Finite Group Databasebeta

← #10588665⋊C3 #13142655

Subgroups — #10588665⋊C3 #13142655

Order 1728.

448 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
33−1+2⋊#95C211yesmaximal, characteristic
43−1+2⋊#95C212yesmaximal
53−1+2⋊#95C212yesmaximal
6C3×#4985—31nomaximal
7#10588665C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2212yes—
9C22×C4×3−1+2C2211yescharacteristic
10C22×3−1+2⋊C4C2212yes—
11C22×3−1+2⋊C4C2211yescharacteristic
12C22×3−1+2⋊C4C2211yescharacteristic
13C4⋊C4×3−1+2—22no—
14(3−1+2⋊C4)⋊C4—22no—
15C2×C32×C4⋊C4D311yescharacteristic
16C2×C3×Dic3⋊C4—31no—
17C3×C3⋊#95—32no—
18C3×C3⋊#95—32no—
19C3×C3⋊#95—31no—
20C2×C9×C4⋊C4—21no—
21C2×C9×C4⋊C4C2×C311yescharacteristic
22C2×Dic9⋊C4C2×C311yescharacteristic
23C9⋊#95C2×C311yescharacteristic
24C9⋊#95C2×C312yes—
25C9⋊#95C2×C312yes—
26C23×3−1+2C2311yescharacteristic
27C2×C4×3−1+2—22no—
28C2×C4×3−1+2—22no—
29C2×C4×3−1+2—24no—
30C2×C4×3−1+2D412yes—
31C2×3−1+2⋊C4—21no—
32C2×3−1+2⋊C4—21no—
33C2×3−1+2⋊C4—21no—
34C2×3−1+2⋊C4—22no—
35C2×3−1+2⋊C4—22no—
36C2×3−1+2⋊C4—22no—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4Q812yes—
39C3×#400—91no—
40#4985—31no—
41C22×C32×C4C2×D312yes—
42C22×C32×C4C2×D311yescharacteristic
43C22×C3×Dic3—31no—
44C22×C3×Dic3—31no—
45C22×C3×Dic3—32no—
46C32×C4⋊C4—22no—
47C3×Dic3⋊C4—62no—
48C22×C4×C9—21no—
49C22×C4×C9—22no—
50C22×C4×C9C22×C312yes—
51C22×C4×C9C22×C311yescharacteristic
52C22×Dic9C22×C312yes—
53C22×Dic9C22×C311yescharacteristic
54C22×Dic9C22×C311yescharacteristic
55C9×C4⋊C4—24no—
56C9×C4⋊C4—22no—
57Dic9⋊C4—22no—
58C22×3−1+2C2×D411yescharacteristic
59C22×3−1+2C2×Q811yescharacteristic
60C22×3−1+2#4111yescharacteristic
61C22×3−1+2#4112yes—
62C22×3−1+2#4112yes—
63C4×3−1+2—22no—
64C4×3−1+2—42no—
653−1+2⋊C4—22no—
663−1+2⋊C4—41no—
673−1+2⋊C4—42no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C4⋊C4—91no—
70C2×C3×C4⋊C4C3×D311yescharacteristic
71C2×Dic3⋊C4—31no—
72C3×#95—91no—
73C3×#95—92no—
74C3×#95—92no—
75C3⋊#95—32no—
76C3⋊#95—31no—
77C3⋊#95—32no—
78C23×C32C22×D311yescharacteristic
79C2×C32×C4—22no—
80C2×C32×C4—24no—
81C2×C32×C4—22no—
82C2×C32×C4#6712yes—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—61no—
91C23×C9—21no—
92C23×C9C23×C311yescharacteristic
93C2×C4×C9—24no—
94C2×C4×C9—22no—
95C2×C4×C9—24no—
96C2×C4×C9—22no—
97C2×C4×C9—24no—
98C2×C4×C9—22no—
99C2×C4×C9—44no—
100C2×C4×C9C3×D412yes—
101C2×Dic9—21no—
102C2×Dic9—21no—
103C2×Dic9—21no—
104C2×Dic9—22no—
105C2×Dic9—22no—
106C2×Dic9—22no—
107C2×Dic9—22no—
108C2×Dic9C3×Q812yes—
109#400—91noSylow
110C2×3−1+2C22⋊Q811yescharacteristic
111C2×3−1+2#12311yescharacteristic
112C2×3−1+2#12411yescharacteristic
113C2×3−1+2#12512yes—
114C2×3−1+2#12612yes—
115C22×C3×C4—31no—
116C22×C3×C4—32no—
117C22×C3×C4—91no—
118C22×C3×C4—92no—
119C22×C3×C4—91no—
120C22×C3×C4C2×C3×D312yes—
121C22×C3×C4C2×C3×D311yescharacteristic
122C22×Dic3—31no—
123C22×Dic3—32no—
124C22×Dic3—31no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—62no—
127C3×C4⋊C4—182no—
128Dic3⋊C4—62no—
129C22×C32#23512yes—
130C22×C32#23712yes—
131C22×C32D3×Q811yescharacteristic
132C22×C32#23911yescharacteristic
133C22×C32C2×#6711yescharacteristic
134C32×C4—22no—
135C32×C4—42no—
136C3×Dic3—62no—
137C3×Dic3—122no—
138C3×Dic3—121no—
139C22×C9—22no—
140C22×C9—22no—
141C22×C9—21no—
142C22×C9—21no—
143C22×C9—21no—
144C22×C9C2×C3×D411yescharacteristic
145C22×C9C2×C3×Q811yescharacteristic
146C22×C9C3×#4112yes—
147C22×C9C3×#4112yes—
148C22×C9C3×#4111yescharacteristic
149C4×C9—24no—
150C4×C9—22no—
151C4×C9—44no—
152C4×C9—42no—
153Dic9—22no—
154Dic9—41no—
155Dic9—42no—
156C2×C4⋊C4—91no—
157C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
158#95—92no—
159#95—91no—
160#95—92no—
1613−1+2#40011yescharacteristic, complemented, Sylow
162C23×C3—31no—
163C23×C3C22×C3×D311yescharacteristic
164C2×C3×C4—22no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—32no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—64no—
171C2×C3×C4—92no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4—182no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×C3×C4C3×#6712yes—
180C2×Dic3—32no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—62no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—62no—
187C2×Dic3—61no—
188C2×C32#88912yes—
189C2×C32#89812yes—
190C2×C32#92911yescharacteristic
191C2×C32#93611yescharacteristic
192C2×C32#94611yescharacteristic
193C2×C9—22no—
194C2×C9—22no—
195C2×C9—21no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9C3×C22⋊Q811yescharacteristic
199C2×C9C3×#12311yescharacteristic
200C2×C9C3×#12411yescharacteristic
201C2×C9C3×#12512yes—
202C2×C9C3×#12612yes—
203C22×C4—91no—
204C22×C4—92no—
205C22×C4—91no—
206C22×C4C2×AGL(1,ℤ/9)12yes—
207C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—182no—
210C22×C3—32no—
211C22×C3—32no—
212C22×C3—31no—
213C22×C3—31no—
214C22×C3—31no—
215C22×C3C3×#23512yes—
216C22×C3C3×#23712yes—
217C22×C3C3×D3×Q811yescharacteristic
218C22×C3C3×#23911yescharacteristic
219C22×C3C2×C3×#6711yescharacteristic
220C3×C4—22no—
221C3×C4—42no—
222C3×C4—62no—
223C3×C4—122no—
224C3×C4—182no—
225C3×C4—362no—
226C3×C4—361no—
227Dic3—62no—
228Dic3—122no—
229Dic3—121no—
230C32#498511yescharacteristic
231C9—21no—
232C9C3×#40011yescharacteristic, complemented
233C23C22×AGL(1,ℤ/9)11yescharacteristic
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—92no—
238C2×C4—181no—
239C2×C4—181no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4—182no—
243C2×C4—182no—
244C2×C4—181no—
245C2×C4#611⋊C312yes—
246C2×C3—32no—
247C2×C3—32no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3C3×#88912yes—
252C2×C3C3×#89812yes—
253C2×C3C3×#92911yescharacteristic
254C2×C3C3×#93611yescharacteristic
255C2×C3C3×#94611yescharacteristic
256C22#3697⋊C312yes—
257C22#3699⋊C312yes—
258C22Q8×AGL(1,ℤ/9)11yescharacteristic
259C22#8872411yescharacteristic
260C22C2×#611⋊C311yescharacteristic
261C4—22no—
262C4—42no—
263C4—182no—
264C4—362no—
265C4—361no—
266C3—31no—
267C3C3×#498511yescharacteristic
268C2#63451⋊C312yes—
269C2#63460⋊C312yes—
270C2#63491⋊C311yescharacteristic
271C2#63498⋊C311yescharacteristic
272C2#63508⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#400
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588665C3semidirect
161. 3−1+2#400semidirect
232. C9C3×#400semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49853
#105886653