Finite Group Databasebeta

← #49130486

Subgroups — #49130486

Order 36224.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D283C211yesmaximal, characteristic, complemented
2C2×#22583635C211yesmaximal, characteristic, complemented
3#31826972C211yesmaximal, characteristic, complemented
4#31826614C214yesmaximal, complemented
5#31826597C214yesmaximal, complemented
6#31826471C211yesmaximal, characteristic, complemented
7#31826470C211yesmaximal, characteristic, complemented
8#31826428C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
10C24×D283—21no—
11C22×D1132C2211yescharacteristic
12C22×C22⋊D283—21no—
13C2×D4×D283—24no—
14C2×D4×D283—22no—
15C2×D4×D283—24no—
16C2×D4×D283C2214yes—
17C22×C4×D283C2211yescharacteristic
18#22583635—24no—
19C2×#17985379C2212yescomplemented
20C2×#17985377C2212yescomplemented
21C2×#17985376C2211yescharacteristic, complemented
22#22583591—21no—
23#22583586C2212yescomplemented
24#22583581C2212yescomplemented
25#22583578—21no—
26C4×C22⋊D283—22no—
27#22583569C2211yescharacteristic, complemented
28#22583532C2214yescomplemented
29#22583531C2214yescomplemented
30(C2×D283)⋊C8C2214yescomplemented
31C2×#41×C283C2211yescharacteristic, complemented
32C22×D4×C283C2211yescharacteristic
33AGL(1,ℤ/8)×C283—24no—
34C2×SD16×C283C2212yescomplemented
35C2×D8×C283C2212yescomplemented
36C2×M16×C283C2211yescharacteristic, complemented
37C23×D283—28no—
38C23×D283—24no—
39C23×D283—21no—
40C23×D283—48no—
41C23×D283D411yescharacteristic
42C2×D1132—21no—
43C2×D1132—22no—
44C2×D1132D414yescomplemented
45C2×D1132C2312yes—
46C2×C22⋊D283—28no—
47C2×C22⋊D283—22no—
48C2×C22⋊D283—24no—
49D4×D283—216no—
50D4×D283—28no—
51D4×D283—28no—
52C22×Dic283D411yescharacteristic
53C2×C4×D283—22no—
54C2×C4×D283—22no—
55C2×C4×D283D412yes—
56C2×C4×D283C2312yes—
57#17985379—24no—
58#17985377—24no—
59#17985376—22no—
60C2×C283⋊C8C2312yescomplemented
61C4⋊Dic283C2312yescomplemented
62Dic283⋊C4—22no—
63C22⋊Dic283—22no—
64C22⋊Dic283—21no—
65(C2×D283)⋊C4—22no—
66C4×Dic283—21no—
67C24×C283—21no—
68#41×C283—22no—
69#41×C283—22no—
70C2×Q8×C283C2311yescharacteristic, complemented
71C2×D4×C283—21no—
72C2×D4×C283—21no—
73C2×D4×C283—22no—
74C2×D4×C283D414yescomplemented
75C2×D4×C283C2312yes—
76C2×D4×C283C2311yescharacteristic, complemented
77C22×C4×C283—21no—
78C22×C4×C283C2311yescharacteristic
79SD16×C283—24no—
80D8×C283—24no—
81M16×C283—22no—
82C2×C8×C283C2312yescomplemented
83C22×D283—22no—
84C22×D283—24no—
85C22×D283—216no—
86C22×D283—24no—
87C22×D283—22no—
88C22×D283—432no—
89C22×D283—48no—
90C22×D283—44no—
91C22×D283—48no—
92C22×D283C2×D412yes—
93D1132—24no—
94D1132—22no—
95D1132C2×D414yes—
96C22⋊D283—216no—
97C22⋊D283—44no—
98C2×Dic283—22no—
99C2×Dic283—22no—
100C2×Dic283C2×D412yes—
101C4×D283—24no—
102C4×D283—22no—
103C4×D283—22no—
104C283⋊C8—22no—
105C23×C283—24no—
106C23×C283—22no—
107C23×C283—21no—
108C23×C283—44no—
109C23×C283C2×D411yescharacteristic
110Q8×C283—21no—
111Q8×C283—21no—
112D4×C283—24no—
113D4×C283—22no—
114D4×C283—21no—
115D4×C283—21no—
116D4×C283—42no—
117D4×C283C2×D414yes—
118C2×C4×C283—21no—
119C2×C4×C283—42no—
120C2×C4×C283C2×D411yescharacteristic
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2411yescharacteristic
124C8×C283—22no—
125C22×C283—22no—
126C22×C283—24no—
127C22×C283—48no—
128C22×C283—44no—
129C22×C283—41no—
130C22×C283—42no—
131C22×C283#12012yes—
132C22×C283C22×D411yescharacteristic
133C2×D283—28no—
134C2×D283—22no—
135C2×D283—22no—
136C2×D283—24no—
137C2×D283—432no—
138C2×D283—48no—
139C2×D283—44no—
140C2×D283AGL(1,ℤ/8)14yescomplemented
141C4×C283—41no—
142C4×C283#12012yes—
143C4×C283C22×D411yescharacteristic
144C4×C283C22×D411yescharacteristic
145Dic283—22no—
146Dic283—42no—
147C2×C283—22no—
148C2×C283—44no—
149C2×C283—81no—
150C2×C283C2×#12011yescharacteristic
151C2×C283C2×AGL(1,ℤ/8)12yes—
152D283—24no—
153D283—44no—
154C283#300311yescharacteristic, complemented, Sylow
155#3003—2831nomaximal, Sylow
156C23×D4—2831no—
157C2×AGL(1,ℤ/8)—2831no—
158C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
159#525—2831no—
160#450—2834no—
161#447—2834no—
162#418—2831no—
163#418—2831no—
164#407—2831no—
165C25—5661no—
166C2×#41C2×D28311yescharacteristic, complemented
167C22×D4—2834no—
168C22×D4—2831no—
169C22×D4—5662no—
170C22×D4—5661no—
171C22×D4—5664no—
172C22×D4—5664no—
173C22×D4C2×D28311yescharacteristic
174C23×C4—2831no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5664no—
177C2×SD16—2832no—
178C2×SD16C2×D28312yescomplemented
179C2×D8—2832no—
180C2×D8C2×D28312yescomplemented
181C2×M16—2831no—
182C2×M16C2×D28311yescharacteristic, complemented
183#123—5661no—
184C22⋊Q8—2832no—
185#121—2832no—
186#120—5661no—
187C4×D4—5662no—
188#117—2831no—
189#102—2834no—
190#102—2834no—
191C22⋊C8—2834no—
192C24—21no—
193C24—2831no—
194C24—5664no—
195C24—5668no—
196C24—5661no—
197C24—11328no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D28311yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—2834no—
205C2×D4—2832no—
206C2×D4—5664no—
207C2×D4—5668no—
208C2×D4—56616no—
209C2×D4—5668no—
210C2×D4—5668no—
211C2×D4—5662no—
212C2×D4—5661no—
213C2×D4—5662no—
214C2×D4C22⋊D28314yescomplemented
215C2×D4C22×D28312yes—
216C2×D4C22×D28311yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—2831no—
219C22×C4—2832no—
220C22×C4—2832no—
221C22×C4—5662no—
222C22×C4—5662no—
223C22×C4C22×D28311yescharacteristic
224SD16—24no—
225SD16—5664no—
226D8—24no—
227D8—5664no—
228M16—22no—
229M16—5662no—
230C2×C8—2832no—
231C2×C8C22×D28312yescomplemented
232C4⋊C4—2832no—
233C4⋊C4—5662no—
234C22⋊C4—5662no—
235C22⋊C4—5662no—
236C22⋊C4—5661no—
237C42—5661no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—2832no—
243C23—5664no—
244C23—56616no—
245C23—5662no—
246C23—5664no—
247C23—5662no—
248C23—11324no—
249C23—11328no—
250C23—113232no—
251C23—11328no—
252C23D4×D28311yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—2834no—
261D4—56616no—
262D4—5664no—
263D4—5662no—
264D4—11324no—
265D4C2×C22⋊D28314yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—2832no—
269C2×C4—5662no—
270C2×C4—5664no—
271C2×C4—5662no—
272C2×C4—5662no—
273C2×C4—5662no—
274C2×C4D4×D28311yescharacteristic
275C2×C4D4×D28312yes—
276C2×C4C2×C22⋊D28312yes—
277C2×C4C23×D28311yescharacteristic
278C8—22no—
279C8—5662no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—2834no—
287C22—5668no—
288C22—5662no—
289C22—5662no—
290C22—5664no—
291C22—11328no—
292C22—113232no—
293C22—11324no—
294C22#2258357812yes—
295C22C2×D4×D28311yescharacteristic
296C4—41no—
297C4—5662no—
298C4—11322no—
299C4#2258357812yes—
300C4C2×D4×D28311yescharacteristic
301C4C22×C22⋊D28311yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—5664no—
306C2—11324no—
307C2C2×#2258357811yescharacteristic
308C2AGL(1,ℤ/8)×D28312yes—
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#3003
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. C22×D4×D283C2semidirect
2. C2×#22583635C2semidirect
3. #31826972C2semidirect
4. #31826614C2semidirect
5. #31826597C2semidirect
6. #31826471C2semidirect
7. #31826470C2semidirect
8. #31826428C2semidirect
9. C2×AGL(1,ℤ/8)×C283C2semidirect
19. C2×#17985379C22semidirect
20. C2×#17985377C22semidirect
21. C2×#17985376C22semidirect
23. #22583586C22semidirect
24. #22583581C22semidirect
27. #22583569C22semidirect
28. #22583532C22semidirect
29. #22583531C22semidirect
30. (C2×D283)⋊C8C22semidirect
31. C2×#41×C283C22semidirect
34. C2×SD16×C283C22semidirect
35. C2×D8×C283C22semidirect
36. C2×M16×C283C22semidirect
44. C2×D1132D4semidirect
60. C2×C283⋊C8C23semidirect
61. C4⋊Dic283C23semidirect
70. C2×Q8×C283C23semidirect
74. C2×D4×C283D4semidirect
76. C2×D4×C283C23semidirect
82. C2×C8×C283C23semidirect
140. C2×D283AGL(1,ℤ/8)semidirect
154. C283#3003semidirect
158. C2×AGL(1,ℤ/8)D283semidirect
166. C2×#41C2×D283semidirect
178. C2×SD16C2×D283semidirect
180. C2×D8C2×D283semidirect
182. C2×M16C2×D283semidirect
200. C2×Q8C22×D283semidirect
214. C2×D4C22⋊D283semidirect
216. C2×D4C22×D283semidirect
231. C2×C8C22×D283semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2832
C2×#225836352
#318269722
#318266142
#318265972
#318264712
#318264702
#318264282
C2×AGL(1,ℤ/8)×C2832
#3003283