Finite Group Databasebeta

← #6510750

Subgroups — #6510750

Order 512.

1567 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#35227C211yesmaximal, characteristic
3#29120C214yesmaximal, complemented
4#27971C214yesmaximal, complemented
5#27721C214yesmaximal, complemented
6#27719C214yesmaximal, complemented
7#27705C214yesmaximal, complemented
8#27703C214yesmaximal, complemented
9#21766C211yesmaximal, characteristic
10#21766C211yesmaximal, characteristic
11#20700C211yesmaximal, characteristic, complemented
12#20700C211yesmaximal, characteristic, complemented
13#20529C211yesmaximal, characteristic, complemented
14C2×#525C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18#2849—21no—
19#2849—21no—
20#2842—21no—
21#2840—21no—
22#2665—22no—
23#2665—22no—
24#2628C2214yes—
25#2623—22no—
26#2607—22no—
27#2504C2212yes—
28#2503C2212yes—
29#2502C2212yes—
30#2501C2212yes—
31#2487—22no—
32#2486—22no—
33#2439C2214yes—
34#2439C2214yes—
35#2317C2212yes—
36#2311C2212yes—
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2101C2214yescomplemented
41#2101C2214yescomplemented
42#2100C2214yescomplemented
43#2099C2214yescomplemented
44#2074C2214yescomplemented
45#2073C2214yescomplemented
46#2024C2214yescomplemented
47#2023C2214yescomplemented
48#1993C2214yescomplemented
49#1990C2214yescomplemented
50#1936C2214yescomplemented
51#1935C2214yescomplemented
52#1934C2214yescomplemented
53#1933C2214yescomplemented
54#1895C2212yes—
55#1895C2212yes—
56#1894C2212yes—
57#1894C2212yes—
58#1875C2212yes—
59#1875C2212yes—
60#1873C2212yes—
61#1873C2212yes—
62#1836C2214yescomplemented
63#1835C2214yescomplemented
64#1825C2212yes—
65#1825C2212yes—
66#1824C2214yes—
67#1824C2214yes—
68#1823C2214yes—
69#1823C2214yes—
70#1788C2212yes—
71#1788C2212yes—
72#1753C2214yes—
73#1753C2214yes—
74#1750C2214yes—
75#1736C2214yes—
76#1736C2214yes—
77C22×#41C2311yescharacteristic
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C22×M16C2312yes—
81C22×M16C2312yes—
82C2×C4⋊Q8C2314yes—
83#525—24no—
84#525—24no—
85C2×#123—21no—
86C2×#123—22no—
87C2×#123—22no—
88C2×C22⋊Q8C2311yescharacteristic
89C2×C22⋊Q8C2311yescharacteristic
90C2×#121C2311yescharacteristic
91C2×#121C2311yescharacteristic
92C2×#120—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×#117C2311yescharacteristic
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111#407—24no—
112#407—24no—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×#97C2314yes—
118C2×C4×C8C2314yes—
119#400—24no—
120#400—24no—
121#399—24no—
122#399—24no—
123#398—24no—
124#397—24no—
125#396—24no—
126#396—24no—
127#395—24no—
128#393—24no—
129#393—24no—
130#393—24no—
131#391—24no—
132#391—24no—
133#391C2314yes—
134#390C2314yes—
135#389—24no—
136#389C2314yes—
137#389C2314yes—
138#387—24no—
139#387—24no—
140#386—22no—
141#386—22no—
142#386—22no—
143#386—22no—
144#385—22no—
145#385—22no—
146#385—22no—
147#385—22no—
148#384C2314yes—
149#382C2314yes—
150#382C2314yes—
151#381—21no—
152#381—21no—
153#381—22no—
154#381—22no—
155#381D412yes—
156#380—24no—
157#380D412yes—
158#380D412yes—
159#380C2312yes—
160#380C2312yes—
161#380C2312yes—
162#380C2312yes—
163#379—21no—
164#379—21no—
165#379D411yescharacteristic
166#379D411yescharacteristic
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4C2312yes—
172C4×C22⋊C4C2312yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187C25C2×D411yescharacteristic
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—44no—
191C22×Q8—24no—
192C22×Q8C2411yescharacteristic
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C2411yescharacteristic
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—21no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—84no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4#4114yes—
208C23×C4#4114yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—44no—
215C2×M16—44no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—44no—
219C4⋊Q8—44no—
220#123—44no—
221#123—84no—
222#123—84no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#120—44no—
232C4×D4—44no—
233C4×D4—44no—
234#117—22no—
235#117—22no—
236#117—44no—
237#117—44no—
238#117—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2411yescharacteristic
261C2×C4⋊C4C2411yescharacteristic
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2414yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42C2414yes—
287#103—44no—
288#103—44no—
289#102—44no—
290#102—44no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295#97—44no—
296C4×C8—44no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D414yes—
310#95C2×D414yes—
311C24—21no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24#12414yes—
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319#41—84no—
320#41—84no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—24no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—24no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—816no—
360C22×C4#12414yes—
361C22×C4#12414yes—
362C22×C4#12414yes—
363C22×C4C22×D414yes—
364C22×C4C22×D414yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2511yescharacteristic
375M16—84no—
376M16—84no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C42—24no—
424C42—24no—
425C42—42no—
426C42—82no—
427C42—82no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—416no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23C2×#12412yes—
446C23C2×#12412yes—
447C23C2×#12412yes—
448C23C23×D411yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—168no—
452D4—42no—
453D4—42no—
454D4—84no—
455D4—84no—
456D4—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4—88no—
494C2×C4C2×#12412yes—
495C2×C4C2×#12412yes—
496C2×C4C2×#12412yes—
497C2×C4C22×#4111yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4#58514yes—
500C8—44no—
501C8—44no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—416no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—84no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22C22×#12411yescharacteristic
520C22#348512yes—
521C22#358512yes—
522C22#358612yes—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—44no—
527C4—44no—
528C4—84no—
529C4—84no—
530C4—81no—
531C4—84no—
532C4—81no—
533C4—81no—
534C4#349112yes—
535C4#349112yes—
536C2—42no—
537C2—42no—
538C2—42no—
539C2—161no—
540C2#3545511yescharacteristic
541C2#6061911yescharacteristic
542C2#6062011yescharacteristic
543C2#6177514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
1. #37480C2semidirect
3. #29120C2semidirect
4. #27971C2semidirect
5. #27721C2semidirect
6. #27719C2semidirect
7. #27705C2semidirect
8. #27703C2semidirect
11. #20700C2semidirect
12. #20700C2semidirect
13. #20529C2semidirect
40. #2101C22semidirect
41. #2101C22semidirect
42. #2100C22semidirect
43. #2099C22semidirect
44. #2074C22semidirect
45. #2073C22semidirect
46. #2024C22semidirect
47. #2023C22semidirect
48. #1993C22semidirect
49. #1990C22semidirect
50. #1936C22semidirect
51. #1935C22semidirect
52. #1934C22semidirect
53. #1933C22semidirect
62. #1836C22semidirect
63. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#352272
#291202
#279712
#277212
#277192
#277052
#277032
#217662
#207002
#205292