Finite Group Databasebeta

← D9×#543 #11901856

Subgroups — D9×#543 #11901856

Order 1152.

1944 conjugacy classes of subgroups; only the 598 normal subgroups are shown.

normal subgroups only

This group has 1944 conjugacy classes of subgroups — too many to draw. Only its 598 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
1C9×#543C211yesmaximal, characteristic, complemented
2#10589736C2112yesmaximal, complemented
3D9×C22⋊Q8C2112yesmaximal, complemented
4C2×C22⋊C4×D9C213yesmaximal
5#10589585C213yesmaximal, complemented
6D3×#543—31nomaximal
7C24×D9C2211yescharacteristic
8C2×Q8×D9C2214yescomplemented
9C22×C4×D9C2216yes—
10C9×C22⋊Q8C22112yescomplemented
11C2×C9×C22⋊C4C2213yes—
12C2×#3676C2213yes—
13#63508C22112yescomplemented
14C2×(C2×D9)⋊C4C2216yes—
15#63486C22112yescomplemented
16#63459C22112yescomplemented
17#63458C22124yescomplemented
18C4⋊C4×D9C22112yescomplemented
19C22⋊C4×D9C22112yes—
20C22⋊C4×D9C22112yes—
21#63440C22124yescomplemented
22C3×#543D311yescharacteristic
23#5634—312no—
24D3×C22⋊Q8—312no—
25C2×D3×C22⋊C4—33no—
26#5483—33no—
27C24×C9C2311yescharacteristic
28C23×D9—44no—
29C23×D9Q818yescomplemented
30C23×D9C2313yes—
31C23×D9C2313yes—
32C2×Q8×C9C2314yescomplemented
33C22×C4×C9C2316yes—
34C22×Dic9C2316yes—
35Q8×D9—48no—
36C2×C4×D9—412no—
37C2×C4×D9C23112yes—
38C2×C4×D9C23112yes—
39C2×Dic18C23112yescomplemented
40C9×C4⋊C4C23112yescomplemented
41C9×C22⋊C4C23112yes—
42#3676C23112yes—
43(C2×D9)⋊C4C23124yes—
44C4⋊Dic9C23112yescomplemented
45Dic9⋊C4C23124yescomplemented
46C2×#543—91noSylow
47C24×D3—31no—
48C2×D3×Q8—34no—
49C22×C4×D3—36no—
50C3×C22⋊Q8C2×D3112yes—
51C2×C3×C22⋊C4C2×D313yes—
52C2×#217—33no—
53#946—312no—
54C2×(C2×D3)⋊C4—36no—
55#924—312no—
56#897—312no—
57#896—324no—
58D3×C4⋊C4—312no—
59D3×C22⋊C4—312no—
60D3×C22⋊C4—312no—
61#878—324no—
62C23×C9—42no—
63C23×C9C2×Q814yescomplemented
64C23×C9C2413yes—
65C22×D9—212no—
66C22×D9—416no—
67C22×D9—46no—
68C22×D9C2×Q8112yes—
69C22×D9C2×Q8112yescomplemented
70C22×D9C2413yes—
71C22×D9C2411yescharacteristic
72Q8×C9—44no—
73C2×C4×C9—46no—
74C2×C4×C9C24112yes—
75C2×Dic9—46no—
76C2×Dic9C24112yes—
77C4×D9—224no—
78C4×D9—412no—
79Dic18—412no—
80#543—912no—
81#543—93no—
82#543D911yescharacteristic, direct product factor
83C2×C22⋊Q8—912no—
84C22×C22⋊C4—93no—
85C24×C3C22×D311yescharacteristic
86C23×D3—33no—
87C23×D3—38no—
88C23×D3—33no—
89C23×D3—124no—
90C2×C3×Q8C22×D314yes—
91C22×C3×C4C22×D316yes—
92C22×Dic3—36no—
93D3×Q8—128no—
94C2×C4×D3—312no—
95C2×C4×D3—312no—
96C2×C4×D3—1212no—
97C2×Dic6—312no—
98C3×C4⋊C4C22×D3112yes—
99C3×C22⋊C4C22×D3112yes—
100#217—312no—
101(C2×D3)⋊C4—324no—
102C4⋊Dic3—312no—
103Dic3⋊C4—324no—
104C22×C9—26no—
105C22×C9—44no—
106C22×C9C22×Q816yes—
107C22×C9C2511yescharacteristic
108C2×D9—26no—
109C2×D9—224no—
110C2×D9—412no—
111C2×D9C22×Q816yes—
112C2×D9C22×Q812yes—
113C2×D9H(5,𝔽2)14yes—
114C4×C9—212no—
115Dic9—212no—
116C25—91no—
117C22×Q8—94no—
118C23×C4—96no—
119C22⋊Q8—912no—
120C22⋊Q8—912no—
121C22⋊Q8—924no—
122C22⋊Q8—912no—
123C22⋊Q8—924no—
124C22⋊Q8C2×D9112yescomplemented
125C2×C4⋊C4—912no—
126C2×C22⋊C4—93no—
127C2×C22⋊C4—96no—
128C2×C22⋊C4—912no—
129C2×C22⋊C4—912no—
130C2×C22⋊C4C2×D913yes—
131C23×C3—42no—
132C23×C3D3×Q814yes—
133C23×C3C23×D313yes—
134C22×D3—31no—
135C22×D3—312no—
136C22×D3—33no—
137C22×D3—312no—
138C22×D3—612no—
139C22×D3—1216no—
140C22×D3—126no—
141C3×Q8—44no—
142C2×C3×C4—46no—
143C2×C3×C4C23×D3112yes—
144C2×Dic3—312no—
145C2×Dic3—126no—
146C4×D3—624no—
147C4×D3—1212no—
148Dic6—1212no—
149C2×C9—26no—
150C2×C9C23×Q811yescharacteristic
151C2×C9C2×H(5,𝔽2)12yes—
152D9—26no—
153D9#54314yesdirect product factor
154C24—93no—
155C24—98no—
156C24—93no—
157C24—364no—
158C24C22×D911yescharacteristic
159C2×Q8—912no—
160C2×Q8—368no—
161C2×Q8C22×D914yescomplemented
162C22×C4—96no—
163C22×C4—912no—
164C22×C4—912no—
165C22×C4—3612no—
166C22×C4C22×D916yes—
167C4⋊C4—924no—
168C4⋊C4—912no—
169C4⋊C4C22×D9112yescomplemented
170C22⋊C4—912no—
171C22⋊C4—924no—
172C22⋊C4C22×D9112yes—
173C22×C3—26no—
174C22×C3—44no—
175C22×C3C2×D3×Q816yes—
176C22×C3C24×D311yescharacteristic
177C2×D3—32no—
178C2×D3—34no—
179C2×D3—36no—
180C2×D3—624no—
181C2×D3—66no—
182C2×D3—1212no—
183C3×C4—212no—
184Dic3—612no—
185C9C2×#54311yescharacteristic, complemented, Sylow
186C23—42no—
187C23—91no—
188C23—93no—
189C23—912no—
190C23—912no—
191C23—1812no—
192C23—366no—
193C23—3616no—
194C23Q8×D914yescomplemented
195C23C23×D913yes—
196Q8—44no—
197Q8—3612no—
198C2×C4—46no—
199C2×C4—912no—
200C2×C4—1824no—
201C2×C4—3612no—
202C2×C4—366no—
203C2×C4C23×D9112yes—
204C2×C3—26no—
205C2×C3C22×D3×Q811yescharacteristic
206C2×C3D3×H(5,𝔽2)12yes—
207D3—34no—
208D3—66no—
209C22—26no—
210C22—44no—
211C22—92no—
212C22—94no—
213C22—96no—
214C22—1824no—
215C22—186no—
216C22—3612no—
217C22C2×Q8×D916yes—
218C22C24×D911yescharacteristic
219C4—212no—
220C4—1812no—
221C3D3×#54311yescharacteristic
222C2—26no—
223C2—94no—
224C2—186no—
225C2C22×Q8×D911yescharacteristic
226C2D9×H(5,𝔽2)12yes—
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#543C2
prime radicalC24×D9C22
hypercenter#543D9
hypocenterC9C2×#543
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. C9×#543C2semidirect
2. #10589736C2semidirect
3. D9×C22⋊Q8C2semidirect
5. #10589585C2semidirect
8. C2×Q8×D9C22semidirect
10. C9×C22⋊Q8C22semidirect
13. #63508C22semidirect
15. #63486C22semidirect
16. #63459C22semidirect
17. #63458C22semidirect
18. C4⋊C4×D9C22semidirect
21. #63440C22semidirect
29. C23×D9Q8semidirect
32. C2×Q8×C9C23semidirect
39. C2×Dic18C23semidirect
40. C9×C4⋊C4C23semidirect
44. C4⋊Dic9C23semidirect
45. Dic9⋊C4C23semidirect
63. C23×C9C2×Q8semidirect
69. C22×D9C2×Q8semidirect
82. #543D9direct (N × H)
124. C22⋊Q8C2×D9semidirect
153. D9#543direct (N × H)
161. C2×Q8C22×D9semidirect
169. C4⋊C4C22×D9semidirect
185. C9C2×#543semidirect
194. C23Q8×D9semidirect

Maximal subgroups

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

subgroupindex
C9×#5432
#105897362
D9×C22⋊Q82
C2×C22⋊C4×D92
#105895852
D3×#5433