Finite Group Databasebeta

← #13160610

Subgroups — #13160610

Order 1728.

1468 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C2×Dic3⋊(C3⋊Dic3)C212yesmaximal
3C2×Dic3×C3⋊Dic3C212yesmaximal
4#11722201C211yesmaximal, characteristic
5C3×#63658C211yesmaximal, characteristic
6C4×C4⋊(C3⋊Dic3)—31nomaximal
7#10591242—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C32×C4×Dic3C414yescomplemented
14Dic3⋊(C3⋊Dic3)C418yescomplemented
15Dic3×C3⋊Dic3—24no—
16C2×C32×C42—31no—
17C2×C4⋊(C3⋊Dic3)—32no—
18C2×C4×C3⋊Dic3—32no—
19C2×C3×C4×Dic3D314yes—
20C2×Dic3⋊Dic3—38no—
21C2×Dic32—38no—
22#63658—31no—
23#63658—34no—
24#63658—64no—
25#63658D311yescharacteristic
26C3×C3⋊#95—34no—
27C23×C33C2311yescharacteristic
28C2×C33×C4—22no—
29C2×C33×C4C2×C412yes—
30C2×C3⋊(C3⋊Dic3)—22no—
31C2×C3⋊(C3⋊Dic3)—22no—
32C2×C3⋊(C3⋊Dic3)C2×C414yes—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C414yes—
36C2×C32×Dic3C2×C414yes—
37C2×C32×Dic3C2×C412yes—
38C2×C32×Dic3C2×C412yes—
39C2×C32×Dic3D412yes—
40C2×C32×Dic3Q812yes—
41C4×C4⋊Dic3—94no—
42#4649—91no—
43C22×C32×C4—24no—
44C22×C32×C4—31no—
45C22×C32×C4—31no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D314yes—
48C22×C3⋊Dic3—32no—
49C22×C3⋊Dic3—38no—
50C22×C3⋊Dic3—68no—
51C22×C3⋊Dic3C2×D312yes—
52C22×C3×Dic3—38no—
53C22×C3×Dic3C2×D314yes—
54C22×C3×Dic3C2×D314yes—
55C32×C42—34no—
56C4⋊(C3⋊Dic3)—38no—
57C4×C3⋊Dic3—64no—
58C3×C4×Dic3Dic3116yescomplemented
59Dic3⋊Dic3—332no—
60Dic32—616no—
61C22×C33C22×C411yescharacteristic
62C22×C33C22×C412yes—
63C22×C33C2×D411yescharacteristic
64C22×C33C2×Q811yescharacteristic
65C22×C33#4112yes—
66C33×C4—22no—
67C3⋊(C3⋊Dic3)—24no—
68C3×C3⋊Dic3—24no—
69C32×Dic3C4214yescomplemented
70C32×Dic3C4⋊C414yescomplemented
71C2×C3×C42—34no—
72C2×C4⋊Dic3—98no—
73C2×Dic3⋊C4—92no—
74C2×C4×Dic3—98no—
75C2×C4×Dic3—92no—
76C2×C4×Dic3C3⋊D311yescharacteristic
77C3×#95—91no—
78C3⋊#95—34no—
79C3⋊#95—94no—
80C3⋊#95—91no—
81C3⋊#95—184no—
82C23×C32—24no—
83C23×C32C22×D314yes—
84C23×C32C22×D311yescharacteristic
85C2×C32×C4—22no—
86C2×C32×C4—28no—
87C2×C32×C4—28no—
88C2×C32×C4—34no—
89C2×C32×C4—32no—
90C2×C32×C4—32no—
91C2×C32×C4—32no—
92C2×C32×C4—32no—
93C2×C32×C4—48no—
94C2×C32×C4C4×D312yes—
95C2×C32×C4C2×Dic318yes—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—34no—
99C2×C3⋊Dic3—316no—
100C2×C3⋊Dic3—616no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—68no—
104C2×C3⋊Dic3—68no—
105C2×C3⋊Dic3—128no—
106C2×C3⋊Dic3—128no—
107C2×C3⋊Dic3C4×D314yes—
108C2×C3×Dic3—316no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3Dic618yes—
112C2×C3×Dic3C4×D3116yes—
113C2×C3×Dic3D1218yes—
114C2×C3×Dic3C2×Dic318yes—
115C2×C3×Dic3C2×Dic318yes—
116C4×C4⋊C4—271noSylow
117C2×C33C2×C4211yescharacteristic
118C2×C33C2×C4⋊C411yescharacteristic
119C2×C33#11711yescharacteristic
120C2×C33C4×D412yes—
121C2×C33C4×Q812yes—
122C22×C3×C4—24no—
123C22×C3×C4—34no—
124C22×C3×C4—34no—
125C22×C3×C4—92no—
126C22×C3×C4D3214yes—
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×Dic3—38no—
129C22×Dic3—98no—
130C22×Dic3—92no—
131C22×Dic3—188no—
132C22×Dic3C2×C3⋊D311yescharacteristic
133C22×Dic3C2×C3⋊D311yescharacteristic
134C3×C42—316no—
135C4⋊Dic3—932no—
136Dic3⋊C4—98no—
137C4×Dic3—1816no—
138C4×Dic3—184no—
139C4×Dic3C3⋊Dic314yescomplemented
140C22×C32—24no—
141C22×C32—28no—
142C22×C32—24no—
143C22×C32—24no—
144C22×C32—28no—
145C22×C32C2×Dic614yes—
146C22×C32C2×C4×D318yes—
147C22×C32C2×C4×D312yes—
148C22×C32C2×C4×D311yescharacteristic
149C22×C32C2×D1214yes—
150C22×C32#23518yes—
151C22×C32D3×D411yescharacteristic
152C22×C32#23712yes—
153C22×C32D3×Q811yescharacteristic
154C22×C32C22×Dic314yes—
155C32×C4—22no—
156C32×C4—28no—
157C32×C4—34no—
158C32×C4—34no—
159C32×C4—48no—
160C3⋊Dic3—24no—
161C3⋊Dic3—64no—
162C3⋊Dic3—616no—
163C3⋊Dic3—1216no—
164C3×Dic3—616no—
165C3×Dic3C4×Dic3116yescomplemented
166C3×Dic3C4⋊Dic3116yescomplemented
167C2×C4⋊C4—272no—
168C2×C42—31no—
169C2×C42—272no—
170#95—91no—
171#95—271no—
172C33C4×C4⋊C411yescharacteristic, complemented, Sylow
173C23×C3—24no—
174C23×C3C2×D3214yes—
175C23×C3C22×C3⋊D311yescharacteristic
176C2×C3×C4—28no—
177C2×C3×C4—22no—
178C2×C3×C4—28no—
179C2×C3×C4—38no—
180C2×C3×C4—316no—
181C2×C3×C4—38no—
182C2×C3×C4—38no—
183C2×C3×C4—38no—
184C2×C3×C4—48no—
185C2×C3×C4—94no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×C3×C4D3×Dic318yes—
189C2×C3×C4C2×C3⋊Dic312yes—
190C2×Dic3—316no—
191C2×Dic3—68no—
192C2×Dic3—68no—
193C2×Dic3—916no—
194C2×Dic3—94no—
195C2×Dic3—1816no—
196C2×Dic3—188no—
197C2×Dic3—188no—
198C2×Dic3—182no—
199C2×Dic3—182no—
200C2×Dic3—368no—
201C2×Dic3—368no—
202C2×Dic3#63412yes—
203C2×Dic3C4×C3⋊D314yes—
204C2×Dic3C4⋊(C3⋊D3)12yes—
205C2×Dic3C2×C3⋊Dic312yes—
206C2×Dic3C2×C3⋊Dic312yes—
207C2×C32—28no—
208C2×C32—24no—
209C2×C32—28no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32C4×Dic618yes—
213C2×C32C42×D311yescharacteristic
214C2×C32C4×D1218yes—
215C2×C32#87711yescharacteristic
216C2×C32#67⋊C412yes—
217C2×C32#88712yes—
218C2×C32D3×C4⋊C411yescharacteristic
219C2×C32C2×C4×Dic314yes—
220C2×C32C2×C4⋊Dic314yes—
221C2×C32#92614yes—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—92no—
225C22×C4—272no—
226C22×C4D3×C3⋊D311yescharacteristic
227C4⋊C4—278no—
228C42—34no—
229C42—544no—
230C22×C3—24no—
231C22×C3—28no—
232C22×C3—24no—
233C22×C3—24no—
234C22×C3—28no—
235C22×C3D3×Dic614yes—
236C22×C3C4×D3218yes—
237C22×C3D3×D1214yes—
238C22×C3C2×D3×Dic314yes—
239C22×C3#380418yes—
240C22×C3C2×#63411yescharacteristic
241C22×C3C2×C4×C3⋊D312yes—
242C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
243C22×C3#382812yes—
244C22×C3C22×C3⋊Dic311yescharacteristic
245C3×C4—22no—
246C3×C4—28no—
247C3×C4—316no—
248C3×C4—316no—
249C3×C4—48no—
250C3×C4—184no—
251Dic3—616no—
252Dic3—184no—
253Dic3—1816no—
254Dic3—3616no—
255Dic3C4×C3⋊Dic314yescomplemented
256Dic3C4⋊(C3⋊Dic3)14yescomplemented
257C32—24no—
258C32#464911yescharacteristic, complemented
259C32C4×C4⋊Dic314yescomplemented
260C23C2×D3×C3⋊D311yescharacteristic
261C2×C4—22no—
262C2×C4—32no—
263C2×C4—34no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—32no—
267C2×C4—94no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—274no—
271C2×C4—542no—
272C2×C4—542no—
273C2×C4D3×C3⋊Dic312yes—
274C2×C3—24no—
275C2×C3—28no—
276C2×C3—24no—
277C2×C3—28no—
278C2×C3—24no—
279C2×C3#6383718yes—
280C2×C3C4×D3×Dic314yes—
281C2×C3#6388018yes—
282C2×C3D3×C4⋊Dic314yes—
283C2×C3#6395514yes—
284C2×C3C4×#63412yes—
285C2×C3C4×C4⋊(C3⋊D3)12yes—
286C2×C3C2×C4×C3⋊Dic311yescharacteristic
287C2×C3C2×C4⋊(C3⋊Dic3)11yescharacteristic
288C2×C3#6413511yescharacteristic
289C22D3×#63411yescharacteristic
290C22C4×D3×C3⋊D312yes—
291C22D3×C4⋊(C3⋊D3)11yescharacteristic
292C22C2×D3×C3⋊Dic311yescharacteristic
293C22#8903112yes—
294C4—22no—
295C4—34no—
296C4—34no—
297C4—184no—
298C4—544no—
299C3—24no—
300C3#1059124214yescomplemented
301C3C4×C4⋊(C3⋊Dic3)11yescharacteristic, complemented
302C2#1172338912yes—
303C2C4×D3×C3⋊Dic311yescharacteristic
304C2#1172347112yes—
305C2D3×C4⋊(C3⋊Dic3)11yescharacteristic
306C2#1172359411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C32×C4×Dic3C4semidirect
14. Dic3⋊(C3⋊Dic3)C4semidirect
58. C3×C4×Dic3Dic3semidirect
69. C32×Dic3C42semidirect
70. C32×Dic3C4⋊C4semidirect
139. C4×Dic3C3⋊Dic3semidirect
165. C3×Dic3C4×Dic3semidirect
166. C3×Dic3C4⋊Dic3semidirect
172. C33C4×C4⋊C4semidirect
255. Dic3C4×C3⋊Dic3semidirect
256. Dic3C4⋊(C3⋊Dic3)semidirect
258. C32#4649semidirect
259. C32C4×C4⋊Dic3semidirect
300. C3#10591242semidirect
301. C3C4×C4⋊(C3⋊Dic3)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C2×Dic3⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
#117222012
C3×#636582
C4×C4⋊(C3⋊Dic3)3
#105912423