Finite Group Databasebeta

← #10588763⋊C3 #13143059

Subgroups — #10588763⋊C3 #13143059

Order 1728.

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

normal subgroups only

This group has 358 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
2C2×#3674⋊C3C212yesmaximal
3C4×3−1+2⋊Q8C211yesmaximal, characteristic
4#63395⋊C3C212yesmaximal
5C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic
6C3×#5090—31nomaximal
7#10588763C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2212yes—
9C2×3−1+2⋊Q8C2211yescharacteristic
10C4⋊C4×3−1+2—21no—
11C4⋊C4×3−1+2—21no—
12C42×3−1+2C2211yescharacteristic
13#3674⋊C3—24no—
14C4⋊(3−1+2⋊C4)C2211yescharacteristic
15(3−1+2⋊C4)⋊C4—21no—
16C4×3−1+2⋊C4—21no—
17C2×3−1+2⋊C8C2212yes—
18C32×C4⋊Q8D311yescharacteristic
19C2×C3×#216—32no—
20C3×C4×Dic6—31no—
21C3×#835—32no—
22C3×C4⋊(C3⋊C8)—31no—
23C9×C4⋊Q8—21no—
24C9×C4⋊Q8C2×C311yescharacteristic
25C2×#3674C2×C312yes—
26C4×Dic18C2×C311yescharacteristic
27#63395C2×C312yes—
28C4⋊(C9⋊C8)C2×C311yescharacteristic
29Q8×3−1+2—22no—
30Q8×3−1+2—22no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2D411yescharacteristic
33C2×C4×3−1+2D411yescharacteristic
34C2×C4×3−1+2C2311yescharacteristic
35C2×3−1+2⋊C4—21no—
36C2×3−1+2⋊C4—21no—
373−1+2⋊Q8—21no—
383−1+2⋊Q8D412yes—
393−1+2⋊C8—22no—
40C3×#462—91no—
41#5090—31no—
42C2×C32×Q8C2×D312yes—
43C2×C3×Dic6—31no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2×D311yescharacteristic
47C3×#216—64no—
48C3×C4⋊Dic3—31no—
49C3×Dic3⋊C4—61no—
50C3×C4×Dic3—61no—
51C2×C3×C3⋊C8—32no—
52C2×Q8×C9—22no—
53C2×Q8×C9C22×C312yes—
54C2×Dic18C22×C311yescharacteristic
55C9×C4⋊C4—22no—
56C9×C4⋊C4—22no—
57C9×C4⋊C4—21no—
58C9×C4⋊C4—21no—
59C42×C9—21no—
60C42×C9C22×C311yescharacteristic
61#3674—24no—
62C4⋊Dic9C22×C311yescharacteristic
63Dic9⋊C4—21no—
64C4×Dic9—21no—
65C2×C9⋊C8C22×C312yes—
66C22×3−1+2C2×D411yescharacteristic
67C4×3−1+2—21no—
68C4×3−1+2—42no—
69C4×3−1+2Q1612yescomplemented
70C4×3−1+2C2×D411yescharacteristic
71C4×3−1+2#4111yescharacteristic
723−1+2⋊C4—22no—
733−1+2⋊C4—41no—
74C2×C3×Q16—92no—
75C3×C4⋊Q8—31no—
76C3×C4⋊Q8C3×D311yescharacteristic
77C3×C4×Q8—91no—
78C2×#216—32no—
79C4×Dic6—31no—
80C3×C4⋊C8—91no—
81C3×#103—92no—
82#835—32no—
83C4⋊(C3⋊C8)—31no—
84C32×Q8—22no—
85C32×Q8—22no—
86C2×C32×C4—22no—
87C2×C32×C4#6711yescharacteristic
88C2×C32×C4#6711yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C3×Dic6—32no—
93C3×Dic6—61no—
94C3×C3⋊C8—62no—
95Q8×C9—22no—
96Q8×C9—22no—
97Q8×C9—42no—
98Q8×C9—42no—
99C2×C4×C9—24no—
100C2×C4×C9—22no—
101C2×C4×C9—21no—
102C2×C4×C9—21no—
103C2×C4×C9—21no—
104C2×C4×C9C3×D411yescharacteristic
105C2×C4×C9C3×D411yescharacteristic
106C2×C4×C9C23×C311yescharacteristic
107C2×Dic9—21no—
108C2×Dic9—21no—
109Dic18—21no—
110Dic18C3×D412yes—
111C9⋊C8—22no—
112#462—91noSylow
113C2×3−1+2#12111yescharacteristic
114C2×3−1+2C2×Q1611yescharacteristic
115C2×3−1+2#13711yescharacteristic
116C2×C3×Q8—32no—
117C2×C3×Q8—91no—
118C2×C3×Q8C2×C3×D312yes—
119C2×Dic6—31no—
120C3×Q16—184no—
121C2×C3×C8—92no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—61no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C42—31no—
129C3×C42—181no—
130C3×C42C2×C3×D311yescharacteristic
131#216—64no—
132C4⋊Dic3—31no—
133Dic3⋊C4—61no—
134C4×Dic3—61no—
135C2×C3⋊C8—32no—
136C22×C32C2×#6711yescharacteristic
137C32×C4—21no—
138C32×C4—42no—
139C32×C4#21612yes—
140C32×C4D3×D411yescharacteristic
141C32×C4#23711yescharacteristic
142C3×Dic3—62no—
143C3×Dic3—121no—
144C22×C9—21no—
145C22×C9C2×C3×D411yescharacteristic
146C4×C9—22no—
147C4×C9—21no—
148C4×C9—22no—
149C4×C9—21no—
150C4×C9—21no—
151C4×C9—44no—
152C4×C9—42no—
153C4×C9C3×Q1612yescomplemented
154C4×C9C2×C3×D411yescharacteristic
155C4×C9C3×#4111yescharacteristic
156Dic9—22no—
157Dic9—41no—
158C2×Q16—92no—
159C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
160C4×Q8—91no—
161C4⋊C8—91no—
162#103—92no—
1633−1+2#46211yescharacteristic, complemented, Sylow
164C3×Q8—22no—
165C3×Q8—22no—
166C3×Q8—62no—
167C3×Q8—62no—
168C3×Q8—92no—
169C3×Q8—181no—
170C2×C3×C4—22no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—62no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4C3×#6711yescharacteristic
178C2×C3×C4C3×#6711yescharacteristic
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—61no—
181C2×Dic3—61no—
182Dic6—32no—
183Dic6—61no—
184C3×C8—182no—
185C3⋊C8—62no—
186C2×C32#93811yescharacteristic
187C2×C32#94211yescharacteristic
188C2×C32C2×#21611yescharacteristic
189C2×C9—21no—
190C2×C9—21no—
191C2×C9—21no—
192C2×C9C3×#12111yescharacteristic
193C2×C9C2×C3×Q1611yescharacteristic
194C2×C9C3×#13711yescharacteristic
195C2×Q8—91no—
196C2×Q8C2×AGL(1,ℤ/9)12yes—
197Q16—184no—
198C2×C8—92no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—91no—
202C4⋊C4—181no—
203C42—181no—
204C42C2×AGL(1,ℤ/9)11yescharacteristic
205C22×C3—31no—
206C22×C3C2×C3×#6711yescharacteristic
207C3×C4—21no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—32no—
211C3×C4—42no—
212C3×C4—61no—
213C3×C4—122no—
214C3×C4—182no—
215C3×C4—361no—
216C3×C4C3×#21612yes—
217C3×C4C3×D3×D411yescharacteristic
218C3×C4C3×#23711yescharacteristic
219Dic3—62no—
220Dic3—121no—
221C32#509011yescharacteristic
222C9—21no—
223C9C3×#46211yescharacteristic, complemented
224Q8—22no—
225Q8—22no—
226Q8—92no—
227Q8—181no—
228C2×C4—22no—
229C2×C4—181no—
230C2×C4—181no—
231C2×C4#611⋊C311yescharacteristic
232C2×C4#611⋊C311yescharacteristic
233C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
234C8—182no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3C3×#93811yescharacteristic
239C2×C3C3×#94211yescharacteristic
240C2×C3C2×C3×#21611yescharacteristic
241C22C2×#611⋊C311yescharacteristic
242C4—21no—
243C4—42no—
244C4—182no—
245C4—361no—
246C4#3674⋊C312yescomplemented
247C4D4×AGL(1,ℤ/9)11yescharacteristic
248C4#3699⋊C311yescharacteristic
249C3—31no—
250C3C3×#509011yescharacteristic
251C2#63500⋊C311yescharacteristic
252C2C2×#3674⋊C311yescharacteristic
253C2#63505⋊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×#462
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. #10588763C3semidirect
69. C4×3−1+2Q16semidirect
153. C4×C9C3×Q16semidirect
163. 3−1+2#462semidirect
223. C9C3×#462semidirect
246. C4#3674⋊C3semidirect

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
C2×#3674⋊C32
C4×3−1+2⋊Q82
#63395⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50903
#105887633