Finite Group Databasebeta

← #10588567⋊C3 #13142248

Subgroups — #10588567⋊C3 #13142248

Order 1728.

360 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 100 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×C8×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×Dic36⋊C3C211yesmaximal, characteristic, complemented
3#63459⋊C3C212yesmaximal
43−1+2×#107C211yesmaximal, characteristic, complemented
5#63368⋊C3C212yesmaximal, complemented
6C3×#4887—31nomaximal
7#10588567C311yesmaximal, characteristic, complemented
8C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
9C2×3−1+2⋊Q8C2212yes—
10C2×C8×3−1+2C2211yescharacteristic
11C4⋊C4×3−1+2C2212yes—
12(C2×AGL(1,ℤ/9))⋊C4—22no—
13C4⋊(3−1+2⋊C4)—22no—
14C2×3−1+2⋊C8C2211yescharacteristic
15C8×AGL(1,ℤ/9)—22no—
16Dic36⋊C3—22no—
17C2×C3×Dic12—31no—
18C2×C3×D3×C8—31no—
19C3×#897—32no—
20C32×#107D311yescharacteristic
21C3×#810—32no—
22C2×C8×D9C2×C311yescharacteristic
23C2×Dic36C2×C311yescharacteristic, complemented
24#63459C2×C312yes—
25C9×#107—21no—
26C9×#107C2×C311yescharacteristic, complemented
27#63368C2×C312yescomplemented
28C22×AGL(1,ℤ/9)D411yescharacteristic
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2C2311yescharacteristic
31C2×3−1+2⋊C4—22no—
32C2×3−1+2⋊C4D411yescharacteristic
33C4×AGL(1,ℤ/9)—21no—
34C4×AGL(1,ℤ/9)—21no—
353−1+2⋊Q8—22no—
363−1+2⋊Q8—22no—
37C8×3−1+2D412yes—
383−1+2⋊C8—21no—
39C3×#467—91no—
40#4887—31no—
41C2×C3×C4×D3—31no—
42C2×C3×Dic6—32no—
43C2×C32×C8C2×D311yescharacteristic
44C32×C4⋊C4C2×D312yes—
45C3×(C2×D3)⋊C4—62no—
46C3×C4⋊Dic3—62no—
47C2×C3×C3⋊C8—31no—
48C3×Dic12—62no—
49C3×D3×C8—62no—
50C2×C4×D9C22×C311yescharacteristic
51C2×Dic18C22×C312yes—
52C2×C8×C9—21no—
53C2×C8×C9C22×C311yescharacteristic
54C9×C4⋊C4—22no—
55C9×C4⋊C4C22×C312yes—
56(C2×D9)⋊C4—22no—
57C4⋊Dic9—22no—
58C2×C9⋊C8C22×C311yescharacteristic
59C8×D9—22no—
60Dic36—22no—
61C22×3−1+2C2×D411yescharacteristic
62C2×AGL(1,ℤ/9)—21no—
63C2×AGL(1,ℤ/9)—21no—
64C2×AGL(1,ℤ/9)Q1612yescomplemented
65C4×3−1+2—42no—
66C4×3−1+2C2×D411yescharacteristic
67C4×3−1+2#4111yescharacteristic
683−1+2⋊C4—21no—
693−1+2⋊C4—42no—
70C2×C3×Q16—91no—
71C22×C3×C8—91no—
72C3×C22⋊Q8—92no—
73C2×Dic12—31no—
74C2×D3×C8—31no—
75#897—32no—
76C3×#107—31no—
77C3×#107C3×D311yescharacteristic
78C3×#103—92no—
79#810—32no—
80C22×C3×D3—31no—
81C2×C32×C4—22no—
82C2×C32×C4C22×D311yescharacteristic
83C2×C3×Dic3—31no—
84C2×C3×Dic3—62no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×Dic6—62no—
88C3×Dic6—62no—
89C22×D9C3×D411yescharacteristic
90C32×C8D1212yes—
91C3×C3⋊C8—61no—
92C2×C4×C9—22no—
93C2×C4×C9—21no—
94C2×C4×C9—42no—
95C2×C4×C9C23×C311yescharacteristic
96C2×Dic9—22no—
97C2×Dic9C3×D411yescharacteristic
98C4×D9—21no—
99C4×D9—21no—
100Dic18—22no—
101Dic18—22no—
102C8×C9—22no—
103C8×C9C3×D412yes—
104C9⋊C8—21no—
105#467—91noSylow
106C2×3−1+2#12111yescharacteristic
107C2×3−1+2C2×Q1611yescharacteristic
108C2×3−1+2#13511yescharacteristic
109AGL(1,ℤ/9)—22no—
110C2×C3×Q8—92no—
111C22×C3×C4—91no—
112C2×C4×D3—31no—
113C2×Dic6—32no—
114C3×Q16—182no—
115C2×C3×C8—31no—
116C2×C3×C8—91no—
117C2×C3×C8—182no—
118C2×C3×C8C2×C3×D311yescharacteristic
119C3×C4⋊C4—32no—
120C3×C4⋊C4—182no—
121C3×C4⋊C4C2×C3×D312yes—
122C3×C22⋊C4—182no—
123(C2×D3)⋊C4—62no—
124C4⋊Dic3—62no—
125C2×C3⋊C8—31no—
126Dic12—62no—
127D3×C8—62no—
128C22×C32D3×D411yescharacteristic
129C2×C3×D3—32no—
130C2×C3×D3—61no—
131C2×C3×D3—61no—
132C32×C4—42no—
133C32×C4C2×D1211yescharacteristic
134C32×C4#23911yescharacteristic
135C3×Dic3—61no—
136C3×Dic3—122no—
137C22×C9—21no—
138C22×C9C2×C3×D411yescharacteristic
139C2×D9—21no—
140C2×D9—21no—
141C2×D9C3×Q1612yescomplemented
142C4×C9—21no—
143C4×C9—21no—
144C4×C9—42no—
145C4×C9—82no—
146C4×C9C2×C3×D411yescharacteristic
147C4×C9C3×#4111yescharacteristic
148Dic9—21no—
149Dic9—42no—
150C2×Q16—91no—
151C22×C8—91no—
152C22⋊Q8—92no—
153#107AGL(1,ℤ/9)11yescharacteristic, complemented
154#103—92no—
1553−1+2#46711yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C22×D3—31no—
158C3×Q8—182no—
159C3×Q8—182no—
160C2×C3×C4—22no—
161C2×C3×C4—31no—
162C2×C3×C4—62no—
163C2×C3×C4—91no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4—182no—
167C2×C3×C4C22×C3×D311yescharacteristic
168C2×Dic3—31no—
169C2×Dic3—62no—
170C4×D3—61no—
171C4×D3—61no—
172Dic6—62no—
173Dic6—62no—
174C3×C8—32no—
175C3×C8—181no—
176C3×C8C3×D1212yes—
177C3⋊C8—61no—
178C2×C32#89511yescharacteristic
179C2×C32#91211yescharacteristic
180C2×C32D3×Q1611yescharacteristic
181C3×D3—62no—
182C2×C9—21no—
183C2×C9—21no—
184C2×C9—21no—
185C2×C9C3×#12111yescharacteristic
186C2×C9C2×C3×Q1611yescharacteristic
187C2×C9C3×#13511yescharacteristic
188D9—22no—
189C2×Q8—92no—
190C22×C4—91no—
191Q16—182no—
192C2×C8—91no—
193C2×C8—182no—
194C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
195C4⋊C4—182no—
196C4⋊C4C2×AGL(1,ℤ/9)12yes—
197C22⋊C4—182no—
198C22×C3—31no—
199C22×C3—92no—
200C22×C3—181no—
201C22×C3—181no—
202C22×C3C3×D3×D411yescharacteristic
203C2×D3—32no—
204C2×D3—61no—
205C2×D3—61no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—42no—
209C3×C4—122no—
210C3×C4—181no—
211C3×C4—362no—
212C3×C4C2×C3×D1211yescharacteristic
213C3×C4C3×#23911yescharacteristic
214Dic3—61no—
215Dic3—122no—
216C32#488711yescharacteristic
217C9—21no—
218C9C3×#46711yescharacteristic, complemented
219C23—91no—
220Q8—182no—
221Q8—182no—
222C2×C4—22no—
223C2×C4—91no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4—182no—
227C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
228C8—181no—
229C8C4⋊AGL(1,ℤ/9)12yes—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—182no—
234C2×C3C3×#89511yescharacteristic
235C2×C3C3×#91211yescharacteristic
236C2×C3C3×D3×Q1611yescharacteristic
237D3—62no—
238C22—92no—
239C22—181no—
240C22—181no—
241C22D4×AGL(1,ℤ/9)11yescharacteristic
242C4—42no—
243C4—181no—
244C4—362no—
245C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
246C4#8872411yescharacteristic
247C3—31no—
248C3C3×#488711yescharacteristic
249C2—182no—
250C2#63457⋊C311yescharacteristic
251C2#63474⋊C311yescharacteristic
252C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×AGL(1,ℤ/9)D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#467
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. C2×Dic36⋊C3C2semidirect
4. 3−1+2×#107C2semidirect
5. #63368⋊C3C2semidirect
7. #10588567C3semidirect
23. C2×Dic36C2×C3semidirect
26. C9×#107C2×C3semidirect
27. #63368C2×C3semidirect
64. C2×AGL(1,ℤ/9)Q16semidirect
141. C2×D9C3×Q16semidirect
153. #107AGL(1,ℤ/9)semidirect
155. 3−1+2#467semidirect
218. C9C3×#467semidirect

Maximal subgroups

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

subgroupindex
C2×C8×AGL(1,ℤ/9)2
C2×Dic36⋊C32
#63459⋊C32
3−1+2×#1072
#63368⋊C32
C3×#48873
#105885673