Finite Group Databasebeta

← #6472865

Subgroups — #6472865

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic
2#37453C211yesmaximal, characteristic
3#33056C212yesmaximal, complemented
4#33046C212yesmaximal, complemented
5#30196C212yesmaximal, complemented
6#30193C212yesmaximal, complemented
7#27713C214yesmaximal, complemented
8#27699C214yesmaximal, complemented
9#27268C214yesmaximal, complemented
10#27258C214yesmaximal, complemented
11#21765C212yesmaximal
12C2×#1892C212yesmaximal
13#20385C211yesmaximal, characteristic
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C2×#491C2212yescomplemented
17C2×#487C2212yescomplemented
18C22×#103C2212yes—
19C22×#103C2212yes—
20C2×#407C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×#405C2211yescharacteristic
23#2841—22no—
24#2817—22no—
25#2659—22no—
26#2646—22no—
27#2629C2212yes—
28#2628C2212yes—
29#2627—22no—
30#2627—22no—
31#2621C2212yes—
32#2621C2212yes—
33#2503C2214yes—
34#2483—22no—
35#2442C2214yes—
36#2440C2214yes—
37#2437—22no—
38#2434C2214yes—
39#2280C2211yescharacteristic
40C2×#380C2212yes—
41#2101C2214yescomplemented
42#2101C2214yescomplemented
43#2074C2214yescomplemented
44#2072C2214yescomplemented
45#2043C2214yescomplemented
46#2043C2214yescomplemented
47#2026C2214yescomplemented
48#2024C2214yescomplemented
49#1993C2212yescomplemented
50#1991C2212yescomplemented
51#1960C2212yescomplemented
52#1956C2212yescomplemented
53#1935C2214yescomplemented
54#1935C2214yescomplemented
55#1933C2214yescomplemented
56#1933C2214yescomplemented
57#1894C2214yes—
58#1894C2214yes—
59#1892C2214yes—
60#1892C2214yes—
61#1892C2214yes—
62#1892C2214yes—
63#1872C2214yes—
64#1872C2214yes—
65#1836C2212yescomplemented
66#1835C2212yescomplemented
67#1823C2214yes—
68#1787C2214yes—
69#1752C2214yes—
70C8×C22⋊C4C2214yes—
71C2×#340C2212yes—
72C2×#340C2212yes—
73#1729C2211yescharacteristic
74#1729C2211yescharacteristic
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—24no—
80C23×C8—42no—
81C23×C8—42no—
82C23×C8C2311yescharacteristic
83C23×C8C2312yes—
84C23×C8C2312yes—
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125C2312yes—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C4⋊C4—22no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105#491—44no—
106#487—44no—
107C2×#103—44no—
108C2×#103—44no—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117#407—24no—
118C2×C22⋊C8C2314yes—
119#405—42no—
120#405—42no—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125#400—22no—
126#400—24no—
127#400—24no—
128#400—24no—
129#399—22no—
130#398—22no—
131#398—24no—
132#398—22no—
133#398—24no—
134#398—24no—
135#398—24no—
136#397—22no—
137#397—22no—
138#397—24no—
139#396—24no—
140#395—22no—
141#395—24no—
142#395—24no—
143#395—24no—
144#393—22no—
145#391C2314yes—
146#391C2314yes—
147#389C2314yes—
148#389C2314yes—
149#385—24no—
150#385—24no—
151#385—24no—
152#385—24no—
153#384C2312yes—
154#382C2312yes—
155#382C2312yes—
156#382C2312yes—
157#381—24no—
158#381—24no—
159#381D412yes—
160#380—22no—
161#380—22no—
162#380—24no—
163#380D412yes—
164#380D412yes—
165#380C2314yes—
166#380C2314yes—
167#379D412yes—
168C4×C22⋊C4C2314yes—
169C2×#95—22no—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2314yes—
179#336C2314yes—
180C25C2×D411yescharacteristic
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8C2414yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4#4114yes—
201C23×C4#4114yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—41no—
207C2×M16—41no—
208C2×M16—41no—
209C2×M16—41no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—22no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—41no—
221C22×C8—41no—
222C22×C8—416no—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2411yescharacteristic
227C22×C8C2411yescharacteristic
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—42no—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234#125—42no—
235#125—42no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C22⋊Q8—48no—
241C22⋊Q8—48no—
242C22⋊Q8—88no—
243#117—41no—
244#117—41no—
245#117—41no—
246#117—41no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—84no—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42C2412yes—
284C2×C42C2412yes—
285#103—28no—
286#103—28no—
287#103—44no—
288#103—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293C22⋊C8—28no—
294#97—24no—
295#97—42no—
296C4×C8—24no—
297C4×C8—42no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95C2×D414yes—
312#95C2×D414yes—
313C24—24no—
314C24—24no—
315C24—41no—
316C24—41no—
317C24#12414yes—
318C24C22×D411yescharacteristic
319C24C22×D412yes—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8—84no—
332C2×Q8—82no—
333C2×Q8—84no—
334C2×Q8—84no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4#12418yes—
368C22×C4#12414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D414yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C2×#4114yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2511yescharacteristic
379M16—44no—
380M16—44no—
381C2×C8—28no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—48no—
396C2×C8—48no—
397C2×C8—44no—
398C2×C8H(5,𝔽2)12yes—
399C2×C8H(5,𝔽2)12yes—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—84no—
421C4⋊C4—82no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—84no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—216no—
439C23—21no—
440C23—21no—
441C23—22no—
442C23—22no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—48no—
447C23C2×#12412yes—
448C23C2×#12414yes—
449C23#57714yes—
450C23#57814yes—
451C23C23×D411yescharacteristic
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—88no—
457Q8—164no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4—164no—
494C2×C4—164no—
495C2×C4C2×#12412yes—
496C2×C4C2×#12414yes—
497C2×C4C22×#4111yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×H(5,𝔽2)11yescharacteristic
500C2×C4C2×H(5,𝔽2)11yescharacteristic
501C2×C4#58511yescharacteristic
502C2×C4#58511yescharacteristic
503C8—22no—
504C8—22no—
505C8—42no—
506C8—42no—
507C8—42no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—22no—
515C22—24no—
516C22—416no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22#313718yes—
521C22C22×#12411yescharacteristic
522C22#349611yescharacteristic
523C22#349611yescharacteristic
524C22C2×#57712yes—
525C22C2×#57812yes—
526C4—22no—
527C4—24no—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—84no—
532C4—84no—
533C4—84no—
534C4—84no—
535C4#345911yescharacteristic
536C4#345911yescharacteristic
537C4#349011yescharacteristic
538C4#349111yescharacteristic
539C2—28no—
540C2—42no—
541C2#3499311yescharacteristic
542C2C2×#313712yes—
543C2#6226811yescharacteristic
544C2#6227111yescharacteristic
545C2#6228112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33056C2semidirect
4. #33046C2semidirect
5. #30196C2semidirect
6. #30193C2semidirect
7. #27713C2semidirect
8. #27699C2semidirect
9. #27268C2semidirect
10. #27258C2semidirect
16. C2×#491C22semidirect
17. C2×#487C22semidirect
41. #2101C22semidirect
42. #2101C22semidirect
43. #2074C22semidirect
44. #2072C22semidirect
45. #2043C22semidirect
46. #2043C22semidirect
47. #2026C22semidirect
48. #2024C22semidirect
49. #1993C22semidirect
50. #1991C22semidirect
51. #1960C22semidirect
52. #1956C22semidirect
53. #1935C22semidirect
54. #1935C22semidirect
55. #1933C22semidirect
56. #1933C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#374772
#374532
#330562
#330462
#301962
#301932
#277132
#276992
#272682
#272582
#217652
C2×#18922
#203852