Finite Group Databasebeta

← #49130507

Subgroups — #49130507

Order 36224.

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

normal subgroups only

This group has 758 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
1C22×#17985393C211yesmaximal, characteristic, complemented
2C2×#22583641C211yesmaximal, characteristic, complemented
3C2×#22583638C211yesmaximal, characteristic, complemented
4#31826977C211yesmaximal, characteristic, complemented
5#31826630C214yesmaximal, complemented
6#31826613C214yesmaximal, complemented
7#31826469C211yesmaximal, characteristic, complemented
8#31826428C211yesmaximal, characteristic, complemented
9#419×C283C211yesmaximal, characteristic, complemented
10C2×#17985393—24no—
11C2×#17985393—22no—
12C2×#17985393—24no—
13C2×#17985393C2214yes—
14C2×#17985392C2211yescharacteristic
15C22×D1132—21no—
16C22×D1132C2211yescharacteristic
17C22×C4×D283—21no—
18C22×C4×D283C2211yescharacteristic
19#22583641—24no—
20#22583638—24no—
21C2×#17985382C2212yescomplemented
22C2×#17985381C2212yescomplemented
23C2×#17985379C2212yescomplemented
24C2×D2264C2212yescomplemented
25C2×#17985376C2211yescharacteristic
26#22583594—21no—
27#22583590C2212yescomplemented
28#22583585C2212yescomplemented
29#22583580—21no—
30C4×D1132—22no—
31#22583530C2214yescomplemented
32(C2×D283)⋊C8C2214yescomplemented
33C22×Q8×C283C2211yescharacteristic
34C2×M16×C283C2211yescharacteristic
35#117×C283C2211yescharacteristic
36#103×C283C2214yescomplemented
37C23×D283—21no—
38C23×D283D411yescharacteristic
39#17985393—216no—
40#17985393—28no—
41#17985393—28no—
42#17985392—22no—
43#17985392—22no—
44C2×Dic566C2311yescharacteristic
45C2×D1132—28no—
46C2×D1132—24no—
47C2×D1132—22no—
48C2×D1132—21no—
49C2×D1132—22no—
50C2×D1132D414yescomplemented
51C2×D1132C2311yescharacteristic
52C2×D1132C2312yes—
53C2×C22⋊D283—21no—
54C22×Dic283D411yescharacteristic
55C2×C4×D283—28no—
56C2×C4×D283—24no—
57C2×C4×D283—22no—
58C2×C4×D283—22no—
59C2×C4×D283—21no—
60C2×C4×D283D412yes—
61C2×C4×D283C2312yes—
62#17985382—24no—
63#17985381—24no—
64#17985379—24no—
65D2264—24no—
66#17985376—22no—
67C2×C283⋊C8C2312yes—
68C4⋊Dic283—22no—
69(C2×D283)⋊C4—22no—
70(C2×D283)⋊C4—22no—
71C2×Q8×C283—22no—
72C2×Q8×C283—21no—
73C2×Q8×C283D414yescomplemented
74C2×Q8×C283C2312yes—
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77M16×C283—22no—
78C2×C8×C283C2312yes—
79C4⋊C4×C283C2312yes—
80C22⋊C4×C283—21no—
81C42×C283—21no—
82C22×D283—22no—
83C22×D283—24no—
84C22×D283—22no—
85C22×D283—21no—
86C22×D283—42no—
87C22×D283—44no—
88C22×D283C2×D412yes—
89Dic566—21no—
90Dic566—21no—
91D1132—216no—
92D1132—24no—
93D1132—22no—
94D1132—21no—
95D1132—21no—
96D1132—44no—
97D1132C2×D414yes—
98C22⋊D283—42no—
99C2×Dic283—22no—
100C2×Dic283—21no—
101C2×Dic283C2×D412yes—
102C4×D283—216no—
103C4×D283—24no—
104C4×D283—22no—
105C4×D283—22no—
106C4×D283—42no—
107C283⋊C8—22no—
108C23×C283C2×D411yescharacteristic
109Q8×C283—24no—
110Q8×C283—22no—
111Q8×C283C2×D414yes—
112C2×C4×C283—24no—
113C2×C4×C283—22no—
114C2×C4×C283—22no—
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2×D412yes—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2411yescharacteristic
119C8×C283—22no—
120C22×C283—22no—
121C22×C283#12012yes—
122C22×C283C22×D411yescharacteristic
123C2×D283—22no—
124C2×D283—24no—
125C2×D283—22no—
126C2×D283—22no—
127C2×D283—44no—
128C2×D283—48no—
129C2×D283—44no—
130C2×D283—41no—
131C2×D283—42no—
132C4×C283—24no—
133C4×C283—42no—
134C4×C283#12012yes—
135C4×C283C22×D411yescharacteristic
136C4×C283C22×D411yescharacteristic
137Dic283—22no—
138Dic283—41no—
139C2×C283—22no—
140C2×C283C2×#12011yescharacteristic
141C2×C283#57512yes—
142D283—42no—
143D283—44no—
144D283—81no—
145C283#300511yescharacteristic, complemented, Sylow
146#3005—2831nomaximal, Sylow
147C22×#41—2831no—
148C2×#137—2831no—
149C2×AGL(1,ℤ/8)—2831no—
150#525—2831no—
151#452—2834no—
152#449—2834no—
153#419D28311yescharacteristic, complemented
154#418—2831no—
155#407—2831no—
156C2×#41—2834no—
157C2×#41—2831no—
158C2×#41—5662no—
159C2×#41—5664no—
160C2×#41—5664no—
161C22×Q8C2×D28311yescharacteristic
162C22×D4—2831no—
163C22×D4—5661no—
164C23×C4—2831no—
165C23×C4—5661no—
166#137—5664no—
167AGL(1,ℤ/8)—5664no—
168C2×Q16—2832no—
169C2×SD16—2832no—
170C2×SD16—2832no—
171C2×D8—2832no—
172C2×M16—2831no—
173C2×M16C2×D28311yescharacteristic
174#123—5661no—
175C22⋊Q8—2832no—
176#121—2832no—
177#120—5661no—
178C4×D4—5662no—
179#117C2×D28311yescharacteristic
180#103C2×D28314yescomplemented
181#102—2834no—
182C22⋊C8—2834no—
183C24—2831no—
184C24—5661no—
185#41—5668no—
186#41—56616no—
187#41—5668no—
188#41—5662no—
189#41—5662no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—2831no—
193C2×Q8D113214yescomplemented
194C2×Q8C22×D28312yes—
195C2×D4—2834no—
196C2×D4—2832no—
197C2×D4—2831no—
198C2×D4—5664no—
199C2×D4—5668no—
200C2×D4—5662no—
201C2×D4—5661no—
202C2×D4—5661no—
203C2×D4—5662no—
204C22×C4—21no—
205C22×C4—2831no—
206C22×C4—2832no—
207C22×C4—2832no—
208C22×C4—5664no—
209C22×C4—5668no—
210C22×C4—5662no—
211C22×C4—5662no—
212C22×C4—5661no—
213C22×C4C22×D28311yescharacteristic
214Q16—5664no—
215SD16—5664no—
216SD16—5664no—
217D8—5664no—
218M16—22no—
219M16—5662no—
220C2×C8—2832no—
221C2×C8C22×D28312yes—
222C4⋊C4—5662no—
223C4⋊C4C22×D28312yes—
224C22⋊C4—21no—
225C22⋊C4—5662no—
226C22⋊C4—5662no—
227C42—21no—
228C23—2832no—
229C23—5664no—
230C23—5662no—
231C23—5662no—
232C23—5661no—
233C23—11322no—
234C23—11324no—
235C23D4×D28311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—5661no—
239Q8—5661no—
240Q8C2×D113214yes—
241D4—2834no—
242D4—56616no—
243D4—5664no—
244D4—5661no—
245D4—5662no—
246D4—5661no—
247D4—11324no—
248D4—11322no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—2832no—
253C2×C4—56616no—
254C2×C4—5662no—
255C2×C4—5664no—
256C2×C4—5662no—
257C2×C4—5662no—
258C2×C4—5661no—
259C2×C4—11322no—
260C2×C4D4×D28311yescharacteristic
261C2×C4D4×D28312yes—
262C2×C4C2×D113212yes—
263C2×C4C23×D28311yescharacteristic
264C8—22no—
265C8—5662no—
266C22—22no—
267C22—5664no—
268C22—5662no—
269C22—5662no—
270C22—5662no—
271C22—11324no—
272C22—11328no—
273C22—11324no—
274C22—11321no—
275C22—11322no—
276C22#2258358012yes—
277C22C2×D4×D28311yescharacteristic
278C4—24no—
279C4—42no—
280C4—5662no—
281C4—11321no—
282C4#2258358012yes—
283C4C2×D4×D28311yescharacteristic
284C4C22×D113211yescharacteristic
285C2—22no—
286C2—11322no—
287C2—11324no—
288C2—22641no—
289C2C2×#2258358011yescharacteristic
290C2#3182726412yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#419×C283C2
socleC22×C283C22×D4
hypercenter#419D283
hypocenterC283#3005
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#17985393C2semidirect
2. C2×#22583641C2semidirect
3. C2×#22583638C2semidirect
4. #31826977C2semidirect
5. #31826630C2semidirect
6. #31826613C2semidirect
7. #31826469C2semidirect
8. #31826428C2semidirect
9. #419×C283C2semidirect
21. C2×#17985382C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985379C22semidirect
24. C2×D2264C22semidirect
27. #22583590C22semidirect
28. #22583585C22semidirect
31. #22583530C22semidirect
32. (C2×D283)⋊C8C22semidirect
36. #103×C283C22semidirect
50. C2×D1132D4semidirect
73. C2×Q8×C283D4semidirect
145. C283#3005semidirect
153. #419D283semidirect
180. #103C2×D283semidirect
193. C2×Q8D1132semidirect

Maximal subgroups

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

subgroupindex
C22×#179853932
C2×#225836412
C2×#225836382
#318269772
#318266302
#318266132
#318264692
#318264282
#419×C2832
#3005283