Finite Group Databasebeta

← #49130528

Subgroups — #49130528

Order 36224.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583656C211yesmaximal, characteristic, complemented
2C22×#17985377C211yesmaximal, characteristic, complemented
3#31827032C211yesmaximal, characteristic, complemented
4#31826609C214yesmaximal, complemented
5#31826597C214yesmaximal, complemented
6#31826471C211yesmaximal, characteristic, complemented
7#31826461C211yesmaximal, characteristic, complemented
8#31826433C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
10#22583656—24no—
11#22583656—24no—
12C2×#17985393C2212yescomplemented
13C2×#17985392—21no—
14C2×#17985392C2211yescharacteristic
15C22×D1132—21no—
16C22×D1132C2211yescharacteristic
17C2×D4×D283—22no—
18C2×D4×D283C2212yescomplemented
19C2×#17985377—24no—
20C2×#17985377C2214yescomplemented
21C22×C283⋊C8C2211yescharacteristic, complemented
22#22583606—21no—
23#22583597—21no—
24#22583581C2214yescomplemented
25#22583579—22no—
26#22583569C2211yescharacteristic, complemented
27#22583534C2212yescomplemented
28#22583532C2214yescomplemented
29#22583531C2212yescomplemented
30(C2×D283)⋊C8C2214yescomplemented
31C2×#41×C283C2211yescharacteristic
32C22×D4×C283C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C283—24no—
34C2×SD16×C283C2212yescomplemented
35C2×D8×C283C2212yescomplemented
36C2×M16×C283C2211yescharacteristic, complemented
37C23×D283—22no—
38C23×D283—21no—
39#17985393—24no—
40#17985393—24no—
41#17985392—28no—
42#17985392—22no—
43#17985392—22no—
44C2×Dic566D411yescharacteristic
45C2×D1132—28no—
46C2×D1132—22no—
47C2×D1132—21no—
48C2×D1132—22no—
49C2×D1132D411yescharacteristic
50C2×D1132D414yescomplemented
51C2×D1132C2312yes—
52C2×C22⋊D283—24no—
53C2×C22⋊D283—22no—
54C2×C22⋊D283D412yes—
55D4×D283—216no—
56D4×D283—24no—
57D4×D283—24no—
58C2×C4×D283—22no—
59C2×C4×D283C2312yes—
60#17985377—28no—
61C2×C283⋊C8—22no—
62C2×C283⋊C8C2312yes—
63C4⋊Dic283C2312yes—
64C22⋊Dic283—24no—
65C22⋊Dic283—21no—
66C4×Dic283—21no—
67C24×C283—21no—
68#41×C283—22no—
69#41×C283D414yescomplemented
70C2×Q8×C283C2311yescharacteristic
71C2×D4×C283—21no—
72C2×D4×C283—22no—
73C2×D4×C283—22no—
74C2×D4×C283—21no—
75C2×D4×C283C2312yes—
76C2×D4×C283C2311yescharacteristic
77C22×C4×C283—21no—
78C22×C4×C283C2311yescharacteristic
79SD16×C283—24no—
80D8×C283—24no—
81M16×C283—22no—
82C2×C8×C283C2312yes—
83C22×D283—22no—
84C22×D283—28no—
85C22×D283—24no—
86C22×D283—84no—
87C22×D283—82no—
88C22×D283C2×D412yes—
89Dic566—22no—
90D1132—28no—
91D1132—22no—
92D1132—24no—
93D1132—22no—
94D1132C2×D414yes—
95C22⋊D283—28no—
96C22⋊D283—24no—
97C22⋊D283—84no—
98C2×Dic283—22no—
99C2×Dic283C2×D412yes—
100C4×D283—28no—
101C4×D283—22no—
102C4×D283—22no—
103C283⋊C8—22no—
104C23×C283—21no—
105C23×C283—22no—
106C23×C283—42no—
107C23×C283—82no—
108C23×C283C2×D411yescharacteristic
109Q8×C283—21no—
110Q8×C283C2×D412yes—
111D4×C283—24no—
112D4×C283—21no—
113D4×C283—22no—
114D4×C283—22no—
115D4×C283—42no—
116D4×C283C2×D412yes—
117C2×C4×C283—21no—
118C2×C4×C283—24no—
119C2×C4×C283C2×D411yescharacteristic
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2411yescharacteristic
123C8×C283—22no—
124C22×C283—22no—
125C22×C283—22no—
126C22×C283—42no—
127C22×C283—42no—
128C22×C283—44no—
129C22×C283—84no—
130C22×C283#12012yes—
131C22×C283C22×D411yescharacteristic
132C2×D283—22no—
133C2×D283—24no—
134C2×D283—24no—
135C2×D283—44no—
136C2×D283—84no—
137C2×D283—88no—
138C4×C283—22no—
139C4×C283#12012yes—
140C4×C283C22×D411yescharacteristic
141C4×C283C22×D411yescharacteristic
142Dic283—22no—
143Dic283—42no—
144C2×C283—22no—
145C2×C283—42no—
146C2×C283—82no—
147C2×C283C2×#12011yescharacteristic
148C2×C283#57612yes—
149D283—42no—
150D283—44no—
151C283#300811yescharacteristic, complemented, Sylow
152#3008—2831nomaximal, Sylow
153C2×H(5,𝔽2)—2831no—
154C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
155C22×D8—2831no—
156#535—2831no—
157#449—2834no—
158#447—2834no—
159#418—2831no—
160#416—2831no—
161#408—2831no—
162H(5,𝔽2)—5664no—
163H(5,𝔽2)—5664no—
164C2×#41—2831no—
165C2×#41—2832no—
166C2×#41—5661no—
167C2×#41C2×D28311yescharacteristic
168C22×D4—2832no—
169C22×D4—2831no—
170C22×D4—5661no—
171C22×D4—5662no—
172C22×D4C2×D28311yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D28312yescomplemented
175C2×D8—2834no—
176C2×D8—5664no—
177C2×D8C2×D28312yescomplemented
178C2×M16C2×D28311yescharacteristic, complemented
179C22×C8—2831no—
180#127—5661no—
181#124—5661no—
182#121—2834no—
183#120—5662no—
184#117—2831no—
185#103—2832no—
186#102—2832no—
187#102—2834no—
188C22⋊C8—2834no—
189C24—21no—
190C24—5662no—
191C24—5661no—
192#41—22no—
193#41—5668no—
194#41—5664no—
195#41—5664no—
196#41—5662no—
197#41—5662no—
198#41C22⋊D28314yescomplemented
199C2×Q8—2831no—
200C2×Q8C22×D28311yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2832no—
206C2×D4—2834no—
207C2×D4—2831no—
208C2×D4—2832no—
209C2×D4—5662no—
210C2×D4—5668no—
211C2×D4—5662no—
212C2×D4—5664no—
213C2×D4—56616no—
214C2×D4—5664no—
215C2×D4—5664no—
216C2×D4—5661no—
217C2×D4—5662no—
218C2×D4C22×D28312yes—
219C2×D4C22×D28311yescharacteristic
220C22×C4—21no—
221C22×C4—2832no—
222C22×C4—5662no—
223C22×C4C22×D28311yescharacteristic
224SD16—24no—
225D8—24no—
226D8—5668no—
227M16—22no—
228C2×C8—2832no—
229C2×C8—5662no—
230C2×C8C22×D28312yes—
231C4⋊C4—2832no—
232C22⋊C4—5664no—
233C22⋊C4—5661no—
234C42—5661no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—2832no—
240C23—5668no—
241C23—5662no—
242C23—5664no—
243C23—22642no—
244C23—22644no—
245C23D4×D28311yescharacteristic
246Q8—21no—
247Q8—5662no—
248Q8C2×C22⋊D28312yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—2834no—
255D4—5668no—
256D4—5668no—
257D4—5664no—
258D4—5664no—
259D4—5662no—
260D4—5662no—
261D4—22644no—
262D4C2×C22⋊D28312yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—2832no—
266C2×C4—5668no—
267C2×C4—5662no—
268C2×C4—5662no—
269C2×C4—5662no—
270C2×C4D4×D28311yescharacteristic
271C2×C4D4×D28312yes—
272C2×C4C2×C22⋊D28312yes—
273C2×C4C23×D28311yescharacteristic
274C8—22no—
275C8—5662no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—5664no—
283C22—5662no—
284C22—5664no—
285C22—11324no—
286C22—22644no—
287C22—22648no—
288C22#2258357812yes—
289C22C2×D4×D28311yescharacteristic
290C4—22no—
291C4—5662no—
292C4—11322no—
293C4#2258357812yes—
294C4C2×D4×D28311yescharacteristic
295C4C22×C22⋊D28311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—11322no—
300C2—11324no—
301C2C2×#2258357811yescharacteristic
302C2#3182726912yes—
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#3008
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×#22583656C2semidirect
2. C22×#17985377C2semidirect
3. #31827032C2semidirect
4. #31826609C2semidirect
5. #31826597C2semidirect
6. #31826471C2semidirect
7. #31826461C2semidirect
8. #31826433C2semidirect
9. C2×AGL(1,ℤ/8)×C283C2semidirect
12. C2×#17985393C22semidirect
18. C2×D4×D283C22semidirect
20. C2×#17985377C22semidirect
21. C22×C283⋊C8C22semidirect
24. #22583581C22semidirect
26. #22583569C22semidirect
27. #22583534C22semidirect
28. #22583532C22semidirect
29. #22583531C22semidirect
30. (C2×D283)⋊C8C22semidirect
32. C22×D4×C283C22semidirect
34. C2×SD16×C283C22semidirect
35. C2×D8×C283C22semidirect
36. C2×M16×C283C22semidirect
50. C2×D1132D4semidirect
69. #41×C283D4semidirect
151. C283#3008semidirect
154. C2×AGL(1,ℤ/8)D283semidirect
172. C22×D4C2×D283semidirect
174. C2×SD16C2×D283semidirect
177. C2×D8C2×D283semidirect
178. C2×M16C2×D283semidirect
198. #41C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C22×#179853772
#318270322
#318266092
#318265972
#318264712
#318264612
#318264332
C2×AGL(1,ℤ/8)×C2832
#3008283