Finite Group Databasebeta

← C3⋊D3×H(3,𝔽4) #11910202

Subgroups — C3⋊D3×H(3,𝔽4) #11910202

Order 1152.

4224 conjugacy classes of subgroups; only the 743 normal subgroups are shown.

normal subgroups only

This group has 4224 conjugacy classes of subgroups — too many to draw. Only its 743 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
1C32×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2#10596154C219yesmaximal, complemented
3C3⋊D3×#124C219yesmaximal, complemented
4#10596065C216yesmaximal, complemented
5C3⋊D3×#120C216yesmaximal, complemented
6D3×H(3,𝔽4)—34nomaximal
7C24×C3⋊D3C2212yescomplemented
8C2×Q8×C3⋊D3C2213yescomplemented
9C2×D4×C3⋊D3C2219yes—
10C32×#124C2219yescomplemented
11C32×#120C2216yescomplemented
12#64162C2216yescomplemented
13#64156C2219yescomplemented
14#64146C22118yescomplemented
15#64145C2219yescomplemented
16#64094C22136yescomplemented
17#64091C22118yescomplemented
18C22⋊C4×C3⋊D3C22118yescomplemented
19#64084C2219yescomplemented
20C42×C3⋊D3C2213yescomplemented
21C3×H(3,𝔽4)D314yescomplemented
22#5683—336no—
23D3×#124—336no—
24#5594—324no—
25D3×#120—324no—
26C24×C32C2312yescomplemented
27C23×C3⋊D3—412no—
28C23×C3⋊D3C2316yes—
29C23×C3⋊D3C2316yescomplemented
30C2×C32×Q8C2313yescomplemented
31C2×C32×D4C2319yes—
32C2×#638C23118yes—
33Q8×C3⋊D3—46no—
34D4×C3⋊D3—418no—
35C2×C4⋊(C3⋊D3)C2319yes—
36C2×C4×C3⋊D3C2319yes—
37C2×#634C2319yescomplemented
38C32×C22⋊C4C23118yescomplemented
39C32×C42C2313yescomplemented
40#3757C23118yescomplemented
41#3752C23136yescomplemented
42C4×C3⋊Dic3C2319yescomplemented
43C2×H(3,𝔽4)—91noSylow
44C24×D3—38no—
45C2×D3×Q8—312no—
46C2×D3×D4—336no—
47C3×#124C2×D3136yescomplemented
48C3×#120C2×D3124yescomplemented
49#953—324no—
50#947—336no—
51#937—372no—
52#936—336no—
53#885—3144no—
54#882—372no—
55D3×C22⋊C4—372no—
56#875—336no—
57C42×D3—312no—
58C23×C32—46no—
59C23×C32C2416yes—
60C22×C3⋊D3—236no—
61C22×C3⋊D3—418no—
62C22×C3⋊D3—432no—
63C22×C3⋊D3C2416yes—
64C22×C3⋊D3C2411yescharacteristic
65C32×Q8—43no—
66C32×D4—49no—
67C2×C32×C4C2419yes—
68#638—418no—
69C2×C3⋊Dic3C2419yes—
70C4⋊(C3⋊D3)—49no—
71C4×C3⋊D3—218no—
72#634—49no—
73H(3,𝔽4)—99no—
74H(3,𝔽4)—96no—
75H(3,𝔽4)C3⋊D311yescharacteristic, direct product factor
76C2×#124—99no—
77C2×#120—96no—
78C24×C3C22×D318yescomplemented
79C23×D3—324no—
80C23×D3—324no—
81C23×D3—1248no—
82C2×C3×Q8C22×D3112yescomplemented
83C2×C3×D4C22×D3136yes—
84C2×#67—372no—
85D3×Q8—1224no—
86D3×D4—1272no—
87C2×D12—336no—
88C2×C4×D3—336no—
89C2×Dic6—336no—
90C3×C22⋊C4C22×D3172yescomplemented
91C3×C42C22×D3112yescomplemented
92#217—372no—
93(C2×D3)⋊C4—3144no—
94C4×Dic3—336no—
95C22×C32—218no—
96C22×C32—48no—
97C22×C32C2511yescharacteristic
98C2×C3⋊D3—218no—
99C2×C3⋊D3—424no—
100C2×C3⋊D3—424no—
101C2×C3⋊D3H(5,𝔽2)16yes—
102C32×C4—29no—
103C3⋊Dic3—29no—
104C25—92no—
105C22×Q8—93no—
106C22×D4—99no—
107#124—99no—
108#124—99no—
109#124—936no—
110#124—99no—
111#124C2×C3⋊D319yescomplemented
112#120—96no—
113#120—918no—
114#120—918no—
115#120C2×C3⋊D316yescomplemented
116C2×C22⋊C4—918no—
117C2×C42—93no—
118C23×C3—424no—
119C23×C3C23×D3124yes—
120C22×D3—34no—
121C22×D3—324no—
122C22×D3—6144no—
123C22×D3—1272no—
124C22×D3—12128no—
125C3×Q8—412no—
126C3×D4—436no—
127C2×C3×C4C23×D3136yes—
128#67—1272no—
129C2×Dic3—336no—
130D12—1236no—
131C4×D3—672no—
132Dic6—1236no—
133C2×C32—46no—
134C2×C32C2×H(5,𝔽2)13yes—
135C3⋊D3—46no—
136C3⋊D3H(3,𝔽4)14yesdirect product factor
137C24—96no—
138C24—96no—
139C24—3612no—
140C24C22×C3⋊D312yescomplemented
141C2×Q8—99no—
142C2×Q8—366no—
143C2×Q8C22×C3⋊D313yescomplemented
144C2×D4—918no—
145C2×D4—99no—
146C2×D4—3618no—
147C2×D4C22×C3⋊D319yes—
148C22×C4—99no—
149C22⋊C4—918no—
150C22⋊C4—936no—
151C22⋊C4C22×C3⋊D3118yescomplemented
152C42—99no—
153C42C22×C3⋊D313yescomplemented
154C22×C3—272no—
155C22×C3—432no—
156C22×C3C24×D314yes—
157C2×D3—324no—
158C2×D3—672no—
159C2×D3—1296no—
160C2×D3—1296no—
161C3×C4—236no—
162Dic3—636no—
163C32C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
164C23—46no—
165C23—91no—
166C23—96no—
167C23—1836no—
168C23—3618no—
169C23—3632no—
170C23C23×C3⋊D316yes—
171Q8—43no—
172Q8—369no—
173D4—49no—
174D4—369no—
175D4—3618no—
176C2×C4—99no—
177C2×C4—1818no—
178C2×C4C23×C3⋊D319yes—
179C2×C3—424no—
180C2×C3D3×H(5,𝔽2)112yes—
181D3—316no—
182D3—1224no—
183C22—218no—
184C22—48no—
185C22—96no—
186C22—1818no—
187C22—3624no—
188C22—3624no—
189C22C24×C3⋊D311yescharacteristic
190C4—29no—
191C4—189no—
192C3D3×H(3,𝔽4)14yescomplemented
193C2—46no—
194C2—94no—
195C2—366no—
196C2C3⋊D3×H(5,𝔽2)13yes—
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×H(3,𝔽4)C2
hypercenterH(3,𝔽4)C3⋊D3
hypocenterC32C2×H(3,𝔽4)
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C32×H(3,𝔽4)C2semidirect
2. #10596154C2semidirect
3. C3⋊D3×#124C2semidirect
4. #10596065C2semidirect
5. C3⋊D3×#120C2semidirect
7. C24×C3⋊D3C22semidirect
8. C2×Q8×C3⋊D3C22semidirect
10. C32×#124C22semidirect
11. C32×#120C22semidirect
12. #64162C22semidirect
13. #64156C22semidirect
14. #64146C22semidirect
15. #64145C22semidirect
16. #64094C22semidirect
17. #64091C22semidirect
18. C22⋊C4×C3⋊D3C22semidirect
19. #64084C22semidirect
20. C42×C3⋊D3C22semidirect
21. C3×H(3,𝔽4)D3semidirect
26. C24×C32C23semidirect
29. C23×C3⋊D3C23semidirect
30. C2×C32×Q8C23semidirect
37. C2×#634C23semidirect
38. C32×C22⋊C4C23semidirect
39. C32×C42C23semidirect
40. #3757C23semidirect
41. #3752C23semidirect
42. C4×C3⋊Dic3C23semidirect
47. C3×#124C2×D3semidirect
48. C3×#120C2×D3semidirect
75. H(3,𝔽4)C3⋊D3direct (N × H)
78. C24×C3C22×D3semidirect
82. C2×C3×Q8C22×D3semidirect
90. C3×C22⋊C4C22×D3semidirect
91. C3×C42C22×D3semidirect
111. #124C2×C3⋊D3semidirect
115. #120C2×C3⋊D3semidirect
136. C3⋊D3H(3,𝔽4)direct (N × H)
140. C24C22×C3⋊D3semidirect
143. C2×Q8C22×C3⋊D3semidirect
151. C22⋊C4C22×C3⋊D3semidirect
153. C42C22×C3⋊D3semidirect
163. C32C2×H(3,𝔽4)semidirect
192. C3D3×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
#105961542
C3⋊D3×#1242
#105960652
C3⋊D3×#1202
D3×H(3,𝔽4)3