Finite Group Databasebeta

← #13160582

Subgroups — #13160582

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×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88807C212yesmaximal
3C2×Dic3×C3⋊Dic3C212yesmaximal
4#11722201C211yesmaximal, characteristic
5C32×C3⋊#95C211yesmaximal, characteristic
6#10592348—31nomaximal
7#10591242—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C3×C4×C3⋊Dic3C414yescomplemented
14#88807C418yescomplemented
15Dic3×C3⋊Dic3—24no—
16C2×#3750—32no—
17C2×C4×C3⋊Dic3—32no—
18C2×C4×C3⋊Dic3D311yescharacteristic
19C2×C3×C4×Dic3—34no—
20C2×Dic3⋊Dic3—38no—
21C2×Dic32—38no—
22C32×#95—31no—
23#63658—31no—
24#63658—34no—
25#63658—64no—
26C3×C3⋊#95D314yes—
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⋊Dic3C2×C414yes—
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3D412yes—
37C2×C3×C3⋊Dic3Q812yes—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3C2×C414yes—
41C4×C4⋊Dic3—91no—
42#4649—94no—
43C22×C32×C4—24no—
44C22×C32×C4—32no—
45C22×C32×C4C2×D314yes—
46C22×C32×C4C2×D311yescharacteristic
47C22×C3⋊Dic3—32no—
48C22×C3⋊Dic3—38no—
49C22×C3⋊Dic3—68no—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3×Dic3—34no—
53C22×C3×Dic3—34no—
54C22×C3×Dic3C2×D318yes—
55#3750—38no—
56C4×C3⋊Dic3—64no—
57C4×C3⋊Dic3Dic314yescomplemented
58C3×C4×Dic3—316no—
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⋊Dic3C4214yescomplemented
69C3×C3⋊Dic3C4⋊C414yescomplemented
70C32×Dic3—24no—
71C2×C3×C42—91no—
72C2×C4⋊Dic3—92no—
73C2×Dic3⋊C4—98no—
74C2×C4×Dic3—34no—
75C2×C4×Dic3—98no—
76C2×C4×Dic3—92no—
77C3×#95—34no—
78C3⋊#95—94no—
79C3⋊#95—91no—
80C3⋊#95—184no—
81C3⋊#95C3⋊D311yescharacteristic
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—48no—
90C2×C32×C4—62no—
91C2×C32×C4—62no—
92C2×C32×C4C4×D318yes—
93C2×C32×C4C2×Dic312yes—
94C2×C3⋊Dic3—34no—
95C2×C3⋊Dic3—316no—
96C2×C3⋊Dic3—616no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—68no—
100C2×C3⋊Dic3—68no—
101C2×C3⋊Dic3—128no—
102C2×C3⋊Dic3—128no—
103C2×C3⋊Dic3Dic612yes—
104C2×C3⋊Dic3C4×D314yes—
105C2×C3⋊Dic3D1212yes—
106C2×C3⋊Dic3C2×Dic312yes—
107C2×C3⋊Dic3C2×Dic312yes—
108C2×C3×Dic3—28no—
109C2×C3×Dic3—28no—
110C2×C3×Dic3—316no—
111C2×C3×Dic3—38no—
112C2×C3×Dic3—38no—
113C2×C3×Dic3—38no—
114C2×C3×Dic3—38no—
115C2×C3×Dic3C4×D3116yes—
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—38no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4D3214yes—
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×Dic3—34no—
129C22×Dic3—34no—
130C22×Dic3—98no—
131C22×Dic3—92no—
132C22×Dic3—188no—
133C22×Dic3C2×C3⋊D312yes—
134C3×C42—94no—
135C4⋊Dic3—98no—
136Dic3⋊C4—932no—
137C4×Dic3—316no—
138C4×Dic3—184no—
139C4×Dic3—1816no—
140C22×C32—24no—
141C22×C32—28no—
142C22×C32—24no—
143C22×C32—24no—
144C22×C32—28no—
145C22×C32C2×Dic611yescharacteristic
146C22×C32C2×C4×D318yes—
147C22×C32C2×C4×D314yes—
148C22×C32C2×C4×D312yes—
149C22×C32C2×D1211yescharacteristic
150C22×C32#23512yes—
151C22×C32D3×D414yes—
152C22×C32#23718yes—
153C22×C32D3×Q814yes—
154C22×C32C22×Dic311yescharacteristic
155C32×C4—22no—
156C32×C4—28no—
157C32×C4—48no—
158C32×C4—64no—
159C3⋊Dic3—64no—
160C3⋊Dic3—616no—
161C3⋊Dic3—1216no—
162C3⋊Dic3C4×Dic314yescomplemented
163C3⋊Dic3C4⋊Dic314yescomplemented
164C3×Dic3—216no—
165C3×Dic3—316no—
166C3×Dic3—316no—
167C2×C4⋊C4—272no—
168C2×C42—91no—
169C2×C42—272no—
170#95—31no—
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—316no—
180C2×C3×C4—48no—
181C2×C3×C4—68no—
182C2×C3×C4—68no—
183C2×C3×C4—94no—
184C2×C3×C4—92no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4D3×Dic318yes—
189C2×C3×C4C4×C3⋊D312yes—
190C2×Dic3—22no—
191C2×Dic3—22no—
192C2×Dic3—38no—
193C2×Dic3—316no—
194C2×Dic3—38no—
195C2×Dic3—38no—
196C2×Dic3—38no—
197C2×Dic3—916no—
198C2×Dic3—94no—
199C2×Dic3—1816no—
200C2×Dic3—188no—
201C2×Dic3—188no—
202C2×Dic3—182no—
203C2×Dic3—182no—
204C2×Dic3—368no—
205C2×Dic3—368no—
206C2×Dic3C4×C3⋊D314yes—
207C2×C32—28no—
208C2×C32—24no—
209C2×C32—28no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32C4×Dic612yes—
213C2×C32C42×D314yes—
214C2×C32C4×D1212yes—
215C2×C32#87714yes—
216C2×C32#67⋊C418yes—
217C2×C32#88718yes—
218C2×C32D3×C4⋊C414yes—
219C2×C32C2×C4×Dic311yescharacteristic
220C2×C32C2×C4⋊Dic311yescharacteristic
221C2×C32#92611yescharacteristic
222C22×C4—32no—
223C22×C4—91no—
224C22×C4—91no—
225C22×C4—272no—
226C22×C4D3×C3⋊D311yescharacteristic
227C4⋊C4—278no—
228C42—94no—
229C42—544no—
230C22×C3—24no—
231C22×C3—28no—
232C22×C3—24no—
233C22×C3—28no—
234C22×C3—24no—
235C22×C3D3×Dic614yes—
236C22×C3C4×D3218yes—
237C22×C3D3×D1214yes—
238C22×C3C2×D3×Dic314yes—
239C22×C3#380418yes—
240C22×C3C2×C4×C3⋊D312yes—
241C22×C3C2×C4×C3⋊D311yescharacteristic
242C22×C3D4×C3⋊D311yescharacteristic
243C22×C3#383012yes—
244C22×C3Q8×C3⋊D311yescharacteristic
245C3×C4—22no—
246C3×C4—28no—
247C3×C4—48no—
248C3×C4—616no—
249C3×C4—94no—
250C3×C4—94no—
251Dic3—24no—
252Dic3—316no—
253Dic3—316no—
254Dic3—184no—
255Dic3—1816no—
256Dic3—3616no—
257C32—24no—
258C32#464914yescomplemented
259C32C4×C4⋊Dic311yescharacteristic, complemented
260C23C2×D3×C3⋊D311yescharacteristic
261C2×C4—22no—
262C2×C4—34no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—92no—
266C2×C4—94no—
267C2×C4—92no—
268C2×C4—92no—
269C2×C4—92no—
270C2×C4—274no—
271C2×C4—542no—
272C2×C4—542no—
273C2×C4Dic3×C3⋊D312yes—
274C2×C3—24no—
275C2×C3—28no—
276C2×C3—28no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3#6383718yes—
280C2×C3C4×D3×Dic314yes—
281C2×C3#6388018yes—
282C2×C3D3×C4⋊Dic314yes—
283C2×C3#6395514yes—
284C2×C3C42×C3⋊D311yescharacteristic
285C2×C3#6408611yescharacteristic
286C2×C3#6409012yes—
287C2×C3#6409612yes—
288C2×C3C4⋊C4×C3⋊D311yescharacteristic
289C22C3⋊D3×Dic611yescharacteristic
290C22C4×D3×C3⋊D312yes—
291C22C3⋊D3×D1211yescharacteristic
292C22#8902812yes—
293C22C2×Dic3×C3⋊D311yescharacteristic
294C4—22no—
295C4—64no—
296C4—94no—
297C4—94no—
298C4—544no—
299C3—24no—
300C3#1059124214yescomplemented
301C3#1059234811yescharacteristic, complemented
302C2#1172339012yes—
303C2#1172346112yes—
304C2C4×Dic3×C3⋊D311yescharacteristic
305C2C3⋊D3×C4⋊Dic311yescharacteristic
306C2#1172359511yescharacteristic
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. C3×C4×C3⋊Dic3C4semidirect
14. #88807C4semidirect
57. C4×C3⋊Dic3Dic3semidirect
68. C3×C3⋊Dic3C42semidirect
69. C3×C3⋊Dic3C4⋊C4semidirect
162. C3⋊Dic3C4×Dic3semidirect
163. C3⋊Dic3C4⋊Dic3semidirect
172. C33C4×C4⋊C4semidirect
258. C32#4649semidirect
259. C32C4×C4⋊Dic3semidirect
300. C3#10591242semidirect
301. C3#10592348semidirect

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
C2×#888072
C2×Dic3×C3⋊Dic32
#117222012
C32×C3⋊#952
#105923483
#105912423