Finite Group Databasebeta

← #49130506

Subgroups — #49130506

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×#17985392C211yesmaximal, characteristic, complemented
2C2×#22583643C211yesmaximal, characteristic, complemented
3C2×#22583638C211yesmaximal, characteristic, complemented
4#31826976C211yesmaximal, characteristic, complemented
5#31826629C214yesmaximal, complemented
6#31826612C214yesmaximal, complemented
7#31826475C211yesmaximal, characteristic, complemented
8#31826471C211yesmaximal, characteristic, complemented
9#407×C283C211yesmaximal, characteristic, complemented
10C2×#17985392—24no—
11C2×#17985392—24no—
12C2×#17985392—22no—
13C2×#17985392C2211yescharacteristic
14C2×#17985392C2214yes—
15C22×Dic566C2211yescharacteristic
16C22×D1132C2211yescharacteristic
17C22×C22⋊D283—21no—
18C22×C4×D283—21no—
19#22583643—24no—
20#22583638—24no—
21C2×Dic1132C2212yescomplemented
22C2×#17985381C2212yescomplemented
23C2×#17985381C2212yescomplemented
24C2×D2264C2212yescomplemented
25#22583593—21no—
26#22583589C2212yescomplemented
27#22583584C2212yescomplemented
28#22583579—21no—
29C4×C22⋊D283—22no—
30#22583569C2211yescharacteristic
31#22583535C2214yescomplemented
32#22583532C2214yescomplemented
33C23×C4×C283C2211yescharacteristic
34C2×M16×C283C2211yescharacteristic
35C2×M16×C283C2211yescharacteristic
36C22⋊C8×C283C2214yescomplemented
37C23×D283—21no—
38#17985392—216no—
39#17985392—28no—
40#17985392—28no—
41#17985392—22no—
42#17985392—22no—
43C2×Dic566—22no—
44C2×Dic566—21no—
45C2×Dic566D414yescomplemented
46C2×Dic566C2311yescharacteristic
47C2×Dic566C2312yes—
48C2×D1132—22no—
49C2×D1132—21no—
50C2×D1132D414yescomplemented
51C2×D1132C2311yescharacteristic
52C2×D1132C2312yes—
53C2×C22⋊D283—28no—
54C2×C22⋊D283—24no—
55C2×C22⋊D283—22no—
56C2×C22⋊D283—21no—
57C22×Dic283—21no—
58C2×C4×D283—28no—
59C2×C4×D283—24no—
60C2×C4×D283—21no—
61Dic1132—24no—
62#17985381—24no—
63#17985381—24no—
64D2264—24no—
65C4⋊Dic283C2312yes—
66Dic283⋊C4—22no—
67C22⋊Dic283—21no—
68C22⋊Dic283—22no—
69(C2×D283)⋊C4—22no—
70C4×Dic283—21no—
71C24×C283D411yescharacteristic
72C22×C4×C283—22no—
73C22×C4×C283—22no—
74C22×C4×C283D412yes—
75C22×C4×C283D411yescharacteristic
76C22×C4×C283C2312yes—
77C22×C4×C283C2311yescharacteristic
78M16×C283—22no—
79M16×C283—22no—
80C2×C8×C283C2312yes—
81C2×C8×C283C2312yes—
82C22×D283—24no—
83C22×D283—22no—
84C22×D283—21no—
85C22×D283—44no—
86Dic566—24no—
87Dic566—22no—
88Dic566—21no—
89Dic566—21no—
90Dic566C2×D414yes—
91D1132—24no—
92D1132—22no—
93D1132—21no—
94D1132—21no—
95D1132C2×D414yes—
96C22⋊D283—216no—
97C22⋊D283—44no—
98C22⋊D283—42no—
99C2×Dic283—24no—
100C2×Dic283—22no—
101C2×Dic283—22no—
102C2×Dic283—21no—
103C4×D283—216no—
104C4×D283—42no—
105C23×C283—22no—
106C23×C283—42no—
107C23×C283C2×D412yes—
108C23×C283C2×D411yescharacteristic
109C2×C4×C283—22no—
110C2×C4×C283—24no—
111C2×C4×C283—22no—
112C2×C4×C283—22no—
113C2×C4×C283C2×D412yes—
114C2×C4×C283C2×D412yes—
115C2×C4×C283C2×D412yes—
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283C2411yescharacteristic
118C8×C283—22no—
119C8×C283—22no—
120C22×C283—22no—
121C22×C283—22no—
122C22×C283—22no—
123C22×C283—22no—
124C22×C283—44no—
125C22×C283#12012yes—
126C22×C283C22×D411yescharacteristic
127C2×D283—24no—
128C2×D283—48no—
129C2×D283—44no—
130C2×D283—41no—
131C2×D283—42no—
132C4×C283—22no—
133C4×C283#12012yes—
134C4×C283C22×D411yescharacteristic
135C4×C283C22×D411yescharacteristic
136Dic283—24no—
137Dic283—41no—
138Dic283—42no—
139C2×C283—22no—
140C2×C283—42no—
141C2×C283C2×#12011yescharacteristic
142C2×C283#57512yes—
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#419—2831no—
154#418—2831no—
155#407D28311yescharacteristic, complemented
156C2×#41—2831no—
157C2×#41—2834no—
158C2×#41—5664no—
159C2×#41—5664no—
160C2×#41—5662no—
161C22×Q8—2831no—
162C22×D4—2831no—
163C22×D4—5661no—
164C23×C4—5661no—
165C23×C4C2×D28311yescharacteristic
166#137—5664no—
167AGL(1,ℤ/8)—5664no—
168C2×Q16—2832no—
169C2×SD16—2832no—
170C2×SD16—2832no—
171C2×D8—2832no—
172C2×M16C2×D28311yescharacteristic
173C2×M16C2×D28311yescharacteristic
174#123—5661no—
175C22⋊Q8—2832no—
176#121—2832no—
177#120—5661no—
178C4×D4—5662no—
179#117—2831no—
180#103—2834no—
181#102—2834no—
182C22⋊C8C2×D28314yescomplemented
183C24—5661no—
184C24D113211yescharacteristic
185#41—5668no—
186#41—5668no—
187#41—56616no—
188#41—5662no—
189#41—5662no—
190C2×Q8—2834no—
191C2×Q8—2832no—
192C2×Q8—2831no—
193C2×Q8—5662no—
194C2×Q8—5661no—
195C2×D4—2834no—
196C2×D4—2832no—
197C2×D4—2831no—
198C2×D4—5668no—
199C2×D4—5664no—
200C2×D4—5662no—
201C2×D4—5661no—
202C2×D4—5662no—
203C2×D4—5661no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—5668no—
207C22×C4—5661no—
208C22×C4—5664no—
209C22×C4—5661no—
210C22×C4D113212yes—
211C22×C4D113211yescharacteristic
212C22×C4C22×D28312yes—
213C22×C4C22×D28311yescharacteristic
214Q16—5664no—
215SD16—5664no—
216SD16—5664no—
217D8—5664no—
218M16—22no—
219M16—22no—
220C2×C8C22×D28312yes—
221C2×C8C22×D28312yes—
222C4⋊C4—2832no—
223C4⋊C4—5662no—
224C22⋊C4—5661no—
225C22⋊C4—5662no—
226C22⋊C4—5662no—
227C42—5661no—
228C23—22no—
229C23—42no—
230C23—5664no—
231C23—5662no—
232C23—5661no—
233C23—11324no—
234C23C2×D113212yes—
235C23C2×D113211yescharacteristic
236Q8—2834no—
237Q8—5664no—
238Q8—5662no—
239Q8—5661no—
240Q8—5661no—
241D4—2834no—
242D4—56616no—
243D4—5664no—
244D4—5662no—
245D4—5661no—
246D4—5661no—
247D4—11324no—
248D4—11322no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—56616no—
254C2×C4—5664no—
255C2×C4—5662no—
256C2×C4—5662no—
257C2×C4—5661no—
258C2×C4—11322no—
259C2×C4D4×D28312yes—
260C2×C4D4×D28312yes—
261C2×C4C2×D113212yes—
262C2×C4C2×D113211yescharacteristic
263C2×C4C23×D28311yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—5664no—
272C22—11328no—
273C22—11324no—
274C22—11321no—
275C22—11322no—
276C22#2258358012yes—
277C22C22×D113211yescharacteristic
278C4—22no—
279C4—5664no—
280C4—11321no—
281C4—11322no—
282C4#2258358012yes—
283C4C2×D4×D28311yescharacteristic
284C4C2×D4×D28311yescharacteristic
285C2—22no—
286C2—42no—
287C2—11324no—
288C2—22641no—
289C2C2×#2258358011yescharacteristic
290C2#3182726512yes—
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#407×C283C2
socleC22×C283C22×D4
hypercenter#407D283
hypocenterC283#3005
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#17985392C2semidirect
2. C2×#22583643C2semidirect
3. C2×#22583638C2semidirect
4. #31826976C2semidirect
5. #31826629C2semidirect
6. #31826612C2semidirect
7. #31826475C2semidirect
8. #31826471C2semidirect
9. #407×C283C2semidirect
21. C2×Dic1132C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985381C22semidirect
24. C2×D2264C22semidirect
26. #22583589C22semidirect
27. #22583584C22semidirect
31. #22583535C22semidirect
32. #22583532C22semidirect
36. C22⋊C8×C283C22semidirect
45. C2×Dic566D4semidirect
50. C2×D1132D4semidirect
145. C283#3005semidirect
155. #407D283semidirect
182. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853922
C2×#225836432
C2×#225836382
#318269762
#318266292
#318266122
#318264752
#318264712
#407×C2832
#3005283