Finite Group Databasebeta

← C3×#10593293 #13166663

Subgroups — C3×#10593293 #13166663

Order 1728.

520 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C3×#64231C212yesmaximal, complemented
2C3×#64225C212yesmaximal, complemented
3C3×#63797C212yesmaximal, complemented
4C3×C8×#157C211yesmaximal, characteristic, complemented
5#10593293C311yesmaximal, characteristic, direct product factor
6C2×C3×GO+(4,𝔽2)—24no—
7C3×#3802C2214yes—
8C3×C4×#157C2211yescharacteristic
9C3×#3787C2211yescharacteristic
10C3×#3774—24no—
11C3×C8×C3⋊D3C2211yescharacteristic
12C32×D24—22no—
13C3×#3713—22no—
14#64231C2×C312yescomplemented
15#64225C2×C312yescomplemented
16C3×D3×D8—62no—
17#63797C2×C312yescomplemented
18C8×#157C2×C311yescharacteristic, complemented
19C2×C3×D32—24no—
20C2×C3×#157D412yes—
21C3×GO+(4,𝔽2)—48no—
22C3×C4×C3⋊D3C2311yescharacteristic
23C32×D12—24no—
24C3×#625—24no—
25C3×#622D412yescomplemented
26C33×C8D411yescharacteristic, complemented
27C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
28C3×#493—91nomaximal
29C2×GO+(4,𝔽2)—24no—
30C3×D3×D4—64no—
31#3802C22×C314yes—
32C4×#157C22×C311yescharacteristic
33#3787C22×C311yescharacteristic
34#3774—24no—
35C32×D8—62no—
36C8×C3⋊D3C22×C311yescharacteristic
37C3×#213—64no—
38C3×D24—22no—
39C3×D24—22no—
40C3×D24—42no—
41C3×D3×C8—62no—
42#3713—22no—
43C2×C3×C3⋊D3C2×D411yescharacteristic
44C2×C32×D3—44no—
45C3×D32—48no—
46C3×#157D814yescomplemented
47C33×C4C2×D411yescharacteristic
48C3×C3⋊Dic3C2×D411yescharacteristic
49C2×C3×D8—92no—
50C2×C3×D8—92no—
51C3×#127—92no—
52D3×D8—62no—
53C3×C4×C8—91no—
54C22×C3×D3—124no—
55C2×D32—24no—
56C2×#157C3×D412yes—
57GO+(4,𝔽2)—48no—
58C32×D4—64no—
59C4×C3⋊D3C23×C311yescharacteristic
60C3×#67—124no—
61C3×D12—24no—
62C3×D12—24no—
63C3×D12—44no—
64C3×C4×D3—62no—
65#625—24no—
66#622C3×D412yescomplemented
67C32×C8—22no—
68C32×C8—42no—
69C32×C8C3×D411yescharacteristic, complemented
70C3⋊(C3⋊C8)C3×D411yescharacteristic, complemented
71C3×C3⋊C8—62no—
72#493—91noSylow
73C2×C33#12711yescharacteristic
74C3×C3⋊D3C2×D812yes—
75C32×D3—84no—
76C2×C3×D4—94no—
77C2×C3×D4—184no—
78D3×D4—64no—
79C3×D8—62no—
80C3×D8—62no—
81C3×D8—122no—
82C3×D8—182no—
83C3×D8—184no—
84C2×C3×C8—91no—
85C2×C3×C8—91no—
86C3×C42—91no—
87#213—64no—
88D24—22no—
89D3×C8—62no—
90C22×C32—124no—
91C2×C3⋊D3C2×C3×D411yescharacteristic
92C2×C3×D3—44no—
93C2×C3×D3—44no—
94C2×C3×D3—62no—
95C2×C3×D3—84no—
96C2×C3×D3—248no—
97D32—48no—
98#157C3×D814yescomplemented
99C32×C4—22no—
100C32×C4—42no—
101C32×C4C2×C3×D411yescharacteristic
102C3⋊Dic3C2×C3×D411yescharacteristic
103C3×Dic3—62no—
104C2×D8—92no—
105C2×D8—92no—
106#127—92no—
107C4×C8—91no—
108C33#49311yescharacteristic, complemented, Sylow
109C23×C3—184no—
110C22×D3—124no—
111C3×D4—64no—
112C3×D4—64no—
113C3×D4—124no—
114C3×D4—184no—
115C3×D4—368no—
116C2×C3×C4—92no—
117C2×C3×C4—91no—
118#67—124no—
119D12—24no—
120C4×D3—62no—
121C3×C8—22no—
122C3×C8—42no—
123C3×C8—92no—
124C3×C8—91no—
125C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
126C3⋊C8—62no—
127C2×C32—22no—
128C2×C32—42no—
129C2×C32—244no—
130C2×C32C3×#12711yescharacteristic
131C3⋊D3C2×C3×D812yes—
132C3×D3—64no—
133C3×D3—84no—
134C3×D3—84no—
135C3×D3—164no—
136C2×D4—94no—
137C2×D4—184no—
138D8—62no—
139D8—182no—
140D8—184no—
141C2×C8—91no—
142C2×C8—91no—
143C42—91no—
144C22×C3—91no—
145C22×C3—124no—
146C22×C3—124no—
147C22×C3—244no—
148C22×C3—368no—
149C2×D3—44no—
150C2×D3—62no—
151C2×D3—248no—
152C3×C4—22no—
153C3×C4—42no—
154C3×C4—94no—
155C3×C4—91no—
156C3×C4C2×GO+(4,𝔽2)11yescharacteristic
157Dic3—62no—
158C32—22no—
159C32—42no—
160C32C3×#49311yescharacteristic, complemented
161C23—184no—
162D4—64no—
163D4—184no—
164D4—368no—
165C2×C4—92no—
166C2×C4—91no—
167C8—92no—
168C8—91no—
169C8C3×GO+(4,𝔽2)11yescharacteristic, complemented
170C2×C3—22no—
171C2×C3—42no—
172C2×C3—92no—
173C2×C3—244no—
174C2×C3—244no—
175C2×C3—484no—
176C2×C3#6423111yescharacteristic
177D3—64no—
178D3—84no—
179C22—91no—
180C22—124no—
181C22—368no—
182C4—94no—
183C4—91no—
184C4C2×C3×GO+(4,𝔽2)11yescharacteristic
185C3—22no—
186C3—42no—
187C3#1059329311yescharacteristic, direct product factor
188C2—92no—
189C2—244no—
190C2C3×#6423111yescharacteristic
subgroupG/H
centerC2×C3#64231
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#127
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#493
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#64231C2semidirect
2. C3×#64225C2semidirect
3. C3×#63797C2semidirect
4. C3×C8×#157C2semidirect
5. #10593293C3direct (N × H)
14. #64231C2×C3semidirect
15. #64225C2×C3semidirect
17. #63797C2×C3semidirect
18. C8×#157C2×C3semidirect
25. C3×#622D4semidirect
26. C33×C8D4semidirect
27. C3×C3⋊(C3⋊C8)D4semidirect
46. C3×#157D8semidirect
66. #622C3×D4semidirect
69. C32×C8C3×D4semidirect
70. C3⋊(C3⋊C8)C3×D4semidirect
98. #157C3×D8semidirect
108. C33#493semidirect
125. C3×C8GO+(4,𝔽2)semidirect
160. C32C3×#493semidirect
169. C8C3×GO+(4,𝔽2)semidirect
187. C3#10593293direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642312
C3×#642252
C3×#637972
C3×C8×#1572
#105932933
C3×#4939