Finite Group Databasebeta

← #13144092

Subgroups — #13144092

Order 1728.

450 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 155 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
1#11721746C211yesmaximal, characteristic
2#11721744C211yesmaximal, characteristic
3C3×C9×#106C211yesmaximal, characteristic
4#11720541C211yesmaximal, characteristic
5#11720538C211yesmaximal, characteristic
6#11720526C211yesmaximal, characteristic
7#11720525C211yesmaximal, characteristic
8#10592555—31nomaximal
9#10588536—33nomaximal
10C2×#6226—21no—
11C2×C3×C8×C9C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17#88534—21no—
18#88534—21no—
19C4×C3⋊Dic9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64099—31no—
22#64097—31no—
23C32×#106D311yescharacteristic
24#63645—31no—
25#63642—31no—
26#63635—31no—
27#63634—31no—
28#63452—33no—
29#63450—33no—
30C9×#106D313yes—
31#63378—33no—
32#63375—33no—
33#63367—33no—
34#63366—33no—
35C2×C3×C4×C9—21no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42#6226—42no—
43C3×C8×C9Q812yescomplemented
44C3⋊(C9⋊C8)Q812yescomplemented
45#4856—93no—
46#4856—91no—
47C2×#634—61no—
48C2×C32×C8C2×D311yescharacteristic
49C32×C4⋊C4C2×D311yescharacteristic
50C32×C4⋊C4C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)—31no—
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—61no—
54#3750—61no—
55#3750—61no—
56C4×C3⋊Dic3—31no—
57C2×C3⋊(C3⋊C8)—31no—
58C2×Dic18—63no—
59C2×C8×C9C2×D313yes—
60C9×C4⋊C4C2×D313yes—
61C9×C4⋊C4C2×D313yes—
62C4⋊Dic9—33no—
63C4⋊Dic9—33no—
64C4⋊Dic9—63no—
65Dic9⋊C4—63no—
66Dic9⋊C4—63no—
67C4×Dic9—33no—
68C2×C9⋊C8—33no—
69C22×C3×C9C2×D411yescharacteristic
70C3×C4×C9—41no—
71C3×C4×C9—41no—
72C3×C4×C9C2×Q811yescharacteristic
73C3×C4×C9C2×Q811yescharacteristic
74C3⋊Dic9—21no—
75C3⋊Dic9—21no—
76C3⋊Dic9—41no—
77C3⋊Dic9—41no—
78#890—93no—
79#890—91no—
80#888—93no—
81#888—91no—
82C3×#106D913yes—
83C3×#106C3⋊D311yescharacteristic
84#818—93no—
85#818—91no—
86#815—93no—
87#815—91no—
88#808—93no—
89#808—91no—
90#807—93no—
91#807—91no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—31no—
96C2×C3⋊Dic3—31no—
97C2×C3⋊Dic3—61no—
98C2×C3⋊Dic3—61no—
99#634—122no—
100C32×C8Dic612yes—
101C3⋊(C3⋊C8)—32no—
102C2×C4×C9—23no—
103C2×C4×C9—23no—
104C2×C4×C9C22×D313yes—
105C2×Dic9—33no—
106C2×Dic9—33no—
107C2×Dic9—63no—
108C2×Dic9—63no—
109Dic18—126no—
110C8×C9Dic616yescomplemented
111C9⋊C8—36no—
112#501—271noSylow
113C2×C3×C9C4⋊Q811yescharacteristic
114C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
115C2×C3×C9#13711yescharacteristic
116C2×Dic6—181no—
117C2×Dic6—183no—
118C2×C3×C8C2×D913yes—
119C2×C3×C8C2×C3⋊D311yescharacteristic
120C3×C4⋊C4C2×D913yes—
121C3×C4⋊C4C2×D913yes—
122C3×C4⋊C4C2×C3⋊D311yescharacteristic
123C3×C4⋊C4C2×C3⋊D311yescharacteristic
124C4⋊Dic3—93no—
125C4⋊Dic3—93no—
126C4⋊Dic3—91no—
127C4⋊Dic3—91no—
128C4⋊Dic3—181no—
129C4⋊Dic3—183no—
130Dic3⋊C4—181no—
131Dic3⋊C4—181no—
132Dic3⋊C4—183no—
133Dic3⋊C4—183no—
134C4×Dic3—93no—
135C4×Dic3—91no—
136C2×C3⋊C8—93no—
137C2×C3⋊C8—91no—
138C22×C32D3×D411yescharacteristic
139C32×C4—41no—
140C32×C4—41no—
141C32×C4C2×Dic611yescharacteristic
142C32×C4D3×Q811yescharacteristic
143C3⋊Dic3—61no—
144C3⋊Dic3—61no—
145C3⋊Dic3—121no—
146C3⋊Dic3—121no—
147C22×C9D3×D413yes—
148C4×C9—43no—
149C4×C9—43no—
150C4×C9C2×Dic613yes—
151C4×C9D3×Q813yes—
152Dic9—63no—
153Dic9—63no—
154Dic9—123no—
155Dic9—123no—
156C4⋊Q8—271no—
157#125—271no—
158#107—271no—
159#107—271no—
160#106—271no—
161#106C3⋊D911yescharacteristic
162#97—271no—
163C3×C9#50111yescharacteristic, complemented, Sylow
164C2×C3×C4—23no—
165C2×C3×C4—23no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4C22×D913yes—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—93no—
171C2×Dic3—93no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—183no—
175C2×Dic3—183no—
176C2×Dic3—181no—
177C2×Dic3—181no—
178Dic6—362no—
179Dic6—366no—
180C3×C8Dic1816yescomplemented
181C3×C8#63412yes—
182C3⋊C8—96no—
183C3⋊C8—92no—
184C2×C32#88811yescharacteristic
185C2×C32#91411yescharacteristic
186C2×C32#91511yescharacteristic
187C2×C9#88813yes—
188C2×C9#91413yes—
189C2×C9#91513yes—
190C2×Q8—541no—
191C2×C8—271no—
192C2×C8C2×C3⋊D911yescharacteristic
193C4⋊C4—271no—
194C4⋊C4—271no—
195C4⋊C4—541no—
196C4⋊C4—541no—
197C4⋊C4—541no—
198C4⋊C4C2×C3⋊D911yescharacteristic
199C4⋊C4C2×C3⋊D911yescharacteristic
200C42—271no—
201C22×C3D4×D913yes—
202C22×C3D4×C3⋊D311yescharacteristic
203C3×C4—41no—
204C3×C4—41no—
205C3×C4—43no—
206C3×C4—43no—
207C3×C4C2×Dic1813yes—
208C3×C4Q8×D913yes—
209C3×C4C2×#63411yescharacteristic
210C3×C4Q8×C3⋊D311yescharacteristic
211Dic3—181no—
212Dic3—181no—
213Dic3—183no—
214Dic3—183no—
215Dic3—361no—
216Dic3—361no—
217Dic3—363no—
218Dic3—363no—
219C32#485611yescharacteristic
220C9#485613yescomplemented
221Q8—1082no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—271no—
225C2×C4—271no—
226C2×C4—541no—
227C2×C4—541no—
228C2×C4C22×C3⋊D911yescharacteristic
229C8—272no—
230C8#622612yescomplemented
231C2×C3#6345013yes—
232C2×C3#6347613yes—
233C2×C3#6347713yes—
234C2×C3#6409711yescharacteristic
235C2×C3#6412311yescharacteristic
236C2×C3#6412411yescharacteristic
237C22D4×C3⋊D911yescharacteristic
238C4—41no—
239C4—41no—
240C4—541no—
241C4—541no—
242C4—1081no—
243C4—1081no—
244C4C2×#622611yescharacteristic
245C4Q8×C3⋊D911yescharacteristic
246C3#1058853613yescomplemented
247C3#1059255511yescharacteristic
248C2#1172174411yescharacteristic
249C2#1172179611yescharacteristic
250C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socle, prime radicalC22×C32D3×D4
hypercenter#106C3⋊D9
hypocenterC3×C9#501
LCSG ▷ C2×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
43. C3×C8×C9Q8semidirect
44. C3⋊(C9⋊C8)Q8semidirect
110. C8×C9Dic6semidirect
163. C3×C9#501semidirect
180. C3×C8Dic18semidirect
220. C9#4856semidirect
230. C8#6226semidirect
246. C3#10588536semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117217442
C3×C9×#1062
#117205412
#117205382
#117205262
#117205252
#105925553
#105885363