Finite Group Databasebeta

← #13147066

Subgroups — #13147066

Order 1728.

402 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 69 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×C3×#41⋊C9C211yesmaximal, characteristic
2C2×C3⋊#3688C211yesmaximal, characteristic, complemented
3C3⋊#63421C211yesmaximal, characteristic, complemented
4#10592938—31nomaximal
5#10589527—33nomaximal
6C3×#41⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3⋊#3688—21no—
9#88534—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×#3756—31no—
12C2×#3754—31no—
13#64133—31no—
14C2×#41⋊C9D313yes—
15C2×#3688—33no—
16#63662—31no—
17#63659—31no—
18#63657—31no—
19#63421—33no—
20C2×C3×C4×C9—41no—
21C2×C3⋊Dic9—41no—
22C2×C3⋊Dic9—41no—
23C3×Q8⋊C9D411yescharacteristic
24#5239—93no—
25#5239—91no—
26C32×#41—21no—
27C32×#41—61no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C2×#634—31no—
32#3757—61no—
33#3756—61no—
34#3756—61no—
35#3754—62no—
36C4⋊(C3⋊Dic3)—31no—
37#3750—61no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—31no—
40#41⋊C9—23no—
41C2×Q8⋊C9C2×D313yes—
42#3688—63no—
43Dic9⋊C4—123no—
44C22×C3×C9—41no—
45C3×C4×C9—81no—
46C3⋊Dic9—42no—
47C3⋊Dic9—81no—
48C2×C3×#41D913yes—
49C2×C3×#41C3⋊D311yescharacteristic
50C2×#216—93no—
51C2×#216—91no—
52C2×#214—93no—
53C2×#214—91no—
54#924—93no—
55#924—91no—
56#835—93no—
57#835—91no—
58#832—93no—
59#832—91no—
60#830—93no—
61#830—91no—
62C23×C32—31no—
63C32×Q8—31no—
64C32×Q8#6711yescharacteristic
65C32×D4—32no—
66C32×D4—61no—
67C2×C32×C4—31no—
68C2×C32×C4—61no—
69C2×C32×C4—61no—
70C2×C32×C4S411yescharacteristic
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—61no—
73#634—61no—
74#634—61no—
75C3⋊(C3⋊C8)—61no—
76C3⋊(C3⋊C8)—61no—
77C2×C4×C9—43no—
78C2×Dic9—123no—
79C2×Dic9—123no—
80Q8⋊C9#6713yes—
81#452—271noSylow
82C2×C3×C9—41no—
83C2×C3×C9—41no—
84C2×C3×C9—41no—
85C3×#41—23no—
86C3×#41—21no—
87C3×#41—63no—
88C3×#41—61no—
89C2×C3×Q8C2×D913yes—
90C2×C3×Q8C2×C3⋊D311yescharacteristic
91C2×C3×D4—33no—
92C2×C3×D4—31no—
93C22×C3×C4—33no—
94C22×C3×C4—31no—
95C2×Dic6—91no—
96C2×Dic6—93no—
97#217—181no—
98#217—183no—
99#216—183no—
100#216—181no—
101#216—183no—
102#216—181no—
103#214—182no—
104#214—186no—
105C4⋊Dic3—91no—
106C4⋊Dic3—93no—
107Dic3⋊C4—183no—
108Dic3⋊C4—181no—
109C2×C3⋊C8—91no—
110C2×C3⋊C8—93no—
111C2×C3⋊C8—91no—
112C2×C3⋊C8—93no—
113C22×C32—61no—
114C22×C32—61no—
115C22×C32—61no—
116C22×C32C2×S411yescharacteristic
117C32×C4—21no—
118C32×C4—31no—
119C32×C4—31no—
120C3⋊Dic3—121no—
121C3⋊Dic3—121no—
122C22×C9—43no—
123C4×C9—83no—
124Dic9—126no—
125Dic9—243no—
126C2×#41C3⋊D911yescharacteristic
127C2×Q16—271no—
128C2×SD16—271no—
129C22⋊Q8—271no—
130#103—271no—
131#102—271no—
132C22⋊C8—271no—
133C3×C9—41noSylow
134C23×C3—31no—
135C23×C3—33no—
136C3×Q8—33no—
137C3×Q8—31no—
138C3×Q8#61113yes—
139C3×Q8#63811yescharacteristic
140C3×D4—32no—
141C3×D4—36no—
142C3×D4—61no—
143C3×D4—63no—
144C2×C3×C4—31no—
145C2×C3×C4—33no—
146C2×C3×C4—61no—
147C2×C3×C4—63no—
148C2×C3×C4—61no—
149C2×C3×C4—63no—
150C2×C3×C4(C22⋊C9)⋊C213yes—
151C2×C3×C4C3⋊S411yescharacteristic
152C2×Dic3—181no—
153C2×Dic3—181no—
154C2×Dic3—183no—
155C2×Dic3—183no—
156Dic6—181no—
157Dic6—183no—
158Dic6—181no—
159Dic6—183no—
160C3⋊C8—181no—
161C3⋊C8—183no—
162C3⋊C8—181no—
163C3⋊C8—183no—
164C2×C32—121no—
165C2×C32#98411yescharacteristic
166C2×C32#98511yescharacteristic
167C2×C32GL(2,ℤ/4)11yescharacteristic
168C2×C9—43no—
169C2×C9—43no—
170C2×C9—43no—
171#41—21no—
172#41—61no—
173C2×Q8—271no—
174C2×Q8C2×C3⋊D911yescharacteristic
175C2×D4—31no—
176C22×C4—31no—
177Q16—541no—
178Q16—541no—
179SD16—542no—
180C2×C8—271no—
181C2×C8—271no—
182C4⋊C4—271no—
183C4⋊C4—541no—
184C22⋊C4—541no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—63no—
189C22×C3—63no—
190C22×C3—63no—
191C22×C3C2×(C22⋊C9)⋊C213yes—
192C22×C3C2×C3⋊S411yescharacteristic
193C3×C4—21no—
194C3×C4—23no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—33no—
198C3×C4—33no—
199Dic3—361no—
200Dic3—361no—
201Dic3—363no—
202Dic3—363no—
203C32#542511yescharacteristic
204C9—43no—
205C23—31no—
206Q8—31no—
207Q8—541no—
208Q8—541no—
209Q8#623311yescharacteristic
210D4—32no—
211D4—61no—
212C2×C4—31no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—541no—
216C2×C4—541no—
217C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
218C8—541no—
219C8—541no—
220C2×C3—123no—
221C2×C3—121no—
222C2×C3#6369013yes—
223C2×C3#6369113yes—
224C2×C3#6369413yes—
225C2×C3#6426511yescharacteristic
226C2×C3#6426611yescharacteristic
227C2×C3#6426911yescharacteristic
228C22—61no—
229C22—61no—
230C22—61no—
231C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
232C4—21no—
233C4—31no—
234C4—31no—
235C4—1081no—
236C4—1081no—
237C3#1058952713yescomplemented
238C3#1059365911yescharacteristic
239C2—121no—
240C2#1172238911yescharacteristic
241C2#1172239011yescharacteristic
242C2#1172239611yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×Q8⋊C9C22
FrattiniC22×C3C2×C3⋊S4
Fitting, prime radicalC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
hypocenterC3×Q8⋊C9D4
LCSG ▷ C2×C3×Q8⋊C9 ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3688C2semidirect
3. C3⋊#63421C2semidirect
237. C3#10589527semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C2×C3⋊#36882
C3⋊#634212
#105929383
#105895273
#885344