Finite Group Databasebeta

← #10588379⋊C3 #13141486

Subgroups — #10588379⋊C3 #13141486

Order 1728.

310 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#63437⋊C3C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic, complemented
3C4×C8×3−1+2C211yesmaximal, characteristic, complemented
4#63380⋊C3C212yesmaximal
5#63376⋊C3C212yesmaximal, complemented
6C3×#4697—31nomaximal
7#10588379C311yesmaximal, characteristic, complemented
8C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
9C2×3−1+2⋊Q8C2211yescharacteristic
10C2×C8×3−1+2C2212yes—
11C42×3−1+2C2211yescharacteristic
12(C2×AGL(1,ℤ/9))⋊C4—22no—
13C4⋊(3−1+2⋊C4)C2212yes—
14(3−1+2⋊C4)⋊C4—22no—
15C3×#875—31no—
16C3×#870—31no—
17C32×C4×C8D311yescharacteristic
18C3×#821—32no—
19C3×#816—32no—
20#63437C2×C311yescharacteristic
21#63432C2×C311yescharacteristic, complemented
22C4×C8×C9—21no—
23C4×C8×C9C2×C311yescharacteristic, complemented
24#63380C2×C312yes—
25#63376C2×C312yescomplemented
26C22×AGL(1,ℤ/9)—21no—
27C2×C4×3−1+2D412yes—
28C2×C4×3−1+2C2311yescharacteristic
29C2×3−1+2⋊C4—21no—
30C2×3−1+2⋊C4—22no—
31C4⋊AGL(1,ℤ/9)—22no—
323−1+2⋊Q8—22no—
33C8×3−1+2—22no—
34C3×#488—91no—
35#4697—31no—
36C2×C3×D12—31no—
37C2×C3×Dic6—31no—
38C2×C32×C8C2×D312yes—
39C32×C42C2×D311yescharacteristic
40C3×(C2×D3)⋊C4—62no—
41C3×C4⋊Dic3—32no—
42C3×Dic3⋊C4—62no—
43C2×D36C22×C311yescharacteristic
44C2×Dic18C22×C311yescharacteristic
45C2×C8×C9—22no—
46C2×C8×C9C22×C312yes—
47C42×C9—21no—
48C42×C9C22×C311yescharacteristic
49(C2×D9)⋊C4—22no—
50C4⋊Dic9C22×C312yes—
51Dic9⋊C4—22no—
52C22×3−1+2C2×D411yescharacteristic
53C2×AGL(1,ℤ/9)—41no—
54C2×AGL(1,ℤ/9)—42no—
55C4×3−1+2—22no—
56C4×3−1+2#4112yes—
573−1+2⋊C4—41no—
583−1+2⋊C4—42no—
59C3×#125—91no—
60C3×#124—91no—
61#875—31no—
62#870—31no—
63C3×#103—92no—
64C3×#102—92no—
65C3×C4×C8—31no—
66C3×C4×C8C3×D311yescharacteristic
67#821—32no—
68#816—32no—
69C22×C3×D3—61no—
70C2×C32×C4D1212yes—
71C2×C32×C4C22×D311yescharacteristic
72C2×C3×Dic3—62no—
73C2×C3×Dic3—61no—
74C3×D12—62no—
75C3×Dic6—62no—
76C22×D9—21no—
77C32×C8—22no—
78C2×C4×C9—22no—
79C2×C4×C9—21no—
80C2×C4×C9C3×D412yes—
81C2×C4×C9C23×C311yescharacteristic
82C2×Dic9—22no—
83C2×Dic9—21no—
84D36—22no—
85Dic18—22no—
86C8×C9—24no—
87C8×C9—22no—
88#488—91noSylow
89C2×3−1+2#12411yescharacteristic
90C2×3−1+2#13512yes—
91AGL(1,ℤ/9)—81no—
92C2×C3×Q8—91no—
93C2×C3×D4—91no—
94C2×D12—31no—
95C2×Dic6—31no—
96C2×C3×C8—32no—
97C2×C3×C8C2×C3×D312yes—
98C3×C4⋊C4—92no—
99C3×C4⋊C4—182no—
100C3×C22⋊C4—182no—
101C3×C42—31no—
102C3×C42C2×C3×D311yescharacteristic
103(C2×D3)⋊C4—62no—
104C4⋊Dic3—32no—
105Dic3⋊C4—62no—
106C22×C32C2×D1211yescharacteristic
107C2×C3×D3—121no—
108C2×C3×D3—122no—
109C32×C4—22no—
110C32×C4#23512yes—
111C3×Dic3—121no—
112C3×Dic3—122no—
113C22×C9—21no—
114C22×C9C2×C3×D411yescharacteristic
115C2×D9—41no—
116C2×D9—42no—
117C4×C9—24no—
118C4×C9—22no—
119C4×C9—22no—
120C4×C9C3×#4112yes—
121Dic9—41no—
122Dic9—42no—
123#125—91no—
124#124—91no—
125#103—92no—
126#102—92no—
127C4×C8AGL(1,ℤ/9)11yescharacteristic, complemented
1283−1+2#48811yescharacteristic, complemented, Sylow
129C23×C3—181no—
130C22×D3—61no—
131C3×Q8—182no—
132C3×D4—182no—
133C2×C3×C4—32no—
134C2×C3×C4—31no—
135C2×C3×C4—181no—
136C2×C3×C4—182no—
137C2×C3×C4C3×D1212yes—
138C2×C3×C4C22×C3×D311yescharacteristic
139C2×Dic3—62no—
140C2×Dic3—61no—
141D12—62no—
142Dic6—62no—
143C3×C8—22no—
144C3×C8—62no—
145C2×C32#87511yescharacteristic
146C2×C32#90412yes—
147C3×D3—241no—
148C2×C9—22no—
149C2×C9—21no—
150C2×C9C3×#12411yescharacteristic
151C2×C9C3×#13512yes—
152D9—81no—
153C2×Q8—91no—
154C2×D4—91no—
155C2×C8C2×AGL(1,ℤ/9)12yes—
156C4⋊C4—92no—
157C4⋊C4—182no—
158C22⋊C4—182no—
159C42C2×AGL(1,ℤ/9)11yescharacteristic
160C22×C3—31no—
161C22×C3—361no—
162C22×C3—362no—
163C22×C3C2×C3×D1211yescharacteristic
164C2×D3—122no—
165C2×D3—121no—
166C3×C4—22no—
167C3×C4—32no—
168C3×C4—62no—
169C3×C4—361no—
170C3×C4—362no—
171C3×C4C3×#23512yes—
172Dic3—122no—
173Dic3—121no—
174C32#469711yescharacteristic
175C9—21no—
176C9C3×#48811yescharacteristic, complemented
177C23—181no—
178Q8—182no—
179D4—182no—
180C2×C4—181no—
181C2×C4—182no—
182C2×C4C4⋊AGL(1,ℤ/9)12yes—
183C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
184C8—22no—
185C2×C3—32no—
186C2×C3—31no—
187C2×C3—721no—
188C2×C3C3×#87511yescharacteristic
189C2×C3C3×#90412yes—
190D3—241no—
191C22—362no—
192C22—361no—
193C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
194C4—22no—
195C4—362no—
196C4—361no—
197C4#3697⋊C312yes—
198C3—31no—
199C3C3×#469711yescharacteristic
200C2—721no—
201C2#63437⋊C311yescharacteristic
202C2#63467⋊C312yes—
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×C8×3−1+2C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenterC4×C8AGL(1,ℤ/9)
hypocenterC9C3×#488
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #63432⋊C3C2semidirect
3. C4×C8×3−1+2C2semidirect
5. #63376⋊C3C2semidirect
7. #10588379C3semidirect
21. #63432C2×C3semidirect
23. C4×C8×C9C2×C3semidirect
25. #63376C2×C3semidirect
127. C4×C8AGL(1,ℤ/9)semidirect
128. 3−1+2#488semidirect
176. C9C3×#488semidirect

Maximal subgroups

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

subgroupindex
#63437⋊C32
#63432⋊C32
C4×C8×3−1+22
#63380⋊C32
#63376⋊C32
C3×#46973
#105883793