Finite Group Databasebeta

← #6471007

Subgroups — #6471007

Order 512.

1714 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, complemented
2#34201C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#32965C212yesmaximal, complemented
5#29976C212yesmaximal, complemented
6#29964C212yesmaximal, complemented
7#26437C214yesmaximal, complemented
8#26435C214yesmaximal, complemented
9#26359C214yesmaximal, complemented
10#26353C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic, complemented
12#21769C212yesmaximal
13#20846C212yesmaximal, complemented
14C2×#525C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#475C2212yescomplemented
18C2×#476C2212yescomplemented
19C2×#428C2214yes—
20C2×#407C2212yes—
21#2841—22no—
22#2659—22no—
23#2646—22no—
24#2629C2212yes—
25#2628C2212yes—
26#2442C2214yes—
27#2440C2214yes—
28#2437—22no—
29#2363C2214yes—
30#2362C2214yes—
31#2354—22no—
32#2352—22no—
33#2321—22no—
34#2317C2211yescharacteristic
35#2311C2211yescharacteristic
36C4×#123—22no—
37#2280C2211yescharacteristic
38C2×#380C2212yes—
39#2274C2211yescharacteristic
40#2101C2214yescomplemented
41#2095C2212yescomplemented
42#2093C2212yescomplemented
43#2087C2214yescomplemented
44#2086C2214yescomplemented
45#2068C2214yescomplemented
46#2067C2214yescomplemented
47#2057C2214yescomplemented
48#2026C2212yescomplemented
49#2025C2212yescomplemented
50#1989C2214yescomplemented
51#1948C2214yescomplemented
52#1920C2214yescomplemented
53#1919C2214yescomplemented
54#1895C2214yes—
55#1894C2214yes—
56#1874C2212yes—
57#1874C2212yes—
58#1872C2212yes—
59#1872C2212yes—
60#1865C2214yescomplemented
61#1864C2214yescomplemented
62#1854C2214yes—
63#1854C2214yes—
64#1853C2214yes—
65#1852C2214yes—
66#1851C2214yes—
67#1851C2214yes—
68#1825C2214yes—
69#1805C2212yescomplemented
70#1804C2212yescomplemented
71#1798C2212yes—
72#1798C2212yes—
73#1788C2214yes—
74#1736C2214yes—
75C22×#41C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C22×M16C2314yes—
78C2×C4⋊Q8C2312yes—
79C2×#125C2312yes—
80#525—24no—
81#525Q818yescomplemented
82C2×#123—21no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8C2312yes—
85C2×#121C2312yes—
86C2×#120—21no—
87#520—44no—
88C2×C4×Q8C2312yes—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4C2312yes—
92C2×#117—22no—
93C2×#117—21no—
94C2×#117—44no—
95C2×#117C2314yes—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C42—21no—
107#476—44no—
108#475—44no—
109#428—48no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121#407—28no—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125#400—22no—
126#400—24no—
127#399—22no—
128#398—24no—
129#398—24no—
130#398—24no—
131#396—24no—
132#395—22no—
133#395—24no—
134#393—22no—
135#391C2312yes—
136#390—24no—
137#390C2312yes—
138#389C2314yes—
139#389C2314yes—
140#389C2314yes—
141#388—24no—
142#388—24no—
143#388—24no—
144#387—24no—
145#386—22no—
146#386—24no—
147#385—22no—
148#385—24no—
149#385—24no—
150#384—21no—
151#384C2312yes—
152#382—21no—
153#382C2312yes—
154#381D412yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—24no—
159#380D412yes—
160#380D412yes—
161#380C2314yes—
162#380C2314yes—
163#379D412yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4C2314yes—
170C2×#95—22no—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2314yes—
180#336C2312yes—
181#336C2312yes—
182C25C2×D411yescharacteristic
183C2×#41—22no—
184C2×#41—42no—
185C2×#41—42no—
186C2×#41C2×Q814yes—
187C22×Q8—22no—
188C22×Q8C2411yescharacteristic
189C22×D4—21no—
190C22×D4—22no—
191C22×D4C2411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×Q814yes—
203C23×C4#4114yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C2×M16—28no—
207C2×M16—44no—
208C2×M16—44no—
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213#125—44no—
214#123—28no—
215C22⋊Q8—24no—
216C22⋊Q8—88no—
217C22⋊Q8C2×Q818yes—
218#121—24no—
219#121C2×Q818yes—
220#120—28no—
221C4×Q8—24no—
222C4×Q8—42no—
223C4×D4—216no—
224C4×D4—24no—
225C4×D4—42no—
226C4×D4—48no—
227#117—22no—
228#117—28no—
229#117—42no—
230#117—42no—
231#117—44no—
232#117—48no—
233#117—44no—
234#117C2×Q814yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—84no—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D412yes—
272C2×C22⋊C4C2×D414yes—
273C2×C22⋊C4C2×D414yes—
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42C2412yes—
280C2×C42C2412yes—
281#107—44no—
282#107—44no—
283#106—28no—
284#106—28no—
285C4⋊C8—44no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288#103—44no—
289#103—44no—
290#102—44no—
291#102—44no—
292C22⋊C8—28no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—22no—
303#95—24no—
304#95C2×D414yes—
305#95C2×D414yes—
306C24—21no—
307C24—41no—
308C24—42no—
309C24—41no—
310C24—42no—
311C24C22⋊Q814yes—
312C24C22×D412yes—
313C24C22×D411yescharacteristic
314#41—48no—
315#41—48no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—84no—
320C2×Q8—82no—
321C2×Q8C22×Q812yes—
322C2×D4—24no—
323C2×D4—28no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4C22×Q812yes—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—84no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4C22⋊Q818yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22×D412yes—
366C22×C4C22×D414yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×Q812yes—
371C22×C4C22×Q814yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4114yes—
374C22×C4C2511yescharacteristic
375M16—416no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—84no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—82no—
404C4⋊C4C22×Q814yes—
405C4⋊C4H(5,𝔽2)14yes—
406C22⋊C4—24no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—24no—
422C42—48no—
423C42—44no—
424C42—84no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—44no—
437C23—48no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—84no—
443C23—82no—
444C23—82no—
445C23C2×C22⋊Q814yes—
446C23C2×C22⋊Q812yes—
447C23C23×D411yescharacteristic
448Q8—24no—
449Q8—42no—
450Q8—164no—
451D4—24no—
452D4—48no—
453D4—42no—
454D4—88no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—164no—
492C2×C4—164no—
493C2×C4—164no—
494C2×C4C2×C22⋊Q812yes—
495C2×C4C2×C22⋊Q814yes—
496C2×C4C23×Q811yescharacteristic
497C2×C4C22×#4111yescharacteristic
498C2×C4C2×H(5,𝔽2)12yes—
499C2×C4#58512yes—
500C8—44no—
501C8—44no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—41no—
511C22—42no—
512C22—41no—
513C22—42no—
514C22—44no—
515C22—42no—
516C22—88no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22C22×C22⋊Q811yescharacteristic
521C22#348411yescharacteristic
522C22#348511yescharacteristic
523C22#358512yes—
524C22#358612yes—
525C4—22no—
526C4—24no—
527C4—44no—
528C4—44no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—42no—
533C4—84no—
534C4—82no—
535C4—84no—
536C4#345912yes—
537C4#350112yes—
538C2—44no—
539C2—42no—
540C2—82no—
541C2#3498511yescharacteristic
542C2#6049312yes—
543C2#6164911yescharacteristic
544C2#6165111yescharacteristic
545C2#6177712yes—
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. #37477C2semidirect
3. #33157C2semidirect
4. #32965C2semidirect
5. #29976C2semidirect
6. #29964C2semidirect
7. #26437C2semidirect
8. #26435C2semidirect
9. #26359C2semidirect
10. #26353C2semidirect
11. #21935C2semidirect
13. #20846C2semidirect
17. C2×#475C22semidirect
18. C2×#476C22semidirect
40. #2101C22semidirect
41. #2095C22semidirect
42. #2093C22semidirect
43. #2087C22semidirect
44. #2086C22semidirect
45. #2068C22semidirect
46. #2067C22semidirect
47. #2057C22semidirect
48. #2026C22semidirect
49. #2025C22semidirect
50. #1989C22semidirect
51. #1948C22semidirect
52. #1920C22semidirect
53. #1919C22semidirect
60. #1865C22semidirect
61. #1864C22semidirect
69. #1805C22semidirect
70. #1804C22semidirect
81. #525Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#342012
#331572
#329652
#299762
#299642
#264372
#264352
#263592
#263532
#219352
#217692
#208462