Finite Group Databasebeta

← #13138298

Subgroups — #13138298

Order 1728.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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#11721000C212yesmaximal
3C9×#818C211yesmaximal, characteristic, complemented
4C3×#63378C211yesmaximal, characteristic, complemented
5#11720263C212yesmaximal, complemented
6#10590687—31nomaximal
7D9×#107—31nomaximal
8C2×C4×C3⋊D9C2211yescharacteristic
9C2×(C3⋊D9)⋊C4—22no—
10C2×C3×C8×C9C2211yescharacteristic
11C2×C3⋊(C9⋊C8)C2211yescharacteristic
12C8×C3⋊D9C414yescomplemented
13C9×C4⋊Dic3C2212yes—
14C3×C4⋊Dic9C2212yes—
15#88442—22no—
16C2×C8×C3⋊D3—31no—
17#63900—32no—
18C3×#818—31no—
19C3×#818D311yescharacteristic
20#63576—32no—
21C2×C8×D9—31no—
22C2×C8×D9—61no—
23C4⋊C4×D9—32no—
24C9×#107—31no—
25#63378D311yescharacteristic, complemented
26#63366—32no—
27C22×C3⋊D9D411yescharacteristic
28C2×C3×C4×C9C2311yescharacteristic
29C2×C3⋊Dic9D411yescharacteristic
30C4×C3⋊D9C2×C412yes—
31C4×C3⋊D9Q812yes—
32C2×C9×Dic3—22no—
33C2×C3×Dic9—22no—
34(C3⋊D9)⋊C4—44no—
35C3×C8×C9C2×C412yescomplemented
36C3⋊(C9⋊C8)C2×C412yescomplemented
37D3×#107—91no—
38D3×#107—91no—
39C2×C4×C3⋊D3—31no—
40C2×#624—62no—
41C2×C32×C8C2×D311yescharacteristic
42C2×C3⋊(C3⋊C8)—31no—
43C8×C3⋊D3—34no—
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3C2×D312yes—
46#3724—62no—
47C2×C4×D9—31no—
48C2×C4×D9—61no—
49C2×C4×D9—62no—
50C2×C8×C9—21no—
51C2×C8×C9C2×D311yescharacteristic
52C9×C4⋊C4—32no—
53C4⋊Dic9C2×D312yes—
54Dic9⋊C4—62no—
55C2×C9⋊C8—31no—
56C2×C9⋊C8—61no—
57C8×D9—34no—
58C8×D9—64no—
59C22×C3×C9C2×D411yescharacteristic
60C2×C3⋊D9C4⋊C412yes—
61C2×C3⋊D9D812yes—
62C2×C3⋊D9Q1612yes—
63C3×C4×C9C22×C411yescharacteristic
64C3×C4×C9C2×Q811yescharacteristic
65C3⋊Dic9C4⋊C412yes—
66C9×Dic3—42no—
67C3×Dic9—42no—
68C2×D3×C8—91no—
69C2×D3×C8—91no—
70C2×D3×C8—181no—
71D3×C4⋊C4—92no—
72D3×C4⋊C4—92no—
73C3×#107—31no—
74C3×#107—91no—
75#818—31no—
76#818D911yescharacteristic, complemented
77#807—92no—
78#807—92no—
79C22×C3⋊D3—31no—
80C2×C32×C4C22×D311yescharacteristic
81C2×C3⋊Dic3—31no—
82C4×C3⋊D3—32no—
83C4×C3⋊D3—32no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—62no—
86#624—124no—
87C22×D9—31no—
88C22×D9—61no—
89C32×C8C4×D312yes—
90C3⋊(C3⋊C8)—32no—
91C2×C4×C9—21no—
92C2×C4×C9—62no—
93C2×C4×C9C22×D311yescharacteristic
94C2×Dic9—22no—
95C2×Dic9—31no—
96C2×Dic9—61no—
97C4×D9—32no—
98C4×D9—32no—
99C4×D9—62no—
100C4×D9—62no—
101C4×D9—124no—
102C8×C9—22no—
103C8×C9C4×D312yescomplemented
104C9⋊C8—32no—
105C9⋊C8—62no—
106C2×#107—271noSylow
107C2×C3×C9C2×C4⋊C411yescharacteristic
108C2×C3×C9C2×D811yescharacteristic
109C2×C3×C9C2×Q1611yescharacteristic
110C3⋊D9#10714yescomplemented
111C2×C4×D3—91no—
112C2×C4×D3—91no—
113C2×C4×D3—181no—
114C2×C4×D3—182no—
115C2×C4×D3—182no—
116C2×C3×C8—21no—
117C2×C3×C8C2×D911yescharacteristic
118C2×C3×C8D3211yescharacteristic
119C3×C4⋊C4—32no—
120C3×C4⋊C4—92no—
121C4⋊Dic3—32no—
122C4⋊Dic3C2×D912yes—
123Dic3⋊C4—182no—
124Dic3⋊C4—182no—
125C2×C3⋊C8—91no—
126C2×C3⋊C8—91no—
127C2×C3⋊C8—181no—
128D3×C8—94no—
129D3×C8—94no—
130D3×C8—184no—
131C22×C32D3×D411yescharacteristic
132C2×C3⋊D3—32no—
133C2×C3⋊D3—32no—
134C2×C3⋊D3—32no—
135C32×C4C2×C4×D311yescharacteristic
136C32×C4D3×Q811yescharacteristic
137C3⋊Dic3—32no—
138C3×Dic3—42no—
139C3×Dic3—122no—
140C22×C9—21no—
141C22×C9D3×D411yescharacteristic
142C2×D9—32no—
143C2×D9—32no—
144C2×D9—32no—
145C2×D9—62no—
146C2×D9—62no—
147C2×D9—62no—
148C4×C9—21no—
149C4×C9—21no—
150C4×C9—122no—
151C4×C9C2×C4×D311yescharacteristic
152C4×C9D3×Q811yescharacteristic
153Dic9—32no—
154Dic9—42no—
155Dic9—62no—
156C22×C8—271no—
157C2×C4⋊C4—272no—
158#107—31no—
159#107—91no—
160#107—272no—
161C3×C9C2×#10711yescharacteristic, complemented, Sylow
162C22×D3—91no—
163C22×D3—91no—
164C22×D3—181no—
165C2×C3×C4—21no—
166C2×C3×C4—62no—
167C2×C3×C4—182no—
168C2×C3×C4C22×D911yescharacteristic
169C2×C3×C4C2×D3211yescharacteristic
170C2×Dic3—22no—
171C2×Dic3—62no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—181no—
175C4×D3—92no—
176C4×D3—92no—
177C4×D3—92no—
178C4×D3—92no—
179C4×D3—182no—
180C4×D3—182no—
181C4×D3—364no—
182C4×D3—364no—
183C3×C8—22no—
184C3×C8C4×D912yescomplemented
185C3×C8#62412yes—
186C3⋊C8—92no—
187C3⋊C8—92no—
188C3⋊C8—182no—
189C2×C32D3×C4⋊C411yescharacteristic
190C2×C32D3×D811yescharacteristic
191C2×C32D3×Q1611yescharacteristic
192C3⋊D3—34no—
193C2×C9—21no—
194C2×C9—21no—
195C2×C9—21no—
196C2×C9D3×C4⋊C411yescharacteristic
197C2×C9D3×D811yescharacteristic
198C2×C9D3×Q1611yescharacteristic
199D9—34no—
200D9—64no—
201C22×C4—271no—
202C22×C4—542no—
203C2×C8—271no—
204C2×C8—274no—
205C2×C8D3×D911yescharacteristic
206C4⋊C4—32no—
207C4⋊C4—92no—
208C4⋊C4—542no—
209C22×C3—21no—
210C22×C3D4×D911yescharacteristic
211C22×C3#380211yescharacteristic
212C2×D3—92no—
213C2×D3—92no—
214C2×D3—92no—
215C2×D3—92no—
216C2×D3—92no—
217C2×D3—92no—
218C2×D3—182no—
219C2×D3—182no—
220C2×D3—182no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—122no—
224C3×C4—362no—
225C3×C4C2×C4×D911yescharacteristic
226C3×C4Q8×D911yescharacteristic
227C3×C4#379711yescharacteristic
228C3×C4C2×#62411yescharacteristic
229Dic3—42no—
230Dic3—92no—
231Dic3—92no—
232Dic3—122no—
233Dic3—182no—
234C32D3×#10711yescharacteristic
235C9—21no—
236C9D3×#10711yescharacteristic, complemented
237C23—271no—
238C2×C4—62no—
239C2×C4—182no—
240C2×C4—271no—
241C2×C4—272no—
242C2×C4—272no—
243C2×C4—1084no—
244C2×C4C2×D3×D911yescharacteristic
245C8—272no—
246C8(C3⋊D9)⋊C412yescomplemented
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3C4⋊C4×D911yescharacteristic
251C2×C3D8×D911yescharacteristic
252C2×C3Q16×D911yescharacteristic
253C2×C3#6379711yescharacteristic
254C2×C3#6380211yescharacteristic
255C2×C3#6390011yescharacteristic
256D3—94no—
257D3—94no—
258D3—184no—
259C22—272no—
260C22—272no—
261C22—272no—
262C22#8864611yescharacteristic
263C4—122no—
264C4—272no—
265C4—362no—
266C4(C3⋊D9)⋊Q811yescharacteristic
267C4C2×(C3⋊D9)⋊C411yescharacteristic
268C3—21no—
269C3D9×#10711yescharacteristic, complemented
270C3#1059068711yescharacteristic
271C2—274no—
272C2#1172081811yescharacteristic
273C2#1172082911yescharacteristic
274C2#1172100011yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C2×#107
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C9×#818C2semidirect
4. C3×#63378C2semidirect
5. #11720263C2semidirect
12. C8×C3⋊D9C4semidirect
25. #63378D3semidirect
35. C3×C8×C9C2×C4semidirect
36. C3⋊(C9⋊C8)C2×C4semidirect
76. #818D9semidirect
103. C8×C9C4×D3semidirect
110. C3⋊D9#107semidirect
161. C3×C9C2×#107semidirect
184. C3×C8C4×D9semidirect
236. C9D3×#107semidirect
246. C8(C3⋊D9)⋊C4semidirect
269. C3D9×#107semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊D92
#117210002
C9×#8182
C3×#633782
#117202632
#105906873
D9×#1073