Finite Group Databasebeta

← #6357465

Subgroups — #6357465

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60968C211yesmaximal, characteristic, complemented
2C2×#3227C212yesmaximal, complemented
3#60800C211yesmaximal, characteristic, complemented
4#60782C212yesmaximal, complemented
5#19724C214yesmaximal, complemented
6#19721C214yesmaximal, complemented
7#19703C214yesmaximal, complemented
8#19701C214yesmaximal, complemented
9#19656C214yesmaximal, complemented
10#19654C214yesmaximal, complemented
11#17325C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#532C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15#3253—22no—
16#3227—24no—
17#3227—24no—
18#3227C2214yes—
19#3227C2214yes—
20#3227C2214yes—
21#3227C2218yescomplemented
22#3218—24no—
23#3212—24no—
24#3211—24no—
25#3210C2214yescomplemented
26#3209C2212yescomplemented
27#3209C2214yescomplemented
28#3208C2212yescomplemented
29#3207C2214yescomplemented
30#3206C2212yescomplemented
31#3206C2214yescomplemented
32#3205C2214yes—
33#3205C2212yescomplemented
34#3192—22no—
35#3184C2214yes—
36C2×#498C2214yescomplemented
37#3152C2212yescomplemented
38#3147C2214yes—
39#3139C2212yescomplemented
40#3131C2214yes—
41C2×#486C2214yescomplemented
42#3109—24no—
43C2×#481—24no—
44C2×#476—24no—
45#3039—22no—
46#3035—24no—
47#3004—22no—
48#3002—24no—
49#2920C2211yescharacteristic
50C2×#431C2212yes—
51#1721C2214yescomplemented
52#1720C2218yescomplemented
53#1720C2218yescomplemented
54#1717C2218yescomplemented
55#1716C2214yescomplemented
56#1716C2218yescomplemented
57#1715C2218yescomplemented
58#1709C2214yescomplemented
59#1708C2218yescomplemented
60#1705C2214yescomplemented
61#1453C2214yescomplemented
62#1452C2214yescomplemented
63#1450C2214yescomplemented
64C22×#41—22no—
65C23×Q8—21no—
66C2×#137—24no—
67C2×#135—28no—
68#533C2312yes—
69#533C2311yescharacteristic
70#532—22no—
71#532—22no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532C2314yes—
77#532C2312yes—
78#532C2311yescharacteristic
79#532C2314yes—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2312yes—
89#529—22no—
90#529—22no—
91#529—22no—
92C2×#124—22no—
93#525—22no—
94#525—21no—
95C2×#121—22no—
96C2×#121—22no—
97C4×#41—22no—
98C4×#41—21no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C22×C4⋊C4—22no—
102C22×C42C2311yescharacteristic
103#498—28no—
104#498C2318yes—
105#498C2318yes—
106#498C2314yes—
107#497—24no—
108#495—28no—
109#494C2314yes—
110#494C2314yes—
111#493C2314yes—
112#493C2314yes—
113#492C2314yes—
114#492C2314yes—
115#492C2314yes—
116#491—24no—
117#488—28no—
118#487C2314yes—
119#487C2314yes—
120#487C2318yes—
121#486—28no—
122#486C2318yes—
123#486C2318yes—
124#486C2314yes—
125#486C2314yes—
126#486C2318yes—
127#481—28no—
128#481—28no—
129#481—216no—
130#481—28no—
131#476—28no—
132#476—28no—
133#476—28no—
134#476—216no—
135#469—28no—
136#464—28no—
137#463—28no—
138#462—24no—
139#462—28no—
140#461—24no—
141#461—28no—
142#460—24no—
143#459—28no—
144#459—24no—
145#452—28no—
146#451—24no—
147#451—28no—
148#450—24no—
149#449—28no—
150#448—24no—
151#448—28no—
152#447—28no—
153#447—24no—
154#442—28no—
155#432D414yes—
156#431D414yes—
157#431D414yes—
158#431C2312yes—
159#431C2312yes—
160#431C2314yes—
161#431C2314yes—
162C2×#106—28no—
163#423D412yes—
164C2×C4⋊C8D414yes—
165#419—24no—
166#416—28no—
167C2×#102—28no—
168#407D412yes—
169C2×C22⋊C8D414yes—
170C4×M16C2312yes—
171C2×C4×C8C2314yes—
172#322C2318yes—
173#322C2314yes—
174C82C2314yes—
175C2×#41—22no—
176C2×#41—21no—
177C2×#41—24no—
178C2×#41—24no—
179C2×#41—42no—
180C2×#41—42no—
181C2×#41—88no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8—84no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—42no—
192C23×C4—41no—
193C23×C4—42no—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196#137—88no—
197#135—816no—
198C2×Q16—28no—
199C2×Q16—28no—
200C2×SD16—28no—
201C2×SD16—28no—
202C2×SD16—28no—
203C2×D8—28no—
204C2×D8—28no—
205C2×M16C2×D414yes—
206C22×C8C2×D418yes—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—41no—
216C4⋊Q8—42no—
217C4⋊Q8—41no—
218C4⋊Q8—42no—
219C4⋊Q8C2414yes—
220C4⋊Q8C2414yes—
221C4⋊Q8C2411yescharacteristic
222C4⋊Q8C2412yes—
223#127—22no—
224#127—22no—
225#127—22no—
226#127C2414yes—
227#127C2411yescharacteristic
228#126—42no—
229#125—22no—
230#125—21no—
231#124—24no—
232#124—24no—
233#124—22no—
234#124—21no—
235#124—42no—
236#124—42no—
237#123—42no—
238#123—41no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242#121—24no—
243#121—24no—
244#121—24no—
245#121—24no—
246#121—22no—
247#121—22no—
248#121—22no—
249#121—42no—
250#121—42no—
251#121—42no—
252#121—42no—
253#120—41no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—42no—
270C4×D4—42no—
271#117—22no—
272#117—21no—
273#117—42no—
274#117—41no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C42—41no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2412yes—
292C2×C42C2411yescharacteristic
293#106—216no—
294#106—216no—
295#106—28no—
296C4⋊C8C2×D414yes—
297C4⋊C8C2×D414yes—
298C4⋊C8C2×D418yes—
299C4⋊C8C2×D418yes—
300#103—28no—
301#103—28no—
302#103—216no—
303#102—216no—
304#102—216no—
305#102—28no—
306#102—28no—
307#102—216no—
308C22⋊C8C2×D414yes—
309C22⋊C8C2×D414yes—
310C22⋊C8C2×D418yes—
311C22⋊C8C2×D418yes—
312#97C2×D414yes—
313C4×C8C2×D418yes—
314C4×C8C2414yes—
315C4×C8C2414yes—
316C4×C8C2414yes—
317C24—42no—
318C24#12711yescharacteristic
319#41—44no—
320#41—48no—
321#41—416no—
322#41—88no—
323#41—88no—
324#41—88no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—48no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×Q8—88no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—88no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—48no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—88no—
353C22×C4—28no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—88no—
363C22×C4—84no—
364C22×C4#12712yes—
365C22×C4#12712yes—
366C22×C4#12712yes—
367C22×C4#12712yes—
368C22×C4#12711yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374Q16—88no—
375Q16—88no—
376SD16—88no—
377SD16—88no—
378SD16—88no—
379D8—88no—
380D8—88no—
381M16—44no—
382C2×C8—216no—
383C2×C8C22×D418yes—
384C2×C8C22×D418yes—
385C2×C8C22×D418yes—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C22⋊C4—41no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—41no—
410C42—42no—
411C42#12711yescharacteristic
412C42#12712yes—
413C42#12711yescharacteristic
414C42#12712yes—
415C42C22×D412yes—
416C42C22×D411yescharacteristic
417C42C22×D412yes—
418C42C22×D412yes—
419C42C22×D412yes—
420C42C2511yescharacteristic
421C23—24no—
422C23—41no—
423C23—41no—
424C23—44no—
425C23—84no—
426C23—84no—
427C23—84no—
428C23C2×#12711yescharacteristic
429C23C2×#12712yes—
430Q8—48no—
431Q8—42no—
432Q8—44no—
433Q8—48no—
434Q8—84no—
435Q8—88no—
436Q8—84no—
437D4—48no—
438D4—42no—
439D4—44no—
440D4—48no—
441D4—48no—
442D4—84no—
443D4—84no—
444D4—88no—
445D4—82no—
446D4—88no—
447D4—84no—
448D4—84no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—88no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—88no—
467C2×C4—88no—
468C2×C4—84no—
469C2×C4C2×#12711yescharacteristic
470C2×C4C2×#12712yes—
471C2×C4C2×#12712yes—
472C2×C4C2×#12712yes—
473C2×C4C2×#12712yes—
474C2×C4#57718yes—
475C2×C4C23×D411yescharacteristic
476C2×C4C23×D412yes—
477C8—28no—
478C8—28no—
479C8—28no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—21no—
484C22—44no—
485C22—84no—
486C22—82no—
487C22—84no—
488C22—81no—
489C22—84no—
490C22—168no—
491C22#315414yes—
492C22C22×#12711yescharacteristic
493C4—22no—
494C4—22no—
495C4—21no—
496C4—22no—
497C4—84no—
498C4—82no—
499C4—84no—
500C4—81no—
501C4#315414yes—
502C4C2×#57714yes—
503C4#358512yes—
504C2—24no—
505C2—41no—
506C2—161no—
507C2—164no—
508C2C2×#315412yes—
509C2#6064411yescharacteristic
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60968C2semidirect
2. C2×#3227C2semidirect
3. #60800C2semidirect
4. #60782C2semidirect
5. #19724C2semidirect
6. #19721C2semidirect
7. #19703C2semidirect
8. #19701C2semidirect
9. #19656C2semidirect
10. #19654C2semidirect
11. #17325C2semidirect
21. #3227C22semidirect
25. #3210C22semidirect
26. #3209C22semidirect
27. #3209C22semidirect
28. #3208C22semidirect
29. #3207C22semidirect
30. #3206C22semidirect
31. #3206C22semidirect
33. #3205C22semidirect
36. C2×#498C22semidirect
37. #3152C22semidirect
39. #3139C22semidirect
41. C2×#486C22semidirect
51. #1721C22semidirect
52. #1720C22semidirect
53. #1720C22semidirect
54. #1717C22semidirect
55. #1716C22semidirect
56. #1716C22semidirect
57. #1715C22semidirect
58. #1709C22semidirect
59. #1708C22semidirect
60. #1705C22semidirect
61. #1453C22semidirect
62. #1452C22semidirect
63. #1450C22semidirect

Maximal subgroups

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

subgroupindex
#609682
C2×#32272
#608002
#607822
#197242
#197212
#197032
#197012
#196562
#196542
#173252