Finite Group Databasebeta

← #510075

Subgroups — #510075

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28708C211yesmaximal, characteristic, complemented
2#24267C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13492C211yesmaximal, characteristic, complemented
5#13381C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic, complemented
7#13365C211yesmaximal, characteristic, complemented
8#13365C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13061C211yesmaximal, characteristic, complemented
12#13061C211yesmaximal, characteristic, complemented
13#12977C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×M32C2211yescharacteristic, complemented
18C2×#499C2211yescharacteristic, complemented
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2562—21no—
23#2408—21no—
24C2×#367C2211yescharacteristic, complemented
25C2×#366C2211yescharacteristic, 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
37#2077—21no—
38#2077—21no—
39#2054—21no—
40#1999—21no—
41#1989—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic, complemented
48#1808C2211yescharacteristic, complemented
49#1786—21no—
50C4×#107C2211yescharacteristic, complemented
51#1380C2211yescharacteristic, complemented
52#1380C2211yescharacteristic, complemented
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1374C2211yescharacteristic, complemented
58#1374C2211yescharacteristic, 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×#125—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#120—41no—
75C2×C4×D4D411yescharacteristic
76C2×M32—41no—
77C2×M32—41no—
78C2×M32—41no—
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#499—41no—
84#499—41no—
85#466—22no—
86#466—22no—
87#466—41no—
88#466—41no—
89C8×D4—41no—
90C8×D4—41no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#106—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C22⋊C8D411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#400—41no—
107#394—41no—
108#394—41no—
109#392—41no—
110#390—21no—
111#390—21no—
112#388—41no—
113#388—41no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#379—41no—
120C4×C4⋊C4—21no—
121#367—41no—
122#366—41no—
123#357—22no—
124#357—22no—
125#357—41no—
126#357—41no—
127#357—41no—
128#357—41no—
129#357—41no—
130#357—41no—
131C22⋊C16—41no—
132C22⋊C16—41no—
133C22⋊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#125—42no—
191#125—42no—
192#121—42no—
193#121—42no—
194#121—81no—
195#121—81no—
196#120—84no—
197C4×D4—41no—
198C4×D4—41no—
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
223M32—44no—
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—
235C2×C16—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#107—21no—
248#106—42no—
249C4⋊C8—41no—
250#102—42no—
251#102—42no—
252#102—81no—
253#102—81no—
254C22⋊C8—41no—
255C22⋊C8—41no—
256C4×C8—21no—
257C4×C8—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—41no—
265C24—41no—
266C24—82no—
267C24—82no—
268C24—82no—
269C24—82no—
270C24—82no—
271C24—161no—
272C24—161no—
273C24C2×D811yescharacteristic
274C24C2×D811yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—48no—
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—41no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—84no—
295C2×D4—84no—
296C2×D4—84no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—161no—
302C2×D4—161no—
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—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C22×D411yescharacteristic
326D8—22no—
327D8—22no—
328D8—22no—
329D8—22no—
330D8—48no—
331D8—41no—
332D8—41no—
333D8—41no—
334D8—41no—
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—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8C2×#4111yescharacteristic
353C2×C8C2×#4111yescharacteristic
354C2×C8H(5,𝔽2)11yescharacteristic
355C2×C8H(5,𝔽2)11yescharacteristic
356C2×C8H(5,𝔽2)11yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
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—41no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C22⋊C4—161no—
385C22⋊C4—161no—
386C42—21no—
387C42—21no—
388C42—161no—
389C16—41no—
390C16—41no—
391C16—41no—
392C16—41no—
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—21no—
406C23—42no—
407C23—42no—
408C23—88no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—164no—
422C23—164no—
423C23—164no—
424C23—164no—
425C23—164no—
426C23—164no—
427C23#50912yes—
428C23C22×D811yescharacteristic
429D4—44no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—88no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—164no—
440D4—164no—
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—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
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—81no—
478C2×C4—81no—
479C2×C4#53411yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#54611yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57711yescharacteristic
485C8—41no—
486C8—41no—
487C8—41no—
488C8#53811yescharacteristic
489C8#53811yescharacteristic
490C8#55111yescharacteristic
491C8H(3,𝔽4)11yescharacteristic
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—1616no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22—161no—
521C22#318611yescharacteristic
522C22#329411yescharacteristic
523C22#329411yescharacteristic
524C22C2×#50911yescharacteristic
525C22#341411yescharacteristic
526C22#341511yescharacteristic
527C22#341611yescharacteristic
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4—161no—
536C4—161no—
537C4—161no—
538C4#268411yescharacteristic
539C4#320011yescharacteristic
540C4#331111yescharacteristic
541C4#331111yescharacteristic
542C2—42no—
543C2—81no—
544C2—162no—
545C2—162no—
546C2#3126011yescharacteristic
547C2#3376011yescharacteristic
548C2#3376411yescharacteristic
549C2#3387711yescharacteristic
550C2#3388511yescharacteristic
551C2#3388911yescharacteristic
552C2#3389011yescharacteristic
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. #28708C2semidirect
3. #13494C2semidirect
4. #13492C2semidirect
5. #13381C2semidirect
6. #13380C2semidirect
7. #13365C2semidirect
8. #13365C2semidirect
9. #13131C2semidirect
10. #13130C2semidirect
11. #13061C2semidirect
12. #13061C2semidirect
13. #12977C2semidirect
14. #12731C2semidirect
15. #12731C2semidirect
17. C22×M32C22semidirect
18. C2×#499C22semidirect
24. C2×#367C22semidirect
25. C2×#366C22semidirect
36. C2×C22⋊C16C22semidirect
47. #1808C22semidirect
48. #1808C22semidirect
50. C4×#107C22semidirect
51. #1380C22semidirect
52. #1380C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1374C22semidirect
58. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#287082
#242672
#134942
#134922
#133812
#133802
#133652
#131312
#131302
#130612
#129772
#127312