Finite Group Databasebeta

← #10588216⋊C3 #13135607

Subgroups — #10588216⋊C3 #13135607

Order 1728.

243 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 56 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×#123C211yesmaximal, characteristic
2#63393⋊C3C211yesmaximal, characteristic
3#63391⋊C3C211yesmaximal, characteristic
4C3×#4539—31nomaximal
5#10588216C311yesmaximal, characteristic, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C22×C4×3−1+2C411yescharacteristic, complemented
8C4⋊C4×3−1+2—21no—
9C22⋊C4×3−1+2—21no—
10C22⋊C4×3−1+2C411yescharacteristic, complemented
11#3676⋊C3—21no—
12#3667⋊C3—21no—
13C32×#123D311yescharacteristic
14C3×#834—31no—
15C3×#833—31no—
16C9×#123—21no—
17C9×#123C2×C311yescharacteristic
18#63393C2×C311yescharacteristic
19#63391C2×C311yescharacteristic
20C23×3−1+2C2×C411yescharacteristic
21C23×3−1+2D411yescharacteristic
22D4×3−1+2—41no—
23C2×C4×3−1+2—21no—
24C2×C4×3−1+2—21no—
25C2×C4×3−1+2—41no—
26C2×C4×3−1+2D411yescharacteristic
27C2×3−1+2⋊C4—41no—
283−1+2⋊C8—41no—
29C3×#352—91no—
30#4539—31no—
31C2×C32×D4C2×D311yescharacteristic
32C22×C32×C4Dic311yescharacteristic
33C32×C4⋊C4—21no—
34C32×C22⋊C4—21no—
35C32×C22⋊C4Dic311yescharacteristic
36C3×#217—61no—
37C3×#208—61no—
38C2×D4×C9—21no—
39C2×D4×C9C22×C311yescharacteristic
40C22×C4×C9—21no—
41C22×C4×C9C3×C411yescharacteristic, complemented
42C9×C4⋊C4—22no—
43C9×C4⋊C4—21no—
44C9×C22⋊C4—22no—
45C9×C22⋊C4—21no—
46C9×C22⋊C4—21no—
47C9×C22⋊C4C3×C411yescharacteristic, complemented
48#3676—21no—
49#3667—21no—
50C22×3−1+2—21no—
51C22×3−1+2—21no—
52C22×3−1+2—41no—
53C22×3−1+2—41no—
54C22×3−1+2C22⋊C411yescharacteristic
55C4×3−1+2—21no—
56C4×3−1+2—41no—
57C4×3−1+2—41no—
583−1+2⋊C4—81no—
59C3×#123—31no—
60C3×#123C3×D311yescharacteristic
61C3×#100—91no—
62C3×#99—91no—
63#834—31no—
64#833—31no—
65C23×C32C2×Dic311yescharacteristic
66C23×C32#6711yescharacteristic
67C32×D4—41no—
68C2×C32×C4—21no—
69C2×C32×C4—21no—
70C2×C32×C4—41no—
71C2×C32×C4#6711yescharacteristic
72C2×C3×Dic3—121no—
73C23×C9—21no—
74C23×C9—21no—
75C23×C9C2×C3×C411yescharacteristic
76C23×C9C3×D411yescharacteristic
77C3×C3⋊C8—121no—
78D4×C9—41no—
79D4×C9—42no—
80C2×C4×C9—22no—
81C2×C4×C9—22no—
82C2×C4×C9—21no—
83C2×C4×C9—21no—
84C2×C4×C9—21no—
85C2×C4×C9—42no—
86C2×C4×C9—41no—
87C2×C4×C9C3×D411yescharacteristic
88C2×Dic9—41no—
89C9⋊C8—41no—
90#352—91noSylow
91C2×3−1+2—21no—
92C2×3−1+2—41no—
93C2×3−1+2—41no—
94C2×3−1+2#9911yescharacteristic
95C2×C3×D4—31no—
96C2×C3×D4C2×C3×D311yescharacteristic
97C22×C3×C4—31no—
98C22×C3×C4C3×Dic311yescharacteristic
99C3×M16—181no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—61no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—31no—
104C3×C22⋊C4—61no—
105C3×C22⋊C4—181no—
106C3×C22⋊C4C3×Dic311yescharacteristic
107#217—61no—
108#208—61no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32#21711yescharacteristic
114C32×C4—21no—
115C32×C4—41no—
116C32×C4—41no—
117C3×Dic3—241no—
118C22×C9—22no—
119C22×C9—21no—
120C22×C9—21no—
121C22×C9—21no—
122C22×C9—41no—
123C22×C9—41no—
124C22×C9—41no—
125C22×C9—42no—
126C22×C9—42no—
127C22×C9C3×C22⋊C411yescharacteristic
128C4×C9—21no—
129C4×C9—41no—
130C4×C9—42no—
131C4×C9—41no—
132C4×C9—42no—
133C4×C9—41no—
134Dic9—81no—
135#123AGL(1,ℤ/9)11yescharacteristic
136#100—91no—
137#99—91no—
1383−1+2#35211yescharacteristic, complemented, Sylow
139C23×C3—31no—
140C23×C3—31no—
141C23×C3C2×C3×Dic311yescharacteristic
142C23×C3C3×#6711yescharacteristic
143C3×D4—41no—
144C3×D4—121no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—31no—
148C2×C3×C4—41no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—121no—
152C2×C3×C4—361no—
153C2×C3×C4C3×#6711yescharacteristic
154C2×Dic3—121no—
155C3×C8—361no—
156C3⋊C8—121no—
157C2×C32—21no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32#83411yescharacteristic
161C2×C9—22no—
162C2×C9—21no—
163C2×C9—21no—
164C2×C9—42no—
165C2×C9—41no—
166C2×C9—41no—
167C2×C9—81no—
168C2×C9C3×#9911yescharacteristic
169C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
170C22×C43−1+2⋊C411yescharacteristic, complemented
171M16—181no—
172C4⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—181no—
175C22⋊C43−1+2⋊C411yescharacteristic, complemented
176C22×C3—21no—
177C22×C3—21no—
178C22×C3—31no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—121no—
184C22×C3—121no—
185C22×C3C3×#21711yescharacteristic
186C3×C4—21no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4—61no—
190C3×C4—121no—
191C3×C4—121no—
192C3×C4—721no—
193Dic3—241no—
194C32#453911yescharacteristic
195C9—21no—
196C9C3×#35211yescharacteristic, complemented
197C23C2×3−1+2⋊C411yescharacteristic
198C23#611⋊C311yescharacteristic
199D4—41no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—361no—
204C2×C4#611⋊C311yescharacteristic
205C8—361no—
206C2×C3—21no—
207C2×C3—31no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—61no—
211C2×C3—121no—
212C2×C3—121no—
213C2×C3C3×#83411yescharacteristic
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22#3676⋊C311yescharacteristic
219C4—21no—
220C4—41no—
221C4—41no—
222C4—721no—
223C3—31no—
224C3C3×#453911yescharacteristic
225C2—21no—
226C2—41no—
227C2—41no—
228C2#63393⋊C311yescharacteristic
subgroupG/H
centerC2#63393⋊C3
commutatorC23×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#123C2
socleC2×C3C3×#834
prime radicalC2×C32×D4C2×D3
cosocleC2×D4×C9C22×C3
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#352
LCSG ▷ C23×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
5. #10588216C3semidirect
7. C22×C4×3−1+2C4semidirect
10. C22⋊C4×3−1+2C4semidirect
41. C22×C4×C9C3×C4semidirect
47. C9×C22⋊C4C3×C4semidirect
138. 3−1+2#352semidirect
170. C22×C43−1+2⋊C4semidirect
175. C22⋊C43−1+2⋊C4semidirect
196. C9C3×#352semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1232
#63393⋊C32
#63391⋊C32
C3×#45393
#105882163