Finite Group Databasebeta

← #13173402

Subgroups — #13173402

Order 1728.

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

normal subgroups only

This group has 544 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×C4×C3⋊Dic3C211yesmaximal, characteristic
2#11723112C212yesmaximal
3#11723104C212yesmaximal, complemented
4#11723102C212yesmaximal, complemented
5#10593804—31nomaximal
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C3⋊#622C2214yes—
10C3×C4×C3⋊Dic3C411yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C4×C3⋊Dic3D311yescharacteristic
15C2×C3×C4×Dic3—62no—
16C22⋊#622—32no—
17#63779—32no—
18#63777—32no—
19C23×C33C2×C411yescharacteristic
20C2×C33×C4—22no—
21C2×C33×C4C2×C411yescharacteristic
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3⋊#622—24no—
30#4926—91nomaximal
31C22×C32×C4—22no—
32C22×C32×C4—42no—
33C22×C32×C4Dic311yescharacteristic
34C22×C3⋊Dic3Dic311yescharacteristic
35C22×C3⋊Dic3C2×D311yescharacteristic
36C22×C3×Dic3—62no—
37C22×C3×Dic3—62no—
38C2×#622—34no—
39C4×C3⋊Dic3Dic311yescharacteristic
40C4×C3⋊Dic3Dic311yescharacteristic
41C4×C3⋊Dic3C2×D311yescharacteristic
42C4×C3⋊Dic3C2×D311yescharacteristic
43C3×C4×Dic3—62no—
44C3×C4×Dic3—62no—
45C3×C4×Dic3—62no—
46C3×C4×Dic3—62no—
47C22×C33—21no—
48C22×C33—21no—
49C22×C33M1612yes—
50C22×C33C22×C411yescharacteristic
51C33×C4—21no—
52C33×C4—21no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3M1612yes—
56C3×C3⋊Dic3#4112yes—
57C2×C3×C42—91no—
58C2×C4×Dic3—62no—
59#830—92no—
60C4⋊(C3⋊C8)—92no—
61#803—92no—
62C23×C32—22no—
63C23×C32—42no—
64C23×C32C2×Dic311yescharacteristic
65C2×C32×C4—22no—
66C2×C32×C4—28no—
67C2×C32×C4—22no—
68C2×C32×C4—22no—
69C2×C32×C4—48no—
70C2×C32×C4—42no—
71C2×C32×C4—42no—
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C32×C4C2×Dic311yescharacteristic
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3C2×Dic311yescharacteristic
77C2×C3⋊Dic3C2×Dic311yescharacteristic
78C2×C3⋊Dic3C2×Dic311yescharacteristic
79C2×C3⋊Dic3C22×D311yescharacteristic
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—68no—
86#622—64no—
87#432—271noSylow
88C2×C33—22no—
89C2×C33#11711yescharacteristic
90C2×C33C2×M1611yescharacteristic
91C2×C33C2×M1611yescharacteristic
92C22×C3×C4—22no—
93C22×C3×C4—42no—
94C22×C3×C4—91no—
95C22×C3×C4—91no—
96C22×C3×C4#15711yescharacteristic
97C22×Dic3—62no—
98C22×Dic3—62no—
99C3×C42—91no—
100C3×C42—91no—
101C3×C42—91no—
102C3×C42—91no—
103C4×Dic3—62no—
104C4×Dic3—62no—
105C4×Dic3—62no—
106C4×Dic3—62no—
107C2×C3⋊C8—94no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—24no—
111C22×C32—24no—
112C22×C32—22no—
113C22×C32—24no—
114C22×C32—44no—
115C22×C32—44no—
116C22×C32—44no—
117C22×C32—42no—
118C22×C32#20812yes—
119C22×C32C22×Dic311yescharacteristic
120C32×C4—21no—
121C32×C4—21no—
122C32×C4—24no—
123C32×C4—24no—
124C32×C4—44no—
125C32×C4—44no—
126C3⋊Dic3—21no—
127C3⋊Dic3—21no—
128C3⋊Dic3#20812yes—
129C3⋊Dic3#23512yes—
130C3×Dic3—64no—
131C3×Dic3—64no—
132C3×Dic3—64no—
133C3×Dic3—64no—
134C2×C42—91no—
135C4⋊C8—272no—
136C22⋊C8—272no—
137#97—272no—
138C33#43211yescharacteristic, complemented, Sylow
139C23×C3—22no—
140C23×C3—42no—
141C23×C3C2×#15711yescharacteristic
142C2×C3×C4—28no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—48no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—182no—
154C2×C3×C4—182no—
155C2×C3×C4C2×#15711yescharacteristic
156C2×C3×C4C2×#15711yescharacteristic
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—68no—
162C2×Dic3—68no—
163C3⋊C8—184no—
164C2×C32—22no—
165C2×C32—28no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—48no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32C2×#20811yescharacteristic
174C2×C32C2×#20811yescharacteristic
175C2×C32#92611yescharacteristic
176C22×C4—91no—
177C22×C4—91no—
178C22×C4C3⋊#15711yescharacteristic
179C2×C8—274no—
180C42—91no—
181C42—91no—
182C42—91no—
183C42—91no—
184C22×C3—24no—
185C22×C3—24no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—22no—
189C22×C3—24no—
190C22×C3—44no—
191C22×C3—44no—
192C22×C3—42no—
193C22×C3—44no—
194C22×C3#379212yes—
195C22×C3C22×#15711yescharacteristic
196C3×C4—24no—
197C3×C4—24no—
198C3×C4—21no—
199C3×C4—21no—
200C3×C4—44no—
201C3×C4—44no—
202C3×C4—92no—
203C3×C4—92no—
204C3×C4—181no—
205C3×C4—181no—
206Dic3—64no—
207Dic3—64no—
208Dic3—64no—
209Dic3—64no—
210C32—22no—
211C32—42no—
212C32#492611yescharacteristic, complemented
213C23C2×C3⋊#15711yescharacteristic
214C2×C4—22no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—182no—
220C2×C4—182no—
221C2×C4C2×C3⋊#15711yescharacteristic
222C2×C4C2×C3⋊#15711yescharacteristic
223C8—544no—
224C2×C3—28no—
225C2×C3—22no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—48no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3#6428311yescharacteristic
234C2×C3#6428611yescharacteristic
235C2×C3C2×#379211yescharacteristic
236C22—21no—
237C22—21no—
238C22#8899312yes—
239C22C22×C3⋊#15711yescharacteristic
240C4—21no—
241C4—21no—
242C4—92no—
243C4—92no—
244C4—181no—
245C4—181no—
246C3—22no—
247C3—42no—
248C3#1059380411yescharacteristic, complemented
249C2—22no—
250C2#1172451111yescharacteristic
251C2#1172451411yescharacteristic
252C2C2×#8899311yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723104C2semidirect
4. #11723102C2semidirect
138. C33#432semidirect
212. C32#4926semidirect
248. C3#10593804semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
#117231122
#117231042
#117231022
#105938043
#49269