Finite Group Databasebeta

← #13147051

Subgroups — #13147051

Order 1728.

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

normal subgroups only

This group has 384 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
2#11722377C211yesmaximal, characteristic
3C3⋊((C22⋊C9)⋊C8)C211yesmaximal, characteristic
4#10592927—31nomaximal
5#10589518—33nomaximal
6C3×C4×C22⋊C9—21no—
7C3×C4×C22⋊C9C2211yescharacteristic
8C3⋊((C22⋊C9)⋊C4)—21no—
9#88547—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×#3756—31no—
12C2×#3756—31no—
13#64133—31no—
14Q8×C22⋊C9D313yes—
15(C22⋊C9)⋊Q8—33no—
16#63662—31no—
17#63662—31no—
18#63657—31no—
19(C22⋊C9)⋊C8—33no—
20C2×C3×C22⋊C9D411yescharacteristic
21C3×Q8×C9—41no—
22#6226—41no—
23C3⋊(C9⋊C8)—41no—
24#5228—93no—
25#5228—91no—
26C2×C32×Q8—31no—
27C2×C32×Q8—31no—
28C2×C32×Q8—61no—
29C22×C32×C4—21no—
30C22×C32×C4C2×D311yescharacteristic
31C2×#634—31no—
32#3757—61no—
33#3756—62no—
34#3756—61no—
35#3756—61no—
36C4⋊(C3⋊Dic3)—31no—
37#3750—61no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—31no—
40C4×C22⋊C9—23no—
41C4×C22⋊C9C2×D313yes—
42(C22⋊C9)⋊C4—63no—
43#3674—123no—
44C3×C22⋊C9Q1611yescharacteristic, complemented
45C3×C4×C9—41no—
46C3×C4×C9—81no—
47C3⋊Dic9—81no—
48C22×C3×Q8D913yes—
49C22×C3×Q8C3⋊D311yescharacteristic
50C2×#216—93no—
51C2×#216—91no—
52C2×#216—93no—
53C2×#216—91no—
54#924—93no—
55#924—91no—
56#835—93no—
57#835—91no—
58#835—93no—
59#835—91no—
60#830—93no—
61#830—91no—
62C23×C32#6711yescharacteristic
63C32×Q8—32no—
64C32×Q8—31no—
65C32×Q8—61no—
66C32×Q8S411yescharacteristic
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—61no—
70C2×C32×C4—61no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—61no—
73#634—61no—
74#634—61no—
75C2×C22⋊C9#6713yes—
76C3⋊(C3⋊C8)—61no—
77C3⋊(C3⋊C8)—61no—
78Q8×C9—43no—
79Dic18—123no—
80C9⋊C8—123no—
81#451—271noSylow
82C2×C3×C9—41no—
83C2×C3×Q8—33no—
84C2×C3×Q8—31no—
85C2×C3×Q8—33no—
86C2×C3×Q8—31no—
87C2×C3×Q8—63no—
88C2×C3×Q8—61no—
89C22×C3×C4—23no—
90C22×C3×C4—21no—
91C22×C3×C4C2×D913yes—
92C22×C3×C4C2×C3⋊D311yescharacteristic
93C2×Dic6—91no—
94C2×Dic6—93no—
95#217—183no—
96#217—181no—
97#216—182no—
98#216—186no—
99#216—181no—
100#216—183no—
101#216—183no—
102#216—181no—
103C4⋊Dic3—91no—
104C4⋊Dic3—93no—
105Dic3⋊C4—183no—
106Dic3⋊C4—181no—
107C2×C3⋊C8—91no—
108C2×C3⋊C8—93no—
109C2×C3⋊C8—91no—
110C2×C3⋊C8—93no—
111C22×C32—31no—
112C22×C32—31no—
113C22×C32#21611yescharacteristic
114C32×C4—21no—
115C32×C4—31no—
116C32×C4—61no—
117C32×C4C2×S411yescharacteristic
118C3⋊Dic3—121no—
119C3⋊Dic3—121no—
120C22⋊C9#21613yescomplemented
121C4×C9—43no—
122C4×C9—83no—
123Dic9—243no—
124C22×Q8C3⋊D911yescharacteristic
125C2×Q16—271no—
126C2×Q16—271no—
127C22⋊Q8—271no—
128#103—271no—
129#103—271no—
130C22⋊C8—271no—
131C3×C9—41noSylow
132C23×C3#61113yes—
133C23×C3#63811yescharacteristic
134C3×Q8—32no—
135C3×Q8—36no—
136C3×Q8—31no—
137C3×Q8—33no—
138C3×Q8—61no—
139C3×Q8—63no—
140C3×Q8(C22⋊C9)⋊C213yes—
141C3×Q8C3⋊S411yescharacteristic
142C2×C3×C4—31no—
143C2×C3×C4—33no—
144C2×C3×C4—33no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—63no—
148C2×C3×C4—61no—
149C2×C3×C4—63no—
150C2×Dic3—181no—
151C2×Dic3—183no—
152C2×Dic3—181no—
153C2×Dic3—183no—
154Dic6—181no—
155Dic6—183no—
156Dic6—181no—
157Dic6—183no—
158C3⋊C8—181no—
159C3⋊C8—183no—
160C3⋊C8—181no—
161C3⋊C8—183no—
162C2×C32—31no—
163C2×C32—31no—
164C2×C32GL(2,ℤ/4)11yescharacteristic
165C2×C9—43no—
166C2×Q8—31no—
167C2×Q8—31no—
168C2×Q8—61no—
169C2×Q8—271no—
170C22×C4—21no—
171C22×C4C2×C3⋊D911yescharacteristic
172Q16—542no—
173Q16—541no—
174Q16—541no—
175C2×C8—271no—
176C2×C8—271no—
177C4⋊C4—271no—
178C4⋊C4—541no—
179C22⋊C4—541no—
180C22×C3—33no—
181C22×C3—31no—
182C22×C3—31no—
183C22×C3—33no—
184C22×C3#367413yescomplemented
185C22×C3#375611yescharacteristic
186C3×C4—21no—
187C3×C4—23no—
188C3×C4—31no—
189C3×C4—33no—
190C3×C4—61no—
191C3×C4—63no—
192C3×C4C2×(C22⋊C9)⋊C213yes—
193C3×C4C2×C3⋊S411yescharacteristic
194Dic3—363no—
195Dic3—361no—
196Dic3—361no—
197Dic3—363no—
198C32#541611yescharacteristic
199C9—43no—
200C23#623311yescharacteristic
201Q8—32no—
202Q8—31no—
203Q8—61no—
204Q8—541no—
205Q8—541no—
206Q8C3⋊((C22⋊C9)⋊C2)11yescharacteristic
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—541no—
212C2×C4—541no—
213C8—541no—
214C8—541no—
215C2×C3—33no—
216C2×C3—33no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3#6369413yes—
220C2×C3#6426911yescharacteristic
221C22—31no—
222C22—31no—
223C22#8854711yescharacteristic, complemented
224C4—21no—
225C4—31no—
226C4—61no—
227C4—1081no—
228C4—1081no—
229C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
230C3#1058951813yescomplemented
231C3#1059365011yescharacteristic
232C2—31no—
233C2—31no—
234C2#1172239611yescharacteristic
subgroupG/H
centerC2#11722396
commutator, cosocleC3×C4×C22⋊C9C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×Q8D3
socle, prime radicalC23×C32#67
hypercenterQ8C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9Q16
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
44. C3×C22⋊C9Q16semidirect
120. C22⋊C9#216semidirect
184. C22×C3#3674semidirect
223. C22#88547semidirect
230. C3#10589518semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
#117223772
C3⋊((C22⋊C9)⋊C8)2
#105929273
#105895183
#885474