Finite Group Databasebeta

← #49132875

Subgroups — #49132875

Order 36224.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2#31827134C211yesmaximal, characteristic, complemented
3D4×C22⋊D283C211yesmaximal, characteristic, complemented
4#31826840C211yesmaximal, characteristic, complemented
5#31826718C211yesmaximal, characteristic, complemented
6#31826715C211yesmaximal, characteristic, complemented
7#31826663C211yesmaximal, characteristic, complemented
8#31826658C211yesmaximal, characteristic, complemented
9#31826610C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826599C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13SD16×Dic283C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C22×C22⋊D283—21no—
18C2×D4×D283C2211yescharacteristic
19#22583637—24no—
20C2×#17985379C2211yescharacteristic, complemented
21C2×D2264C2211yescharacteristic, complemented
22C2×#17985377C2211yescharacteristic
23C2×#17985377C2211yescharacteristic, complemented
24C2×#17985377C2211yescharacteristic, complemented
25C2×#17985375C2211yescharacteristic, complemented
26#22583606—21no—
27#22583606C2211yescharacteristic
28#22583606C2211yescharacteristic, complemented
29#22583584—21no—
30#22583584C2211yescharacteristic
31#22583584C2211yescharacteristic, complemented
32#22583582—21no—
33#22583581—21no—
34#22583581C2211yescharacteristic
35#22583581C2211yescharacteristic, complemented
36#22583579—21no—
37#22583578—21no—
38Q8×Dic283C2211yescharacteristic, complemented
39D4×Dic283C2211yescharacteristic
40C4×C22⋊D283—21no—
41C4×C22⋊D283C2211yescharacteristic
42#22583542C2211yescharacteristic, complemented
43Dic283⋊C8C2211yescharacteristic, complemented
44#22583534C2211yescharacteristic, complemented
45#22583532C2211yescharacteristic, complemented
46#22583532C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic, complemented
49#22583531C2211yescharacteristic, complemented
50C22⋊(C283⋊C8)C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52C8×Dic283C2211yescharacteristic, complemented
53C2×#41×C283C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic
55AGL(1,ℤ/8)×C283—24no—
56C2×SD16×C283C2211yescharacteristic
57C2×SD16×C283C2211yescharacteristic, complemented
58C2×D8×C283C2211yescharacteristic
59C2×D8×C283C2211yescharacteristic, complemented
60C2×M16×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985393—22no—
63#17985393—41no—
64C2×D1132—21no—
65C2×D1132C2311yescharacteristic
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—44no—
77C2×C22⋊D283D411yescharacteristic
78D4×D283—22no—
79D4×D283—41no—
80C22×Dic283—21no—
81C2×C4×D283—21no—
82C2×C4×D283C2311yescharacteristic
83#17985379—22no—
84D2264—22no—
85#17985377—22no—
86#17985377—41no—
87#17985377—41no—
88#17985375—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C4⋊Dic283—21no—
92C4⋊Dic283C2311yescharacteristic
93C4⋊Dic283C2311yescharacteristic
94Dic283⋊C4D411yescharacteristic
95C22⋊Dic283—21no—
96C22⋊Dic283—21no—
97C22⋊Dic283—21no—
98C22⋊Dic283D411yescharacteristic
99(C2×D283)⋊C4—21no—
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4D411yescharacteristic
102C4×Dic283—21no—
103C4×Dic283C2311yescharacteristic
104C24×C283—21no—
105#41×C283—22no—
106#41×C283—41no—
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283—22no—
109C2×D4×C283—21no—
110C2×D4×C283—21no—
111C2×D4×C283—41no—
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C22×C4×C283—21no—
116C22×C4×C283C2311yescharacteristic
117SD16×C283—22no—
118SD16×C283D414yescomplemented
119D8×C283—22no—
120D8×C283—22no—
121M16×C283—22no—
122C2×C8×C283C2311yescharacteristic
123C2×C8×C283C2311yescharacteristic
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—81no—
129C22×D283C2×D411yescharacteristic
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—42no—
135C22⋊D283—22no—
136C22⋊D283—42no—
137C22⋊D283—42no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—81no—
142C22⋊D283—81no—
143C22⋊D283—81no—
144C22⋊D283—81no—
145C2×Dic283—21no—
146C2×Dic283—21no—
147C2×Dic283—42no—
148C2×Dic283C2×D411yescharacteristic
149C2×Dic283C2×D411yescharacteristic
150C2×Dic283C2×D411yescharacteristic
151C4×D283—21no—
152C4×D283—21no—
153C4×D283—42no—
154C283⋊C8—21no—
155C283⋊C8—21no—
156C23×C283—21no—
157C23×C283—21no—
158C23×C283—21no—
159C23×C283—42no—
160C23×C283—42no—
161C23×C283—81no—
162C23×C283C2×D411yescharacteristic
163Q8×C283—21no—
164Q8×C283C2×D412yescomplemented
165D4×C283—21no—
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283—42no—
171D4×C283—42no—
172D4×C283C2×D412yes—
173C2×C4×C283—21no—
174C2×C4×C283—21no—
175C2×C4×C283—21no—
176C2×C4×C283—42no—
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283C2×D412yescomplemented
181C22×C283—21no—
182C22×C283—22no—
183C22×C283—21no—
184C22×C283—21no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—41no—
190C22×C283—41no—
191C22×C283—41no—
192C22×C283—41no—
193C22×C283—84no—
194C22×C283C22×D411yescharacteristic
195C2×D283—21no—
196C2×D283—21no—
197C2×D283—21no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—41no—
203C2×D283—41no—
204C2×D283—84no—
205C4×C283—21no—
206C4×C283—22no—
207C4×C283C22×D411yescharacteristic
208C4×C283H(5,𝔽2)11yescharacteristic
209Dic283—21no—
210Dic283—21no—
211Dic283—41no—
212Dic283—41no—
213Dic283AGL(1,ℤ/8)12yescomplemented
214C2×C283—42no—
215C2×C283—41no—
216C2×C283—81no—
217C2×C283—81no—
218C2×C283D4211yescharacteristic
219C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C283#57611yescharacteristic
221D283—41no—
222D283—81no—
223D283—81no—
224C283#327411yescharacteristic, complemented, Sylow
225#3274—2831nomaximal, Sylow
226C2×AGL(1,ℤ/8)—2831no—
227C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
228#550—2831no—
229D42—2831no—
230#493—2831no—
231#469—2831no—
232#469—2831no—
233#460—2831no—
234#459—2831no—
235#449—2831no—
236#449—2831no—
237#447—2831no—
238#447—2831no—
239C4×SD16—2831no—
240#436—2831no—
241C2×#41—2831no—
242C2×#41C2×D28311yescharacteristic, complemented
243C22×D4—2831no—
244C22×D4—5661no—
245C22×D4C2×D28311yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16C2×D28311yescharacteristic
250C2×SD16C2×D28311yescharacteristic, complemented
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8C2×D28311yescharacteristic
256C2×D8C2×D28311yescharacteristic, complemented
257C2×M16—2831no—
258C2×M16C2×D28311yescharacteristic, complemented
259#127—2831no—
260#127—2831no—
261#127—5661no—
262#121—2831no—
263#121—2831no—
264#121—2831no—
265#121—2831no—
266#121—5661no—
267#121—5661no—
268#121—5661no—
269#120—5661no—
270#120—5661no—
271C4×Q8—2831no—
272C4×D4—2831no—
273C4×D4—2831no—
274C4×D4—5661no—
275#106—2831no—
276C4⋊C8—2831no—
277#103—2831no—
278#102—2831no—
279#102—2831no—
280#102—2831no—
281#102—2831no—
282#102—2831no—
283C22⋊C8—2831no—
284C22⋊C8—2831no—
285C4×C8—2831no—
286C24—21no—
287C24—5661no—
288#41—22no—
289#41—41no—
290#41—5662no—
291#41—11321no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—2831no—
298C2×D4—2831no—
299C2×D4—2831no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—5661no—
304C2×D4—5661no—
305C2×D4—5661no—
306C2×D4—5662no—
307C2×D4—5661no—
308C2×D4—5661no—
309C2×D4—5661no—
310C2×D4—5661no—
311C2×D4—11321no—
312C2×D4—11324no—
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C2×D4C22×D28311yescharacteristic
316C22×C4—21no—
317C22×C4—2831no—
318C22×C4—5661no—
319C22×C4—5661no—
320C22×C4C22×D28311yescharacteristic
321SD16—22no—
322SD16—5662no—
323SD16C22⋊D28314yescomplemented
324D8—22no—
325D8—22no—
326D8—5662no—
327D8—5662no—
328D8—11321no—
329D8—11321no—
330M16—22no—
331M16—5662no—
332C2×C8—2831no—
333C2×C8—2831no—
334C2×C8C22×D28311yescharacteristic
335C2×C8C22×D28311yescharacteristic
336C4⋊C4—2831no—
337C4⋊C4—2831no—
338C4⋊C4—2831no—
339C4⋊C4—5661no—
340C22⋊C4—2831no—
341C22⋊C4—2831no—
342C22⋊C4—5661no—
343C22⋊C4—5661no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C42—2831no—
348C42—5661no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2831no—
356C23—5661no—
357C23—5661no—
358C23—11322no—
359C23—11322no—
360C23—22641no—
361C23D4×D28311yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D28312yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—5662no—
372D4—5661no—
373D4—5661no—
374D4—5661no—
375D4—5661no—
376D4—11322no—
377D4—11322no—
378D4—11322no—
379D4—11322no—
380D4—11322no—
381D4—11322no—
382D4—22641no—
383D4—22641no—
384D4—22641no—
385D4—22641no—
386D4C2×C22⋊D28312yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2831no—
392C2×C4—2831no—
393C2×C4—2831no—
394C2×C4—5661no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—5661no—
398C2×C4—11322no—
399C2×C4—11322no—
400C2×C4D4×D28311yescharacteristic
401C2×C4C23×D28311yescharacteristic
402C8—21no—
403C8—5661no—
404C8—5661no—
405C8C2×C22⋊D28312yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—5661no—
420C22—5661no—
421C22—5661no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22C2×D4×D28311yescharacteristic
430C4—21no—
431C4—22no—
432C4—2832no—
433C4—5661no—
434C4—5661no—
435C4—11321no—
436C4—11321no—
437C4C22×C22⋊D28311yescharacteristic
438C4#2258365511yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—11321no—
444C2—22641no—
445C2—22641no—
446C2D4×C22⋊D28311yescharacteristic
447C2AGL(1,ℤ/8)×D28311yescharacteristic
448C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3274
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. #31827134C2semidirect
3. D4×C22⋊D283C2semidirect
4. #31826840C2semidirect
5. #31826718C2semidirect
6. #31826715C2semidirect
7. #31826663C2semidirect
8. #31826658C2semidirect
9. #31826610C2semidirect
10. #31826609C2semidirect
11. #31826599C2semidirect
12. #31826597C2semidirect
13. SD16×Dic283C2semidirect
14. #31826541C2semidirect
15. C2×AGL(1,ℤ/8)×C283C2semidirect
16. C2×#17985393C22semidirect
20. C2×#17985379C22semidirect
21. C2×D2264C22semidirect
23. C2×#17985377C22semidirect
24. C2×#17985377C22semidirect
25. C2×#17985375C22semidirect
28. #22583606C22semidirect
31. #22583584C22semidirect
35. #22583581C22semidirect
38. Q8×Dic283C22semidirect
42. #22583542C22semidirect
43. Dic283⋊C8C22semidirect
44. #22583534C22semidirect
45. #22583532C22semidirect
46. #22583532C22semidirect
48. #22583531C22semidirect
49. #22583531C22semidirect
50. C22⋊(C283⋊C8)C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. C8×Dic283C22semidirect
53. C2×#41×C283C22semidirect
57. C2×SD16×C283C22semidirect
59. C2×D8×C283C22semidirect
60. C2×M16×C283C22semidirect
118. SD16×C283D4semidirect
164. Q8×C283C2×D4semidirect
180. C8×C283C2×D4semidirect
213. Dic283AGL(1,ℤ/8)semidirect
224. C283#3274semidirect
227. C2×AGL(1,ℤ/8)D283semidirect
242. C2×#41C2×D283semidirect
250. C2×SD16C2×D283semidirect
256. C2×D8C2×D283semidirect
258. C2×M16C2×D283semidirect
323. SD16C22⋊D283semidirect
363. Q8C2×C22⋊D283semidirect
405. C8C2×C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
#318271342
D4×C22⋊D2832
#318268402
#318267182
#318267152
#318266632
#318266582
#318266102
#318266092
#318265992
#318265972
SD16×Dic2832
#318265412
C2×AGL(1,ℤ/8)×C2832
#3274283