Finite Group Databasebeta

← #49131202

Subgroups — #49131202

Order 36224.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22583584C211yesmaximal, characteristic
2C2×#22583569C211yesmaximal, characteristic, complemented
3#31826790C214yesmaximal, complemented
4#31826784C212yesmaximal, complemented
5#31826779C212yesmaximal, complemented
6#31826507C212yesmaximal, complemented
7#31826471C212yesmaximal, complemented
8#407×C283C211yesmaximal, characteristic, complemented
9C22×D1132C2211yescharacteristic
10C22×C22⋊D283—21no—
11#22583584—21no—
12#22583584—21no—
13#22583584C2212yes—
14#22583584C2212yes—
15#22583569—21no—
16#22583569—21no—
17#22583569C2212yes—
18#22583569C2212yescomplemented
19C2×C4⋊Dic283C2211yescharacteristic
20C2×C4⋊Dic283C2211yescharacteristic, complemented
21C2×C4⋊Dic283C2211yescharacteristic, complemented
22C2×C22⋊Dic283—21no—
23C2×(C2×D283)⋊C4—21no—
24C2×C4×Dic283—21no—
25#22583544C2214yescomplemented
26#22583542C2214yescomplemented
27#22583532C2214yescomplemented
28#22583532C2214yescomplemented
29C23×C4×C283C2211yescharacteristic
30C2×M16×C283C2212yes—
31C22⋊C8×C283C2212yescomplemented
32C22⋊C8×C283C2212yescomplemented
33C23×D283—21no—
34C2×D1132—21no—
35C2×D1132—24no—
36C2×D1132C2312yes—
37C2×C22⋊D283—22no—
38C2×C22⋊D283—22no—
39C2×C22⋊D283—44no—
40C22×Dic283—21no—
41C22×Dic283—21no—
42C22×Dic283—21no—
43C4⋊Dic283—21no—
44C4⋊Dic283—21no—
45C4⋊Dic283—21no—
46C4⋊Dic283C2312yes—
47C4⋊Dic283C2312yes—
48C4⋊Dic283C2312yes—
49C22⋊Dic283—22no—
50C22⋊Dic283—22no—
51(C2×D283)⋊C4—22no—
52(C2×D283)⋊C4—22no—
53C4×Dic283—22no—
54C4×Dic283—22no—
55C24×C283D411yescharacteristic
56C22×C4×C283—24no—
57C22×C4×C283D411yescharacteristic
58C22×C4×C283D411yescharacteristic
59C22×C4×C283D411yescharacteristic
60C22×C4×C283C2311yescharacteristic
61C22×C4×C283C2311yescharacteristic
62C22×C4×C283C2311yescharacteristic
63M16×C283—24no—
64C2×C8×C283C2314yes—
65C22×D283—22no—
66C22×D283—42no—
67C22×D283—42no—
68C22×D283—42no—
69D1132—24no—
70D1132—42no—
71C22⋊D283—44no—
72C22⋊D283—44no—
73C2×Dic283—22no—
74C2×Dic283—22no—
75C2×Dic283—22no—
76C2×Dic283—42no—
77C2×Dic283—42no—
78C2×Dic283—42no—
79C23×C283—22no—
80C23×C283—22no—
81C23×C283—41no—
82C23×C283C2×D411yescharacteristic
83C23×C283C2×D411yescharacteristic
84C23×C283C2×D411yescharacteristic
85C2×C4×C283—22no—
86C2×C4×C283—22no—
87C2×C4×C283—22no—
88C2×C4×C283—22no—
89C2×C4×C283—22no—
90C2×C4×C283C2×D411yescharacteristic
91C2×C4×C283C2×D411yescharacteristic
92C2×C4×C283C2×D411yescharacteristic
93C2×C4×C283#4114yes—
94C2×C4×C283C2411yescharacteristic
95C8×C283—24no—
96C22×C283—21no—
97C22×C283—21no—
98C22×C283—21no—
99C22×C283—21no—
100C22×C283—21no—
101C22×C283—21no—
102C22×C283—21no—
103C22×C283—44no—
104C22×C283#12312yes—
105C22×C283AGL(1,ℤ/8)12yes—
106C22×C283C22×D411yescharacteristic
107C2×D283—42no—
108C2×D283—42no—
109C2×D283—42no—
110C2×D283—84no—
111C4×C283—21no—
112C4×C283—21no—
113C4×C283#12312yes—
114C4×C283C2×#4112yes—
115Dic283—42no—
116Dic283—42no—
117Dic283—42no—
118C2×C283—22no—
119C2×C283—22no—
120C2×C283—41no—
121C2×C283C2×#12311yescharacteristic
122C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C283#57511yescharacteristic
124D283—82no—
125C283#309411yescharacteristic, complemented, Sylow
126#3094—2831nomaximal, Sylow
127C2×#121—2831no—
128C2×#117—2831no—
129#482—2834no—
130#481—2832no—
131#480—2832no—
132#428—2832no—
133#418—2832no—
134#407D28311yescharacteristic, complemented
135C22×D4—2831no—
136C22×D4—5661no—
137C23×C4C2×D28311yescharacteristic
138C2×M16C2×D28312yes—
139#121—2832no—
140#121—2832no—
141#121—5661no—
142#121—5661no—
143#117—2832no—
144#117—2832no—
145#117—5661no—
146#117—5661no—
147C2×C4⋊C4—2831no—
148C2×C4⋊C4—2831no—
149C2×C4⋊C4—2831no—
150C2×C22⋊C4—5661no—
151C2×C22⋊C4—5661no—
152C2×C42—5661no—
153#107—2834no—
154#106—2834no—
155#102—2834no—
156#102—2834no—
157C22⋊C8C2×D28312yescomplemented
158C22⋊C8C2×D28312yescomplemented
159C24—5661no—
160C24D113211yescharacteristic
161C2×D4—2832no—
162C2×D4—5662no—
163C2×D4—5662no—
164C2×D4—5661no—
165C2×D4—5664no—
166C2×D4—11324no—
167C22×C4—24no—
168C22×C4—5661no—
169C22×C4—5661no—
170C22×C4—5661no—
171C22×C4D113211yescharacteristic
172C22×C4D113211yescharacteristic
173C22×C4D113211yescharacteristic
174C22×C4C22×D28311yescharacteristic
175C22×C4C22×D28311yescharacteristic
176C22×C4C22×D28311yescharacteristic
177M16—24no—
178C2×C8C22×D28314yes—
179C4⋊C4—2832no—
180C4⋊C4—2832no—
181C4⋊C4—2832no—
182C4⋊C4—5661no—
183C4⋊C4—5661no—
184C4⋊C4—5661no—
185C22⋊C4—5662no—
186C22⋊C4—5662no—
187C22⋊C4—5662no—
188C22⋊C4—5662no—
189C42—5662no—
190C42—5662no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—5662no—
195C23—11322no—
196C23—11322no—
197C23—11322no—
198C23C2×D113211yescharacteristic
199C23C2×D113211yescharacteristic
200C23C2×D113211yescharacteristic
201D4—5664no—
202D4—11324no—
203D4—11324no—
204D4—11322no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—5662no—
211C2×C4—5662no—
212C2×C4—5662no—
213C2×C4—11322no—
214C2×C4—11322no—
215C2×C4—11322no—
216C2×C4C2×D113211yescharacteristic
217C2×C4C2×D113211yescharacteristic
218C2×C4C2×D113211yescharacteristic
219C2×C4#1798539114yes—
220C2×C4C23×D28311yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—11322no—
231C22—11322no—
232C22—11322no—
233C22—22644no—
234C22#2258359412yes—
235C22#2258363812yes—
236C22C22×D113211yescharacteristic
237C4—21no—
238C4—21no—
239C4—11322no—
240C4—11322no—
241C4—11322no—
242C4#2258359412yes—
243C4C2×#1798539112yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—22642no—
248C2C2×#2258359411yescharacteristic
249C2C2×#2258363811yescharacteristic
250C2#3182726511yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#407×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583584C2
hypercenter#407D283
hypocenterC283#3094
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
2. C2×#22583569C2semidirect
3. #31826790C2semidirect
4. #31826784C2semidirect
5. #31826779C2semidirect
6. #31826507C2semidirect
7. #31826471C2semidirect
8. #407×C283C2semidirect
18. #22583569C22semidirect
20. C2×C4⋊Dic283C22semidirect
21. C2×C4⋊Dic283C22semidirect
25. #22583544C22semidirect
26. #22583542C22semidirect
27. #22583532C22semidirect
28. #22583532C22semidirect
31. C22⋊C8×C283C22semidirect
32. C22⋊C8×C283C22semidirect
125. C283#3094semidirect
134. #407D283semidirect
157. C22⋊C8C2×D283semidirect
158. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835842
C2×#225835692
#318267902
#318267842
#318267792
#318265072
#318264712
#407×C2832
#3094283