Finite Group Databasebeta

← #10588152⋊C3 #13135354

Subgroups — #10588152⋊C3 #13135354

Order 1728.

227 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 50 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#63497⋊C3C211yesmaximal, characteristic
23−1+2×#100C211yesmaximal, characteristic
3#63393⋊C3C211yesmaximal, characteristic
4C3×#4476—31nomaximal
5#10588152C311yesmaximal, characteristic, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C22×3−1+2⋊C4C411yescharacteristic, complemented
8M16×3−1+2—21no—
9#3676⋊C3—21no—
10#3676⋊C3—21no—
11#3676⋊C3C411yescharacteristic, complemented
12(3−1+2⋊C4)⋊C4—21no—
13C3×#935—31no—
14C32×#100D311yescharacteristic
15C3×#834—31no—
16#63497C2×C311yescharacteristic
17C9×#100—21no—
18C9×#100C2×C311yescharacteristic
19#63393C2×C311yescharacteristic
20C23×3−1+2C2×C411yescharacteristic
21C23×3−1+2D411yescharacteristic
22D4×3−1+2—41no—
23C2×C4×3−1+2D411yescharacteristic
24C2×3−1+2⋊C4—21no—
25C2×3−1+2⋊C4—21no—
26C2×3−1+2⋊C4—41no—
27C2×3−1+2⋊C4—41no—
28C8×3−1+2—41no—
29C3×#352—91no—
30#4476—31no—
31C2×C32×D4C2×D311yescharacteristic
32C22×C3×Dic3—31no—
33C32×M16—21no—
34C3×#217—31no—
35C3×#217—61no—
36C3×#217—61no—
37C3×Dic3⋊C4—61no—
38C2×D4×C9—21no—
39C2×D4×C9C22×C311yescharacteristic
40C22×Dic9C3×C411yescharacteristic, complemented
41C9×M16—22no—
42C9×M16—21no—
43#3676—21no—
44#3676—21no—
45#3676C3×C411yescharacteristic, complemented
46Dic9⋊C4—21no—
47C22×3−1+2—21no—
48C22×3−1+2—21no—
49C22×3−1+2—41no—
50C22×3−1+2—41no—
51C22×3−1+2C22⋊C411yescharacteristic
52C4×3−1+2—21no—
533−1+2⋊C4—41no—
543−1+2⋊C4—41no—
553−1+2⋊C4—81no—
56C3×#123—91no—
57#935—31no—
58C3×#100—31no—
59C3×#100C3×D311yescharacteristic
60C3×#99—91no—
61#834—31no—
62C23×C32C4×D311yescharacteristic
63C23×C32D1211yescharacteristic
64C32×D4—41no—
65C2×C32×C4#6711yescharacteristic
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—121no—
69C2×C3×Dic3—121no—
70C23×C9—21no—
71C23×C9—21no—
72C23×C9C2×C3×C411yescharacteristic
73C23×C9C3×D411yescharacteristic
74C32×C8—41no—
75D4×C9—41no—
76D4×C9—42no—
77C2×C4×C9—21no—
78C2×C4×C9C3×D411yescharacteristic
79C2×Dic9—21no—
80C2×Dic9—21no—
81C2×Dic9—41no—
82C2×Dic9—41no—
83C8×C9—42no—
84C8×C9—41no—
85#352—91noSylow
86C2×3−1+2—21no—
87C2×3−1+2—41no—
88C2×3−1+2—41no—
89C2×3−1+2#9911yescharacteristic
90C2×C3×D4—31no—
91C2×C3×D4C2×C3×D311yescharacteristic
92C22×C3×C4—91no—
93C22×Dic3—31no—
94C3×M16—21no—
95C3×M16—61no—
96C3×C4⋊C4—181no—
97C3×C22⋊C4—91no—
98C3×C22⋊C4—181no—
99C3×C22⋊C4—181no—
100#217—31no—
101#217—61no—
102#217—61no—
103Dic3⋊C4—61no—
104C22×C32—21no—
105C22×C32—21no—
106C22×C32—41no—
107C22×C32—41no—
108C22×C32(C2×D3)⋊C411yescharacteristic
109C32×C4—21no—
110C3×Dic3—121no—
111C3×Dic3—121no—
112C3×Dic3—241no—
113C22×C9—21no—
114C22×C9—21no—
115C22×C9—21no—
116C22×C9—41no—
117C22×C9—41no—
118C22×C9—41no—
119C22×C9—41no—
120C22×C9—81no—
121C22×C9—81no—
122C22×C9C3×C22⋊C411yescharacteristic
123C4×C9—22no—
124C4×C9—21no—
125Dic9—41no—
126Dic9—41no—
127Dic9—81no—
128#123—91no—
129#100AGL(1,ℤ/9)11yescharacteristic
130#99—91no—
1313−1+2#35211yescharacteristic, complemented, Sylow
132C23×C3—31no—
133C23×C3—31no—
134C23×C3C3×C4×D311yescharacteristic
135C23×C3C3×D1211yescharacteristic
136C3×D4—41no—
137C3×D4—121no—
138C2×C3×C4—31no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4—361no—
142C2×C3×C4—361no—
143C2×C3×C4C3×#6711yescharacteristic
144C2×Dic3—61no—
145C2×Dic3—61no—
146C2×Dic3—121no—
147C2×Dic3—121no—
148C3×C8—41no—
149C3×C8—121no—
150C2×C32—21no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32#80611yescharacteristic
154C2×C9—21no—
155C2×C9—21no—
156C2×C9—41no—
157C2×C9—41no—
158C2×C9—41no—
159C2×C9—81no—
160C2×C9—81no—
161C2×C9C3×#9911yescharacteristic
162C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
163C22×C4—91no—
164M16—21no—
165C4⋊C4—181no—
166C22⋊C4—91no—
167C22⋊C4—181no—
168C22⋊C4—181no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—31no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—121no—
177C22×C3—121no—
178C22×C3C3×(C2×D3)⋊C411yescharacteristic
179C3×C4—21no—
180C3×C4—61no—
181C3×C4—361no—
182C3×C4—361no—
183C3×C4—721no—
184Dic3—121no—
185Dic3—121no—
186Dic3—241no—
187C32#447611yescharacteristic
188C9—21no—
189C9C3×#35211yescharacteristic, complemented
190C23C4×AGL(1,ℤ/9)11yescharacteristic
191C23C4⋊AGL(1,ℤ/9)11yescharacteristic
192D4—41no—
193C2×C4—181no—
194C2×C4—181no—
195C2×C4—361no—
196C2×C4—361no—
197C2×C4#611⋊C311yescharacteristic
198C8—41no—
199C2×C3—21no—
200C2×C3—31no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—61no—
204C2×C3—121no—
205C2×C3—121no—
206C2×C3C3×#80611yescharacteristic
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22(C2×AGL(1,ℤ/9))⋊C411yescharacteristic
212C4—21no—
213C4—361no—
214C4—361no—
215C4—721no—
216C3—31no—
217C3C3×#447611yescharacteristic
218C2—21no—
219C2—41no—
220C2—41no—
221C2#63365⋊C311yescharacteristic
subgroupG/H
centerC2#63365⋊C3
commutatorC23×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#100C2
socleC2×C3C3×#806
prime radicalC2×C32×D4C2×D3
cosocleC2×D4×C9C22×C3
hypercenter#100AGL(1,ℤ/9)
hypocenterC9C3×#352
LCSG ▷ C23×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
5. #10588152C3semidirect
7. C22×3−1+2⋊C4C4semidirect
11. #3676⋊C3C4semidirect
40. C22×Dic9C3×C4semidirect
45. #3676C3×C4semidirect
131. 3−1+2#352semidirect
189. C9C3×#352semidirect

Maximal subgroups

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

subgroupindex
#63497⋊C32
3−1+2×#1002
#63393⋊C32
C3×#44763
#105881523