Finite Group Databasebeta

← #10588361⋊C3 #13141468

Subgroups — #10588361⋊C3 #13141468

Order 1728.

316 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 112 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#63431⋊C3C211yesmaximal, characteristic
2#63431⋊C3C211yesmaximal, characteristic
3C4×C8×3−1+2C211yesmaximal, characteristic
4#63376⋊C3C214yesmaximal
5C3×#4681—31nomaximal
6#10588361C311yesmaximal, characteristic, complemented
7C2×3−1+2⋊Q8—21no—
8C2×3−1+2⋊Q8C2212yes—
9C2×C8×3−1+2C2212yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)—21no—
12C4⋊(3−1+2⋊C4)—21no—
13C4⋊(3−1+2⋊C4)—21no—
14C4⋊(3−1+2⋊C4)C2212yes—
15C3×#869—31no—
16C3×#869—31no—
17C32×C4×C8D311yescharacteristic
18C3×#816—34no—
19#63431C2×C311yescharacteristic
20#63431C2×C311yescharacteristic
21C4×C8×C9—21no—
22C4×C8×C9C2×C311yescharacteristic
23#63376C2×C314yes—
24C2×C4×3−1+2D411yescharacteristic
25C2×C4×3−1+2D411yescharacteristic
26C2×C4×3−1+2C2311yescharacteristic
27C2×3−1+2⋊C4—22no—
28C2×3−1+2⋊C4—22no—
293−1+2⋊Q8—24no—
303−1+2⋊Q8—42no—
31C8×3−1+2—22no—
32C3×#487—91no—
33#4681—31no—
34C2×C3×Dic6—32no—
35C2×C3×Dic6—61no—
36C2×C32×C8C2×D312yes—
37C32×C42C2×D311yescharacteristic
38C3×C4⋊Dic3—32no—
39C3×C4⋊Dic3—61no—
40C3×C4⋊Dic3—61no—
41C3×C4⋊Dic3—61no—
42C2×Dic18—21no—
43C2×Dic18C22×C312yes—
44C2×C8×C9—22no—
45C2×C8×C9C22×C312yes—
46C42×C9—21no—
47C42×C9C22×C311yescharacteristic
48C4⋊Dic9—21no—
49C4⋊Dic9—21no—
50C4⋊Dic9—21no—
51C4⋊Dic9C22×C312yes—
52C22×3−1+2C2×D411yescharacteristic
53C4×3−1+2SD1612yes—
54C4×3−1+2Q1612yes—
55C4×3−1+2#4112yes—
563−1+2⋊C4—42no—
573−1+2⋊C4—42no—
58C3×C4⋊Q8—91no—
59C3×C4⋊Q8—91no—
60#869—31no—
61#869—31no—
62C3×#103—94no—
63C3×C4×C8—31no—
64C3×C4×C8C3×D311yescharacteristic
65#816—34no—
66C2×C32×C4D1211yescharacteristic
67C2×C32×C4D1211yescharacteristic
68C2×C32×C4C22×D311yescharacteristic
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C3×Dic6—64no—
72C3×Dic6—122no—
73C32×C8—22no—
74C2×C4×C9—21no—
75C2×C4×C9—21no—
76C2×C4×C9—21no—
77C2×C4×C9C3×D411yescharacteristic
78C2×C4×C9C3×D411yescharacteristic
79C2×C4×C9C23×C311yescharacteristic
80C2×Dic9—22no—
81C2×Dic9—22no—
82Dic18—24no—
83Dic18—42no—
84C8×C9—24no—
85C8×C9—22no—
86#487—91noSylow
87C2×3−1+2#12411yescharacteristic
88C2×3−1+2C2×SD1611yescharacteristic
89C2×3−1+2C2×Q1611yescharacteristic
90C2×C3×Q8—92no—
91C2×C3×Q8—181no—
92C2×Dic6—32no—
93C2×Dic6—61no—
94C2×C3×C8—32no—
95C2×C3×C8C2×C3×D312yes—
96C3×C4⋊C4—92no—
97C3×C4⋊C4—181no—
98C3×C4⋊C4—181no—
99C3×C4⋊C4—181no—
100C3×C42—31no—
101C3×C42C2×C3×D311yescharacteristic
102C4⋊Dic3—32no—
103C4⋊Dic3—61no—
104C4⋊Dic3—61no—
105C4⋊Dic3—61no—
106C22×C32C2×D1211yescharacteristic
107C32×C4#20412yes—
108C32×C4Dic1212yes—
109C32×C4#23512yes—
110C3×Dic3—122no—
111C3×Dic3—122no—
112C22×C9—21no—
113C22×C9C2×C3×D411yescharacteristic
114C4×C9—22no—
115C4×C9—22no—
116C4×C9—22no—
117C4×C9C3×SD1612yes—
118C4×C9C3×Q1612yes—
119C4×C9C3×#4112yes—
120Dic9—42no—
121Dic9—42no—
122C4⋊Q8—91no—
123C4⋊Q8—91no—
124#103—94no—
125C4×C8AGL(1,ℤ/9)11yescharacteristic
1263−1+2#48711yescharacteristic, complemented, Sylow
127C3×Q8—184no—
128C3×Q8—362no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—182no—
133C2×C3×C4—182no—
134C2×C3×C4C3×D1211yescharacteristic
135C2×C3×C4C3×D1211yescharacteristic
136C2×C3×C4C22×C3×D311yescharacteristic
137C2×Dic3—62no—
138C2×Dic3—62no—
139Dic6—64no—
140Dic6—122no—
141C3×C8—22no—
142C3×C8—62no—
143C2×C32#87511yescharacteristic
144C2×C32C2×#20411yescharacteristic
145C2×C32C2×Dic1211yescharacteristic
146C2×C9—21no—
147C2×C9—21no—
148C2×C9—21no—
149C2×C9C3×#12411yescharacteristic
150C2×C9C2×C3×SD1611yescharacteristic
151C2×C9C2×C3×Q1611yescharacteristic
152C2×Q8—92no—
153C2×Q8—181no—
154C2×C8C2×AGL(1,ℤ/9)12yes—
155C4⋊C4—92no—
156C4⋊C4—181no—
157C4⋊C4—181no—
158C4⋊C4—181no—
159C42C2×AGL(1,ℤ/9)11yescharacteristic
160C22×C3—31no—
161C22×C3C2×C3×D1211yescharacteristic
162C3×C4—32no—
163C3×C4—32no—
164C3×C4—32no—
165C3×C4—362no—
166C3×C4—362no—
167C3×C4C3×#20412yes—
168C3×C4C3×Dic1212yes—
169C3×C4C3×#23512yes—
170Dic3—122no—
171Dic3—122no—
172C32#468111yescharacteristic
173C9—21no—
174C9C3×#48711yescharacteristic, complemented
175Q8—184no—
176Q8—362no—
177C2×C4—182no—
178C2×C4—182no—
179C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
180C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
181C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
182C8—22no—
183C2×C3—31no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3C3×#87511yescharacteristic
187C2×C3C2×C3×#20411yescharacteristic
188C2×C3C2×C3×Dic1211yescharacteristic
189C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
190C4—362no—
191C4—362no—
192C4Dic36⋊C312yes—
193C4#3664⋊C312yes—
194C4#3697⋊C312yes—
195C3—31no—
196C3C3×#468111yescharacteristic
197C2#63437⋊C311yescharacteristic
198C2C2×Dic36⋊C311yescharacteristic
199C2C2×#3664⋊C311yescharacteristic
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 radicalC22×C32C2×D12
hypercenterC4×C8AGL(1,ℤ/9)
hypocenterC9C3×#487
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
6. #10588361C3semidirect
126. 3−1+2#487semidirect
174. C9C3×#487semidirect

Maximal subgroups

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

subgroupindex
#63431⋊C32
C4×C8×3−1+22
#63376⋊C32
C3×#46813
#105883613