Finite Group Databasebeta

← #10588382⋊C3 #13141541

Subgroups — #10588382⋊C3 #13141541

Order 1728.

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

normal subgroups only

This group has 290 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#63432⋊C3C211yesmaximal, characteristic
2#63431⋊C3C211yesmaximal, characteristic
33−1+2×#97C211yesmaximal, characteristic
4#63378⋊C3C212yesmaximal
5#63377⋊C3C212yesmaximal
6C3×#4702—31nomaximal
7#10588382C311yesmaximal, characteristic, complemented
8C2×3−1+2⋊Q8—21no—
9C2×C8×3−1+2C2212yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)—21no—
12C4⋊(3−1+2⋊C4)C2212yes—
13C4⋊(3−1+2⋊C4)C2211yescharacteristic
14C4⋊(3−1+2⋊C4)C2211yescharacteristic
15(3−1+2⋊C4)⋊C4—21no—
16(3−1+2⋊C4)⋊C4—21no—
17C3×#870—31no—
18C3×#869—31no—
19C32×#97D311yescharacteristic
20C3×#818—32no—
21C3×#817—32no—
22#63432C2×C311yescharacteristic
23#63431C2×C311yescharacteristic
24C9×#97—21no—
25C9×#97C2×C311yescharacteristic
26#63378C2×C312yes—
27#63377C2×C312yes—
28C2×C4×3−1+2D411yescharacteristic
29C2×C4×3−1+2D411yescharacteristic
30C2×C4×3−1+2C2311yescharacteristic
31C2×3−1+2⋊C4—22no—
32C2×3−1+2⋊C4—21no—
33C2×3−1+2⋊C4—21no—
343−1+2⋊Q8—42no—
35C8×3−1+2Q814yescomplemented
36C3×#501—91no—
37#4702—31no—
38C2×C3×Dic6—61no—
39C2×C32×C8C2×D312yes—
40C32×C42C2×D311yescharacteristic
41C3×C4⋊Dic3—31no—
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—31no—
44C3×C4⋊Dic3—61no—
45C3×Dic3⋊C4—61no—
46C3×Dic3⋊C4—61no—
47C2×Dic18—21no—
48C2×C8×C9—22no—
49C2×C8×C9C22×C312yes—
50C42×C9—21no—
51C42×C9C22×C311yescharacteristic
52C4⋊Dic9—21no—
53C4⋊Dic9C22×C311yescharacteristic
54C4⋊Dic9C22×C312yes—
55C4⋊Dic9C22×C311yescharacteristic
56Dic9⋊C4—21no—
57Dic9⋊C4—21no—
58C22×3−1+2C2×D411yescharacteristic
59C4×3−1+2—21no—
60C4×3−1+2—21no—
61C4×3−1+2C2×Q812yes—
623−1+2⋊C4—42no—
633−1+2⋊C4—41no—
643−1+2⋊C4—41no—
65C3×C4⋊Q8—91no—
66C3×#125—91no—
67#870—31no—
68#869—31no—
69C3×#107—92no—
70C3×#106—92no—
71C3×#97—31no—
72C3×#97C3×D311yescharacteristic
73#818—32no—
74#817—32no—
75C2×C32×C4D1211yescharacteristic
76C2×C32×C4D1211yescharacteristic
77C2×C32×C4C22×D311yescharacteristic
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—62no—
81C3×Dic6—122no—
82C32×C8Dic614yes—
83C2×C4×C9—21no—
84C2×C4×C9—21no—
85C2×C4×C9—21no—
86C2×C4×C9C3×D411yescharacteristic
87C2×C4×C9C3×D411yescharacteristic
88C2×C4×C9C23×C311yescharacteristic
89C2×Dic9—21no—
90C2×Dic9—21no—
91C2×Dic9—22no—
92Dic18—42no—
93C8×C9—24no—
94C8×C9C3×Q814yescomplemented
95#501—91noSylow
96C2×3−1+2C4⋊Q811yescharacteristic
97C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
98C2×3−1+2#13711yescharacteristic
99C2×C3×Q8—181no—
100C2×Dic6—61no—
101C2×C3×C8—32no—
102C2×C3×C8C2×C3×D312yes—
103C3×C4⋊C4—91no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4—181no—
108C3×C4⋊C4—181no—
109C3×C42—31no—
110C3×C42C2×C3×D311yescharacteristic
111C4⋊Dic3—31no—
112C4⋊Dic3—32no—
113C4⋊Dic3—31no—
114C4⋊Dic3—61no—
115Dic3⋊C4—61no—
116Dic3⋊C4—61no—
117C22×C32C2×D1211yescharacteristic
118C32×C4—21no—
119C32×C4—21no—
120C32×C4C2×Dic612yes—
121C3×Dic3—122no—
122C3×Dic3—121no—
123C3×Dic3—121no—
124C22×C9—21no—
125C22×C9C2×C3×D411yescharacteristic
126C4×C9—21no—
127C4×C9—21no—
128C4×C9—22no—
129C4×C9—41no—
130C4×C9—41no—
131C4×C9C2×C3×Q812yes—
132Dic9—42no—
133Dic9—41no—
134Dic9—41no—
135C4⋊Q8—91no—
136#125—91no—
137#107—92no—
138#106—92no—
139#97AGL(1,ℤ/9)11yescharacteristic
1403−1+2#50111yescharacteristic, complemented, Sylow
141C3×Q8—362no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4—182no—
148C2×C3×C4C3×D1211yescharacteristic
149C2×C3×C4C3×D1211yescharacteristic
150C2×C3×C4C22×C3×D311yescharacteristic
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—62no—
154Dic6—122no—
155C3×C8—34no—
156C3×C8C3×Dic614yes—
157C2×C32#86911yescharacteristic
158C2×C32#90811yescharacteristic
159C2×C32#90911yescharacteristic
160C2×C9—21no—
161C2×C9—21no—
162C2×C9—21no—
163C2×C9C3×C4⋊Q811yescharacteristic
164C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
165C2×C9C3×#13711yescharacteristic
166C2×Q8—181no—
167C2×C8C2×AGL(1,ℤ/9)12yes—
168C4⋊C4—91no—
169C4⋊C4—92no—
170C4⋊C4—91no—
171C4⋊C4—181no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C42C2×AGL(1,ℤ/9)11yescharacteristic
175C22×C3—31no—
176C22×C3C2×C3×D1211yescharacteristic
177C3×C4—21no—
178C3×C4—21no—
179C3×C4—32no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—362no—
183C3×C4—361no—
184C3×C4—361no—
185C3×C4C2×C3×Dic612yes—
186Dic3—122no—
187Dic3—121no—
188Dic3—121no—
189C32#470211yescharacteristic
190C9—21no—
191C9C3×#50111yescharacteristic, complemented
192Q8—362no—
193C2×C4—181no—
194C2×C4—181no—
195C2×C4—182no—
196C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
197C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
198C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
199C83−1+2⋊Q814yescomplemented
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3C3×#86911yescharacteristic
204C2×C3C3×#90811yescharacteristic
205C2×C3C3×#90911yescharacteristic
206C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
207C4—21no—
208C4—21no—
209C4—362no—
210C4—361no—
211C4—361no—
212C4C2×3−1+2⋊Q812yes—
213C3—31no—
214C3C3×#470211yescharacteristic
215C2#63431⋊C311yescharacteristic
216C2#63470⋊C311yescharacteristic
217C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#97C2
socleC22×C3C2×C3×D12
prime radicalC22×C32C2×D12
hypercenter#97AGL(1,ℤ/9)
hypocenterC9C3×#501
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
7. #10588382C3semidirect
35. C8×3−1+2Q8semidirect
94. C8×C9C3×Q8semidirect
140. 3−1+2#501semidirect
191. C9C3×#501semidirect
199. C83−1+2⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#63432⋊C32
#63431⋊C32
3−1+2×#972
#63378⋊C32
#63377⋊C32
C3×#47023
#105883823