Finite Group Databasebeta

← #49130899

Subgroups — #49130899

Order 36224.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17985380C211yesmaximal, characteristic, complemented
2C2×#22583618C211yesmaximal, characteristic, complemented
3#31827041C211yesmaximal, characteristic, complemented
4#31827032C211yesmaximal, characteristic
5#31826710C214yesmaximal, complemented
6#31826693C214yesmaximal, complemented
7#31826501C211yesmaximal, characteristic, complemented
8#31826475C211yesmaximal, characteristic, complemented
9#418×C283C211yesmaximal, characteristic, complemented
10C2×#17985392C2211yescharacteristic
11C22×Dic566C2211yescharacteristic
12C2×#17985380—24no—
13C2×#17985380C2212yes—
14C2×#17985380C2212yes—
15#22583618—24no—
16C2×#17985375C2212yescomplemented
17C22×C283⋊C8C2211yescharacteristic
18C2×C8×D283C2212yescomplemented
19#22583610—21no—
20#22583606—21no—
21#22583598—21no—
22#22583597—21no—
23#22583594—22no—
24#22583590C2214yescomplemented
25#22583581C2214yes—
26#22583579—22no—
27#22583569C2211yescharacteristic
28#22583543C2212yescomplemented
29#22583541C2212yescomplemented
30#22583535C2214yescomplemented
31C22×D4×C283C2211yescharacteristic
32C2×M16×C283C2211yescharacteristic
33#117×C283C2211yescharacteristic
34#102×C283C2214yes—
35#17985392—22no—
36#17985392—22no—
37C2×Dic566—21no—
38C2×Dic566—22no—
39C2×Dic566—22no—
40C2×Dic566D411yescharacteristic
41C2×Dic566C2312yes—
42C2×D1132D411yescharacteristic
43C2×C22⋊D283—24no—
44C2×C22⋊D283D412yes—
45C22×Dic283—21no—
46C2×C4×D283C2312yes—
47#17985380—24no—
48#17985380—24no—
49#17985375—24no—
50C2×C283⋊C8D414yescomplemented
51C2×C283⋊C8C2311yescharacteristic
52C2×C283⋊C8C2311yescharacteristic
53C8×D283—24no—
54C4⋊Dic283—24no—
55C4⋊Dic283C2312yes—
56C22⋊Dic283—24no—
57C22⋊Dic283—21no—
58(C2×D283)⋊C4—24no—
59C4×Dic283—21no—
60C24×C283—21no—
61C2×D4×C283—22no—
62C2×D4×C283—21no—
63C2×D4×C283—22no—
64C2×D4×C283C2312yes—
65C22×C4×C283C2311yescharacteristic
66M16×C283—22no—
67C2×C8×C283C2312yes—
68C4⋊C4×C283C2312yes—
69C22⋊C4×C283—21no—
70C42×C283—21no—
71C22×D283C2×D412yes—
72Dic566—22no—
73Dic566—22no—
74Dic566—22no—
75Dic566—42no—
76D1132—22no—
77C22⋊D283—24no—
78C22⋊D283—84no—
79C2×Dic283—22no—
80C2×Dic283—22no—
81C2×Dic283—42no—
82C2×Dic283C2×D412yes—
83C4×D283—22no—
84C4×D283—22no—
85C283⋊C8C2×D412yes—
86C283⋊C8C2×D412yes—
87C23×C283—22no—
88C23×C283—42no—
89C23×C283—82no—
90C23×C283C2×D411yescharacteristic
91D4×C283—22no—
92D4×C283—22no—
93D4×C283—42no—
94C2×C4×C283—22no—
95C2×C4×C283C2×D411yescharacteristic
96C2×C4×C283C2×D412yes—
97C2×C4×C283#4112yes—
98C2×C4×C283C2411yescharacteristic
99C8×C283—22no—
100C22×C283—22no—
101C22×C283—42no—
102C22×C283—44no—
103C22×C283—84no—
104C22×C283#12112yes—
105C22×C283C22×D411yescharacteristic
106C2×D283—22no—
107C2×D283—24no—
108C4×C283—42no—
109C4×C283#12112yes—
110C4×C283C22×D411yescharacteristic
111C4×C283C2×#4111yescharacteristic
112Dic283—22no—
113Dic283—42no—
114Dic283—42no—
115C2×C283—22no—
116C2×C283—82no—
117C2×C283C2×#12111yescharacteristic
118C2×C283#57712yes—
119D283—42no—
120C283#305711yescharacteristic, complemented, Sylow
121#3057—2831nomaximal, Sylow
122C22×SD16—2831no—
123C2×#131—2831no—
124#536—2831no—
125#535—2831no—
126#468—2834no—
127#465—2834no—
128#427—2831no—
129#419—2831no—
130#418D28311yescharacteristic, complemented
131C2×#41—2831no—
132C22×Q8—2831no—
133C22×D4C2×D28311yescharacteristic
134C2×SD16—2832no—
135C2×SD16—2832no—
136C2×SD16—5664no—
137#131—5664no—
138C2×M16—2832no—
139C2×M16C2×D28311yescharacteristic
140C22×C8—2831no—
141C22×C8—2832no—
142C4⋊Q8—5661no—
143#127—5661no—
144#124—5661no—
145#124—5661no—
146#123—5662no—
147C22⋊Q8—2834no—
148#121—2834no—
149#120—5662no—
150#117—2831no—
151#117C2×D28311yescharacteristic
152#107—2832no—
153#106—2832no—
154#103—2834no—
155#102C2×D28314yes—
156C24—21no—
157#41—5662no—
158#41—5662no—
159C2×Q8—2831no—
160C2×Q8—2832no—
161C2×Q8—5661no—
162C2×Q8—5662no—
163C2×Q8—5662no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—2832no—
168C2×D4—2831no—
169C2×D4—5664no—
170C2×D4C22×D28312yes—
171C22×C4—2832no—
172C22×C4—5661no—
173C22×C4C22×D28311yescharacteristic
174SD16—5664no—
175SD16—5664no—
176M16—22no—
177M16—5664no—
178C2×C8—2834no—
179C2×C8—2831no—
180C2×C8—2831no—
181C2×C8—5664no—
182C2×C8C22×D28312yes—
183C4⋊C4—2832no—
184C4⋊C4—5664no—
185C4⋊C4C22×D28312yes—
186C22⋊C4—21no—
187C22⋊C4—5664no—
188C22⋊C4—5664no—
189C22⋊C4—5661no—
190C42—21no—
191C42—5661no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—2832no—
196C23D4×D28311yescharacteristic
197Q8—5662no—
198Q8—5662no—
199Q8—5662no—
200Q8—11322no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—5664no—
205D4—5662no—
206D4—22644no—
207C2×C4—22no—
208C2×C4—2832no—
209C2×C4—5662no—
210C2×C4—5662no—
211C2×C4—5662no—
212C2×C4—5662no—
213C2×C4—11322no—
214C2×C4D4×D28311yescharacteristic
215C2×C4D4×D28312yes—
216C2×C4#1798539212yes—
217C2×C4C23×D28311yescharacteristic
218C8—22no—
219C8—2832no—
220C8—2832no—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—5662no—
226C22—5664no—
227C22#2258358312yes—
228C22C2×D4×D28311yescharacteristic
229C4—42no—
230C4—5662no—
231C4—11322no—
232C4—11322no—
233C4#2258358312yes—
234C4C2×D4×D28311yescharacteristic
235C4C2×#1798539211yescharacteristic
236C2—22no—
237C2—82no—
238C2—11322no—
239C2C2×#2258358311yescharacteristic
240C2#3182727512yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#418×C283C2
socleC22×C283C22×D4
prime radical#31827032C2
hypercenter#418D283
hypocenterC283#3057
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#17985380C2semidirect
2. C2×#22583618C2semidirect
3. #31827041C2semidirect
5. #31826710C2semidirect
6. #31826693C2semidirect
7. #31826501C2semidirect
8. #31826475C2semidirect
9. #418×C283C2semidirect
16. C2×#17985375C22semidirect
18. C2×C8×D283C22semidirect
24. #22583590C22semidirect
28. #22583543C22semidirect
29. #22583541C22semidirect
30. #22583535C22semidirect
50. C2×C283⋊C8D4semidirect
120. C283#3057semidirect
130. #418D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853802
C2×#225836182
#318270412
#318270322
#318267102
#318266932
#318265012
#318264752
#418×C2832
#3057283