Finite Group Databasebeta

← #482231

Subgroups — #482231

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60707C211yesmaximal, characteristic
2#33422C211yesmaximal, characteristic, complemented
3#20768C211yesmaximal, characteristic, complemented
4#11866C212yesmaximal, complemented
5#11864C212yesmaximal, complemented
6#11862C212yesmaximal, complemented
7#11861C212yesmaximal, 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—
28#2905—21no—
29C2×#405C2211yescharacteristic
30#2164—22no—
31#2163C2212yescomplemented
32#2162C2212yescomplemented
33#2151—21no—
34#2112C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36#2108—21no—
37#1838—22no—
38#1837C2212yescomplemented
39#1837C2212yescomplemented
40#1738—21no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1350C2211yescharacteristic, complemented
44#1350C2212yescomplemented
45#1350C2211yescharacteristic, complemented
46#1349C2212yes—
47#1349C2212yes—
48#1349C2212yes—
49#1349C2212yes—
50#1349C2211yescharacteristic, complemented
51#1349C2212yescomplemented
52#1349C2211yescharacteristic, complemented
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#428—21no—
114#427C2×C411yescharacteristic
115C2×#107C2×C411yescharacteristic
116#424—42no—
117#424—41no—
118#424—41no—
119#423—22no—
120#423—22no—
121#418—21no—
122#417—21no—
123#416—21no—
124C2×#102—21no—
125#405C2×C411yescharacteristic
126#405C2×C412yes—
127#405C2×C411yescharacteristic
128#405D411yescharacteristic
129#405D411yescharacteristic
130#405D412yes—
131C2×#97D412yes—
132C2×C4×C8C2312yes—
133C4⋊C16—22no—
134C4⋊C16—22no—
135C22⋊C16—21no—
136C22⋊C16—21no—
137C22⋊C16C2311yescharacteristic
138C22⋊C16C2311yescharacteristic
139C22⋊C16C2312yes—
140#346—22no—
141C4×C16—22no—
142#341—22no—
143#341—22no—
144#341—21no—
145#341—21no—
146#341C2311yescharacteristic
147#341C2312yes—
148#341C2311yescharacteristic
149C2×#41—21no—
150C2×#41—42no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4C2×D411yescharacteristic
154AGL(1,ℤ/8)—44no—
155#135—42no—
156C2×Q16C22×C411yescharacteristic
157C2×SD16—21no—
158C2×SD16—22no—
159C2×D8—21no—
160C2×D8—22no—
161C2×D8—42no—
162C2×D8C22×C412yes—
163C2×D8C22×C411yescharacteristic
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—42no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16C22⋊C411yescharacteristic
175C2×M16C22⋊C411yescharacteristic
176C2×M16C22⋊C412yes—
177C22×C8—42no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D412yes—
185C22×C8C2×D411yescharacteristic
186C22×C8C2411yescharacteristic
187C4⋊Q8—22no—
188C4⋊Q8—41no—
189#127—22no—
190#126—42no—
191#126—42no—
192#125—22no—
193#125—41no—
194#124—22no—
195#124—42no—
196#124—42no—
197#123—41no—
198#123—42no—
199#123—42no—
200#123—42no—
201#123—42no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206#121—22no—
207#121—21no—
208#121—22no—
209#121—21no—
210#121—41no—
211#121—41no—
212#121—42no—
213#121—41no—
214#121—41no—
215#121—42no—
216#121—42no—
217#120—41no—
218#120—42no—
219C4×Q8—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—42no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—42no—
229C4×D4—42no—
230#117—21no—
231#117—41no—
232#117—42no—
233#117C22⋊C411yescharacteristic
234#117C22⋊C411yescharacteristic
235#117C22⋊C412yes—
236#117D812yes—
237#117D811yescharacteristic
238#117D811yescharacteristic
239C2×C4⋊C4—21no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4SD1611yescharacteristic
242C2×C4⋊C4SD1611yescharacteristic
243C2×C22⋊C4—41no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4SD1612yes—
246C2×C42C2×D412yes—
247M32—42no—
248C2×C16—21no—
249C2×C16—22no—
250C2×C16—21no—
251C2×C16—42no—
252#107—21no—
253#107—22no—
254#107C22×C412yes—
255#107C22×C411yescharacteristic
256#107C22×C411yescharacteristic
257#106—21no—
258#106—22no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263#103—21no—
264#103—22no—
265#103—21no—
266#102—22no—
267#102—21no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—21no—
273#97C22⋊C412yes—
274#97C22⋊C412yes—
275#97C22⋊C412yes—
276#97C22⋊C412yes—
277C4×C8C22×C412yes—
278C4×C8C22×C412yes—
279C4×C8C2×D412yes—
280C4×C8C2×D412yes—
281C24—41no—
282C24C2×SD1611yescharacteristic
283#41—42no—
284#41—42no—
285#41—164no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—162no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C2×SD1612yes—
314C22×C4C2×SD1612yes—
315C22×C4C22×D411yescharacteristic
316Q16—41no—
317SD16—42no—
318SD16—44no—
319D8—42no—
320D8—41no—
321D8—42no—
322D8—44no—
323M16—22no—
324M16—22no—
325M16—42no—
326M16—42no—
327M16—41no—
328M16—41no—
329M16—42no—
330M16—41no—
331M16—41no—
332M16—82no—
333M16—82no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—82no—
345C2×C8—82no—
346C2×C8C2×C22⋊C412yes—
347C2×C8C2×C22⋊C412yes—
348C2×C8C2×C22⋊C411yescharacteristic
349C2×C8C2×C22⋊C411yescharacteristic
350C2×C8C23×C411yescharacteristic
351C2×C8C22×D411yescharacteristic
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4#10211yescharacteristic
363C4⋊C4#10211yescharacteristic
364C4⋊C4#10211yescharacteristic
365C4⋊C4#10211yescharacteristic
366C4⋊C4#10212yes—
367C4⋊C4#10212yes—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4#10212yes—
381C22⋊C4#10212yes—
382C22⋊C4#10212yes—
383C22⋊C4#10212yes—
384C42—41no—
385C42—42no—
386C42C2×C22⋊C412yes—
387C42C2×C22⋊C412yes—
388C42C2×D812yes—
389C42C2×D812yes—
390C16—41no—
391C16—41no—
392C16—42no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—82no—
399C23—161no—
400C23—161no—
401C23C2×#10211yescharacteristic
402C23C2×#10211yescharacteristic
403C23C22×SD1611yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—162no—
407D4—42no—
408D4—41no—
409D4—41no—
410D4—42no—
411D4—84no—
412D4—82no—
413D4—82no—
414D4—82no—
415D4—162no—
416D4—162no—
417D4—161no—
418D4—162no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4—161no—
437C2×C4#41012yes—
438C2×C4C2×#10212yes—
439C2×C4C2×#10212yes—
440C2×C4C2×#10212yes—
441C2×C4C2×#10212yes—
442C2×C4C2×#10211yescharacteristic
443C2×C4C2×#10211yescharacteristic
444C2×C4#42112yes—
445C2×C4C22×C22⋊C411yescharacteristic
446C2×C4C22×D811yescharacteristic
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—22no—
452C8—22no—
453C8—82no—
454C8—81no—
455C8—81no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—81no—
468C22—82no—
469C22—81no—
470C22—82no—
471C22—322no—
472C22#151712yes—
473C22C22×#10211yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—81no—
481C4—82no—
482C4—81no—
483C4—81no—
484C4#151712yes—
485C4C2×#41011yescharacteristic
486C4C2×#42111yescharacteristic
487C2—22no—
488C2—41no—
489C2—41no—
490C2—161no—
491C2—162no—
492C2C2×#151711yescharacteristic
493C2#3338511yescharacteristic
494C2#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. #20768C2semidirect
4. #11866C2semidirect
5. #11864C2semidirect
6. #11862C2semidirect
7. #11861C2semidirect
8. #11711C2semidirect
9. #11710C2semidirect
10. #11709C2semidirect
11. C2×#1349C2semidirect
31. #2163C22semidirect
32. #2162C22semidirect
38. #1837C22semidirect
39. #1837C22semidirect
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
#207682
#118662
#118642
#118622
#118612
#117112
#117102
#117092
C2×#13492