Finite Group Databasebeta

← #13173427

Subgroups — #13173427

Order 1728.

720 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3832C211yesmaximal, characteristic
2#11724514C212yesmaximal, complemented
3#11723099C214yesmaximal, complemented
4#10593832—31nomaximal
5C2×C33×Q8C411yescharacteristic, complemented
6C3×#3832—22no—
7C3×#3832—22no—
8C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×C4×C3⋊D3C412yescomplemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12#88994C2214yes—
13C4⋊(C3⋊#157)—22no—
14C4×C3⋊#157—22no—
15C2×#3832D311yescharacteristic
16C2×C3×#239—62no—
17#64286—32no—
18#63774—34no—
19C22×C3×C3⋊D3C2×C412yes—
20C22×C3×C3⋊D3C2311yescharacteristic
21C2×C3⋊#157—24no—
22C33×Q8—22no—
23C2×C33×C4C2×C412yes—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C3×C4⋊(C3⋊D3)—24no—
27C3×C4⋊(C3⋊D3)—22no—
28C3×C4×C3⋊D3—22no—
29C3×C4×C3⋊D3—22no—
30C3×C4×C3⋊D3D412yes—
31C3×C4×C3⋊D3D412yes—
32#5240—91nomaximal
33C2×C32×Q8—22no—
34C2×C32×Q8—42no—
35C2×C32×Q8Dic311yescharacteristic, complemented
36#3832—22no—
37#3832—22no—
38C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C4×C3⋊D3Dic312yescomplemented
41C2×C4×C3⋊D3C2×D311yescharacteristic
42C3×#239—68no—
43C3×#239—68no—
44C2×C3×D12—62no—
45C2×C3×D12—64no—
46C2×C3×C4×D3—64no—
47C2×C3×C4×D3—62no—
48C22⋊#157—34no—
49C4⋊#157—62no—
50C4×#157—62no—
51C22×C33C22×C411yescharacteristic
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3—41no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3C2×D412yes—
56C3⋊#157—44no—
57C33×C4—22no—
58C33×C4C22⋊C412yes—
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C22⋊C411yescharacteristic
61C2×C3×#41—91no—
62C2×#239—62no—
63#926—92no—
64#834—94no—
65C22×C3⋊D3C2×Dic312yes—
66C22×C3⋊D3C22×D311yescharacteristic
67C22×C3×D3—64no—
68C22×C3×D3—62no—
69C2×#157—64no—
70C32×Q8—22no—
71C32×Q8—28no—
72C32×Q8—48no—
73C2×C32×C4—24no—
74C2×C32×C4—22no—
75C2×C32×C4—44no—
76C2×C32×C4—42no—
77C2×C32×C4C2×Dic312yes—
78C2×C32×C4C2×Dic311yescharacteristic
79C2×C3⋊Dic3C2×Dic311yescharacteristic
80C4⋊(C3⋊D3)—24no—
81C4⋊(C3⋊D3)—22no—
82C4×C3⋊D3—22no—
83C4×C3⋊D3—22no—
84C4×C3⋊D3#6712yes—
85C4×C3⋊D3#6712yes—
86C2×C3×Dic3—62no—
87C3×D12—616no—
88C3×D12—68no—
89C3×C4×D3—64no—
90C3×C4×D3—64no—
91C3×C4×D3—68no—
92C3×C4×D3—68no—
93#410—271noSylow
94C2×C33—21no—
95C2×C33C2×C22⋊C411yescharacteristic
96C3×C3⋊D3—22no—
97C3×C3⋊D3—42no—
98C3×#41—182no—
99C3×#41—182no—
100C2×C3×Q8—22no—
101C2×C3×Q8—42no—
102C2×C3×Q8#15711yescharacteristic, complemented
103C2×C3×D4—92no—
104C2×C3×D4—91no—
105C22×C3×C4—91no—
106C22×C3×C4—92no—
107#239—68no—
108#239—68no—
109C2×D12—62no—
110C2×D12—64no—
111C2×C4×D3—64no—
112C2×C4×D3—62no—
113#217—94no—
114C4⋊Dic3—182no—
115C4×Dic3—182no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32C22×Dic311yescharacteristic
119C2×C3⋊D3—22no—
120C2×C3⋊D3—41no—
121C2×C3⋊D3—42no—
122C2×C3⋊D3C2×#6712yes—
123C2×C3×D3—64no—
124C2×C3×D3—68no—
125C2×C3×D3—124no—
126C2×C3×D3—128no—
127#157—124no—
128C32×C4—22no—
129C32×C4—28no—
130C32×C4—24no—
131C32×C4—48no—
132C32×C4—44no—
133C32×C4#21712yes—
134C3⋊Dic3#21711yescharacteristic
135C3⋊Dic3#21711yescharacteristic
136C3×Dic3—62no—
137C3×Dic3—62no—
138C2×#41—91no—
139#117—272no—
140#99—274no—
141C33#41011yescharacteristic, complemented, Sylow
142C23×C3—91no—
143C23×C3—92no—
144C22×D3—64no—
145C22×D3—62no—
146C3×Q8—28no—
147C3×Q8—22no—
148C3×Q8—48no—
149C3×D4—184no—
150C3×D4—182no—
151C2×C3×C4—24no—
152C2×C3×C4—22no—
153C2×C3×C4—44no—
154C2×C3×C4—42no—
155C2×C3×C4—91no—
156C2×C3×C4—92no—
157C2×C3×C4—92no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4C2×#15712yes—
161C2×C3×C4C2×#15711yescharacteristic
162C2×Dic3—62no—
163C2×Dic3—184no—
164D12—616no—
165D12—68no—
166C4×D3—64no—
167C4×D3—64no—
168C4×D3—68no—
169C4×D3—68no—
170C2×C32—21no—
171C2×C32—24no—
172C2×C32—22no—
173C2×C32—44no—
174C2×C32—42no—
175C2×C32C2×#21711yescharacteristic
176C3⋊D3—22no—
177C3⋊D3—42no—
178C3×D3—124no—
179C3×D3—128no—
180#41—182no—
181#41—182no—
182C2×Q8C3⋊#15711yescharacteristic, complemented
183C2×D4—92no—
184C2×D4—91no—
185C22×C4—91no—
186C22×C4—92no—
187C4⋊C4—542no—
188C22⋊C4—274no—
189C42—542no—
190C22×C3—22no—
191C22×C3—42no—
192C22×C3—92no—
193C22×C3—182no—
194C22×C3—361no—
195C22×C3—362no—
196C22×C3C22×#15711yescharacteristic
197C2×D3—64no—
198C2×D3—68no—
199C2×D3—124no—
200C2×D3—128no—
201C3×C4—28no—
202C3×C4—22no—
203C3×C4—24no—
204C3×C4—48no—
205C3×C4—44no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4C22⋊#15712yes—
209Dic3—62no—
210Dic3—62no—
211Dic3—364no—
212C32—22no—
213C32—42no—
214C32#524011yescharacteristic, complemented
215C23—91no—
216C23—92no—
217Q8—22no—
218D4—184no—
219D4—182no—
220C2×C4—91no—
221C2×C4—92no—
222C2×C4—92no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4—544no—
226C2×C4C2×C3⋊#15712yes—
227C2×C4C2×C3⋊#15711yescharacteristic
228C2×C3—24no—
229C2×C3—21no—
230C2×C3—22no—
231C2×C3—44no—
232C2×C3—42no—
233C2×C3—182no—
234C2×C3—362no—
235C2×C3C2×C22⋊#15711yescharacteristic
236D3—124no—
237D3—128no—
238C22—92no—
239C22—182no—
240C22—361no—
241C22—362no—
242C22C22×C3⋊#15711yescharacteristic
243C4—22no—
244C4—91no—
245C4—91no—
246C4—1084no—
247C4#8899412yes—
248C3—22no—
249C3—42no—
250C3#1059383211yescharacteristic, complemented
251C2—21no—
252C2—182no—
253C2—362no—
254C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3832C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. #11723099C2semidirect
5. C2×C33×Q8C4semidirect
8. C2×C3×C4⋊(C3⋊D3)C4semidirect
10. C2×C3×C4×C3⋊D3C4semidirect
35. C2×C32×Q8Dic3semidirect
38. C2×C4⋊(C3⋊D3)Dic3semidirect
40. C2×C4×C3⋊D3Dic3semidirect
102. C2×C3×Q8#157semidirect
141. C33#410semidirect
182. C2×Q8C3⋊#157semidirect
214. C32#5240semidirect
250. C3#10593832semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117245142
#117230992
#105938323
#52409