Finite Group Databasebeta

← #481591

Subgroups — #481591

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60707C211yesmaximal, characteristic
2#33422C211yesmaximal, characteristic, complemented
3#20459C211yesmaximal, characteristic, complemented
4#11751C212yesmaximal, complemented
5#11750C212yesmaximal, complemented
6#11749C212yesmaximal, complemented
7#11747C212yesmaximal, complemented
8#11711C211yesmaximal, characteristic, complemented
9#11710C211yesmaximal, characteristic, complemented
10#11709C211yesmaximal, characteristic, complemented
11C2×#1349C211yesmaximal, characteristic, complemented
12#3494—21no—
13#3180—22no—
14#3179—22no—
15#3176C412yes—
16#3175C412yes—
17#3169C2212yes—
18#3168C2212yes—
19#3164C411yescharacteristic
20#3155C411yescharacteristic
21#3136—21no—
22#3055—21no—
23#3050C2211yescharacteristic
24C2×#466C2211yescharacteristic
25#2951C411yescharacteristic
26#2948C411yescharacteristic
27#2946—21no—
28C2×#430—21no—
29#2873—21no—
30C2×#405C2211yescharacteristic
31#2164—22no—
32#2163C2212yescomplemented
33#2162C2212yescomplemented
34#2151—21no—
35#2112C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37#2108—21no—
38#1778—22no—
39C4×#108C2212yescomplemented
40C4×#108C2212yescomplemented
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1350C2211yescharacteristic, complemented
44#1350C2211yescharacteristic, complemented
45#1350C2212yescomplemented
46#1349C2212yes—
47#1349C2212yes—
48#1349C2212yes—
49#1349C2212yes—
50#1349C2211yescharacteristic, complemented
51#1349C2211yescharacteristic, complemented
52#1349C2212yescomplemented
53C2×AGL(1,ℤ/8)—21no—
54C2×#135C2×C411yescharacteristic
55C22×D8C2×C411yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C22×M16D411yescharacteristic
59C23×C8C2311yescharacteristic
60#555—41no—
61#553—41no—
62#553—41no—
63#551—42no—
64#547—41no—
65#546—42no—
66#544—21no—
67#542—22no—
68#540—22no—
69#535—21no—
70#532—22no—
71#529—22no—
72#525—21no—
73C2×#121—21no—
74#520—21no—
75#518—21no—
76C2×#117D411yescharacteristic
77C2×M32—21no—
78C22×C16—21no—
79#501—22no—
80#500C2×C412yes—
81#499C2×C412yes—
82#496—22no—
83#495C2×C412yes—
84#493C2×C412yes—
85#490—22no—
86#489—22no—
87#488—22no—
88#486—22no—
89#469—22no—
90#468—22no—
91#467C2×C411yescharacteristic
92#467C2311yescharacteristic
93#466—22no—
94#466—21no—
95#466—21no—
96#466C2×C412yes—
97#466C2×C411yescharacteristic
98#466C2312yes—
99#466C2311yescharacteristic
100#465—22no—
101#465—21no—
102#465—21no—
103#442—22no—
104#442—22no—
105#440—22no—
106#440—22no—
107C4×Q16C2×C412yes—
108C4×SD16—22no—
109C4×D8—22no—
110C4×D8C2×C412yes—
111C4×D8C2×C412yes—
112C4×D8C2×C412yes—
113#430—28no—
114C2×#108—22no—
115C2×#108—22no—
116#428—21no—
117#427C2×C411yescharacteristic
118C2×#107C2×C411yescharacteristic
119#418—21no—
120#417—21no—
121#416—21no—
122C2×#102—21no—
123#405—42no—
124#405—41no—
125#405—41no—
126#405C2×C411yescharacteristic
127#405C2×C412yes—
128#405C2×C411yescharacteristic
129#405D411yescharacteristic
130#405D411yescharacteristic
131#405D412yes—
132C4×M16—22no—
133C4×M16—22no—
134C2×#97D412yes—
135C2×C4×C8C2312yes—
136C4⋊C16—22no—
137C4⋊C16—22no—
138C22⋊C16—21no—
139C22⋊C16—21no—
140C22⋊C16C2311yescharacteristic
141C22⋊C16C2311yescharacteristic
142C22⋊C16C2312yes—
143#346—22no—
144C4×C16—22no—
145#341—21no—
146#341—21no—
147#341C2312yes—
148#341C2311yescharacteristic
149#341C2311yescharacteristic
150C2×#41—21no—
151C2×#41—42no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4C2×D411yescharacteristic
155AGL(1,ℤ/8)—44no—
156#135—42no—
157C2×Q16C22×C411yescharacteristic
158C2×SD16—21no—
159C2×SD16—22no—
160C2×D8—21no—
161C2×D8—22no—
162C2×D8—42no—
163C2×D8C22×C412yes—
164C2×D8C22×C411yescharacteristic
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—22no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16C22⋊C411yescharacteristic
176C2×M16C22⋊C412yes—
177C2×M16C22⋊C411yescharacteristic
178C22×C8—42no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C2×D412yes—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2411yescharacteristic
188C4⋊Q8—22no—
189C4⋊Q8—41no—
190#127—22no—
191#126—42no—
192#126—42no—
193#125—22no—
194#125—41no—
195#124—22no—
196#124—42no—
197#124—42no—
198#123—41no—
199#123—42no—
200#123—42no—
201#123—42no—
202#123—42no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207#121—22no—
208#121—21no—
209#121—22no—
210#121—21no—
211#121—41no—
212#121—41no—
213#121—42no—
214#121—41no—
215#121—41no—
216#121—42no—
217#121—42no—
218#120—41no—
219#120—42no—
220C4×Q8—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—42no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—42no—
230C4×D4—42no—
231#117—21no—
232#117—41no—
233#117—42no—
234#117C22⋊C411yescharacteristic
235#117C22⋊C411yescharacteristic
236#117C22⋊C412yes—
237#117D812yes—
238#117D811yescharacteristic
239#117D811yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4SD1611yescharacteristic
243C2×C4⋊C4SD1611yescharacteristic
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4SD1612yes—
247C2×C42C2×D412yes—
248M32—42no—
249C2×C16—21no—
250C2×C16—22no—
251C2×C16—21no—
252C2×C16—42no—
253#108—28no—
254#108—28no—
255#107—21no—
256#107—22no—
257#107C22×C412yes—
258#107C22×C411yescharacteristic
259#107C22×C411yescharacteristic
260#106—21no—
261#106—22no—
262#103—21no—
263#103—22no—
264#103—21no—
265#102—22no—
266#102—21no—
267#102—22no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—21no—
272#97—42no—
273#97—42no—
274#97C22⋊C412yes—
275#97C22⋊C412yes—
276#97C22⋊C412yes—
277#97C22⋊C412yes—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8C22×C412yes—
281C4×C8C22×C412yes—
282C4×C8C2×D412yes—
283C4×C8C2×D412yes—
284C24—41no—
285C24C2×SD1611yescharacteristic
286#41—42no—
287#41—42no—
288#41—164no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—162no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×SD1611yescharacteristic
316C22×C4C2×SD1612yes—
317C22×C4C2×SD1612yes—
318C22×C4C22×D411yescharacteristic
319Q16—41no—
320SD16—42no—
321SD16—44no—
322D8—42no—
323D8—41no—
324D8—42no—
325D8—44no—
326M16—22no—
327M16—24no—
328M16—24no—
329M16—22no—
330M16—22no—
331M16—44no—
332M16—42no—
333M16—41no—
334M16—41no—
335M16—44no—
336M16#10414yes—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8C2×C22⋊C412yes—
350C2×C8C2×C22⋊C411yescharacteristic
351C2×C8C2×C22⋊C411yescharacteristic
352C2×C8C2×C22⋊C412yes—
353C2×C8C23×C411yescharacteristic
354C2×C8C22×D411yescharacteristic
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4#10211yescharacteristic
366C4⋊C4#10211yescharacteristic
367C4⋊C4#10211yescharacteristic
368C4⋊C4#10211yescharacteristic
369C4⋊C4#10212yes—
370C4⋊C4#10212yes—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—41no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—41no—
383C22⋊C4#10212yes—
384C22⋊C4#10212yes—
385C22⋊C4#10212yes—
386C22⋊C4#10212yes—
387C42—41no—
388C42—42no—
389C42C2×C22⋊C412yes—
390C42C2×C22⋊C412yes—
391C42C2×D812yes—
392C42C2×D812yes—
393C16—41no—
394C16—41no—
395C16—42no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—82no—
402C23—161no—
403C23—161no—
404C23C2×#10211yescharacteristic
405C23C2×#10211yescharacteristic
406C23C22×SD1611yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—162no—
410D4—42no—
411D4—41no—
412D4—41no—
413D4—42no—
414D4—84no—
415D4—82no—
416D4—82no—
417D4—82no—
418D4—162no—
419D4—162no—
420D4—161no—
421D4—162no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—162no—
437C2×C4—162no—
438C2×C4—162no—
439C2×C4—161no—
440C2×C4#41012yes—
441C2×C4C2×#10212yes—
442C2×C4C2×#10212yes—
443C2×C4C2×#10212yes—
444C2×C4C2×#10212yes—
445C2×C4C2×#10211yescharacteristic
446C2×C4C2×#10211yescharacteristic
447C2×C4C2×#10412yes—
448C2×C4C22×C22⋊C411yescharacteristic
449C2×C4C22×D811yescharacteristic
450C8—21no—
451C8—21no—
452C8—21no—
453C8—21no—
454C8—22no—
455C8—44no—
456C8—42no—
457C8—42no—
458C8C2×#10414yes—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22—82no—
472C22—81no—
473C22—82no—
474C22—322no—
475C22#150212yes—
476C22C22×#10211yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—81no—
484C4—82no—
485C4—81no—
486C4—81no—
487C4#150212yes—
488C4C2×#41011yescharacteristic
489C4C22×#10411yescharacteristic
490C2—22no—
491C2—41no—
492C2—41no—
493C2—161no—
494C2—162no—
495C2C2×#150211yescharacteristic
496C2#3338511yescharacteristic
497C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#60707C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33422C2semidirect
3. #20459C2semidirect
4. #11751C2semidirect
5. #11750C2semidirect
6. #11749C2semidirect
7. #11747C2semidirect
8. #11711C2semidirect
9. #11710C2semidirect
10. #11709C2semidirect
11. C2×#1349C2semidirect
32. #2163C22semidirect
33. #2162C22semidirect
39. C4×#108C22semidirect
40. C4×#108C22semidirect
43. #1350C22semidirect
44. #1350C22semidirect
45. #1350C22semidirect
50. #1349C22semidirect
51. #1349C22semidirect
52. #1349C22semidirect

Maximal subgroups

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

subgroupindex
#607072
#334222
#204592
#117512
#117502
#117492
#117472
#117112
#117102
#117092
C2×#13492