Finite Group Databasebeta

← #10588764⋊C3 #13143056

Subgroups — #10588764⋊C3 #13143056

Order 1728.

322 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 106 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×C4⋊Q8C211yesmaximal, characteristic
2C4×3−1+2⋊Q8C211yesmaximal, characteristic
3#63368⋊C3C212yesmaximal
4#63366⋊C3C212yesmaximal
5C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic
6C3×#5091—31nomaximal
7#10588764C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—21no—
9C2×3−1+2⋊Q8C2211yescharacteristic
10C4⋊C4×3−1+2—21no—
11C4⋊C4×3−1+2C2212yes—
12C42×3−1+2C2211yescharacteristic
13C4⋊(3−1+2⋊C4)C2211yescharacteristic
14(3−1+2⋊C4)⋊C4—21no—
15C4×3−1+2⋊C4—21no—
16C2×3−1+2⋊C8C2212yes—
17C32×C4⋊Q8D311yescharacteristic
18C3×C4×Dic6—31no—
19C3×#810—32no—
20C3×#807—32no—
21C3×C4⋊(C3⋊C8)—31no—
22C9×C4⋊Q8—21no—
23C9×C4⋊Q8C2×C311yescharacteristic
24C4×Dic18C2×C311yescharacteristic
25#63368C2×C312yes—
26#63366C2×C312yes—
27C4⋊(C9⋊C8)C2×C311yescharacteristic
28Q8×3−1+2—42no—
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2D411yescharacteristic
31C2×C4×3−1+2D411yescharacteristic
32C2×C4×3−1+2C2311yescharacteristic
33C2×3−1+2⋊C4—21no—
34C2×3−1+2⋊C4—21no—
353−1+2⋊Q8—21no—
363−1+2⋊Q8Q812yescomplemented
373−1+2⋊C8—22no—
38C3×#477—91no—
39#5091—31no—
40C2×C32×Q8—21no—
41C2×C3×Dic6—31no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4C2×D312yes—
44C32×C42C2×D311yescharacteristic
45C3×C4⋊Dic3—31no—
46C3×Dic3⋊C4—61no—
47C3×C4×Dic3—61no—
48C2×C3×C3⋊C8—32no—
49C2×Q8×C9—22no—
50C2×Q8×C9—21no—
51C2×Dic18C22×C311yescharacteristic
52C9×C4⋊C4—22no—
53C9×C4⋊C4—22no—
54C9×C4⋊C4—21no—
55C9×C4⋊C4C22×C312yes—
56C42×C9—21no—
57C42×C9C22×C311yescharacteristic
58C4⋊Dic9C22×C311yescharacteristic
59Dic9⋊C4—21no—
60C4×Dic9—21no—
61C2×C9⋊C8C22×C312yes—
62C22×3−1+2C2×D411yescharacteristic
63C4×3−1+2—21no—
64C4×3−1+2—42no—
65C4×3−1+2Q1612yes—
66C4×3−1+2C2×Q811yescharacteristic
67C4×3−1+2#4111yescharacteristic
683−1+2⋊C4—22no—
693−1+2⋊C4—41no—
70C3×C4⋊Q8—31no—
71C3×C4⋊Q8C3×D311yescharacteristic
72C3×C4×Q8—91no—
73C4×Dic6—31no—
74C3×#107—92no—
75C3×C4⋊C8—91no—
76C3×#103—92no—
77#810—32no—
78#807—32no—
79C4⋊(C3⋊C8)—31no—
80C32×Q8—42no—
81C2×C32×C4—22no—
82C2×C32×C4#6711yescharacteristic
83C2×C32×C4#6711yescharacteristic
84C2×C32×C4C22×D311yescharacteristic
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C3×Dic6—32no—
88C3×Dic6—61no—
89C3×C3⋊C8—62no—
90Q8×C9—44no—
91Q8×C9—42no—
92C2×C4×C9—24no—
93C2×C4×C9—22no—
94C2×C4×C9—21no—
95C2×C4×C9—21no—
96C2×C4×C9—21no—
97C2×C4×C9C3×D411yescharacteristic
98C2×C4×C9C3×D411yescharacteristic
99C2×C4×C9C23×C311yescharacteristic
100C2×Dic9—21no—
101C2×Dic9—21no—
102Dic18—21no—
103Dic18C3×Q812yescomplemented
104C9⋊C8—22no—
105#477—91noSylow
106C2×3−1+2C22⋊Q811yescharacteristic
107C2×3−1+2C2×Q1611yescharacteristic
108C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—61no—
111C2×C3×Q8—91no—
112C2×Dic6—31no—
113C2×C3×C8—92no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—61no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4—181no—
119C3×C4⋊C4C2×C3×D312yes—
120C3×C42—31no—
121C3×C42—181no—
122C3×C42C2×C3×D311yescharacteristic
123C4⋊Dic3—31no—
124Dic3⋊C4—61no—
125C4×Dic3—61no—
126C2×C3⋊C8—32no—
127C22×C32C2×#6711yescharacteristic
128C32×C4—21no—
129C32×C4—42no—
130C32×C4#21612yes—
131C32×C4D3×Q811yescharacteristic
132C32×C4#23911yescharacteristic
133C3×Dic3—62no—
134C3×Dic3—121no—
135C22×C9—21no—
136C22×C9C2×C3×D411yescharacteristic
137C4×C9—22no—
138C4×C9—21no—
139C4×C9—22no—
140C4×C9—21no—
141C4×C9—21no—
142C4×C9—44no—
143C4×C9—42no—
144C4×C9C3×Q1612yes—
145C4×C9C2×C3×Q811yescharacteristic
146C4×C9C3×#4111yescharacteristic
147Dic9—22no—
148Dic9—41no—
149C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
150C4×Q8—91no—
151#107—92no—
152C4⋊C8—91no—
153#103—92no—
1543−1+2#47711yescharacteristic, complemented, Sylow
155C3×Q8—42no—
156C3×Q8—92no—
157C3×Q8—122no—
158C3×Q8—181no—
159C2×C3×C4—22no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—62no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4C3×#6711yescharacteristic
167C2×C3×C4C3×#6711yescharacteristic
168C2×C3×C4C22×C3×D311yescharacteristic
169C2×Dic3—61no—
170C2×Dic3—61no—
171Dic6—32no—
172Dic6—61no—
173C3×C8—182no—
174C3⋊C8—62no—
175C2×C32#93211yescharacteristic
176C2×C32C2×#21611yescharacteristic
177C2×C32#94611yescharacteristic
178C2×C9—21no—
179C2×C9—21no—
180C2×C9—21no—
181C2×C9C3×C22⋊Q811yescharacteristic
182C2×C9C2×C3×Q1611yescharacteristic
183C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
184C2×Q8—21no—
185C2×Q8—91no—
186C2×C8—92no—
187C4⋊C4—21no—
188C4⋊C4—91no—
189C4⋊C4—181no—
190C4⋊C4C2×AGL(1,ℤ/9)12yes—
191C42—181no—
192C42C2×AGL(1,ℤ/9)11yescharacteristic
193C22×C3—31no—
194C22×C3C2×C3×#6711yescharacteristic
195C3×C4—21no—
196C3×C4—31no—
197C3×C4—31no—
198C3×C4—32no—
199C3×C4—42no—
200C3×C4—61no—
201C3×C4—122no—
202C3×C4—182no—
203C3×C4—361no—
204C3×C4C3×#21612yes—
205C3×C4C3×D3×Q811yescharacteristic
206C3×C4C3×#23911yescharacteristic
207Dic3—62no—
208Dic3—121no—
209C32#509111yescharacteristic
210C9—21no—
211C9C3×#47711yescharacteristic, complemented
212Q8—42no—
213Q8—92no—
214Q8—181no—
215C2×C4—22no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C4#611⋊C311yescharacteristic
219C2×C4#611⋊C311yescharacteristic
220C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
221C8—182no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3C3×#93211yescharacteristic
226C2×C3C2×C3×#21611yescharacteristic
227C2×C3C3×#94611yescharacteristic
228C22C2×#611⋊C311yescharacteristic
229C4—21no—
230C4—42no—
231C4—182no—
232C4—361no—
233C4#3674⋊C312yes—
234C4Q8×AGL(1,ℤ/9)11yescharacteristic
235C4#8872411yescharacteristic
236C3—31no—
237C3C3×#509111yescharacteristic
238C2#63495⋊C311yescharacteristic
239C2C2×#3674⋊C311yescharacteristic
240C2#63508⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C4⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenterC4⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#477
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
7. #10588764C3semidirect
36. 3−1+2⋊Q8Q8semidirect
103. Dic18C3×Q8semidirect
154. 3−1+2#477semidirect
211. C9C3×#477semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C4⋊Q82
C4×3−1+2⋊Q82
#63368⋊C32
#63366⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50913
#105887643