3386311yescharacteristic
549C2#3386411yescharacteristic
550C2#3389111yescharacteristic
551C2#3389211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#24267C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27660C2semidirect
3. #13494C2semidirect
4. #13493C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13189C2semidirect
10. #13147C2semidirect
11. #13147C2semidirect
12. #12992C2semidirect
13. #12992C2semidirect
14. #12731C2semidirect
15. #12731C2semidirect
20. C2×#427C22semidirect
23. C2×#367C22semidirect
24. C2×#366C22semidirect
25. C2×C4⋊C16C22semidirect
36. C2×C22⋊C16C22semidirect
37. C2×C22⋊C16C22semidirect
38. C2×C4×C16C22semidirect
45. #1942C22semidirect
46. #1942C22semidirect
51. #1808C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#276602
#242672
#134942
#134932
#134762
#134742
#133812
#133802
#131892
#131472
#129922
#127312