Finite Group Databasebeta

← #10588743⋊C3 #13142957

Subgroups — #10588743⋊C3 #13142957

Order 1728.

298 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 100 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
13−1+2×#125C211yesmaximal, characteristic
2#63431⋊C3C211yesmaximal, characteristic
3#63367⋊C3C212yesmaximal
4#63366⋊C3C212yesmaximal
5#63362⋊C3C211yesmaximal, characteristic
6C3×#5064—31nomaximal
7#10588743C311yesmaximal, characteristic, complemented
8C2×3−1+2⋊Q8—21no—
9C4⋊C4×3−1+2—21no—
10C4⋊C4×3−1+2—21no—
11C4⋊C4×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2C2211yescharacteristic
13C42×3−1+2C2211yescharacteristic
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2212yes—
16C2×3−1+2⋊C8C2212yes—
17C32×#125D311yescharacteristic
18C3×#869—31no—
19C3×#808—32no—
20C3×#807—32no—
21C3×#803—31no—
22C9×#125—21no—
23C9×#125C2×C311yescharacteristic
24#63431C2×C311yescharacteristic
25#63367C2×C312yes—
26#63366C2×C312yes—
27#63362C2×C311yescharacteristic
28C2×C4×3−1+2—21no—
29C2×C4×3−1+2—21no—
30C2×C4×3−1+2D411yescharacteristic
31C2×C4×3−1+2D411yescharacteristic
32C2×C4×3−1+2C2311yescharacteristic
33C2×3−1+2⋊C4—22no—
343−1+2⋊Q8—42no—
353−1+2⋊C8Q814yescomplemented
36C3×#501—91no—
37#5064—31no—
38C2×C3×Dic6—61no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4C2×D311yescharacteristic
42C32×C4⋊C4C2×D311yescharacteristic
43C32×C42C2×D311yescharacteristic
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3—61no—
46C2×C3×C3⋊C8—32no—
47C2×Dic18—21no—
48C9×C4⋊C4—22no—
49C9×C4⋊C4—22no—
50C9×C4⋊C4—21no—
51C9×C4⋊C4—21no—
52C9×C4⋊C4—21no—
53C9×C4⋊C4—21no—
54C9×C4⋊C4C22×C311yescharacteristic
55C9×C4⋊C4C22×C311yescharacteristic
56C42×C9—21no—
57C42×C9C22×C311yescharacteristic
58C4⋊Dic9—21no—
59C4⋊Dic9C22×C312yes—
60C2×C9⋊C8C22×C312yes—
61C22×3−1+2C2×D411yescharacteristic
62C4×3−1+2—21no—
63C4×3−1+2—21no—
64C4×3−1+2—41no—
65C4×3−1+2—41no—
66C4×3−1+2C2×Q812yes—
673−1+2⋊C4—42no—
68C3×C4⋊Q8—91no—
69C3×#125—31no—
70C3×#125C3×D311yescharacteristic
71#869—31no—
72C3×#107—92no—
73C3×#106—92no—
74C3×#97—91no—
75#808—32no—
76#807—32no—
77#803—31no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4#6711yescharacteristic
81C2×C32×C4#6711yescharacteristic
82C2×C32×C4C22×D311yescharacteristic
83C2×C3×Dic3—62no—
84C3×Dic6—122no—
85C3×C3⋊C8—34no—
86C2×C4×C9—22no—
87C2×C4×C9—22no—
88C2×C4×C9—21no—
89C2×C4×C9—21no—
90C2×C4×C9—21no—
91C2×C4×C9—21no—
92C2×C4×C9—21no—
93C2×C4×C9C3×D411yescharacteristic
94C2×C4×C9C3×D411yescharacteristic
95C2×C4×C9C23×C311yescharacteristic
96C2×Dic9—22no—
97Dic18—42no—
98C9⋊C8C3×Q814yescomplemented
99#501—91noSylow
100C2×3−1+2C4⋊Q811yescharacteristic
101C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
102C2×3−1+2#13711yescharacteristic
103C2×C3×Q8—181no—
104C2×Dic6—61no—
105C2×C3×C8—92no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—181no—
114C3×C4⋊C4C2×C3×D311yescharacteristic
115C3×C4⋊C4C2×C3×D311yescharacteristic
116C3×C42—31no—
117C3×C42C2×C3×D311yescharacteristic
118C4⋊Dic3—32no—
119C4⋊Dic3—61no—
120C2×C3⋊C8—32no—
121C22×C32C2×#6711yescharacteristic
122C32×C4—21no—
123C32×C4—21no—
124C32×C4—41no—
125C32×C4—41no—
126C32×C4D3×Q812yes—
127C3×Dic3—122no—
128C22×C9—21no—
129C22×C9C2×C3×D411yescharacteristic
130C4×C9—21no—
131C4×C9—21no—
132C4×C9—22no—
133C4×C9—41no—
134C4×C9—41no—
135C4×C9—42no—
136C4×C9—42no—
137C4×C9—41no—
138C4×C9—41no—
139C4×C9C2×C3×Q812yes—
140Dic9—42no—
141C4⋊Q8—91no—
142#125AGL(1,ℤ/9)11yescharacteristic
143#107—92no—
144#106—92no—
145#97—91no—
1463−1+2#50111yescharacteristic, complemented, Sylow
147C3×Q8—362no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—182no—
156C2×C3×C4C3×#6711yescharacteristic
157C2×C3×C4C3×#6711yescharacteristic
158C2×C3×C4C22×C3×D311yescharacteristic
159C2×Dic3—62no—
160Dic6—122no—
161C3×C8—94no—
162C3⋊C8—34no—
163C2×C32#94411yescharacteristic
164C2×C32#94911yescharacteristic
165C2×C32#95111yescharacteristic
166C2×C9—21no—
167C2×C9—21no—
168C2×C9—21no—
169C2×C9C3×C4⋊Q811yescharacteristic
170C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
171C2×C9C3×#13711yescharacteristic
172C2×Q8—181no—
173C2×C8—92no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—92no—
177C4⋊C4—181no—
178C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
179C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
180C42C2×AGL(1,ℤ/9)11yescharacteristic
181C22×C3—31no—
182C22×C3C2×C3×#6711yescharacteristic
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—32no—
186C3×C4—41no—
187C3×C4—41no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—121no—
191C3×C4—121no—
192C3×C4—362no—
193C3×C4C3×D3×Q812yes—
194Dic3—122no—
195C32#506411yescharacteristic
196C9—21no—
197C9C3×#50111yescharacteristic, complemented
198Q8—362no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—182no—
202C2×C4#611⋊C311yescharacteristic
203C2×C4#611⋊C311yescharacteristic
204C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
205C8—94no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3C3×#94411yescharacteristic
210C2×C3C3×#94911yescharacteristic
211C2×C3C3×#95111yescharacteristic
212C22C2×#611⋊C311yescharacteristic
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—362no—
218C4Q8×AGL(1,ℤ/9)12yes—
219C3—31no—
220C3C3×#506411yescharacteristic
221C2#63506⋊C311yescharacteristic
222C2#63511⋊C311yescharacteristic
223C2#63512⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#501
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
7. #10588743C3semidirect
35. 3−1+2⋊C8Q8semidirect
98. C9⋊C8C3×Q8semidirect
146. 3−1+2#501semidirect
197. C9C3×#501semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63431⋊C32
#63367⋊C32
#63366⋊C32
#63362⋊C32
C3×#50643
#105887433