Finite Group Databasebeta

← #49130901

Subgroups — #49130901

Order 36224.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17985381C211yesmaximal, characteristic, complemented
2C2×#22583616C211yesmaximal, characteristic, complemented
3#31827041C211yesmaximal, characteristic, complemented
4#31827034C211yesmaximal, characteristic
5#31826712C214yesmaximal, complemented
6#31826695C214yesmaximal, complemented
7#31826473C211yesmaximal, characteristic, complemented
8#31826469C211yesmaximal, characteristic, complemented
9#427×C283C211yesmaximal, characteristic, complemented
10C2×#17985392C2211yescharacteristic
11C22×Dic566C2211yescharacteristic
12C22×D1132C2211yescharacteristic
13C2×#17985381—24no—
14C2×#17985381C2212yes—
15C2×#17985381C2212yes—
16#22583616—24no—
17C2×#17985376C2211yescharacteristic
18C2×#17985375C2212yescomplemented
19C2×C8×D283C2212yescomplemented
20#22583610—21no—
21#22583607—21no—
22#22583598—21no—
23#22583598—21no—
24#22583594—22no—
25#22583590C2214yescomplemented
26#22583585C2214yes—
27#22583580—22no—
28#22583533C2214yescomplemented
29#22583530C2214yes—
30C22×C8×C283C2211yescharacteristic
31#117×C283C2211yescharacteristic
32#117×C283C2211yescharacteristic
33#107×C283C2212yescomplemented
34#106×C283C2212yescomplemented
35C23×D283—21no—
36#17985392—22no—
37#17985392—22no—
38C2×Dic566—21no—
39C2×Dic566—22no—
40C2×Dic566—22no—
41C2×Dic566D411yescharacteristic
42C2×Dic566C2312yes—
43C2×D1132—24no—
44C2×D1132—22no—
45C2×D1132—21no—
46C2×D1132—22no—
47C2×D1132D411yescharacteristic
48C2×D1132C2312yes—
49C2×C22⋊D283D412yes—
50C22×Dic283—21no—
51C2×C4×D283C2312yes—
52#17985381—24no—
53#17985381—24no—
54#17985376—22no—
55#17985375—24no—
56C2×C283⋊C8C2312yes—
57C8×D283—24no—
58C4⋊Dic283—24no—
59(C2×D283)⋊C4—24no—
60(C2×D283)⋊C4—24no—
61C22×C4×C283C2311yescharacteristic
62C2×C8×C283D414yescomplemented
63C2×C8×C283C2311yescharacteristic
64C2×C8×C283C2311yescharacteristic
65C4⋊C4×C283C2312yes—
66C4⋊C4×C283C2312yes—
67C22⋊C4×C283—21no—
68C22⋊C4×C283—21no—
69C42×C283—21no—
70C42×C283—21no—
71C22×D283—22no—
72C22×D283—42no—
73C22×D283—82no—
74C22×D283C2×D412yes—
75Dic566—22no—
76Dic566—22no—
77Dic566—22no—
78Dic566—42no—
79D1132—22no—
80D1132—22no—
81D1132—22no—
82D1132—42no—
83D1132—84no—
84C22⋊D283—24no—
85C2×Dic283—22no—
86C2×Dic283—42no—
87C2×Dic283C2×D412yes—
88C4×D283—22no—
89C4×D283—22no—
90C283⋊C8—22no—
91C23×C283C2×D411yescharacteristic
92C2×C4×C283—22no—
93C2×C4×C283—22no—
94C2×C4×C283C2×D411yescharacteristic
95C2×C4×C283C2×D412yes—
96C2×C4×C283#4112yes—
97C2×C4×C283C2411yescharacteristic
98C8×C283C2×D412yes—
99C8×C283C2×D412yes—
100C22×C283—22no—
101C22×C283#12112yes—
102C22×C283C22×D411yescharacteristic
103C2×D283—22no—
104C2×D283—24no—
105C2×D283—42no—
106C2×D283—44no—
107C2×D283—84no—
108C4×C283—42no—
109C4×C283—42no—
110C4×C283#12112yes—
111C4×C283C22×D411yescharacteristic
112C4×C283C2×#4111yescharacteristic
113Dic283—22no—
114Dic283—42no—
115C2×C283—22no—
116C2×C283C2×#12111yescharacteristic
117C2×C283#57712yes—
118D283—42no—
119D283—82no—
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#427D28311yescharacteristic, complemented
129#419—2831no—
130#418—2831no—
131C2×#41—2831no—
132C22×Q8—2831no—
133C22×D4—2831no—
134C2×SD16—2832no—
135C2×SD16—2832no—
136C2×SD16—5664no—
137#131—5664no—
138C2×M16—2831no—
139C2×M16—2832no—
140C22×C8—2832no—
141C22×C8C2×D28311yescharacteristic
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#117C2×D28311yescharacteristic
151#117C2×D28311yescharacteristic
152#107C2×D28312yescomplemented
153#106C2×D28312yescomplemented
154#103—2834no—
155#102—2834no—
156C24—5661no—
157#41—5662no—
158#41—5662no—
159C2×Q8—2831no—
160C2×Q8—2832no—
161C2×Q8—5661no—
162C2×Q8—5662no—
163C2×Q8—5662no—
164C2×D4—2832no—
165C2×D4—2831no—
166C2×D4—2832no—
167C2×D4—5664no—
168C2×D4—5661no—
169C2×D4—5662no—
170C2×D4—5662no—
171C22×C4—2832no—
172C22×C4—5661no—
173C22×C4C22×D28311yescharacteristic
174SD16—5664no—
175SD16—5664no—
176M16—5662no—
177M16—5664no—
178C2×C8—2832no—
179C2×C8—5664no—
180C2×C8D113214yescomplemented
181C2×C8C22×D28311yescharacteristic
182C2×C8C22×D28311yescharacteristic
183C4⋊C4—5664no—
184C4⋊C4C22×D28312yes—
185C4⋊C4C22×D28312yes—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—5664no—
189C22⋊C4—5664no—
190C42—21no—
191C42—21no—
192C23—2832no—
193C23—5662no—
194C23—11322no—
195C23—22642no—
196C23D4×D28311yescharacteristic
197Q8—5662no—
198Q8—5662no—
199Q8—5662no—
200Q8—11322no—
201D4—5664no—
202D4—5662no—
203D4—5662no—
204D4—5662no—
205D4—11322no—
206D4—22644no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—2832no—
210C2×C4—5662no—
211C2×C4—5662no—
212C2×C4—5662no—
213C2×C4—11322no—
214C2×C4D4×D28311yescharacteristic
215C2×C4C2×D113212yes—
216C2×C4#1798539312yes—
217C2×C4C23×D28311yescharacteristic
218C8—5662no—
219C8C2×D113212yes—
220C8C2×D113212yes—
221C22—22no—
222C22—5662no—
223C22—5664no—
224C22—11322no—
225C22—11324no—
226C22—22644no—
227C22#2258358512yes—
228C22C2×D4×D28311yescharacteristic
229C4—42no—
230C4—42no—
231C4—5662no—
232C4—11322no—
233C4#2258358512yes—
234C4C22×D113211yescharacteristic
235C4C2×#1798539311yescharacteristic
236C2—22no—
237C2—11322no—
238C2—22642no—
239C2C2×#2258358511yescharacteristic
240C2#3182727212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#427×C283C2
socleC22×C283C22×D4
prime radical#31827034C2
hypercenter#427D283
hypocenterC283#3057
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C22×#17985381C2semidirect
2. C2×#22583616C2semidirect
3. #31827041C2semidirect
5. #31826712C2semidirect
6. #31826695C2semidirect
7. #31826473C2semidirect
8. #31826469C2semidirect
9. #427×C283C2semidirect
18. C2×#17985375C22semidirect
19. C2×C8×D283C22semidirect
25. #22583590C22semidirect
28. #22583533C22semidirect
33. #107×C283C22semidirect
34. #106×C283C22semidirect
62. C2×C8×C283D4semidirect
120. C283#3057semidirect
128. #427D283semidirect
152. #107C2×D283semidirect
153. #106C2×D283semidirect
180. C2×C8D1132semidirect

Maximal subgroups

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

subgroupindex
C22×#179853812
C2×#225836162
#318270412
#318270342
#318267122
#318266952
#318264732
#318264692
#427×C2832
#3057283