Finite Group Databasebeta

← #13137194

Subgroups — #13137194

Order 1728.

540 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721711C211yesmaximal, characteristic, complemented
2C3×#63437C211yesmaximal, characteristic, complemented
3C2×#88417C212yesmaximal, complemented
4C2×#88416C212yesmaximal, complemented
5C9×#803C211yesmaximal, characteristic, complemented
6#10590351—31nomaximal
7#10588393—31nomaximal
8C2×C4⋊(C3⋊D9)—22no—
9C2×C4⋊(C3⋊D9)C2212yes—
10C2×C3×D36C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C3×C42×C9C2211yescharacteristic
13C3×(C2×D9)⋊C4—22no—
14C2×C9×C3⋊C8C2212yes—
15#88417—24no—
16#88416—24no—
17#64083—31no—
18C3×#875—31no—
19C2×#3717—32no—
20C2×#3714—32no—
21C3×#803D311yescharacteristic
22C2×D72—32no—
23C2×#3664—32no—
24#63437D311yescharacteristic, complemented
25#63436—31no—
26#63436—61no—
27C9×#97—31no—
28C22×C3⋊D9—22no—
29C22×C3×D9—21no—
30C2×C3×C4×C9D412yes—
31C2×C3×C4×C9C2311yescharacteristic
32C4⋊(C3⋊D9)—22no—
33C4⋊(C3⋊D9)—22no—
34C4⋊(C3⋊D9)—44no—
35C2×C3×Dic9—21no—
36C3×D36—22no—
37C3×Dic18—22no—
38C9×C3⋊C8D414yescomplemented
39#5058—91no—
40#4712—91no—
41C2×C4⋊(C3⋊D3)—32no—
42C2×C4⋊(C3⋊D3)—62no—
43C2×C3×D12—31no—
44C2×C3×Dic6—31no—
45C32×C42C2×D311yescharacteristic
46C3×(C2×D3)⋊C4—62no—
47C2×C3×C3⋊C8C2×D312yes—
48#3717—64no—
49#3714—64no—
50C2×D36—32no—
51C2×D36—62no—
52C2×D36—64no—
53C2×D36—62no—
54C2×D36C2×D311yescharacteristic
55C2×Dic18C2×D311yescharacteristic
56C2×C8×C9—32no—
57C42×C9—21no—
58C42×C9C2×D311yescharacteristic
59(C2×D9)⋊C4—22no—
60D72—64no—
61#3664—64no—
62C22×C3×C9C2×D411yescharacteristic
63C2×C3⋊D9—42no—
64C2×C3⋊D9—44no—
65C2×C3×D9—41no—
66C2×C3×D9—42no—
67C3×C4×C9—22no—
68C3×C4×C9C2×D412yes—
69C3×Dic9—41no—
70C3×#124—91no—
71C2×#215—92no—
72C2×#213—92no—
73C2×D24—92no—
74C2×#204—92no—
75#875—31no—
76#874—91no—
77#874—91no—
78#874—181no—
79C3×#97—31no—
80#803D911yescharacteristic, complemented
81C22×C3⋊D3—62no—
82C22×C3×D3—61no—
83C2×C32×C4D1212yes—
84C2×C32×C4C22×D311yescharacteristic
85C4⋊(C3⋊D3)—62no—
86C4⋊(C3⋊D3)—62no—
87C4⋊(C3⋊D3)—124no—
88C2×C3×Dic3—61no—
89C3×D12—62no—
90C3×Dic6—62no—
91C22×D9—21no—
92C22×D9—64no—
93C22×D9—62no—
94C3×C3⋊C8D1214yes—
95C2×C4×C9—22no—
96C2×C4×C9—21no—
97C2×C4×C9#6712yes—
98C2×C4×C9C22×D311yescharacteristic
99C2×Dic9—21no—
100D36—22no—
101D36—62no—
102D36—62no—
103D36—122no—
104D36—122no—
105D36—128no—
106D36—124no—
107Dic18—22no—
108C8×C9—34no—
109#496—271noSylow
110C2×C3×C9#12711yescharacteristic
111C2×C3×C9AGL(1,ℤ/8)12yes—
112C3⋊D9—82no—
113C3×D9—81no—
114C2×C3×Q8—91no—
115C2×C3×D4—91no—
116C2×D12—31no—
117C2×D12—92no—
118C2×D12—92no—
119C2×D12—182no—
120C2×D12—184no—
121C2×D12—182no—
122C2×D12—182no—
123C2×Dic6—31no—
124C2×C3×C8—32no—
125C3×C22⋊C4—182no—
126C3×C42—21no—
127C3×C42C2×D911yescharacteristic
128C3×C42D3211yescharacteristic
129#215—184no—
130#213—184no—
131(C2×D3)⋊C4—62no—
132C2×C3⋊C8C2×D912yes—
133D24—184no—
134#204—184no—
135C22×C32C2×D1211yescharacteristic
136C2×C3⋊D3—122no—
137C2×C3⋊D3—124no—
138C2×C3×D3—121no—
139C2×C3×D3—122no—
140C32×C4—22no—
141C32×C4C2×D1212yes—
142C3×Dic3—121no—
143C22×C9—21no—
144C22×C9C2×#6711yescharacteristic
145C2×D9—41no—
146C2×D9—42no—
147C2×D9—128no—
148C2×D9—124no—
149C2×D9—122no—
150C2×D9—124no—
151C4×C9—24no—
152C4×C9—22no—
153C4×C9—22no—
154C4×C9D3×D412yes—
155Dic9—41no—
156C2×SD16—272no—
157C2×D8—272no—
158#127—271no—
159#124—91no—
160#97—31no—
161C3×C9#49611yescharacteristic, complemented, Sylow
162C23×C3—181no—
163C22×D3—61no—
164C22×D3—184no—
165C22×D3—182no—
166C22×D3—182no—
167C3×Q8—182no—
168C3×D4—182no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—181no—
172C2×C3×C4D3612yes—
173C2×C3×C4C22×D911yescharacteristic
174C2×C3×C4#62612yes—
175C2×C3×C4C2×D3211yescharacteristic
176C2×Dic3—61no—
177D12—62no—
178D12—182no—
179D12—182no—
180D12—182no—
181D12—182no—
182D12—362no—
183D12—362no—
184D12—364no—
185D12—368no—
186D12—364no—
187Dic6—62no—
188C3×C8—34no—
189C3⋊C8D3614yescomplemented
190C2×C32#87411yescharacteristic
191C2×C32#90812yes—
192C3⋊D3—242no—
193C3×D3—241no—
194C2×C9—22no—
195C2×C9—21no—
196C2×C9#94011yescharacteristic
197C2×C9#94912yes—
198D9—81no—
199D9—244no—
200D9—242no—
201C2×Q8—91no—
202C2×D4—91no—
203C2×D4—272no—
204C2×D4—542no—
205SD16—544no—
206D8—544no—
207C2×C8—32no—
208C22⋊C4—182no—
209C42D3×D911yescharacteristic
210C22×C3—21no—
211C22×C3—361no—
212C22×C3—362no—
213C22×C3C2×D3611yescharacteristic
214C22×C3C2×#62611yescharacteristic
215C2×D3—122no—
216C2×D3—121no—
217C2×D3—364no—
218C2×D3—368no—
219C2×D3—364no—
220C2×D3—362no—
221C2×D3—362no—
222C2×D3—364no—
223C3×C4—24no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—361no—
228C3×C4C2×D3612yes—
229C3×C4D3×D1212yes—
230Dic3—121no—
231C32#471211yescharacteristic
232C9—21no—
233C9#505811yescharacteristic, complemented
234C23—181no—
235C23—542no—
236Q8—182no—
237D4—182no—
238D4—542no—
239D4—542no—
240D4—1084no—
241C2×C4—181no—
242C2×C4#619212yes—
243C2×C4C2×D3×D911yescharacteristic
244C8—34no—
245C2×C3—21no—
246C2×C3—22no—
247C2×C3—721no—
248C2×C3#6343611yescharacteristic
249C2×C3#6347012yes—
250C2×C3#6382512yes—
251C2×C3#6391111yescharacteristic
252D3—241no—
253D3—724no—
254D3—722no—
255D3—722no—
256C22—361no—
257C22—362no—
258C22—1082no—
259C22—1084no—
260C22C2×#619211yescharacteristic
261C4—22no—
262C4—361no—
263C4D3×D3612yes—
264C3—21no—
265C3#1058839311yescharacteristic, complemented
266C3#1059035111yescharacteristic
267C2—721no—
268C2—2162no—
269C2#1172087312yes—
270C2#1172100611yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×D12
hypercenterC42D3×D9
hypocenterC3×C9#496
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721711C2semidirect
2. C3×#63437C2semidirect
3. C2×#88417C2semidirect
4. C2×#88416C2semidirect
5. C9×#803C2semidirect
24. #63437D3semidirect
38. C9×C3⋊C8D4semidirect
80. #803D9semidirect
161. C3×C9#496semidirect
189. C3⋊C8D36semidirect
233. C9#5058semidirect
265. C3#10588393semidirect

Maximal subgroups

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

subgroupindex
#117217112
C3×#634372
C2×#884172
C2×#884162
C9×#8032
#105903513
#105883933