Finite Group Databasebeta

← #20286201

Subgroups — #20286201

Order 6784.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133244C211yesmaximal, characteristic, complemented
2C2×D8×D53C211yesmaximal, characteristic, complemented
3#16811266C211yesmaximal, characteristic, complemented
4D4×D212C211yesmaximal, characteristic, complemented
5#16811013C211yesmaximal, characteristic, complemented
6#16810875C211yesmaximal, characteristic, complemented
7#16810855C211yesmaximal, characteristic, complemented
8#16810819C211yesmaximal, characteristic, complemented
9#16810772C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14#16810693C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C22×D212—21no—
18C2×D4×D53C2211yescharacteristic
19C2×D4×D53C2211yescharacteristic, complemented
20C2×D4×D53C2211yescharacteristic, complemented
21#13133244—24no—
22C2×#11719924C2211yescharacteristic, complemented
23C2×#11719924C2211yescharacteristic, complemented
24C2×D424C2211yescharacteristic, complemented
25C2×D424C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic
27C2×#11719922C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic, complemented
29D8×D53—24no—
30C2×#11719912C2211yescharacteristic, complemented
31C2×C8×D53C2211yescharacteristic, complemented
32#13133214C2211yescharacteristic, complemented
33#13133199—21no—
34#13133193C2211yescharacteristic, complemented
35#13133192C2211yescharacteristic
36#13133192C2211yescharacteristic, complemented
37#13133191—21no—
38#13133190—21no—
39#13133188C2211yescharacteristic, complemented
40#13133187—21no—
41#13133187—21no—
42#13133187—21no—
43C4×D212C2211yescharacteristic
44C4×D212C2211yescharacteristic, complemented
45C22⋊C4×D53—21no—
46#13133148C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133138C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53#13133128C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×D8×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic, complemented
57C4×D4×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C23×D53—21no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72#11719949—22no—
73#11719949—41no—
74D4×D53—22no—
75D4×D53—22no—
76D4×D53—22no—
77D4×D53—41no—
78D4×D53—41no—
79D4×D53—41no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—44no—
86C2×D212D411yescharacteristic
87C2×D212C2311yescharacteristic
88C2×D212C2311yescharacteristic
89C2×D212C2311yescharacteristic
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53C2311yescharacteristic
93C2×C4×D53C2311yescharacteristic
94SD16×C53—22no—
95D8×C53—22no—
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C4⋊C4×C53C2311yescharacteristic
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C42×C53C2311yescharacteristic
103C22⋊Dic53—21no—
104#11719924—22no—
105#11719924—41no—
106#11719922—22no—
107#11719922—41no—
108#11719922D414yescomplemented
109#11719921—21no—
110#11719921—21no—
111#11719921—21no—
112#11719921—21no—
113#11719921D411yescharacteristic
114#11719921D411yescharacteristic
115C4⋊Dic53D411yescharacteristic
116C4⋊Dic53C2311yescharacteristic
117#11719919—21no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120D424—22no—
121D424—22no—
122#11719912—22no—
123C8×D53—22no—
124C23×C53—21no—
125C23×C53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—81no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53C2×D411yescharacteristic
139C22×D53C2×D411yescharacteristic
140Q8×C53—21no—
141Q8×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53C2×D412yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—42no—
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2411yescharacteristic
152C22⋊D53—42no—
153C22⋊D53—42no—
154C22⋊D53—42no—
155C22⋊D53—81no—
156C2×Dic53—21no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159D212—22no—
160D212—21no—
161D212—21no—
162D212—21no—
163D212—21no—
164D212—21no—
165D212—42no—
166D212—42no—
167D212—42no—
168D212—42no—
169D212—81no—
170D212C2×D412yescomplemented
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—21no—
174C4×D53—21no—
175C4×D53—42no—
176C4×D53—81no—
177C8×C53—21no—
178C8×C53—21no—
179#88322—21no—
180#88322C2×D412yescomplemented
181C22×C53—22no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53C22×D411yescharacteristic
188C2×D53—21no—
189C2×D53—21no—
190C2×D53—21no—
191C2×D53—22no—
192C2×D53—21no—
193C2×D53—21no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—84no—
203C2×D53—84no—
204C2×D53—84no—
205C2×D53AGL(1,ℤ/8)12yescomplemented
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53C22×D411yescharacteristic
211C4×C53H(5,𝔽2)11yescharacteristic
212Dic53—21no—
213Dic53—21no—
214Dic53—41no—
215#3264—531nomaximal, Sylow
216C2×C53—42no—
217C2×C53—81no—
218C2×C53D4211yescharacteristic
219C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C53#57611yescharacteristic
221D53—22no—
222D53—41no—
223D53—42no—
224D53—81no—
225D53—81no—
226C2×AGL(1,ℤ/8)—531no—
227C22×D8—531no—
228#546—531no—
229D42—531no—
230#496—531no—
231#469—531no—
232#465—531no—
233#463D5311yescharacteristic, complemented
234#459—531no—
235#450—531no—
236#449—531no—
237#447—531no—
238#447—531no—
239#442—531no—
240#435—531no—
241C53#326411yescharacteristic, complemented, Sylow
242C2×#41—531no—
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—531no—
246C22×D4—1061no—
247AGL(1,ℤ/8)—1064no—
248C2×SD16—531no—
249C2×SD16—531no—
250C2×SD16C2×D5311yescharacteristic, complemented
251C2×D8—531no—
252C2×D8—531no—
253C2×D8—531no—
254C2×D8—531no—
255C2×D8—531no—
256C2×D8—1064no—
257C2×D8C2×D5311yescharacteristic
258C2×M16—531no—
259C22×C8—531no—
260#127—531no—
261#124C2×D5311yescharacteristic, complemented
262#123—1061no—
263C22⋊Q8—531no—
264#121—531no—
265#121—531no—
266#121—531no—
267#121—1061no—
268#121—1061no—
269#120—1061no—
270#120—1061no—
271#120—1061no—
272C4×D4—531no—
273C4×D4—531no—
274C4×D4C2×D5311yescharacteristic, complemented
275C2×C22⋊C4—1061no—
276#106—531no—
277C4⋊C8C2×D5311yescharacteristic, complemented
278#103C2×D5311yescharacteristic, complemented
279#102—531no—
280#102—531no—
281#102—531no—
282#102—531no—
283#102C2×D5311yescharacteristic, complemented
284C22⋊C8—531no—
285C22⋊C8—531no—
286#97—531no—
287C24—1061no—
288C24—1061no—
289C24—1061no—
290#41—1062no—
291#41—2121no—
292C2×Q8C22×D5311yescharacteristic
293C2×D4—531no—
294C2×D4—531no—
295C2×D4—531no—
296C2×D4—531no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1062no—
304C2×D4—1062no—
305C2×D4—1062no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—2124no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4C22×D5311yescharacteristic
314C2×D4C22×D5311yescharacteristic
315C22×C4—21no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—1061no—
319C22×C4—1061no—
320SD16—22no—
321SD16—1062no—
322SD16—2121no—
323D8—22no—
324D8—534no—
325D8—1062no—
326D8—1062no—
327D8—1062no—
328D8—2121no—
329M16—1062no—
330C2×C8—531no—
331C2×C8—531no—
332C2×C8—1062no—
333C2×C8C22×D5311yescharacteristic
334C2×C8C22×D5311yescharacteristic
335C4⋊C4—531no—
336C4⋊C4—531no—
337C4⋊C4—1061no—
338C4⋊C4C22×D5311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—531no—
343C22⋊C4—531no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4—1061no—
348C22⋊C4—1061no—
349C42C22×D5311yescharacteristic
350C23—21no—
351C23—21no—
352C23—531no—
353C23—531no—
354C23—1061no—
355C23—1061no—
356C23—1061no—
357C23—4241no—
358C23—4241no—
359C23—4241no—
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23—2122no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—532no—
372D4—1062no—
373D4—1061no—
374D4—1061no—
375D4—1061no—
376D4—1061no—
377D4—1061no—
378D4—4241no—
379D4—4241no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387D4D4×D5312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—531no—
392C2×C4—531no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—4241no—
399C2×C4—2122no—
400C2×C4D4×D5311yescharacteristic
401C2×C4D4×D5311yescharacteristic
402C2×C4C23×D5311yescharacteristic
403C8—21no—
404C8—21no—
405C8—532no—
406C8—1061no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—532no—
414C22—1061no—
415C22—1061no—
416C22—1061no—
417C22—1061no—
418C22—1061no—
419C22—1062no—
420C22—4244no—
421C22—4244no—
422C22—4244no—
423C22—2121no—
424C22—2121no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22—2121no—
431C22C2×D4×D5311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—1061no—
437C4—1061no—
438C4—2121no—
439C4C2×D4×D5311yescharacteristic
440C4#1313326311yescharacteristic
441C2—42no—
442C2—81no—
443C2—1062no—
444C2—4241no—
445C2—4241no—
446C2—2121no—
447C2—2122no—
448C2#1681125711yescharacteristic
449C2AGL(1,ℤ/8)×D5311yescharacteristic
450C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3264
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C2×D8×D53C2semidirect
3. #16811266C2semidirect
4. D4×D212C2semidirect
5. #16811013C2semidirect
6. #16810875C2semidirect
7. #16810855C2semidirect
8. #16810819C2semidirect
9. #16810772C2semidirect
10. #16810771C2semidirect
11. #16810758C2semidirect
12. #16810755C2semidirect
13. #16810733C2semidirect
14. #16810693C2semidirect
15. C53×#463C2semidirect
16. C2×#11719949C22semidirect
19. C2×D4×D53C22semidirect
20. C2×D4×D53C22semidirect
22. C2×#11719924C22semidirect
23. C2×#11719924C22semidirect
24. C2×D424C22semidirect
25. C2×D424C22semidirect
27. C2×#11719922C22semidirect
28. C2×#11719922C22semidirect
30. C2×#11719912C22semidirect
31. C2×C8×D53C22semidirect
32. #13133214C22semidirect
34. #13133193C22semidirect
36. #13133192C22semidirect
39. #13133188C22semidirect
44. C4×D212C22semidirect
46. #13133148C22semidirect
47. #13133139C22semidirect
48. #13133139C22semidirect
49. #13133138C22semidirect
50. #13133137C22semidirect
51. #13133129C22semidirect
52. #13133129C22semidirect
53. #13133128C22semidirect
54. C2×SD16×C53C22semidirect
56. #124×C53C22semidirect
57. C4×D4×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
108. #11719922D4semidirect
170. D212C2×D4semidirect
180. #88322C2×D4semidirect
205. C2×D53AGL(1,ℤ/8)semidirect
233. #463D53semidirect
241. C53#3264semidirect
250. C2×SD16C2×D53semidirect
261. #124C2×D53semidirect
274. C4×D4C2×D53semidirect
277. C4⋊C8C2×D53semidirect
278. #103C2×D53semidirect
283. #102C2×D53semidirect