Finite Group Databasebeta

← #483245

Subgroups — #483245

Order 512.

734 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#62419C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#17835C211yesmaximal, characteristic, complemented
4#11980C212yesmaximal, complemented
5#11977C211yesmaximal, characteristic, complemented
6#11959C212yesmaximal, complemented
7#11958C211yesmaximal, characteristic, complemented
8#11946C212yesmaximal, complemented
9#11943C211yesmaximal, characteristic, complemented
10#11926C212yesmaximal, complemented
11#11924C211yesmaximal, characteristic, complemented
12#3514—21no—
13#3396C411yescharacteristic
14#3394—21no—
15#3393C411yescharacteristic
16D4×Q16C411yescharacteristic
17#3285C412yes—
18#3282—22no—
19D4×SD16—21no—
20#3280C412yes—
21D4×D8C411yescharacteristic
22#3212—21no—
23#3211—22no—
24#3168C2211yescharacteristic
25#3168C2212yes—
26#3147C2211yescharacteristic
27C2×#495C2212yes—
28#2991—21no—
29C8×#41C2211yescharacteristic
30#2183—21no—
31#2169—21no—
32#2168—22no—
33#2163C2212yescomplemented
34#2162C2211yescharacteristic, complemented
35#2150C2212yescomplemented
36C2×C4⋊C16C2211yescharacteristic, complemented
37#1574—22no—
38#1574—21no—
39#1566C2211yescharacteristic, complemented
40#1566C2212yescomplemented
41#1564C2211yescharacteristic, complemented
42#1564C2212yescomplemented
43#1363C2211yescharacteristic, complemented
44#1363C2211yescharacteristic, complemented
45#1363C2212yescomplemented
46#1360C2211yescharacteristic
47#1360C2212yes—
48#1360C2211yescharacteristic
49#1360C2212yescomplemented
50#1360C2212yescomplemented
51#1360C2212yescomplemented
52#1360C2212yescomplemented
53C2×#135—21no—
54C2×#135C2×C412yes—
55C22×Q16C2×C412yes—
56C22×SD16—22no—
57C22×D8C2×C412yes—
58#559—21no—
59#556—21no—
60#555—41no—
61#551—42no—
62D4×Q8—21no—
63#548—22no—
64#546—22no—
65#546—41no—
66D42—21no—
67#542—41no—
68#538—42no—
69#532—21no—
70#529—22no—
71#529—21no—
72C2×#124—22no—
73#518—41no—
74C4×#41D411yescharacteristic
75C2×M32—22no—
76C22×C16—21no—
77#499C2311yescharacteristic
78#495—22no—
79#495—22no—
80#495—21no—
81#495C2×C412yes—
82#495C2×C412yes—
83#495C2311yescharacteristic
84#495C2312yes—
85#494C2×C411yescharacteristic
86#493C2×C411yescharacteristic
87#492—21no—
88#488—21no—
89#488—22no—
90#467C2×C412yes—
91#467C2×C411yescharacteristic
92#466C2×C412yes—
93#466C2×C411yescharacteristic
94#465—22no—
95#465—21no—
96#464—22no—
97#464—22no—
98#463—22no—
99#463—22no—
100#462—21no—
101#461—21no—
102#460—21no—
103#459—21no—
104#452—22no—
105#451—22no—
106#450—22no—
107#449—22no—
108#448—22no—
109#447—22no—
110C8×Q8—41no—
111C8×Q8C2×C411yescharacteristic
112C8×Q8D411yescharacteristic
113C4×Q16C2×C412yes—
114C4×Q16C2×C411yescharacteristic
115C4×SD16—21no—
116C4×SD16—22no—
117C4×D8C2×C411yescharacteristic
118C4×D8C2×C412yes—
119#436—41no—
120#436—42no—
121C8×D4C2×C411yescharacteristic
122C8×D4C2×C412yes—
123C8×D4D412yes—
124C8×D4D411yescharacteristic
125#431—22no—
126#431—21no—
127#431D412yes—
128#431D411yescharacteristic
129#423—22no—
130#423—21no—
131C2×C4×C8C2312yes—
132C2×C4×C8C2311yescharacteristic
133C4⋊C16—21no—
134C4⋊C16—22no—
135C4⋊C16C2312yes—
136C4⋊C16C2311yescharacteristic
137C4⋊C16C2311yescharacteristic
138C22⋊C16—21no—
139C22⋊C16—22no—
140#346—22no—
141C4×C16—21no—
142#335—21no—
143#335—22no—
144#334—21no—
145#334—22no—
146#334C2311yescharacteristic
147#334C2311yescharacteristic
148#334C2312yes—
149C2×#41—22no—
150C2×#41SD1611yescharacteristic
151C22×Q8—22no—
152C22×D4—22no—
153C22×D4—41no—
154#135—81no—
155#135—81no—
156#135—82no—
157C2×Q16—22no—
158C2×Q16—21no—
159C2×Q16—82no—
160C2×Q16C22×C411yescharacteristic
161C2×Q16C22×C412yes—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16—21no—
166C2×SD16—22no—
167C2×SD16—82no—
168C2×SD16—82no—
169C2×D8—22no—
170C2×D8—21no—
171C2×D8—82no—
172C2×D8C22×C411yescharacteristic
173C2×D8C22×C412yes—
174C2×M16—42no—
175C2×M16—41no—
176C22×C8C22×C412yes—
177C22×C8C22×C411yescharacteristic
178C22×C8C2×D412yes—
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—41no—
182#127—21no—
183#126—42no—
184#126—42no—
185#126—41no—
186#125—21no—
187#124—22no—
188#124—22no—
189#124—21no—
190#124—22no—
191#124—41no—
192#124—42no—
193#124—42no—
194#124—42no—
195#123—41no—
196#123—42no—
197#123—42no—
198#123—42no—
199C22⋊Q8—22no—
200C22⋊Q8—21no—
201C22⋊Q8—41no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204#121—22no—
205#121—21no—
206#121—42no—
207#121—41no—
208#121—41no—
209#121—41no—
210#121—42no—
211#120—42no—
212#120—42no—
213C4×Q8—22no—
214C4×Q8—21no—
215C4×Q8C22⋊C411yescharacteristic
216C4×Q8D811yescharacteristic
217C4×D4—21no—
218C4×D4—22no—
219C4×D4—41no—
220C4×D4—42no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—42no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4C22⋊C411yescharacteristic
227C4×D4C22⋊C412yes—
228C4×D4D811yescharacteristic
229C4×D4D812yes—
230#117—41no—
231#117—41no—
232#117—42no—
233#117SD1612yes—
234#117SD1611yescharacteristic
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240M32—82no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—22no—
244C2×C16—81no—
245#107C22×C411yescharacteristic
246#107C22×C411yescharacteristic
247#106—21no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—41no—
252C4⋊C8—42no—
253C4⋊C8C22⋊C411yescharacteristic
254C4⋊C8C22⋊C412yes—
255C4⋊C8C22⋊C411yescharacteristic
256C4⋊C8C22⋊C412yes—
257#103—21no—
258#103—22no—
259#103—22no—
260#103—21no—
261#102—21no—
262#102—22no—
263#102—22no—
264#102—21no—
265C22⋊C8—42no—
266C22⋊C8—41no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C4×C8—41no—
271C4×C8—42no—
272C4×C8C22×C411yescharacteristic
273C4×C8C22×C412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D412yes—
276C4×C8C2×D411yescharacteristic
277C4×C8C2×D412yes—
278C4×C8C2411yescharacteristic
279C24—42no—
280#41—42no—
281#41—82no—
282#41—82no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8#10211yescharacteristic
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—84no—
300C2×D4#10211yescharacteristic
301C2×D4#10212yes—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×D812yes—
309C22×C4C2×D811yescharacteristic
310C22×C4C2×SD1611yescharacteristic
311C22×C4C2×SD1612yes—
312Q16—41no—
313Q16—42no—
314Q16—82no—
315Q16—81no—
316SD16—41no—
317SD16—42no—
318SD16—41no—
319SD16—42no—
320SD16—82no—
321SD16—82no—
322SD16—82no—
323D8—41no—
324D8—42no—
325D8—82no—
326D8—81no—
327M16—162no—
328M16—161no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8C2×C22⋊C411yescharacteristic
339C2×C8C2×C22⋊C412yes—
340C2×C8C2×C22⋊C412yes—
341C2×C8C2×C22⋊C411yescharacteristic
342C2×C8C23×C411yescharacteristic
343C2×C8C22×D411yescharacteristic
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4#10211yescharacteristic
355C4⋊C4#10211yescharacteristic
356C4⋊C4#10212yes—
357C4⋊C4#10212yes—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—42no—
364C22⋊C4—41no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4#10212yes—
373C22⋊C4#10212yes—
374C22⋊C4#10212yes—
375C42—42no—
376C42—41no—
377C42C2×C22⋊C412yes—
378C42C2×C22⋊C411yescharacteristic
379C42C2×D811yescharacteristic
380C42C2×D812yes—
381C42C2×SD1612yes—
382C42C2×SD1611yescharacteristic
383C42C22×D411yescharacteristic
384C16—42no—
385C16—41no—
386C16—41no—
387C23—41no—
388C23—42no—
389C23—84no—
390C23—162no—
391C23—162no—
392C23C2×#10212yes—
393C23C2×#10211yescharacteristic
394Q8—22no—
395Q8—42no—
396Q8—41no—
397Q8—41no—
398Q8—42no—
399Q8—162no—
400D4—22no—
401D4—42no—
402D4—41no—
403D4—42no—
404D4—42no—
405D4—41no—
406D4—84no—
407D4—82no—
408D4—161no—
409D4—162no—
410D4—161no—
411D4—162no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—162no—
426C2×C4—162no—
427C2×C4—162no—
428C2×C4C2×#10212yes—
429C2×C4C2×#10212yes—
430C2×C4C2×#10211yescharacteristic
431C2×C4C2×#10211yescharacteristic
432C2×C4C2×#10212yes—
433C2×C4C2×#10211yescharacteristic
434C2×C4C22×C22⋊C411yescharacteristic
435C2×C4C22×D811yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C8—21no—
438C8—22no—
439C8—21no—
440C8—21no—
441C8—22no—
442C8—81no—
443C8—81no—
444C8—82no—
445C8#57611yescharacteristic
446C8#57711yescharacteristic
447C22—22no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—21no—
452C22—22no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22—82no—
457C22—82no—
458C22—81no—
459C22—324no—
460C22C22×#10211yescharacteristic
461C4—21no—
462C4—22no—
463C4—21no—
464C4—22no—
465C4—21no—
466C4—22no—
467C4—21no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#214312yes—
473C4#288811yescharacteristic
474C4#289711yescharacteristic
475C2—41no—
476C2—42no—
477C2—162no—
478C2—161no—
479C2#1772011yescharacteristic
480C2C2×#214311yescharacteristic
481C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62419C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #33503C2semidirect
3. #17835C2semidirect
4. #11980C2semidirect
5. #11977C2semidirect
6. #11959C2semidirect
7. #11958C2semidirect
8. #11946C2semidirect
9. #11943C2semidirect
10. #11926C2semidirect
11. #11924C2semidirect
33. #2163C22semidirect
34. #2162C22semidirect
35. #2150C22semidirect
36. C2×C4⋊C16C22semidirect
39. #1566C22semidirect
40. #1566C22semidirect
41. #1564C22semidirect
42. #1564C22semidirect
43. #1363C22semidirect
44. #1363C22semidirect
45. #1363C22semidirect
49. #1360C22semidirect
50. #1360C22semidirect
51. #1360C22semidirect
52. #1360C22semidirect

Maximal subgroups

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

subgroupindex
#624192
#335032
#178352
#119802
#119772
#119592
#119582
#119462
#119432
#119262
#119242