Finite Group Databasebeta

← #13173328

Subgroups — #13173328

Order 1728.

496 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2#11723104C212yesmaximal, complemented
3#11723103C212yesmaximal
4#11723102C212yesmaximal, complemented
5#10593731—31nomaximal
6C2×C3×C4×C3⋊D3C411yescharacteristic
7C2×C3×C4×C3⋊D3C411yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3⋊#622C2214yes—
10C33×C42C411yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C42×C3⋊D3D311yescharacteristic
15C3×C42×D3—62no—
16#63779—32no—
17#63778—32no—
18#63777—32no—
19C22×C3×C3⋊D3C2×C411yescharacteristic
20C2×C33×C4C2×C411yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3—22no—
27C3×C4×C3⋊D3—22no—
28C3×C4×C3⋊D3—22no—
29C3⋊#622—24no—
30#4926—91nomaximal
31C2×C4×C3⋊D3Dic311yescharacteristic
32C2×C4×C3⋊D3Dic311yescharacteristic
33C2×C4×C3⋊D3C2×D311yescharacteristic
34C2×C3×C4×D3—62no—
35C2×C3×C4×D3—62no—
36C2×C3×C4×D3—62no—
37C2×#622—34no—
38C32×C42—22no—
39C32×C42—42no—
40C32×C42Dic311yescharacteristic
41C4×C3⋊Dic3Dic311yescharacteristic
42C4×C3⋊Dic3C2×D311yescharacteristic
43C4×C3⋊Dic3C2×D311yescharacteristic
44C3×C4×Dic3—62no—
45C3×C4×Dic3—62no—
46C3×C4×Dic3—62no—
47C22×C33C22×C411yescharacteristic
48C2×C3×C3⋊D3—21no—
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3M1612yes—
51C33×C4—21no—
52C33×C4—21no—
53C33×C4—21no—
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3M1612yes—
56C3×C3⋊Dic3#4112yes—
57C2×C3×C42—91no—
58C42×D3—62no—
59#830—92no—
60C4⋊(C3⋊C8)—92no—
61#803—92no—
62C22×C3⋊D3C2×Dic311yescharacteristic
63C22×C3×D3—62no—
64C2×C32×C4—22no—
65C2×C32×C4—22no—
66C2×C32×C4—22no—
67C2×C32×C4—42no—
68C2×C32×C4—42no—
69C2×C32×C4—42no—
70C2×C32×C4C2×Dic311yescharacteristic
71C2×C32×C4C2×Dic311yescharacteristic
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C3⋊Dic3C2×Dic311yescharacteristic
74C2×C3⋊Dic3C2×Dic311yescharacteristic
75C2×C3⋊Dic3C22×D311yescharacteristic
76C4×C3⋊D3—22no—
77C4×C3⋊D3—22no—
78C4×C3⋊D3—22no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C3×C4×D3—68no—
83C3×C4×D3—68no—
84C3×C4×D3—68no—
85#622—64no—
86#432—271noSylow
87C2×C33#11711yescharacteristic
88C2×C33C2×M1611yescharacteristic
89C2×C33C2×M1611yescharacteristic
90C3×C3⋊D3—22no—
91C22×C3×C4—91no—
92C22×C3×C4—91no—
93C22×C3×C4—91no—
94C2×C4×D3—62no—
95C2×C4×D3—62no—
96C2×C4×D3—62no—
97C3×C42—22no—
98C3×C42—42no—
99C3×C42—91no—
100C3×C42—91no—
101C3×C42—91no—
102C3×C42#15711yescharacteristic
103C4×Dic3—62no—
104C4×Dic3—62no—
105C4×Dic3—62no—
106C2×C3⋊C8—94no—
107C22×C32—22no—
108C22×C32—42no—
109C22×C32C22×Dic311yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3#20812yes—
113C2×C3×D3—64no—
114C2×C3×D3—64no—
115C2×C3×D3—64no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—24no—
120C32×C4—24no—
121C32×C4—24no—
122C32×C4—44no—
123C32×C4—44no—
124C32×C4—44no—
125C3⋊Dic3—21no—
126C3⋊Dic3#20812yes—
127C3⋊Dic3#23512yes—
128C3×Dic3—64no—
129C3×Dic3—64no—
130C3×Dic3—64no—
131C2×C42—91no—
132C4⋊C8—272no—
133C22⋊C8—272no—
134#97—272no—
135C33#43211yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—62no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—182no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4C2×#15711yescharacteristic
151C2×C3×C4C2×#15711yescharacteristic
152C2×C3×C4C2×#15711yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C4×D3—68no—
157C4×D3—68no—
158C4×D3—68no—
159C3⋊C8—184no—
160C2×C32—22no—
161C2×C32—22no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32—42no—
165C2×C32—42no—
166C2×C32C2×#20811yescharacteristic
167C2×C32C2×#20811yescharacteristic
168C2×C32#92611yescharacteristic
169C3⋊D3—22no—
170C3×D3—68no—
171C22×C4—91no—
172C22×C4—91no—
173C22×C4—91no—
174C2×C8—274no—
175C42—91no—
176C42—91no—
177C42—91no—
178C42C3⋊#15711yescharacteristic
179C22×C3—22no—
180C22×C3—42no—
181C22×C3—92no—
182C22×C3—181no—
183C22×C3—181no—
184C22×C3C22×#15711yescharacteristic
185C2×D3—64no—
186C2×D3—64no—
187C2×D3—64no—
188C3×C4—24no—
189C3×C4—24no—
190C3×C4—24no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—21no—
194C3×C4—44no—
195C3×C4—44no—
196C3×C4—44no—
197C3×C4—92no—
198C3×C4—92no—
199C3×C4—181no—
200Dic3—64no—
201Dic3—64no—
202Dic3—64no—
203C32—22no—
204C32—42no—
205C32#492611yescharacteristic, complemented
206C23—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—182no—
211C2×C4—182no—
212C2×C4—182no—
213C2×C4C2×C3⋊#15711yescharacteristic
214C2×C4C2×C3⋊#15711yescharacteristic
215C2×C4C2×C3⋊#15711yescharacteristic
216C8—544no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—22no—
220C2×C3—42no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3—182no—
224C2×C3#6428311yescharacteristic
225C2×C3#6428311yescharacteristic
226C2×C3#6428611yescharacteristic
227D3—68no—
228C22—92no—
229C22—181no—
230C22—181no—
231C22C22×C3⋊#15711yescharacteristic
232C4—21no—
233C4—21no—
234C4—21no—
235C4—92no—
236C4—92no—
237C4—181no—
238C3—22no—
239C3—42no—
240C3#1059373111yescharacteristic, complemented
241C2—182no—
242C2#1172451111yescharacteristic
243C2#1172451111yescharacteristic
244C2#1172451411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C42C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC42C3⋊#157
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723104C2semidirect
4. #11723102C2semidirect
135. C33#432semidirect
205. C32#4926semidirect
240. C3#10593731semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
#117231042
#117231032
#117231022
#105937313
#49269