Finite Group Databasebeta

← #13173422

Subgroups — #13173422

Order 1728.

952 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 89 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×D4×C3⋊D3C211yesmaximal, characteristic
2#11724511C212yesmaximal, complemented
3#11723113C214yesmaximal, complemented
4#10593825—31nomaximal
5C23×C3×C3⋊D3C412yes—
6C2×C33×D4C411yescharacteristic
7C2×C3×#638C2212yes—
8C3×D4×C3⋊D3—22no—
9C3×D4×C3⋊D3—22no—
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12#88993C2214yescomplemented
13#88989—22no—
14#88988—22no—
15C2×D4×C3⋊D3D311yescharacteristic
16C2×C3×D3×D4—62no—
17#64283—32no—
18#63788—34no—
19C23×C33C2×C412yes—
20C22×C3×C3⋊D3—22no—
21C22×C3×C3⋊D3—22no—
22C22×C3×C3⋊D3—44no—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C22×C3×C3⋊D3C2×C412yes—
25C33×D4—22no—
26C2×C33×C4C2×C411yescharacteristic
27C3×#638—24no—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4⋊(C3⋊D3)—22no—
30C3×C4×C3⋊D3D412yes—
31C3×C4×C3⋊D3D412yes—
32C3⋊#622—24no—
33C2×#833—91nomaximal
34C23×C3⋊D3Dic312yes—
35C23×C3×D3—64no—
36C2×C32×D4—22no—
37C2×C32×D4—42no—
38C2×C32×D4Dic311yescharacteristic
39C2×#638C2×D312yes—
40D4×C3⋊D3—22no—
41D4×C3⋊D3—22no—
42C2×C4⋊(C3⋊D3)Dic311yescharacteristic
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×C3×#67—64no—
45C3×D3×D4—68no—
46C3×D3×D4—68no—
47C2×C3×D12—62no—
48C2×C3×C4×D3—62no—
49#3792—34no—
50#3788—62no—
51#3787—62no—
52C22×C33—22no—
53C22×C33—42no—
54C22×C33C22×C411yescharacteristic
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3—44no—
58C2×C3×C3⋊D3—44no—
59C2×C3×C3⋊D3—42no—
60C2×C3×C3⋊D3C22⋊C411yescharacteristic
61C2×C3×C3⋊D3C22⋊C411yescharacteristic
62C33×C4C22⋊C412yes—
63C3×C3⋊Dic3C2×D412yes—
64C22×C3×D4—91no—
65C2×D3×D4—62no—
66C2×#208—92no—
67#833—94no—
68C23×C32—24no—
69C23×C32—44no—
70C23×C32C2×Dic312yes—
71C22×C3⋊D3—22no—
72C22×C3⋊D3—22no—
73C22×C3⋊D3—44no—
74C22×C3⋊D3C2×Dic311yescharacteristic
75C22×C3⋊D3C2×Dic312yes—
76C22×C3×D3—62no—
77C22×C3×D3—64no—
78C22×C3×D3—68no—
79C22×C3×D3—68no—
80C22×C3×D3—1216no—
81C32×D4—22no—
82C32×D4—28no—
83C32×D4—48no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4C2×Dic311yescharacteristic
87#638—24no—
88C2×C3⋊Dic3C22×D311yescharacteristic
89C4⋊(C3⋊D3)—22no—
90C4×C3⋊D3#6712yes—
91C4×C3⋊D3#6712yes—
92C3×#67—616no—
93C2×C3×Dic3—62no—
94C3×D12—68no—
95C3×C4×D3—64no—
96C3×C4×D3—64no—
97#622—64no—
98C2×#100—271noSylow
99C2×C33—21no—
100C2×C33—42no—
101C2×C33C2×C22⋊C411yescharacteristic
102C3×C3⋊D3—21no—
103C3×C3⋊D3—42no—
104C3×C3⋊D3#10012yescomplemented
105C24×C3—92no—
106C23×D3—64no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C2×C3×D4—92no—
110C2×C3×D4—91no—
111C2×C3×D4—182no—
112C2×C3×D4—182no—
113C2×C3×D4#15711yescharacteristic
114C22×C3×C4—91no—
115C2×#67—64no—
116D3×D4—68no—
117D3×D4—68no—
118C2×D12—62no—
119C2×C4×D3—62no—
120#208—94no—
121#208—182no—
122C2×C3⋊C8—182no—
123C22×C32—22no—
124C22×C32—28no—
125C22×C32—22no—
126C22×C32—48no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—48no—
130C22×C32—88no—
131C22×C32C22×Dic311yescharacteristic
132C2×C3⋊D3—22no—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—44no—
135C2×C3⋊D3—44no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3#21711yescharacteristic
138C2×C3⋊D3#21711yescharacteristic
139C2×C3×D3—62no—
140C2×C3×D3—62no—
141C2×C3×D3—68no—
142C2×C3×D3—68no—
143C2×C3×D3—1216no—
144C2×C3×D3—1216no—
145C2×C3×D3—128no—
146C32×C4—24no—
147C32×C4—44no—
148C32×C4#21712yes—
149C3⋊Dic3C2×#6712yes—
150C3×Dic3—64no—
151C22×D4—91no—
152C2×M16—272no—
153#100—274no—
154C33C2×#10011yescharacteristic, complemented, Sylow
155C23×C3—24no—
156C23×C3—44no—
157C23×C3—91no—
158C23×C3—92no—
159C23×C3—182no—
160C23×C3—182no—
161C23×C3—364no—
162C23×C3C2×#15712yes—
163C22×D3—62no—
164C22×D3—64no—
165C22×D3—68no—
166C22×D3—68no—
167C22×D3—1216no—
168C3×D4—28no—
169C3×D4—22no—
170C3×D4—48no—
171C3×D4—184no—
172C3×D4—182no—
173C2×C3×C4—22no—
174C2×C3×C4—42no—
175C2×C3×C4—91no—
176C2×C3×C4—92no—
177C2×C3×C4—92no—
178C2×C3×C4C2×#15711yescharacteristic
179#67—616no—
180C2×Dic3—62no—
181D12—68no—
182C4×D3—64no—
183C4×D3—64no—
184C3⋊C8—184no—
185C2×C32—21no—
186C2×C32—24no—
187C2×C32—22no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—48no—
192C2×C32—88no—
193C2×C32C2×#21711yescharacteristic
194C3⋊D3—21no—
195C3⋊D3—42no—
196C3⋊D3#83312yescomplemented
197C3×D3—64no—
198C3×D3—64no—
199C3×D3—128no—
200C24—92no—
201C2×D4—92no—
202C2×D4—91no—
203C2×D4—182no—
204C2×D4—182no—
205C2×D4C3⋊#15711yescharacteristic
206C22×C4—91no—
207M16—274no—
208M16—542no—
209C2×C8—542no—
210C22×C3—28no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—48no—
214C22×C3—42no—
215C22×C3—48no—
216C22×C3—42no—
217C22×C3—88no—
218C22×C3—91no—
219C22×C3—91no—
220C22×C3—182no—
221C22×C3—182no—
222C22×C3—362no—
223C22×C3—364no—
224C22×C3—364no—
225C22×C3C22×#15711yescharacteristic
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—68no—
229C2×D3—68no—
230C2×D3—1216no—
231C2×D3—1216no—
232C2×D3—128no—
233C3×C4—24no—
234C3×C4—44no—
235C3×C4—92no—
236C3×C4C22⋊#15712yes—
237Dic3—64no—
238C32—22no—
239C32—42no—
240C32C2×#83311yescharacteristic, complemented
241C23—91no—
242C23—92no—
243C23—182no—
244C23—182no—
245C23—364no—
246C23C2×C3⋊#15712yes—
247D4—22no—
248D4—184no—
249D4—182no—
250C2×C4—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4C2×C3⋊#15711yescharacteristic
254C8—544no—
255C2×C3—24no—
256C2×C3—21no—
257C2×C3—22no—
258C2×C3—44no—
259C2×C3—48no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—88no—
263C2×C3—92no—
264C2×C3—181no—
265C2×C3—362no—
266C2×C3C2×C22⋊#15711yescharacteristic
267D3—64no—
268D3—64no—
269D3—128no—
270C22—22no—
271C22—42no—
272C22—91no—
273C22—91no—
274C22—182no—
275C22—182no—
276C22—362no—
277C22—364no—
278C22—364no—
279C22C22×C3⋊#15711yescharacteristic
280C4—92no—
281C4#8899412yes—
282C3—22no—
283C3—42no—
284C3#1059382511yescharacteristic, complemented
285C2—21no—
286C2—42no—
287C2—92no—
288C2—181no—
289C2—362no—
290C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33C2×#100
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. #11724511C2semidirect
3. #11723113C2semidirect
12. #88993C22semidirect
104. C3×C3⋊D3#100semidirect
154. C33C2×#100semidirect
196. C3⋊D3#833semidirect
240. C32C2×#833semidirect
284. C3#10593825semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
#117245112
#117231132
#105938253
C2×#8339