Finite Group Databasebeta

← #13147052

Subgroups — #13147052

Order 1728.

422 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊((C22⋊C9)⋊C2))C211yesmaximal, characteristic
3C3⋊((C22⋊C9)⋊C8)C211yesmaximal, characteristic, complemented
4#10592926—31nomaximal
5#10589519—33nomaximal
6C2×C3⋊((C22⋊C9)⋊C2)—21no—
7C3×C4×C22⋊C9—21no—
8C3×C4×C22⋊C9C2211yescharacteristic
9#88548—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×#3755—31no—
12C2×#3755—31no—
13#64139—31no—
14Q8×C22⋊C9D313yescomplemented
15C4⋊((C22⋊C9)⋊C2)—33no—
16#63662—31no—
17#63662—31no—
18#63657—31no—
19(C22⋊C9)⋊C8—33no—
20C2×C3×C22⋊C9D411yescharacteristic
21C3⋊((C22⋊C9)⋊C2)—41no—
22C3×Q8×C9—41no—
23C4⋊(C3⋊D9)—41no—
24C3⋊(C9⋊C8)—41no—
25#5227—93no—
26#5227—91no—
27C2×C32×Q8—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C22×C32×C4—21no—
31C22×C32×C4C2×D311yescharacteristic
32C2×#638—61no—
33C2×C4⋊(C3⋊D3)—31no—
34C2×(C22⋊C9)⋊C2—63no—
35#3755—62no—
36#3755—61no—
37#3755—61no—
38#3752—61no—
39C4⋊(C3⋊Dic3)—31no—
40C2×C3⋊(C3⋊C8)—31no—
41C2×C3⋊(C3⋊C8)—31no—
42C4×C22⋊C9—23no—
43C4×C22⋊C9C2×D313yes—
44#3675—123no—
45C2×C3⋊D9—81no—
46C3×C22⋊C9SD1611yescharacteristic, complemented
47C3×C4×C9—41no—
48C3×C4×C9—81no—
49C22×C3×Q8D913yescomplemented
50C22×C3×Q8C3⋊D311yescharacteristic
51C2×#215—93no—
52C2×#215—91no—
53C2×#215—93no—
54C2×#215—91no—
55#930—93no—
56#930—91no—
57#835—93no—
58#835—91no—
59#835—93no—
60#835—91no—
61#830—93no—
62#830—91no—
63C23×C32#6711yescharacteristic
64C22×C3⋊D3—61no—
65C32×Q8—32no—
66C32×Q8—31no—
67C32×Q8—61no—
68C32×Q8S411yescharacteristic
69C2×C32×C4—31no—
70C2×C32×C4—31no—
71C2×C32×C4—61no—
72C2×C32×C4—61no—
73#638—121no—
74#638—121no—
75C2×C3⋊Dic3—61no—
76C4⋊(C3⋊D3)—61no—
77C4⋊(C3⋊D3)—61no—
78C2×C22⋊C9#6713yes—
79(C22⋊C9)⋊C2—123no—
80C3⋊(C3⋊C8)—61no—
81C3⋊(C3⋊C8)—61no—
82Q8×C9—43no—
83D36—123no—
84C9⋊C8—123no—
85#448—271noSylow
86C2×C3×C9—41no—
87C3⋊D9—161no—
88C2×C3×Q8—33no—
89C2×C3×Q8—31no—
90C2×C3×Q8—33no—
91C2×C3×Q8—31no—
92C2×C3×Q8—63no—
93C2×C3×Q8—61no—
94C22×C3×C4—23no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D913yes—
97C22×C3×C4C2×C3⋊D311yescharacteristic
98C2×#67—183no—
99C2×#67—181no—
100C2×D12—91no—
101C2×D12—93no—
102#215—182no—
103#215—186no—
104#215—181no—
105#215—183no—
106#215—183no—
107#215—181no—
108(C2×D3)⋊C4—183no—
109(C2×D3)⋊C4—181no—
110C4⋊Dic3—91no—
111C4⋊Dic3—93no—
112C2×C3⋊C8—91no—
113C2×C3⋊C8—93no—
114C2×C3⋊C8—91no—
115C2×C3⋊C8—93no—
116C22×C32—31no—
117C22×C32—31no—
118C22×C32#21511yescharacteristic
119C2×C3⋊D3—121no—
120C2×C3⋊D3—121no—
121C2×C3⋊D3—121no—
122C32×C4—21no—
123C32×C4—31no—
124C32×C4—61no—
125C32×C4C2×S411yescharacteristic
126C3⋊Dic3—121no—
127C2×D9—243no—
128C22⋊C9#21513yescomplemented
129C4×C9—43no—
130C4×C9—83no—
131C22×Q8C3⋊D911yescharacteristic, complemented
132C2×SD16—271no—
133C2×SD16—271no—
134#121—271no—
135#103—271no—
136#103—271no—
137C22⋊C8—271no—
138C3×C9—41noSylow
139C23×C3#61113yes—
140C23×C3#63811yescharacteristic
141C22×D3—181no—
142C22×D3—183no—
143C3×Q8—32no—
144C3×Q8—36no—
145C3×Q8—31no—
146C3×Q8—33no—
147C3×Q8—61no—
148C3×Q8—63no—
149C3×Q8(C22⋊C9)⋊C213yescomplemented
150C3×Q8C3⋊S411yescharacteristic
151C2×C3×C4—31no—
152C2×C3×C4—33no—
153C2×C3×C4—33no—
154C2×C3×C4—31no—
155C2×C3×C4—61no—
156C2×C3×C4—63no—
157C2×C3×C4—61no—
158C2×C3×C4—63no—
159#67—363no—
160#67—361no—
161#67—363no—
162#67—361no—
163C2×Dic3—181no—
164C2×Dic3—183no—
165D12—181no—
166D12—183no—
167D12—181no—
168D12—183no—
169C3⋊C8—181no—
170C3⋊C8—183no—
171C3⋊C8—181no—
172C3⋊C8—183no—
173C2×C32—31no—
174C2×C32—31no—
175C2×C32GL(2,ℤ/4)11yescharacteristic
176C3⋊D3—241no—
177C2×C9—43no—
178D9—483no—
179C2×Q8—31no—
180C2×Q8—31no—
181C2×Q8—61no—
182C2×D4—271no—
183C2×D4—541no—
184C22×C4—21no—
185C22×C4C2×C3⋊D911yescharacteristic
186SD16—542no—
187SD16—541no—
188SD16—541no—
189C2×C8—271no—
190C2×C8—271no—
191C4⋊C4—271no—
192C22⋊C4—541no—
193C22×C3—33no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—33no—
197C22×C3#367513yescomplemented
198C22×C3#375511yescharacteristic
199C2×D3—363no—
200C2×D3—361no—
201C2×D3—363no—
202C2×D3—361no—
203C2×D3—361no—
204C2×D3—363no—
205C3×C4—21no—
206C3×C4—23no—
207C3×C4—31no—
208C3×C4—33no—
209C3×C4—61no—
210C3×C4—63no—
211C3×C4C2×(C22⋊C9)⋊C213yes—
212C3×C4C2×C3⋊S411yescharacteristic
213Dic3—363no—
214Dic3—361no—
215C32#541711yescharacteristic
216C9—43no—
217C23—541no—
218C23#623311yescharacteristic
219Q8—32no—
220Q8—31no—
221Q8—61no—
222Q8C3⋊((C22⋊C9)⋊C2)11yescharacteristic, complemented
223D4—541no—
224D4—541no—
225D4—1081no—
226D4—1081no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—541no—
232C8—541no—
233C8—541no—
234C2×C3—33no—
235C2×C3—33no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3#6369413yes—
239C2×C3#6426911yescharacteristic
240D3—723no—
241D3—721no—
242C22—31no—
243C22—31no—
244C22—1081no—
245C22—1081no—
246C22—1081no—
247C22#8854811yescharacteristic, complemented
248C4—21no—
249C4—31no—
250C4—61no—
251C4—1081no—
252C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
253C3#1058951913yescomplemented
254C3#1059365111yescharacteristic
255C2—31no—
256C2—31no—
257C2—2161no—
258C2#1172239611yescharacteristic
subgroupG/H
centerC2#11722396
commutator, cosocleC3×C4×C22⋊C9C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C32#67
prime radicalC4⋊(C3⋊((C22⋊C9)⋊C2))C2
hypercenterQ8C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9SD16
LCSG ▷ C3×C4×C22⋊C9 ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C22⋊C9C2semidirect
3. C3⋊((C22⋊C9)⋊C8)C2semidirect
14. Q8×C22⋊C9D3semidirect
46. C3×C22⋊C9SD16semidirect
49. C22×C3×Q8D9semidirect
128. C22⋊C9#215semidirect
131. C22×Q8C3⋊D9semidirect
149. C3×Q8(C22⋊C9)⋊C2semidirect
197. C22×C3#3675semidirect
222. Q8C3⋊((C22⋊C9)⋊C2)semidirect
247. C22#88548semidirect
253. C3#10589519semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C4⋊(C3⋊((C22⋊C9)⋊C2))2
C3⋊((C22⋊C9)⋊C8)2
#105929263
#105895193
#885484