Finite Group Databasebeta

← #482969

Subgroups — #482969

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62251C211yesmaximal, characteristic
2#33486C211yesmaximal, characteristic, complemented
3#22717C211yesmaximal, characteristic, complemented
4#11893C212yesmaximal, complemented
5#11891C212yesmaximal, complemented
6#11862C211yesmaximal, characteristic, complemented
7#11861C211yesmaximal, characteristic, complemented
8#11749C211yesmaximal, characteristic, complemented
9#11747C211yesmaximal, characteristic, complemented
10#11709C212yesmaximal, complemented
11C2×#1349C212yesmaximal, complemented
12#3519—21no—
13#3341C411yescharacteristic
14#3339C412yes—
15#3338C411yescharacteristic
16#3305—21no—
17#3297—22no—
18#3295—21no—
19#3287C411yescharacteristic
20#3280C412yes—
21D4×D8C411yescharacteristic
22#3247—21no—
23#3188—22no—
24#3169C2211yescharacteristic
25#3168C2211yescharacteristic
26#3050C2212yes—
27C2×#466C2212yes—
28D4×M16—21no—
29C2×C8×D4C2211yescharacteristic
30D4×C16—21no—
31D4×C16—22no—
32D4×C16—21no—
33#2162C2211yescharacteristic, complemented
34#2162C2211yescharacteristic, complemented
35C2×C22⋊C16C2212yescomplemented
36C2×C22⋊C16C2212yescomplemented
37#1950—21no—
38#1855—22no—
39#1837C2211yescharacteristic, complemented
40#1810C2212yescomplemented
41C4×#108C2211yescharacteristic, complemented
42C2×#341C2212yescomplemented
43#1350C2212yescomplemented
44#1350C2211yescharacteristic, complemented
45#1350C2211yescharacteristic, complemented
46#1349C2212yes—
47#1349C2211yescharacteristic
48#1349C2211yescharacteristic
49#1349C2212yescomplemented
50#1349C2212yescomplemented
51#1349C2212yescomplemented
52#1349C2212yescomplemented
53C2×#135C2×C412yes—
54C22×D8C2×C412yes—
55C22×M16—22no—
56C23×C8C2312yes—
57#555—21no—
58#554—21no—
59#553—22no—
60#553—41no—
61#550—21no—
62#546—22no—
63#546—42no—
64D42—21no—
65D42—41no—
66#540—41no—
67#538—42no—
68#532—21no—
69#529—21no—
70#525—22no—
71C2×#121—22no—
72#518—41no—
73C2×C4×D4D411yescharacteristic
74C22×C16—22no—
75C22×C16—22no—
76#500C2×C411yescharacteristic
77#499C2×C411yescharacteristic
78#495C2×C411yescharacteristic
79#493C2×C411yescharacteristic
80#488—21no—
81#486—21no—
82#484—22no—
83#482—22no—
84#482—22no—
85#481—22no—
86#478—22no—
87#477—21no—
88#476—21no—
89#467C2×C412yes—
90#467C2311yescharacteristic
91#466—21no—
92#466—22no—
93#466—22no—
94#466C2×C412yes—
95#466C2×C412yes—
96#466C2×C412yes—
97#466C2311yescharacteristic
98#466C2312yes—
99#465—21no—
100#465—22no—
101#465—22no—
102#463—22no—
103#460—21no—
104#459—21no—
105#450—22no—
106#449—22no—
107#449—22no—
108#447—22no—
109C4×Q16C2×C411yescharacteristic
110C4×SD16—21no—
111C4×D8—21no—
112C4×D8C2×C412yes—
113C4×D8C2×C411yescharacteristic
114#436—41no—
115#435—42no—
116C8×D4—41no—
117C8×D4C2×C411yescharacteristic
118C8×D4C2×C412yes—
119C8×D4C2×C411yescharacteristic
120C8×D4D411yescharacteristic
121C8×D4D412yes—
122C8×D4D411yescharacteristic
123C2×#108—21no—
124C2×#108—22no—
125#427C2×C412yes—
126C2×#107C2×C412yes—
127#423—21no—
128C2×C4⋊C8D411yescharacteristic
129#416—22no—
130C2×#102—22no—
131#407—22no—
132C2×C22⋊C8D412yes—
133C4×M16—21no—
134C2×C4×C8C2311yescharacteristic
135C4⋊C16—21no—
136C4⋊C16—21no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139C22⋊C16C2311yescharacteristic
140C22⋊C16C2312yes—
141C22⋊C16C2311yescharacteristic
142C4×C16—21no—
143C4×C16—21no—
144#341—21no—
145#341—22no—
146#341C2311yescharacteristic
147#341C2312yes—
148#341C2311yescharacteristic
149C2×#41—22no—
150C2×#41—41no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—41no—
154C22×D4SD1611yescharacteristic
155C23×C4C2×D412yes—
156#135—82no—
157C2×Q16C22×C411yescharacteristic
158C2×SD16—22no—
159C2×SD16—21no—
160C2×D8—22no—
161C2×D8—21no—
162C2×D8—82no—
163C2×D8C22×C411yescharacteristic
164C2×D8C22×C412yes—
165C2×M16—21no—
166C2×M16—22no—
167C2×M16—21no—
168C2×M16—42no—
169C2×M16—44no—
170C2×M16—82no—
171C22×C8—42no—
172C22×C8—82no—
173C22×C8C22×C412yes—
174C22×C8C22×C412yes—
175C22×C8C2×D412yes—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2411yescharacteristic
180C4⋊Q8—21no—
181#127—21no—
182#127—41no—
183#126—42no—
184#126—41no—
185#125—21no—
186#125—41no—
187#124—21no—
188#124—42no—
189#124—41no—
190#123—42no—
191#123—41no—
192#123—42no—
193#123—42no—
194C22⋊Q8—22no—
195C22⋊Q8—21no—
196C22⋊Q8—42no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—21no—
201#121—22no—
202#121—42no—
203#121—41no—
204#121—42no—
205#121—42no—
206#121—41no—
207#121—41no—
208#121—42no—
209#121—42no—
210#120—42no—
211#120—42no—
212#120—42no—
213C4×Q8—21no—
214C4×D4—21no—
215C4×D4—22no—
216C4×D4—41no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—42no—
222C4×D4—41no—
223C4×D4C22⋊C411yescharacteristic
224C4×D4C22⋊C412yes—
225C4×D4C22⋊C411yescharacteristic
226C4×D4D811yescharacteristic
227C4×D4D812yes—
228C4×D4D811yescharacteristic
229#117—22no—
230#117—41no—
231#117—41no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4SD1611yescharacteristic
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4SD1612yes—
237C2×C42C2×D411yescharacteristic
238C2×C16—22no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—82no—
242C2×C16—82no—
243#108—42no—
244#108—82no—
245#107—22no—
246#107—21no—
247#107C22×C411yescharacteristic
248#107C22×C412yes—
249#107C22×C411yescharacteristic
250#106—22no—
251#106—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8C22⋊C411yescharacteristic
255C4⋊C8C22⋊C411yescharacteristic
256C4⋊C8C22⋊C412yes—
257#103—22no—
258#103—21no—
259#103—21no—
260#102—22no—
261#102—22no—
262#102—22no—
263#102—21no—
264#102—22no—
265#102—21no—
266#102—22no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271C22⋊C8C22⋊C412yes—
272C22⋊C8C22⋊C412yes—
273#97—41no—
274C4×C8—41no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D411yescharacteristic
279C24—42no—
280C24C2×SD1612yes—
281#41—82no—
282#41—82no—
283#41—162no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×D4—24no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—84no—
301C2×D4—164no—
302C2×D4#10212yes—
303C2×D4#10211yescharacteristic
304C2×D4#10211yescharacteristic
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×D812yes—
314C22×C4C2×D812yes—
315C22×C4C2×SD1611yescharacteristic
316C22×C4C2×SD1611yescharacteristic
317C22×C4C2×SD1612yes—
318C22×C4C22×D411yescharacteristic
319Q16—41no—
320SD16—81no—
321SD16—82no—
322D8—41no—
323D8—42no—
324D8—81no—
325D8—82no—
326M16—22no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—42no—
331M16—42no—
332M16—84no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—84no—
344C2×C8C2×C22⋊C411yescharacteristic
345C2×C8C2×C22⋊C411yescharacteristic
346C2×C8C2×C22⋊C412yes—
347C2×C8C2×C22⋊C412yes—
348C2×C8C23×C411yescharacteristic
349C2×C8C22×D411yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4#10211yescharacteristic
361C4⋊C4#10211yescharacteristic
362C4⋊C4#10212yes—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4#10212yes—
376C22⋊C4#10212yes—
377C22⋊C4#10212yes—
378C22⋊C4#10212yes—
379C42—42no—
380C42—41no—
381C42C2×C22⋊C411yescharacteristic
382C42C2×C22⋊C411yescharacteristic
383C42C2×D811yescharacteristic
384C42C2×D811yescharacteristic
385C16—42no—
386C16—41no—
387C16—41no—
388C23—24no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—84no—
394C23—162no—
395C23—162no—
396C23C2×#10212yes—
397C23C2×#10212yes—
398C23C22×SD1611yescharacteristic
399Q8—41no—
400Q8—41no—
401Q8—161no—
402D4—24no—
403D4—22no—
404D4—41no—
405D4—42no—
406D4—42no—
407D4—41no—
408D4—82no—
409D4—84no—
410D4—84no—
411D4—82no—
412D4—162no—
413D4—161no—
414D4—162no—
415D4—162no—
416D4—162no—
417D4#35914yes—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—82no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4—162no—
437C2×C4C2×#10212yes—
438C2×C4C2×#10211yescharacteristic
439C2×C4C2×#10211yescharacteristic
440C2×C4C2×#10211yescharacteristic
441C2×C4C2×#10212yes—
442C2×C4C2×#10211yescharacteristic
443C2×C4C22×C22⋊C411yescharacteristic
444C2×C4C22×D811yescharacteristic
445C8—21no—
446C8—21no—
447C8—22no—
448C8—22no—
449C8—21no—
450C8—21no—
451C8—42no—
452C8—82no—
453C8—81no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—84no—
463C22—81no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—82no—
468C22—81no—
469C22—324no—
470C22C2×#35914yes—
471C22C22×#10211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—81no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4C2×#35912yes—
483C4#288311yescharacteristic
484C4#290111yescharacteristic
485C2—24no—
486C2—42no—
487C2—41no—
488C2—162no—
489C2—161no—
490C2#1761311yescharacteristic
491C2C22×#35911yescharacteristic
492C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62251C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33486C2semidirect
3. #22717C2semidirect
4. #11893C2semidirect
5. #11891C2semidirect
6. #11862C2semidirect
7. #11861C2semidirect
8. #11749C2semidirect
9. #11747C2semidirect
10. #11709C2semidirect
11. C2×#1349C2semidirect
33. #2162C22semidirect
34. #2162C22semidirect
35. C2×C22⋊C16C22semidirect
36. C2×C22⋊C16C22semidirect
39. #1837C22semidirect
40. #1810C22semidirect
41. C4×#108C22semidirect
42. C2×#341C22semidirect
43. #1350C22semidirect
44. #1350C22semidirect
45. #1350C22semidirect
49. #1349C22semidirect
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
#622512
#334862
#227172
#118932
#118912
#118622
#118612
#117492
#117472
#117092
C2×#13492