Finite Group Databasebeta

← #13173413

Subgroups — #13173413

Order 1728.

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

normal subgroups only

This group has 752 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×#3830C211yesmaximal, characteristic
2#11724514C212yesmaximal, complemented
3#11723111C214yesmaximal, complemented
4#10593818—31nomaximal
5C2×C33×D4C411yescharacteristic, complemented
6C2×C3×#638C2212yes—
7C22×C3×C3⋊Dic3C412yescomplemented
8C3×#3830—22no—
9C3×#3830—22no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C411yescharacteristic, complemented
12#88994C2214yes—
13C4⋊(C3⋊#157)—22no—
14C4×C3⋊#157—22no—
15C2×#3830D311yescharacteristic
16C2×C3×#237—62no—
17#64286—32no—
18#63786—34no—
19C23×C33C2×C412yes—
20C22×C3×C3⋊D3C2311yescharacteristic
21C2×C3⋊#157—24no—
22C33×D4—22no—
23C2×C33×C4C2×C411yescharacteristic
24C3×#638—24no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2×C412yes—
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C3×C4×C3⋊D3D412yes—
30C3×C4×C3⋊D3D412yes—
31C3×#634—22no—
32#5240—91nomaximal
33C2×C32×D4—22no—
34C2×C32×D4—42no—
35C2×C32×D4Dic311yescharacteristic, complemented
36C2×#638C2×D312yes—
37C22×C3⋊Dic3Dic312yescomplemented
38#3830—22no—
39#3830—22no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×#634Dic311yescharacteristic, complemented
42C2×C3×#67—64no—
43C22×C3×Dic3—64no—
44C3×#237—68no—
45C3×#237—68no—
46C2×C3×C4×D3—62no—
47C2×C3×Dic6—62no—
48C22⋊#157—34no—
49C4⋊#157—62no—
50C4×#157—62no—
51C22×C33—22no—
52C22×C33—42no—
53C22×C33C22×C411yescharacteristic
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3C2×D412yes—
56C3⋊#157—44no—
57C33×C4C22⋊C412yes—
58C3×C3⋊Dic3—22no—
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C22⋊C411yescharacteristic
61C2×C3×#41—91no—
62C2×#237—62no—
63#926—92no—
64#834—94no—
65C23×C32—24no—
66C23×C32—44no—
67C23×C32C2×Dic312yes—
68C22×C3⋊D3C22×D311yescharacteristic
69C22×C3×D3—62no—
70C2×#157—64no—
71C32×D4—22no—
72C32×D4—28no—
73C32×D4—48no—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4C2×Dic311yescharacteristic
77#638—24no—
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3C2×Dic312yes—
81C2×C3⋊Dic3C2×Dic311yescharacteristic
82C4×C3⋊D3#6712yes—
83C4×C3⋊D3#6712yes—
84#634—22no—
85C3×#67—616no—
86C2×C3×Dic3—64no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—68no—
89C2×C3×Dic3—68no—
90C3×C4×D3—64no—
91C3×C4×D3—64no—
92C3×Dic6—68no—
93#410—271noSylow
94C2×C33—21no—
95C2×C33—42no—
96C2×C33C2×C22⋊C411yescharacteristic
97C3×C3⋊D3—22no—
98C3×#41—182no—
99C3×#41—182no—
100C2×C3×Q8—91no—
101C2×C3×D4—22no—
102C2×C3×D4—42no—
103C2×C3×D4—92no—
104C2×C3×D4#15711yescharacteristic, complemented
105C22×C3×C4—91no—
106C22×C3×C4—92no—
107C2×#67—64no—
108C22×Dic3—64no—
109#237—68no—
110#237—68no—
111C2×C4×D3—62no—
112C2×Dic6—62no—
113#217—94no—
114C4⋊Dic3—182no—
115C4×Dic3—182no—
116C22×C32—22no—
117C22×C32—28no—
118C22×C32—22no—
119C22×C32—48no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32—48no—
123C22×C32—88no—
124C22×C32C22×Dic311yescharacteristic
125C2×C3⋊D3—41no—
126C2×C3⋊D3C2×#6712yes—
127C2×C3×D3—64no—
128C2×C3×D3—124no—
129#157—124no—
130C32×C4—24no—
131C32×C4—44no—
132C32×C4#21712yes—
133C3⋊Dic3—22no—
134C3⋊Dic3#21711yescharacteristic
135C3⋊Dic3#21711yescharacteristic
136C3×Dic3—62no—
137C3×Dic3—62no—
138C3×Dic3—68no—
139C2×#41—91no—
140#117—272no—
141#99—274no—
142C33#41011yescharacteristic, complemented, Sylow
143C23×C3—24no—
144C23×C3—44no—
145C23×C3—91no—
146C23×C3C2×#15712yes—
147C22×D3—62no—
148C3×Q8—182no—
149C3×D4—28no—
150C3×D4—22no—
151C3×D4—48no—
152C3×D4—184no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—91no—
156C2×C3×C4—92no—
157C2×C3×C4—92no—
158C2×C3×C4—92no—
159C2×C3×C4—182no—
160C2×C3×C4—182no—
161C2×C3×C4C2×#15711yescharacteristic
162#67—616no—
163C2×Dic3—64no—
164C2×Dic3—62no—
165C2×Dic3—68no—
166C2×Dic3—68no—
167C2×Dic3—184no—
168C4×D3—64no—
169C4×D3—64no—
170Dic6—68no—
171C2×C32—21no—
172C2×C32—24no—
173C2×C32—22no—
174C2×C32—44no—
175C2×C32—42no—
176C2×C32—42no—
177C2×C32—48no—
178C2×C32—88no—
179C2×C32C2×#21711yescharacteristic
180C3⋊D3—22no—
181C3×D3—124no—
182#41—182no—
183#41—182no—
184C2×Q8—91no—
185C2×D4—92no—
186C2×D4C3⋊#15711yescharacteristic, complemented
187C22×C4—91no—
188C22×C4—92no—
189C4⋊C4—542no—
190C22⋊C4—274no—
191C42—542no—
192C22×C3—28no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—48no—
196C22×C3—42no—
197C22×C3—48no—
198C22×C3—42no—
199C22×C3—88no—
200C22×C3—92no—
201C22×C3—361no—
202C22×C3C22×#15711yescharacteristic
203C2×D3—64no—
204C2×D3—124no—
205C3×C4—24no—
206C3×C4—44no—
207C3×C4—91no—
208C3×C4—91no—
209C3×C4—182no—
210C3×C4C22⋊#15712yes—
211Dic3—62no—
212Dic3—62no—
213Dic3—68no—
214Dic3—364no—
215C32—22no—
216C32—42no—
217C32#524011yescharacteristic, complemented
218C23—91no—
219C23C2×C3⋊#15712yes—
220Q8—182no—
221D4—22no—
222D4—184no—
223C2×C4—91no—
224C2×C4—92no—
225C2×C4—92no—
226C2×C4—92no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—544no—
230C2×C4C2×C3⋊#15711yescharacteristic
231C2×C3—24no—
232C2×C3—21no—
233C2×C3—22no—
234C2×C3—44no—
235C2×C3—48no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—88no—
239C2×C3—182no—
240C2×C3C2×C22⋊#15711yescharacteristic
241D3—124no—
242C22—22no—
243C22—42no—
244C22—92no—
245C22—361no—
246C22C22×C3⋊#15711yescharacteristic
247C4—91no—
248C4—91no—
249C4—182no—
250C4—1084no—
251C4#8899412yes—
252C3—22no—
253C3—42no—
254C3#1059381811yescharacteristic, complemented
255C2—21no—
256C2—42no—
257C2—182no—
258C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. #11723111C2semidirect
5. C2×C33×D4C4semidirect
7. C22×C3×C3⋊Dic3C4semidirect
11. C2×C3×#634C4semidirect
35. C2×C32×D4Dic3semidirect
37. C22×C3⋊Dic3Dic3semidirect
41. C2×#634Dic3semidirect
104. C2×C3×D4#157semidirect
142. C33#410semidirect
186. C2×D4C3⋊#157semidirect
217. C32#5240semidirect
254. C3#10593818semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
#117245142
#117231112
#105938183
#52409