Finite Group Databasebeta

← #10588401⋊C3 #13141626

Subgroups — #10588401⋊C3 #13141626

Order 1728.

337 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 102 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⋊(3−1+2⋊C4)C211yesmaximal, characteristic
2#63486⋊C3C211yesmaximal, characteristic
33−1+2×C22⋊C8C211yesmaximal, characteristic
4#63377⋊C3C212yesmaximal, complemented
5#63376⋊C3C212yesmaximal, complemented
6C3×#4721—31nomaximal
7#10588401C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C22×3−1+2⋊C4—21no—
10C2×3−1+2⋊Q8C2211yescharacteristic
11C2×C8×3−1+2C2212yes—
12#3676⋊C3—21no—
13C4⋊(3−1+2⋊C4)—21no—
14C4⋊(3−1+2⋊C4)C2211yescharacteristic
15C4⋊(3−1+2⋊C4)C2212yes—
16(3−1+2⋊C4)⋊C4—21no—
17C2×C3×C4⋊Dic3—31no—
18C3×#924—31no—
19C32×C22⋊C8D311yescharacteristic
20C3×#817—32no—
21C3×#816—32no—
22C2×C4⋊Dic9C2×C311yescharacteristic
23#63486C2×C311yescharacteristic
24C9×C22⋊C8—21no—
25C9×C22⋊C8C2×C311yescharacteristic
26#63377C2×C312yescomplemented
27#63376C2×C312yescomplemented
28C23×3−1+2D411yescharacteristic
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2D411yescharacteristic
31C2×C4×3−1+2C2311yescharacteristic
32C2×3−1+2⋊C4—21no—
33C2×3−1+2⋊C4—22no—
34C2×3−1+2⋊C4—21no—
35C2×3−1+2⋊C4—42no—
363−1+2⋊Q8—22no—
37C8×3−1+2—22no—
38C3×#483—91no—
39#4721—31no—
40C22×C32×C4C2×D311yescharacteristic
41C22×C3×Dic3—61no—
42C2×C3×Dic6—31no—
43C2×C32×C8C2×D312yes—
44C3×#217—61no—
45C3×C4⋊Dic3—31no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—61no—
48C3×Dic3⋊C4—61no—
49C22×C4×C9—21no—
50C22×C4×C9C22×C311yescharacteristic
51C22×Dic9—21no—
52C2×Dic18C22×C311yescharacteristic
53C2×C8×C9—22no—
54C2×C8×C9C22×C312yes—
55#3676—21no—
56C4⋊Dic9—21no—
57C4⋊Dic9C22×C312yes—
58C4⋊Dic9C22×C311yescharacteristic
59Dic9⋊C4—21no—
60C22×3−1+2—21no—
61C22×3−1+2—21no—
62C22×3−1+2SD1612yes—
63C22×3−1+2C2×D411yescharacteristic
64C4×3−1+2—21no—
65C4×3−1+2#4112yes—
663−1+2⋊C4—41no—
673−1+2⋊C4—42no—
683−1+2⋊C4—41no—
69C3×C22⋊Q8—91no—
70C2×C3×C4⋊C4—91no—
71C2×C4⋊Dic3—31no—
72#924—31no—
73C3×#106—92no—
74C3×#103—92no—
75C3×C22⋊C8—31no—
76C3×C22⋊C8C3×D311yescharacteristic
77#817—32no—
78#816—32no—
79C23×C32D1211yescharacteristic
80C2×C32×C4—22no—
81C2×C32×C4D1211yescharacteristic
82C2×C32×C4C22×D311yescharacteristic
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—122no—
87C3×Dic6—62no—
88C23×C9—21no—
89C23×C9C3×D411yescharacteristic
90C32×C8—22no—
91C2×C4×C9—22no—
92C2×C4×C9—21no—
93C2×C4×C9—21no—
94C2×C4×C9—42no—
95C2×C4×C9C3×D411yescharacteristic
96C2×C4×C9C23×C311yescharacteristic
97C2×Dic9—21no—
98C2×Dic9—21no—
99C2×Dic9—22no—
100C2×Dic9—42no—
101Dic18—22no—
102C8×C9—22no—
103C8×C9—42no—
104#483—91noSylow
105C2×3−1+2—22no—
106C2×3−1+2#12311yescharacteristic
107C2×3−1+2C2×SD1611yescharacteristic
108C2×3−1+2#13711yescharacteristic
109C2×C3×Q8—91no—
110C22×C3×C4—31no—
111C22×C3×C4—181no—
112C22×C3×C4C2×C3×D311yescharacteristic
113C22×Dic3—61no—
114C2×Dic6—31no—
115C2×C3×C8—32no—
116C2×C3×C8C2×C3×D312yes—
117C3×C4⋊C4—92no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4—181no—
120C3×C4⋊C4—181no—
121C3×C22⋊C4—181no—
122#217—61no—
123C4⋊Dic3—32no—
124C4⋊Dic3—31no—
125C4⋊Dic3—61no—
126Dic3⋊C4—61no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32#20412yes—
130C22×C32C2×D1211yescharacteristic
131C32×C4—21no—
132C32×C4#23712yes—
133C3×Dic3—121no—
134C3×Dic3—122no—
135C3×Dic3—121no—
136C22×C9—21no—
137C22×C9—21no—
138C22×C9—22no—
139C22×C9—21no—
140C22×C9—41no—
141C22×C9—41no—
142C22×C9C3×SD1612yes—
143C22×C9C2×C3×D411yescharacteristic
144C4×C9—21no—
145C4×C9—22no—
146C4×C9—41no—
147C4×C9C3×#4112yes—
148Dic9—41no—
149Dic9—42no—
150Dic9—41no—
151C22⋊Q8—91no—
152C2×C4⋊C4—91no—
153#106—92no—
154#103—92no—
155C22⋊C8AGL(1,ℤ/9)11yescharacteristic
1563−1+2#48311yescharacteristic, complemented, Sylow
157C23×C3—31no—
158C23×C3C3×D1211yescharacteristic
159C3×Q8—182no—
160C2×C3×C4—22no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—62no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4—182no—
167C2×C3×C4—362no—
168C2×C3×C4C3×D1211yescharacteristic
169C2×C3×C4C22×C3×D311yescharacteristic
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—62no—
173C2×Dic3—122no—
174Dic6—62no—
175C3×C8—22no—
176C3×C8—62no—
177C2×C32—22no—
178C2×C32#88611yescharacteristic
179C2×C32C2×#20411yescharacteristic
180C2×C32#90911yescharacteristic
181C2×C9—22no—
182C2×C9—21no—
183C2×C9—21no—
184C2×C9—21no—
185C2×C9—42no—
186C2×C9C3×#12311yescharacteristic
187C2×C9C2×C3×SD1611yescharacteristic
188C2×C9C3×#13711yescharacteristic
189C2×Q8—91no—
190C22×C4—181no—
191C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
192C2×C8C2×AGL(1,ℤ/9)12yes—
193C4⋊C4—92no—
194C4⋊C4—91no—
195C4⋊C4—181no—
196C4⋊C4—181no—
197C22⋊C4—181no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—31no—
201C22×C3—32no—
202C22×C3—61no—
203C22×C3—61no—
204C22×C3C3×#20412yes—
205C22×C3C2×C3×D1211yescharacteristic
206C3×C4—21no—
207C3×C4—32no—
208C3×C4—61no—
209C3×C4—362no—
210C3×C4—361no—
211C3×C4—361no—
212C3×C4C3×#23712yes—
213Dic3—121no—
214Dic3—122no—
215Dic3—121no—
216C32#472111yescharacteristic
217C9—21no—
218C9C3×#48311yescharacteristic, complemented
219C23C4⋊AGL(1,ℤ/9)11yescharacteristic
220Q8—182no—
221C2×C4—22no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—182no—
225C2×C4—362no—
226C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
227C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
228C8—22no—
229C2×C3—22no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—62no—
234C2×C3C3×#88611yescharacteristic
235C2×C3C2×C3×#20411yescharacteristic
236C2×C3C3×#90911yescharacteristic
237C22—21no—
238C22—21no—
239C22#3664⋊C312yes—
240C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
241C4—21no—
242C4—362no—
243C4—361no—
244C4—361no—
245C4#3699⋊C312yes—
246C3—31no—
247C3C3×#472111yescharacteristic
248C2—22no—
249C2#63448⋊C311yescharacteristic
250C2C2×#3664⋊C311yescharacteristic
251C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC23×C32D12
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#483
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63377⋊C3C2semidirect
5. #63376⋊C3C2semidirect
7. #10588401C3semidirect
26. #63377C2×C3semidirect
27. #63376C2×C3semidirect
156. 3−1+2#483semidirect
218. C9C3×#483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(3−1+2⋊C4)2
#63486⋊C32
3−1+2×C22⋊C82
#63377⋊C32
#63376⋊C32
C3×#47213
#105884013