Finite Group Databasebeta

← #481410

Subgroups — #481410

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61431C211yesmaximal, characteristic
2#33479C211yesmaximal, characteristic, complemented
3#17837C211yesmaximal, characteristic, complemented
4#11674C212yesmaximal, complemented
5#11671C211yesmaximal, characteristic, complemented
6#11660C211yesmaximal, characteristic, complemented
7#11657C212yesmaximal, complemented
8#11649C211yesmaximal, characteristic, complemented
9#11647C212yesmaximal, complemented
10#11593C212yesmaximal, complemented
11#11591C211yesmaximal, characteristic, complemented
12#3481—21no—
13#3396C411yescharacteristic
14#3380C411yescharacteristic
15#3361—21no—
16#3339C412yes—
17#3338C411yescharacteristic
18#3297—22no—
19#3295—21no—
20#3280C412yes—
21D4×D8C411yescharacteristic
22#3169C2211yescharacteristic
23#3168C2212yes—
24#3106—21no—
25#3105—22no—
26#2991—21no—
27C8×#41C2211yescharacteristic
28C4×#135C2211yescharacteristic
29C2×C4×D8C2212yes—
30#2183—21no—
31#2169—21no—
32#2168—22no—
33#2162C2212yescomplemented
34#2162C2211yescharacteristic, complemented
35C4×M32C2212yescomplemented
36C2×C4×C16C2211yescharacteristic, complemented
37#1574—22no—
38#1574—21no—
39#1567C2212yescomplemented
40#1567C2211yescharacteristic, complemented
41#1564C2211yescharacteristic, complemented
42#1564C2212yescomplemented
43#1332C2211yescharacteristic, complemented
44#1332C2211yescharacteristic, complemented
45#1332C2212yescomplemented
46#1331C2211yescharacteristic
47#1331C2211yescharacteristic
48#1331C2212yes—
49#1331C2212yescomplemented
50#1331C2212yescomplemented
51#1331C2212yescomplemented
52#1331C2212yescomplemented
53C2×#135C2×C411yescharacteristic
54C22×D8C2×C412yes—
55#559—21no—
56#554—21no—
57#554—21no—
58#553—22no—
59#553—41no—
60#551—42no—
61#546—22no—
62#546—41no—
63D42—21no—
64#532—21no—
65#529—22no—
66#529—41no—
67#525—42no—
68C4×#41—21no—
69C4×#41D411yescharacteristic
70C2×C4×D4—22no—
71C2×C4×D4—41no—
72C2×M32—22no—
73C22×C16—21no—
74#500C2×C411yescharacteristic
75#499C2×C412yes—
76#495C2×C412yes—
77#493C2×C411yescharacteristic
78#488—22no—
79#486—21no—
80#484—22no—
81#482—22no—
82#481—22no—
83#478—22no—
84#478—22no—
85#477—21no—
86#476—21no—
87#467C2×C412yes—
88#466—21no—
89#466C2×C412yes—
90#466C2×C412yes—
91#465—22no—
92#463—22no—
93#463—22no—
94#460—21no—
95#459—21no—
96#450—22no—
97#449—22no—
98#447—22no—
99C8×Q8—41no—
100C8×Q8C2×C411yescharacteristic
101C8×Q8D411yescharacteristic
102C4×Q16C2×C411yescharacteristic
103C4×Q16C2311yescharacteristic
104C4×SD16—21no—
105C4×SD16—22no—
106C4×SD16—21no—
107C4×D8—21no—
108C4×D8—22no—
109C4×D8—22no—
110C4×D8C2×C412yes—
111C4×D8C2×C412yes—
112C4×D8C2×C411yescharacteristic
113C4×D8C2311yescharacteristic
114C4×D8C2312yes—
115#436—41no—
116#436—42no—
117C8×D4C2×C411yescharacteristic
118C8×D4C2×C412yes—
119C8×D4D412yes—
120C8×D4D411yescharacteristic
121#431—22no—
122#431—21no—
123#431D412yes—
124#431D411yescharacteristic
125#427C2×C411yescharacteristic
126C2×#107C2×C412yes—
127#423—22no—
128#423—21no—
129#416—21no—
130C2×#102—22no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—22no—
134C4⋊C16—21no—
135C22⋊C16—21no—
136C22⋊C16—22no—
137#346—22no—
138C4×C16—21no—
139C4×C16C2311yescharacteristic
140C4×C16C2311yescharacteristic
141C4×C16C2312yes—
142#335—22no—
143#335—21no—
144#335C2311yescharacteristic
145#335C2311yescharacteristic
146#335C2312yes—
147#334—21no—
148#334—22no—
149C2×#41—21no—
150C2×#41—41no—
151C2×#41SD1611yescharacteristic
152C22×D4—22no—
153C22×D4—41no—
154C23×C4—42no—
155#135—22no—
156#135—41no—
157C2×Q16C22×C411yescharacteristic
158C2×SD16—21no—
159C2×SD16—22no—
160C2×D8—24no—
161C2×D8—22no—
162C2×D8—21no—
163C2×D8—42no—
164C2×D8C22×C412yes—
165C2×D8C22×C411yescharacteristic
166C2×M16—42no—
167C2×M16—41no—
168C22×C8C22×C412yes—
169C22×C8C22×C411yescharacteristic
170C22×C8C2×D412yes—
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—21no—
173#127—21no—
174#126—42no—
175#126—42no—
176#125—22no—
177#125—41no—
178#124—22no—
179#124—41no—
180#124—41no—
181#123—42no—
182#123—42no—
183#123—42no—
184#123—42no—
185C22⋊Q8—22no—
186C22⋊Q8—41no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189#121—22no—
190#121—22no—
191#121—42no—
192#121—41no—
193#121—41no—
194#121—42no—
195#120—42no—
196#120—42no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8C22⋊C411yescharacteristic
200C4×Q8D811yescharacteristic
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—22no—
206C4×D4—41no—
207C4×D4—42no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4C22⋊C411yescharacteristic
216C4×D4C22⋊C412yes—
217C4×D4D811yescharacteristic
218C4×D4D812yes—
219#117—21no—
220#117—42no—
221#117—41no—
222#117SD1612yes—
223#117SD1611yescharacteristic
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C42—41no—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231M32—44no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—22no—
235C2×C16—42no—
236#107—22no—
237#107—21no—
238#107C22×C411yescharacteristic
239#107C22×C412yes—
240#107C22×C411yescharacteristic
241#106—21no—
242#106—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—41no—
247C4⋊C8—42no—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C412yes—
250C4⋊C8C22⋊C411yescharacteristic
251C4⋊C8C22⋊C412yes—
252#103—21no—
253#103—22no—
254#103—21no—
255#102—22no—
256#102—22no—
257#102—21no—
258#102—22no—
259#102—22no—
260#102—21no—
261#102—22no—
262C22⋊C8—42no—
263C22⋊C8—41no—
264C22⋊C8C22⋊C412yes—
265C22⋊C8C22⋊C412yes—
266C22⋊C8C22⋊C412yes—
267C4×C8—41no—
268C4×C8—42no—
269C4×C8C22×C412yes—
270C4×C8C22×C411yescharacteristic
271C4×C8C2×D411yescharacteristic
272C4×C8C2×D412yes—
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2411yescharacteristic
276C24—42no—
277#41—42no—
278#41—42no—
279#41—81no—
280#41—161no—
281#41—161no—
282C2×Q8—21no—
283C2×Q8—41no—
284C2×Q8#10211yescharacteristic
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—82no—
295C2×D4—84no—
296C2×D4#10211yescharacteristic
297C2×D4#10212yes—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—162no—
306C22×C4—162no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×D812yes—
310C22×C4C2×D811yescharacteristic
311C22×C4C2×SD1611yescharacteristic
312C22×C4C2×SD1612yes—
313Q16—21no—
314Q16—21no—
315SD16—42no—
316SD16—82no—
317D8—22no—
318D8—21no—
319D8—22no—
320D8—44no—
321D8—81no—
322D8#13112yes—
323M16—162no—
324M16—161no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8C2×C22⋊C412yes—
335C2×C8C2×C22⋊C411yescharacteristic
336C2×C8C2×C22⋊C411yescharacteristic
337C2×C8C2×C22⋊C412yes—
338C2×C8C23×C411yescharacteristic
339C2×C8C22×D411yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4#10211yescharacteristic
351C4⋊C4#10211yescharacteristic
352C4⋊C4#10212yes—
353C4⋊C4#10212yes—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4#10212yes—
366C22⋊C4#10212yes—
367C22⋊C4#10212yes—
368C42—41no—
369C42—41no—
370C42—41no—
371C42C2×C22⋊C412yes—
372C42C2×C22⋊C411yescharacteristic
373C42C2×D811yescharacteristic
374C42C2×D812yes—
375C42C2×SD1611yescharacteristic
376C42C2×SD1612yes—
377C42C22×D411yescharacteristic
378C16—22no—
379C16—41no—
380C16—42no—
381C23—41no—
382C23—42no—
383C23—84no—
384C23—84no—
385C23—162no—
386C23C2×#10212yes—
387C23C2×#10211yescharacteristic
388Q8—22no—
389Q8—41no—
390Q8—41no—
391Q8—161no—
392D4—22no—
393D4—22no—
394D4—42no—
395D4—42no—
396D4—41no—
397D4—42no—
398D4—84no—
399D4—82no—
400D4—84no—
401D4—82no—
402D4—161no—
403D4—162no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—162no—
425C2×C4—162no—
426C2×C4—162no—
427C2×C4C2×#10211yescharacteristic
428C2×C4C2×#10212yes—
429C2×C4C2×#10212yes—
430C2×C4C2×#10211yescharacteristic
431C2×C4C2×#10212yes—
432C2×C4C2×#10211yescharacteristic
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C22×D811yescharacteristic
435C2×C4C22×SD1611yescharacteristic
436C8—22no—
437C8—21no—
438C8—21no—
439C8—21no—
440C8—22no—
441C8—81no—
442C8—82no—
443C8—81no—
444C8C2×#13111yescharacteristic
445C8#56811yescharacteristic
446C22—21no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—168no—
459C22C22×#10211yescharacteristic
460C4—21no—
461C4—22no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—81no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4#214312yes—
471C4#288011yescharacteristic
472C4#290111yescharacteristic
473C4#341511yescharacteristic
474C4#341611yescharacteristic
475C2—41no—
476C2—42no—
477C2—82no—
478C2—162no—
479C2#1750711yescharacteristic
480C2#3338111yescharacteristic
481C2C2×#214311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#61431C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33479C2semidirect
3. #17837C2semidirect
4. #11674C2semidirect
5. #11671C2semidirect
6. #11660C2semidirect
7. #11657C2semidirect
8. #11649C2semidirect
9. #11647C2semidirect
10. #11593C2semidirect
11. #11591C2semidirect
33. #2162C22semidirect
34. #2162C22semidirect
35. C4×M32C22semidirect
36. C2×C4×C16C22semidirect
39. #1567C22semidirect
40. #1567C22semidirect
41. #1564C22semidirect
42. #1564C22semidirect
43. #1332C22semidirect
44. #1332C22semidirect
45. #1332C22semidirect
49. #1331C22semidirect
50. #1331C22semidirect
51. #1331C22semidirect
52. #1331C22semidirect

Maximal subgroups

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

subgroupindex
#614312
#334792
#178372
#116742
#116712
#116602
#116572
#116492
#116472
#115932
#115912