Finite Group Databasebeta

← D9×H(3,𝔽4) #11910201

Subgroups — D9×H(3,𝔽4) #11910201

Order 1152.

2112 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 2112 conjugacy classes of subgroups — too many to draw. Only its 530 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×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2#10589785C219yesmaximal, complemented
3D9×#124C219yesmaximal, complemented
4#10589696C216yesmaximal, complemented
5D9×#120C216yesmaximal, complemented
6D3×H(3,𝔽4)—31nomaximal
7C24×D9C2212yescomplemented
8C2×Q8×D9C2213yescomplemented
9C2×D4×D9C2219yes—
10C9×#124C2219yescomplemented
11C9×#120C2216yescomplemented
12#63515C2216yescomplemented
13#63509C2219yescomplemented
14#63499C22118yescomplemented
15#63498C2219yescomplemented
16#63447C22136yescomplemented
17#63444C22118yescomplemented
18C22⋊C4×D9C22118yescomplemented
19#63437C2219yescomplemented
20C42×D9C2213yescomplemented
21C3×H(3,𝔽4)D311yescharacteristic
22#5683—39no—
23D3×#124—39no—
24#5594—36no—
25D3×#120—36no—
26C24×C9C2312yescomplemented
27C23×D9—412no—
28C23×D9C2316yes—
29C23×D9C2316yescomplemented
30C2×Q8×C9C2313yescomplemented
31C2×D4×C9C2319yes—
32C2×#611C23118yes—
33Q8×D9—46no—
34D4×D9—418no—
35C2×D36C2319yes—
36C2×C4×D9C2319yes—
37C2×Dic18C2319yescomplemented
38C9×C22⋊C4C23118yescomplemented
39C42×C9C2313yescomplemented
40#3676C23118yescomplemented
41(C2×D9)⋊C4C23136yescomplemented
42C4×Dic9C2319yescomplemented
43C2×H(3,𝔽4)—91noSylow
44C24×D3—32no—
45C2×D3×Q8—33no—
46C2×D3×D4—39no—
47C3×#124C2×D319yes—
48C3×#120C2×D316yes—
49#953—36no—
50#947—39no—
51#937—318no—
52#936—39no—
53#885—336no—
54#882—318no—
55D3×C22⋊C4—318no—
56#875—39no—
57C42×D3—33no—
58C23×C9—46no—
59C23×C9C2416yes—
60C22×D9—236no—
61C22×D9—418no—
62C22×D9—432no—
63C22×D9C2416yes—
64C22×D9C2411yescharacteristic
65Q8×C9—43no—
66D4×C9—49no—
67C2×C4×C9C2419yes—
68#611—418no—
69C2×Dic9C2419yes—
70D36—49no—
71C4×D9—218no—
72Dic18—49no—
73H(3,𝔽4)—99no—
74H(3,𝔽4)—96no—
75H(3,𝔽4)D911yescharacteristic, direct product factor
76C2×#124—99no—
77C2×#120—96no—
78C24×C3C22×D312yes—
79C23×D3—36no—
80C23×D3—36no—
81C23×D3—1212no—
82C2×C3×Q8C22×D313yes—
83C2×C3×D4C22×D319yes—
84C2×#67—318no—
85D3×Q8—126no—
86D3×D4—1218no—
87C2×D12—39no—
88C2×C4×D3—39no—
89C2×Dic6—39no—
90C3×C22⋊C4C22×D3118yes—
91C3×C42C22×D313yes—
92#217—318no—
93(C2×D3)⋊C4—336no—
94C4×Dic3—39no—
95C22×C9—218no—
96C22×C9—48no—
97C22×C9C2511yescharacteristic
98C2×D9—218no—
99C2×D9—424no—
100C2×D9—424no—
101C2×D9H(5,𝔽2)16yes—
102C4×C9—29no—
103Dic9—29no—
104C25—92no—
105C22×Q8—93no—
106C22×D4—99no—
107#124—99no—
108#124—99no—
109#124—936no—
110#124—99no—
111#124C2×D919yescomplemented
112#120—96no—
113#120—918no—
114#120—918no—
115#120C2×D916yescomplemented
116C2×C22⋊C4—918no—
117C2×C42—93no—
118C23×C3—46no—
119C23×C3C23×D316yes—
120C22×D3—31no—
121C22×D3—36no—
122C22×D3—636no—
123C22×D3—1218no—
124C22×D3—1232no—
125C3×Q8—43no—
126C3×D4—49no—
127C2×C3×C4C23×D319yes—
128#67—1218no—
129C2×Dic3—39no—
130D12—129no—
131C4×D3—618no—
132Dic6—129no—
133C2×C9—46no—
134C2×C9C2×H(5,𝔽2)13yes—
135D9—46no—
136D9H(3,𝔽4)14yesdirect product factor
137C24—96no—
138C24—96no—
139C24—3612no—
140C24C22×D912yescomplemented
141C2×Q8—99no—
142C2×Q8—366no—
143C2×Q8C22×D913yescomplemented
144C2×D4—918no—
145C2×D4—99no—
146C2×D4—3618no—
147C2×D4C22×D919yes—
148C22×C4—99no—
149C22⋊C4—918no—
150C22⋊C4—936no—
151C22⋊C4C22×D9118yescomplemented
152C42—99no—
153C42C22×D913yescomplemented
154C22×C3—218no—
155C22×C3—48no—
156C22×C3C24×D311yescharacteristic
157C2×D3—36no—
158C2×D3—618no—
159C2×D3—1224no—
160C2×D3—1224no—
161C3×C4—29no—
162Dic3—69no—
163C9C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
164C23—46no—
165C23—91no—
166C23—96no—
167C23—1836no—
168C23—3618no—
169C23—3632no—
170C23C23×D916yes—
171Q8—43no—
172Q8—369no—
173D4—49no—
174D4—369no—
175D4—3618no—
176C2×C4—99no—
177C2×C4—1818no—
178C2×C4C23×D919yes—
179C2×C3—46no—
180C2×C3D3×H(5,𝔽2)13yes—
181D3—34no—
182D3—126no—
183C22—218no—
184C22—48no—
185C22—96no—
186C22—1818no—
187C22—3624no—
188C22—3624no—
189C22C24×D911yescharacteristic
190C4—29no—
191C4—189no—
192C3D3×H(3,𝔽4)11yescharacteristic
193C2—46no—
194C2—94no—
195C2—366no—
196C2D9×H(5,𝔽2)13yes—
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×H(3,𝔽4)C2
hypercenterH(3,𝔽4)D9
hypocenterC9C2×H(3,𝔽4)
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C9×H(3,𝔽4)C2semidirect
2. #10589785C2semidirect
3. D9×#124C2semidirect
4. #10589696C2semidirect
5. D9×#120C2semidirect
7. C24×D9C22semidirect
8. C2×Q8×D9C22semidirect
10. C9×#124C22semidirect
11. C9×#120C22semidirect
12. #63515C22semidirect
13. #63509C22semidirect
14. #63499C22semidirect
15. #63498C22semidirect
16. #63447C22semidirect
17. #63444C22semidirect
18. C22⋊C4×D9C22semidirect
19. #63437C22semidirect
20. C42×D9C22semidirect
26. C24×C9C23semidirect
29. C23×D9C23semidirect
30. C2×Q8×C9C23semidirect
37. C2×Dic18C23semidirect
38. C9×C22⋊C4C23semidirect
39. C42×C9C23semidirect
40. #3676C23semidirect
41. (C2×D9)⋊C4C23semidirect
42. C4×Dic9C23semidirect
75. H(3,𝔽4)D9direct (N × H)
111. #124C2×D9semidirect
115. #120C2×D9semidirect
136. D9H(3,𝔽4)direct (N × H)
140. C24C22×D9semidirect
143. C2×Q8C22×D9semidirect
151. C22⋊C4C22×D9semidirect
153. C42C22×D9semidirect
163. C9C2×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
C9×H(3,𝔽4)2
#105897852
D9×#1242
#105896962
D9×#1202
D3×H(3,𝔽4)3