Finite Group Databasebeta

← #13161833

Subgroups — #13161833

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88813C212yesmaximal
3C3×#63658C212yesmaximal
4#11722082C212yesmaximal
5#10591252—31nomaximal
6#10591239—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2214yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C3×C4×C3⋊Dic3C414yescomplemented
12#88813C418yescomplemented
13C2×C4×C3⋊Dic3D311yescharacteristic
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C2×Dic32—34no—
17C2×Dic32—32no—
18#63658D312yes—
19C3×C3⋊#95—32no—
20C3×C3⋊#95—32no—
21C3×C3⋊#95—62no—
22#63579—34no—
23#63579—32no—
24C23×C33C2311yescharacteristic
25C2×C33×C4—22no—
26C2×C33×C4C2×C412yes—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3—24no—
29C2×C3×C3⋊Dic3—24no—
30C2×C3×C3⋊Dic3C2×C418yes—
31C2×C3×C3⋊Dic3C2×C412yes—
32C2×C3×C3⋊Dic3C2×C414yes—
33C2×C3×C3⋊Dic3C2×C412yes—
34#4932—91no—
35#4648—92no—
36C22×C32×C4—21no—
37C22×C32×C4—22no—
38C22×C32×C4—41no—
39C22×C32×C4C2×D312yes—
40C22×C32×C4C2×D311yescharacteristic
41C22×C3⋊Dic3C2×D314yes—
42C22×C3⋊Dic3C2×D311yescharacteristic
43C22×C3⋊Dic3C2×D311yescharacteristic
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—34no—
47C22×C3×Dic3—34no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—64no—
51C4×C3⋊Dic3Dic314yescomplemented
52C3×C4×Dic3—38no—
53C3×C4×Dic3—64no—
54Dic32—316no—
55Dic32—38no—
56C22×C33C22×C412yes—
57C22×C33C22×C411yescharacteristic
58C22×C33#4112yes—
59C22×C33#4112yes—
60C33×C4—22no—
61C3×C3⋊Dic3—22no—
62C3×C3⋊Dic3—28no—
63C3×C3⋊Dic3C4214yes—
64C2×C3×C42—91no—
65C2×C4×Dic3—32no—
66C2×C4×Dic3—61no—
67C2×C4×Dic3—94no—
68C2×C4×Dic3—92no—
69C3×#95—92no—
70C3⋊#95—32no—
71C3⋊#95—32no—
72C3⋊#95—62no—
73C3⋊#95—94no—
74C3⋊#95—92no—
75C23×C32—22no—
76C23×C32—21no—
77C23×C32—41no—
78C23×C32C22×D312yes—
79C23×C32C22×D311yescharacteristic
80C2×C32×C4—22no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4—24no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4—44no—
87C2×C32×C4—44no—
88C2×C32×C4C4×D314yes—
89C2×C32×C4C2×Dic312yes—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—24no—
92C2×C3⋊Dic3—24no—
93C2×C3⋊Dic3C4×D318yes—
94C2×C3⋊Dic3C4×D314yes—
95C2×C3⋊Dic3C2×Dic312yes—
96C2×C3⋊Dic3C2×Dic312yes—
97C2×C3×Dic3—38no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—38no—
100C2×C3×Dic3—38no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—68no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—124no—
114C2×C3×Dic3—124no—
115#376—271noSylow
116C2×C33C2×C4211yescharacteristic
117C2×C33#11714yes—
118C2×C33#11711yescharacteristic
119C2×C33#11711yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—41no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—94no—
126C22×C3×C4D3212yes—
127C22×C3×C4D3211yescharacteristic
128C22×Dic3—32no—
129C22×Dic3—34no—
130C22×Dic3—34no—
131C22×Dic3—32no—
132C22×Dic3—61no—
133C22×Dic3—61no—
134C22×Dic3—64no—
135C3×C42—94no—
136C4×Dic3—38no—
137C4×Dic3—64no—
138C4×Dic3—916no—
139C4×Dic3—98no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—22no—
143C22×C32—24no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—24no—
147C22×C32—22no—
148C22×C32—21no—
149C22×C32—41no—
150C22×C32—42no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32C2×C4×D314yes—
154C22×C32C2×C4×D312yes—
155C22×C32C2×C4×D312yes—
156C22×C32#23512yes—
157C22×C32#23512yes—
158C22×C32#23712yes—
159C22×C32#23714yes—
160C22×C32#23912yes—
161C22×C32C22×Dic311yescharacteristic
162C32×C4—22no—
163C32×C4—24no—
164C32×C4—24no—
165C32×C4—44no—
166C32×C4—44no—
167C3⋊Dic3—22no—
168C3⋊Dic3—28no—
169C3⋊Dic3C4×Dic314yes—
170C3×Dic3—38no—
171C3×Dic3—64no—
172C3×Dic3—64no—
173C3×Dic3—68no—
174C3×Dic3—64no—
175C3×Dic3—68no—
176C3×Dic3—128no—
177C2×C42—91no—
178C2×C42—272no—
179#95—92no—
180#95—272no—
181C33#37611yescharacteristic, complemented, Sylow
182C23×C3—21no—
183C23×C3—22no—
184C23×C3—41no—
185C23×C3C2×D3212yes—
186C23×C3C2×D3211yescharacteristic
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—44no—
193C2×C3×C4—44no—
194C2×C3×C4—42no—
195C2×C3×C4—94no—
196C2×C3×C4—92no—
197C2×C3×C4—98no—
198C2×C3×C4—92no—
199C2×C3×C4—182no—
200C2×C3×C4—184no—
201C2×C3×C4—184no—
202C2×C3×C4D3×Dic314yes—
203C2×C3×C4#62412yes—
204C2×Dic3—38no—
205C2×Dic3—34no—
206C2×Dic3—38no—
207C2×Dic3—38no—
208C2×Dic3—34no—
209C2×Dic3—64no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—68no—
213C2×Dic3—64no—
214C2×Dic3—62no—
215C2×Dic3—64no—
216C2×Dic3—64no—
217C2×Dic3—62no—
218C2×Dic3—64no—
219C2×Dic3—64no—
220C2×Dic3—124no—
221C2×Dic3—124no—
222C2×C32—24no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—24no—
227C2×C32—24no—
228C2×C32—21no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—44no—
235C2×C32C42×D312yes—
236C2×C32#87214yes—
237C2×C32#87714yes—
238C2×C32#87712yes—
239C2×C32#89214yes—
240C2×C32#89212yes—
241C2×C32C2×C4×Dic311yescharacteristic
242C2×C32#92611yescharacteristic
243C2×C32#92611yescharacteristic
244C22×C4—91no—
245C22×C4—94no—
246C22×C4—91no—
247C22×C4#111311yescharacteristic
248C42—94no—
249C42—278no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—24no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—24no—
258C22×C3—22no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3#379512yes—
264C22×C3#379612yes—
265C22×C3#379912yes—
266C22×C3C4×D3214yes—
267C22×C3C4×D3212yes—
268C22×C3C2×D3×Dic312yes—
269C22×C3#380414yes—
270C22×C3#380512yes—
271C22×C3C2×#62411yescharacteristic
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—22no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—94no—
278C3×C4—188no—
279C3×C4—182no—
280Dic3—38no—
281Dic3—64no—
282Dic3—64no—
283Dic3—64no—
284Dic3—68no—
285Dic3—68no—
286Dic3—128no—
287C32—22no—
288C32—21no—
289C32—41no—
290C32#464812yescomplemented
291C32#493211yescharacteristic, complemented
292C23C2×#111311yescharacteristic
293C2×C4—22no—
294C2×C4—94no—
295C2×C4—92no—
296C2×C4—98no—
297C2×C4—92no—
298C2×C4—182no—
299C2×C4—184no—
300C2×C4—184no—
301C2×C4#629412yes—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—21no—
310C2×C3—22no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—44no—
314C2×C3—41no—
315C2×C3#6383614yes—
316C2×C3#6384112yes—
317C2×C3#6384911yescharacteristic
318C2×C3C4×D3×Dic312yes—
319C2×C3#6387714yes—
320C2×C3#6387814yes—
321C2×C3C4×#62411yescharacteristic
322C2×C3#6395512yes—
323C2×C3#6395711yescharacteristic
324C22#8904112yes—
325C22C4×#111312yes—
326C22C2×#629411yescharacteristic
327C22#8904612yes—
328C4—22no—
329C4—94no—
330C4—182no—
331C4—188no—
332C3—22no—
333C3—21no—
334C3—41no—
335C3#1059123912yescomplemented
336C3#1059125211yescharacteristic, complemented
337C2#1172366711yescharacteristic
338C2C4×#629411yescharacteristic
339C2#1172368314yes—
340C2#1172371511yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC2×C33C2×C42
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4#1113
hypocenterC33#376
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
11. C3×C4×C3⋊Dic3C4semidirect
12. #88813C4semidirect
51. C4×C3⋊Dic3Dic3semidirect
181. C33#376semidirect
290. C32#4648semidirect
291. C32#4932semidirect
335. C3#10591239semidirect
336. C3#10591252semidirect

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×#888132
C3×#636582
#117220822
#105912523
#105912393