Finite Group Databasebeta

← #13144097

Subgroups — #13144097

Order 1728.

570 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 167 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×C3⋊D9C211yesmaximal, characteristic
2#11721748C212yesmaximal
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720540C211yesmaximal, characteristic, complemented
5#11720525C212yesmaximal, complemented
6#10592559—31nomaximal
7#10588540—33nomaximal
8C2×C4×C3⋊D9C2211yescharacteristic
9C2×C3×C8×C9C2211yescharacteristic
10C3×C9×C4⋊C4C2212yes—
11#88536—22no—
12C4⋊(C3⋊Dic9)C2212yes—
13C4×C3⋊Dic9—22no—
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C8×C3⋊D9C414yescomplemented
16C2×C8×C3⋊D3—31no—
17#64101—32no—
18C32×#106D311yescharacteristic
19#63644—31no—
20#63634—32no—
21C2×C8×D9—33no—
22#63454—36no—
23C9×#106D313yescomplemented
24#63377—33no—
25#63366—36no—
26C22×C3⋊D9D411yescharacteristic
27C2×C3×C4×C9—22no—
28C2×C3×C4×C9C2311yescharacteristic
29C2×C3⋊Dic9—22no—
30C2×C3⋊Dic9D411yescharacteristic
31C4×C3⋊D9C2×C412yes—
32C4×C3⋊D9Q812yes—
33C3×C8×C9C2×C412yes—
34C3⋊(C9⋊C8)C2×C412yes—
35#4860—93no—
36#4860—91no—
37C2×C4×C3⋊D3—31no—
38C2×C32×C8C2×D311yescharacteristic
39C32×C4⋊C4C2×D312yes—
40#3752—62no—
41C4⋊(C3⋊Dic3)—32no—
42C4×C3⋊Dic3—62no—
43C2×C3⋊(C3⋊C8)—31no—
44C8×C3⋊D3—34no—
45C2×C4×D9—33no—
46C2×C8×C9C2×D313yes—
47C9×C4⋊C4C2×D316yes—
48(C2×D9)⋊C4—66no—
49C4⋊Dic9—36no—
50C4×Dic9—66no—
51C2×C9⋊C8—33no—
52C8×D9—312no—
53C22×C3×C9C2×D411yescharacteristic
54C2×C3⋊D9—22no—
55C2×C3⋊D9C4⋊C412yes—
56C3×C4×C9—42no—
57C3×C4×C9C22×C411yescharacteristic
58C3×C4×C9C2×Q811yescharacteristic
59C3⋊Dic9—42no—
60C3⋊Dic9C4⋊C412yes—
61C2×D3×C8—93no—
62C2×D3×C8—91no—
63#892—96no—
64#892—92no—
65C3×#106D913yescomplemented
66C3×#106C3⋊D311yescharacteristic
67#817—93no—
68#817—91no—
69#807—96no—
70#807—92no—
71C22×C3⋊D3—31no—
72C2×C32×C4—22no—
73C2×C32×C4C22×D311yescharacteristic
74C2×C3⋊Dic3—31no—
75C2×C3⋊Dic3—62no—
76C4×C3⋊D3—32no—
77C4×C3⋊D3—32no—
78C22×D9—33no—
79C32×C8C4×D312yes—
80C3⋊(C3⋊C8)—32no—
81C2×C4×C9—26no—
82C2×C4×C9C22×D313yes—
83C2×Dic9—33no—
84C2×Dic9—66no—
85C4×D9—36no—
86C4×D9—36no—
87C8×C9C4×D316yes—
88C9⋊C8—36no—
89#427—271noSylow
90C2×C3×C9C2×C4⋊C411yescharacteristic
91C2×C3×C9#13512yes—
92C3⋊D9—22no—
93C2×C4×D3—93no—
94C2×C4×D3—91no—
95C2×C3×C8C2×D913yes—
96C2×C3×C8C2×C3⋊D311yescharacteristic
97C3×C4⋊C4C2×D916yes—
98C3×C4⋊C4C2×C3⋊D312yes—
99(C2×D3)⋊C4—182no—
100(C2×D3)⋊C4—186no—
101C4⋊Dic3—96no—
102C4⋊Dic3—92no—
103C4×Dic3—182no—
104C4×Dic3—186no—
105C2×C3⋊C8—93no—
106C2×C3⋊C8—91no—
107D3×C8—912no—
108D3×C8—94no—
109C22×C32D3×D411yescharacteristic
110C2×C3⋊D3—32no—
111C2×C3⋊D3—62no—
112C32×C4—42no—
113C32×C4C2×C4×D311yescharacteristic
114C32×C4D3×Q811yescharacteristic
115C3⋊Dic3—32no—
116C3⋊Dic3—122no—
117C22×C9D3×D413yes—
118C2×D9—36no—
119C2×D9—66no—
120C4×C9—46no—
121C4×C9C2×C4×D313yes—
122C4×C9D3×Q813yes—
123Dic9—36no—
124Dic9—126no—
125C22×C8—271no—
126#117—272no—
127#107—272no—
128#106—271no—
129#106C3⋊D911yescharacteristic, complemented
130C3×C9#42711yescharacteristic, complemented, Sylow
131C22×D3—93no—
132C22×D3—91no—
133C2×C3×C4—26no—
134C2×C3×C4—22no—
135C2×C3×C4C22×D913yes—
136C2×C3×C4C22×C3⋊D311yescharacteristic
137C2×Dic3—93no—
138C2×Dic3—91no—
139C2×Dic3—186no—
140C2×Dic3—182no—
141C4×D3—96no—
142C4×D3—96no—
143C4×D3—92no—
144C4×D3—92no—
145C3×C8C4×D916yes—
146C3×C8C4×C3⋊D312yes—
147C3⋊C8—96no—
148C3⋊C8—92no—
149C2×C32D3×C4⋊C411yescharacteristic
150C2×C32#91612yes—
151C3⋊D3—62no—
152C2×C9D3×C4⋊C413yes—
153C2×C9#91616yes—
154D9—66no—
155C22×C4—271no—
156C2×C8—271no—
157C2×C8—274no—
158C2×C8C2×C3⋊D911yescharacteristic
159C4⋊C4—272no—
160C4⋊C4C2×C3⋊D912yes—
161C22⋊C4—542no—
162C42—542no—
163C22×C3D4×D913yes—
164C22×C3D4×C3⋊D311yescharacteristic
165C2×D3—92no—
166C2×D3—96no—
167C2×D3—182no—
168C2×D3—186no—
169C3×C4—42no—
170C3×C4—46no—
171C3×C4C2×C4×D913yes—
172C3×C4Q8×D913yes—
173C3×C4C2×C4×C3⋊D311yescharacteristic
174C3×C4Q8×C3⋊D311yescharacteristic
175Dic3—92no—
176Dic3—96no—
177Dic3—362no—
178Dic3—366no—
179C32#486011yescharacteristic
180C9#486013yescomplemented
181C23—271no—
182C2×C4—22no—
183C2×C4—271no—
184C2×C4—272no—
185C2×C4—272no—
186C2×C4—542no—
187C2×C4C22×C3⋊D911yescharacteristic
188C8—272no—
189C8C4×C3⋊D912yes—
190C2×C3C4⋊C4×D913yes—
191C2×C3#6347816yes—
192C2×C3C4⋊C4×C3⋊D311yescharacteristic
193C2×C3#6412512yes—
194D3—182no—
195D3—186no—
196C22—272no—
197C22—542no—
198C22D4×C3⋊D911yescharacteristic
199C4—42no—
200C4—272no—
201C4—1082no—
202C4C2×C4×C3⋊D911yescharacteristic
203C4Q8×C3⋊D911yescharacteristic
204C3#1058854013yescomplemented
205C3#1059255911yescharacteristic
206C2—542no—
207C2C4⋊C4×C3⋊D911yescharacteristic
208C2#1172179812yes—
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenter#106C3⋊D9
hypocenterC3×C9#427
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. C3×C9×#106C2semidirect
4. #11720540C2semidirect
5. #11720525C2semidirect
15. C8×C3⋊D9C4semidirect
23. C9×#106D3semidirect
65. C3×#106D9semidirect
129. #106C3⋊D9semidirect
130. C3×C9#427semidirect
180. C9#4860semidirect
204. C3#10588540semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊D92
#117217482
C3×C9×#1062
#117205402
#117205252
#105925593
#105885403