Finite Group Databasebeta

← #49129737

Subgroups — #49129737

Order 36224.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22583655C211yesmaximal, characteristic
2#31827231C212yesmaximal, complemented
3#31826447C212yesmaximal, complemented
4#31826446C214yesmaximal, complemented
5C2×#22583527C211yesmaximal, characteristic, complemented
6#100×D283C214yesmaximal, complemented
7C2×#100×C283C211yesmaximal, characteristic, complemented
8#22583655—24no—
9#22583655—24no—
10C2×#17985392C2211yescharacteristic
11C2×#17985392C2211yescharacteristic
12C2×#17985391C412yes—
13C2×#17985391C2212yes—
14C22×C22⋊D283C414yes—
15C2×D4×D283C412yes—
16C2×D4×D283C2212yes—
17#22583618—24no—
18#22583616—24no—
19C2×#17985376C2212yescomplemented
20M16×D283C2218yescomplemented
21#22583528C2214yescomplemented
22#22583527C2214yescomplemented
23#22583526C2214yescomplemented
24C22×D4×C283C2211yescharacteristic
25C2×M16×C283C2212yescomplemented
26#100×C283C2214yescomplemented
27C23×D283C2×C414yes—
28#17985392—22no—
29#17985392—22no—
30#17985392D414yes—
31#17985392D414yes—
32#17985391—28no—
33#17985391—28no—
34C2×Dic566C2×C411yescharacteristic
35C2×Dic566C2311yescharacteristic
36C2×D1132C2×C411yescharacteristic
37C2×D1132C2311yescharacteristic
38C2×C22⋊D283—28no—
39C2×C22⋊D283—28no—
40C2×C22⋊D283C2×C414yes—
41C2×C22⋊D283C2×C414yes—
42C2×C22⋊D283C2×C412yes—
43C2×C22⋊D283C2×C412yes—
44D4×D283—28no—
45D4×D283—28no—
46C22×Dic283C2×C414yes—
47C2×C4×D283C2×C412yes—
48C2×C4×D283C2312yes—
49#17985376—22no—
50#17985376C2314yes—
51#17985375—28no—
52C2×C283⋊C8—22no—
53C8×D283—28no—
54C24×C283C2×C412yes—
55C2×D4×C283—22no—
56C2×D4×C283—22no—
57C2×D4×C283C2×C412yes—
58C2×D4×C283C2312yes—
59C22×C4×C283C2311yescharacteristic
60M16×C283—22no—
61M16×C283C2314yes—
62C2×C8×C283—22no—
63C22×D283—28no—
64C22×D283—84no—
65C22×D283C22×C412yes—
66C22×D283C22×C412yes—
67Dic566—22no—
68Dic566C22⋊C414yes—
69D1132—22no—
70D1132C22⋊C414yes—
71C22⋊D283—22no—
72C22⋊D283—22no—
73C22⋊D283—28no—
74C22⋊D283—28no—
75C22⋊D283C22⋊C414yes—
76C22⋊D283C22⋊C414yes—
77C2×Dic283—28no—
78C2×Dic283C22×C412yes—
79C2×Dic283C22×C412yes—
80C4×D283—24no—
81C4×D283C2×D418yes—
82C283⋊C8—24no—
83C23×C283—22no—
84C23×C283—22no—
85C23×C283—82no—
86C23×C283C22×C414yes—
87C23×C283C22×C411yescharacteristic
88D4×C283—24no—
89D4×C283—24no—
90C2×C4×C283C22×C411yescharacteristic
91C2×C4×C283C2×D412yes—
92C2×C4×C283C2×D412yes—
93C2×C4×C283C2411yescharacteristic
94C8×C283—24no—
95C22×C283—24no—
96C22×C283—41no—
97C22×C283—44no—
98C22×C283—82no—
99C22×C283—82no—
100C22×C283C2×C22⋊C411yescharacteristic
101C22×C283C2×C22⋊C411yescharacteristic
102C22×C283C23×C411yescharacteristic
103C2×D283—22no—
104C2×D283—42no—
105C2×D283—42no—
106C2×D283—88no—
107C2×D283C2×C22⋊C414yes—
108C4×C283C2×C22⋊C412yes—
109C4×C283C22×D412yes—
110Dic283—22no—
111Dic283C2×C22⋊C414yes—
112C2×C283—21no—
113C2×C283—21no—
114C2×C283—21no—
115C2×C283—44no—
116C2×C283C22×C22⋊C411yescharacteristic
117D283—24no—
118D283—42no—
119C283#288811yescharacteristic, complemented, Sylow
120#2888—2831nomaximal, Sylow
121C2×H(5,𝔽2)—2831no—
122#568—2832no—
123#413—2832no—
124#413—2834no—
125C2×#100—2831no—
126C2×#100—2834no—
127C2×#100D28311yescharacteristic, complemented
128H(5,𝔽2)—5664no—
129H(5,𝔽2)—5664no—
130C2×#41—2832no—
131C2×#41—2832no—
132C2×#41—2831no—
133C2×#41—2831no—
134C22×D4—2834no—
135C22×D4—2832no—
136C22×D4—2832no—
137C22×D4C2×D28311yescharacteristic
138#131—5664no—
139#131—5664no—
140C2×M16—2832no—
141C2×M16—2838no—
142C2×M16C2×D28312yescomplemented
143#101—2834no—
144#100—2834no—
145#100—2834no—
146#100C2×D28314yescomplemented
147C24—2834no—
148C24C4×D28312yes—
149#41—2834no—
150#41—2834no—
151#41—5668no—
152#41—5668no—
153#41—5662no—
154#41—5662no—
155C2×Q8—2831no—
156C2×Q8—2831no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—2832no—
160C2×D4—2834no—
161C2×D4—2834no—
162C2×D4—2832no—
163C2×D4—2831no—
164C2×D4—2831no—
165C2×D4—5668no—
166C2×D4—5668no—
167C2×D4—5668no—
168C2×D4—5668no—
169C2×D4C4×D28312yes—
170C2×D4C22×D28312yes—
171C22×C4—2832no—
172C22×C4—2834no—
173C22×C4—2832no—
174C22×C4C22×D28311yescharacteristic
175M16—22no—
176M16—2834no—
177M16—5668no—
178M16—5662no—
179M16C22×D28314yes—
180C2×C8—22no—
181C2×C8—5668no—
182C2×C8—5662no—
183C23—22no—
184C23—22no—
185C23—82no—
186C23—2832no—
187C23—2832no—
188C23—5668no—
189C23—22644no—
190C23C2×C4×D28314yes—
191C23C2×C4×D28311yescharacteristic
192Q8—2834no—
193Q8—5662no—
194D4—24no—
195D4—24no—
196D4—2834no—
197D4—2834no—
198D4—2834no—
199D4—5668no—
200D4—5668no—
201D4—5662no—
202D4—5662no—
203D4—5662no—
204C2×C4—2832no—
205C2×C4—2838no—
206C2×C4—2832no—
207C2×C4—5668no—
208C2×C4—5664no—
209C2×C4C2×C4×D28311yescharacteristic
210C2×C4D4×D28312yes—
211C2×C4D4×D28312yes—
212C2×C4C23×D28311yescharacteristic
213C8—24no—
214C8—5664no—
215C22—24no—
216C22—41no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—2834no—
221C22—5662no—
222C22—11322no—
223C22—11322no—
224C22—22648no—
225C22C22⋊C4×D28311yescharacteristic
226C22C22⋊C4×D28311yescharacteristic
227C22C22×C4×D28311yescharacteristic
228C4—2834no—
229C4—5662no—
230C4C22⋊C4×D28312yes—
231C4C2×D4×D28312yes—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—5664no—
237C2—11322no—
238C2C2×C22⋊C4×D28311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D283
commutatorC22×C283C23×C4
FrattiniC2×C4C23×D283
FittingC2×#100×C283C2
socleC2×C283C22×C22⋊C4
prime radicalC2×#22583655C2
cosocleC2×C4×C283C24
hypercenterC2×#100D283
hypocenterC283#2888
LCSG ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #31827231C2semidirect
3. #31826447C2semidirect
4. #31826446C2semidirect
5. C2×#22583527C2semidirect
6. #100×D283C2semidirect
7. C2×#100×C283C2semidirect
19. C2×#17985376C22semidirect
20. M16×D283C22semidirect
21. #22583528C22semidirect
22. #22583527C22semidirect
23. #22583526C22semidirect
25. C2×M16×C283C22semidirect
26. #100×C283C22semidirect
119. C283#2888semidirect
127. C2×#100D283semidirect
142. C2×M16C2×D283semidirect
146. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836552
#318272312
#318264472
#318264462
C2×#225835272
#100×D2832
C2×#100×C2832
#2888283