Finite Group Databasebeta

← #6357956

Subgroups — #6357956

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60971C211yesmaximal, characteristic, complemented
2#60966C212yesmaximal
3#60865C212yesmaximal, complemented
4#60862C211yesmaximal, characteristic
5#19752C214yesmaximal, complemented
6#19751C214yesmaximal, complemented
7#19739C214yesmaximal, complemented
8#19738C214yesmaximal, complemented
9#19736C214yesmaximal, complemented
10#19736C214yesmaximal, complemented
11#17339C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#533C2212yes—
14C2×#533C2211yescharacteristic
15#3255—22no—
16#3252—24no—
17#3240—24no—
18#3237—24no—
19#3236—24no—
20#3231C2214yes—
21#3231C2214yes—
22#3231C2214yes—
23#3231C2214yes—
24#3230C2212yes—
25#3230C2214yes—
26#3229C2214yes—
27#3229C2214yes—
28#3228C2214yes—
29#3227C2212yes—
30#3210C2214yes—
31#3209C2212yes—
32#3209C2214yes—
33#3206C2212yes—
34#3196—24no—
35#3195—22no—
36#3185C2214yes—
37#3185C2214yes—
38#3182C2214yes—
39#3165C2212yes—
40#3165C2214yes—
41#3141C2212yes—
42#3139C2214yes—
43#3117—24no—
44#3116—24no—
45#3113—24no—
46#3095—24no—
47#3093—22no—
48#3080—22no—
49#3078—24no—
50#2920C2212yes—
51#2920C2211yescharacteristic
52#1723C2214yescomplemented
53#1723C2214yescomplemented
54#1723C2214yescomplemented
55#1720C2214yescomplemented
56#1719C2214yescomplemented
57#1718C2214yescomplemented
58#1718C2214yescomplemented
59#1718C2214yescomplemented
60#1718C2214yescomplemented
61#1716C2214yescomplemented
62#1716C2214yescomplemented
63#1714C2214yescomplemented
64#1710C2214yescomplemented
65#1708C2214yescomplemented
66#1707C2214yescomplemented
67#1707C2214yescomplemented
68#1455C2214yes—
69#1454C2214yes—
70#1454C2214yes—
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533C2314yes—
81#533C2311yescharacteristic
82#533C2314yes—
83#533C2314yes—
84#533C2312yes—
85#532C2311yescharacteristic
86#532C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90#529—22no—
91#529—22no—
92#529—22no—
93C2×#125—21no—
94C2×#125—22no—
95#525—21no—
96#525—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C4×#41—21no—
101C4×#41—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×#117—21no—
106C2×#117—22no—
107C22×C42C2311yescharacteristic
108#501—28no—
109#501—24no—
110#500C2314yes—
111#500C2314yes—
112#500C2314yes—
113#500C2314yes—
114#498C2314yes—
115#498C2314yes—
116#498C2314yes—
117#498C2314yes—
118#498C2314yes—
119#494C2314yes—
120#494C2314yes—
121#492C2314yes—
122#491—28no—
123#489—24no—
124#487C2314yes—
125#487C2314yes—
126#487C2314yes—
127#487C2314yes—
128#487C2314yes—
129#487C2314yes—
130#487C2314yes—
131#487C2314yes—
132#487C2314yes—
133#487C2314yes—
134#486C2314yes—
135#486C2314yes—
136#485—28no—
137#485—28no—
138#485—28no—
139#485—28no—
140#484—28no—
141#484—24no—
142#483—28no—
143#483—28no—
144#482—28no—
145#481—24no—
146#479—28no—
147#479—28no—
148#479—28no—
149#479—28no—
150#478—28no—
151#477—28no—
152#477—24no—
153#476—24no—
154#475—28no—
155#475—28no—
156#470—28no—
157#470—28no—
158#468—28no—
159#464—28no—
160#462—28no—
161#462—24no—
162#461—24no—
163#452—28no—
164#451—28no—
165#451—24no—
166#448—24no—
167#441—28no—
168#441—28no—
169#440—28no—
170#432D414yes—
171#432D414yes—
172#432D414yes—
173#431C2312yes—
174#431C2312yes—
175#431C2314yes—
176#431C2314yes—
177#428—28no—
178#428—24no—
179#423D414yes—
180#423D412yes—
181#419—28no—
182#417—24no—
183#407D412yes—
184#407D414yes—
185C4×M16C2314yes—
186C4×M16C2312yes—
187#322C2314yes—
188#322C2314yes—
189#322C2314yes—
190C82C2314yes—
191C2×#41—21no—
192C2×#41—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C2×Q16—28no—
199C2×Q16—28no—
200C2×SD16—28no—
201C2×M16C2×D414yes—
202C2×M16C2×D418yes—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8C2414yes—
213C4⋊Q8C2414yes—
214C4⋊Q8C2414yes—
215C4⋊Q8C2411yescharacteristic
216C4⋊Q8C2412yes—
217#127C2411yescharacteristic
218#126—42no—
219#125—24no—
220#125—24no—
221#125—24no—
222#125—22no—
223#125—21no—
224#125—41no—
225#125—42no—
226#125—41no—
227#125—42no—
228#124—22no—
229#124—21no—
230#123—41no—
231#123—42no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247#121—22no—
248#121—22no—
249#121—22no—
250#120—41no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×Q8—42no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—42no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—21no—
276#117—22no—
277#117—42no—
278#117—42no—
279#117—42no—
280#117—41no—
281#117—42no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C42—41no—
294C2×C42—42no—
295C2×C42—41no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2411yescharacteristic
302C2×C42C2412yes—
303#107—28no—
304#107—28no—
305#107—28no—
306#107—28no—
307#106—28no—
308#106—28no—
309#106—28no—
310#106—28no—
311#106—28no—
312C4⋊C8C2×D414yes—
313C4⋊C8C2×D418yes—
314C4⋊C8C2×D414yes—
315C4⋊C8C2×D418yes—
316#103—28no—
317#103—28no—
318#103—28no—
319#103—28no—
320#103—28no—
321#103—28no—
322#103—28no—
323#103—28no—
324#103—28no—
325#103—28no—
326#102—28no—
327#102—28no—
328C22⋊C8C2×D414yes—
329C22⋊C8C2×D414yes—
330C22⋊C8C2×D418yes—
331C22⋊C8C2×D418yes—
332#97C2×D418yes—
333#97C2×D414yes—
334C4×C8C2414yes—
335C4×C8C2414yes—
336C4×C8C2414yes—
337C24#12711yescharacteristic
338#41—44no—
339#41—84no—
340C2×Q8—21no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—48no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4#12711yescharacteristic
363C22×C4#12712yes—
364C22×C4#12712yes—
365C22×C4#12712yes—
366C22×C4#12712yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372Q16—88no—
373Q16—88no—
374SD16—88no—
375M16—48no—
376M16—44no—
377C2×C8C22×D418yes—
378C2×C8C22×D418yes—
379C2×C8C22×D418yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—41no—
413C42—42no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42#12711yescharacteristic
418C42#12712yes—
419C42#12711yescharacteristic
420C42#12712yes—
421C42C22×D412yes—
422C42C22×D411yescharacteristic
423C42C22×D412yes—
424C42C22×D412yes—
425C42C22×D412yes—
426C42C2511yescharacteristic
427C23—42no—
428C23—41no—
429C23—41no—
430C23C2×#12712yes—
431C23C2×#12711yescharacteristic
432Q8—48no—
433Q8—48no—
434Q8—44no—
435Q8—48no—
436Q8—42no—
437Q8—84no—
438Q8—84no—
439Q8—84no—
440D4—44no—
441D4—42no—
442D4—82no—
443D4—84no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4C2×#12712yes—
465C2×C4C2×#12712yes—
466C2×C4C2×#12712yes—
467C2×C4C2×#12711yescharacteristic
468C2×C4C2×#12712yes—
469C2×C4C23×D412yes—
470C2×C4C23×D411yescharacteristic
471C8—28no—
472C8—28no—
473C8—28no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—82no—
480C22—81no—
481C22—82no—
482C22C22×#12711yescharacteristic
483C4—21no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—81no—
489C4—82no—
490C4—84no—
491C4—84no—
492C4—84no—
493C4#358614yes—
494C4#358612yes—
495C2—41no—
496C2—42no—
497C2—161no—
498C2#6064511yescharacteristic
499C2#6064512yes—
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
prime radical#3444C22
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. #60971C2semidirect
3. #60865C2semidirect
5. #19752C2semidirect
6. #19751C2semidirect
7. #19739C2semidirect
8. #19738C2semidirect
9. #19736C2semidirect
10. #19736C2semidirect
11. #17339C2semidirect
52. #1723C22semidirect
53. #1723C22semidirect
54. #1723C22semidirect
55. #1720C22semidirect
56. #1719C22semidirect
57. #1718C22semidirect
58. #1718C22semidirect
59. #1718C22semidirect
60. #1718C22semidirect
61. #1716C22semidirect
62. #1716C22semidirect
63. #1714C22semidirect
64. #1710C22semidirect
65. #1708C22semidirect
66. #1707C22semidirect
67. #1707C22semidirect

Maximal subgroups

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

subgroupindex
#609712
#609662
#608652
#608622
#197522
#197512
#197392
#197382
#197362
#173392