Finite Group Databasebeta

← #13160319

Subgroups — #13160319

Order 1728.

1252 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C22×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
3C32×#926C211yesmaximal, characteristic, complemented
4#11722039C212yesmaximal, complemented
5#11722033C212yesmaximal, complemented
6#10592701—31nomaximal
7#10591083—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×C3×#638—21no—
10C3×#3828—22no—
11C3×#3828C414yescomplemented
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×#634C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
17C32×#217—21no—
18C32×C4⋊Dic3C2212yes—
19C32×C4×Dic3—21no—
20C2×#3828D311yescharacteristic
21C2×C3×#235—34no—
22C32×#117—31no—
23C22×C3⋊(C3⋊C8)—31no—
24C22×C3⋊(C3⋊C8)—34no—
25C22×C3⋊(C3⋊C8)—64no—
26C3×#926D314yescomplemented
27#63637—32no—
28#63636—32no—
29#63565—38no—
30#63562—38no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4D412yes—
36C2×C33×C4C2311yescharacteristic
37C3×#638—24no—
38C2×C3×C3⋊Dic3—21no—
39C3×C4⋊(C3⋊D3)—21no—
40C3×C4⋊(C3⋊D3)C2×C412yes—
41C3×C4×C3⋊D3—24no—
42C3×#634—21no—
43C3×#634C2×C412yes—
44C2×C32×Dic3—22no—
45C3⋊(C3⋊(C3⋊C8))—22no—
46#5233—91no—
47#5002—94no—
48C22×C32×C4—24no—
49C22×C32×C4C2×D314yes—
50C22×C32×C4C2×D311yescharacteristic
51C2×#638—21no—
52#3828—22no—
53#3828Dic314yescomplemented
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634C2×D311yescharacteristic
57C2×C3×#67—64no—
58C3×#235—316no—
59C3×#235—68no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—64no—
62C2×C3×Dic6—34no—
63C32×C4⋊C4—32no—
64C32×C22⋊C4—61no—
65C32×C42—61no—
66C2×C3⋊(C3⋊C8)—32no—
67C2×C3⋊(C3⋊C8)—38no—
68C2×C3⋊(C3⋊C8)—68no—
69C2×C3⋊(C3⋊C8)—616no—
70C2×C3⋊(C3⋊C8)—68no—
71C2×C3⋊(C3⋊C8)—62no—
72C3×#217—24no—
73C3×C4⋊Dic3C2×D318yes—
74C3×C4×Dic3—24no—
75C22×C33—22no—
76C22×C33C22⋊C412yes—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—42no—
80C33×C4C22⋊C412yes—
81C33×C4C22×C411yescharacteristic
82C33×C4C2×D411yescharacteristic
83C3×C3⋊Dic3—22no—
84C32×Dic3—42no—
85C2×C3×#41—91no—
86C2×#235—34no—
87C3×#117—34no—
88#926C3⋊D311yescharacteristic, complemented
89C22×C3⋊C8—94no—
90C22×C3⋊C8—91no—
91C22×C3⋊C8—184no—
92#835—92no—
93#832—92no—
94#810—98no—
95#809—98no—
96C23×C32—24no—
97C23×C32#6714yes—
98C23×C32#6711yescharacteristic
99C22×C3⋊D3—21no—
100C22×C3×D3—64no—
101C2×C32×C4—28no—
102C2×C32×C4—24no—
103C2×C32×C4—28no—
104C2×C32×C4—24no—
105C2×C32×C4—62no—
106C2×C32×C4C4×D318yes—
107C2×C32×C4D1218yes—
108C2×C32×C4C2×Dic312yes—
109C2×C32×C4#6714yes—
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4#6712yes—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114#638—24no—
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)C2×Dic312yes—
118C4×C3⋊D3—24no—
119#634—21no—
120#634C2×Dic312yes—
121C3×#67—616no—
122C2×C3×Dic3—28no—
123C2×C3×Dic3—64no—
124C3×D12—38no—
125C3×D12—64no—
126C3×C4×D3—616no—
127C3×Dic6—38no—
128C3×Dic6—64no—
129C3⋊(C3⋊C8)—616no—
130C3⋊(C3⋊C8)—68no—
131C3⋊(C3⋊C8)—62no—
132#416—271noSylow
133C2×C33—22no—
134C2×C33C2×C22⋊C411yescharacteristic
135C2×C33#13512yes—
136C3×C3⋊D3—42no—
137C3×#41—94no—
138C3×#41—182no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C2×C3×D4—181no—
142C22×C3×C4—24no—
143C22×C3×C4—181no—
144C22×C3×C4D3214yes—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C2×#67—64no—
147#235—316no—
148#235—68no—
149C2×D12—34no—
150C2×C4×D3—64no—
151C2×Dic6—34no—
152C3×C4⋊C4—38no—
153C3×C22⋊C4—64no—
154C3×C42—64no—
155#217—21no—
156C4⋊Dic3C2×C3⋊D312yes—
157C4×Dic3—21no—
158C2×C3⋊C8—98no—
159C2×C3⋊C8—92no—
160C2×C3⋊C8—188no—
161C2×C3⋊C8—1816no—
162C2×C3⋊C8—188no—
163C2×C3⋊C8—182no—
164C22×C32—216no—
165C22×C32—28no—
166C22×C32—22no—
167C22×C32—28no—
168C22×C32—24no—
169C22×C32(C2×D3)⋊C418yes—
170C22×C32#21712yes—
171C22×C32C2×#6714yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3⋊D3—22no—
174C2×C3⋊D3—42no—
175C2×C3×D3—68no—
176C2×C3×D3—128no—
177C32×C4—28no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—122no—
181C32×C4(C2×D3)⋊C418yes—
182C32×C4#21712yes—
183C32×C4C2×C4×D314yes—
184C32×C4C2×D1214yes—
185C32×C4C22×Dic311yescharacteristic
186C32×C4C2×#6711yescharacteristic
187C3⋊Dic3—22no—
188C3×Dic3—48no—
189C3×Dic3—68no—
190C2×#41—91no—
191C22×C8—271no—
192#117—31no—
193#103—272no—
194#102—272no—
195C33#41611yescharacteristic, complemented, Sylow
196C23×C3—24no—
197C23×C3—181no—
198C23×C3#62514yes—
199C23×C3#63811yescharacteristic
200C22×D3—64no—
201C3×Q8—92no—
202C3×Q8—181no—
203C3×D4—92no—
204C3×D4—184no—
205C3×D4—181no—
206C2×C3×C4—28no—
207C2×C3×C4—24no—
208C2×C3×C4—28no—
209C2×C3×C4—24no—
210C2×C3×C4—68no—
211C2×C3×C4—181no—
212C2×C3×C4—184no—
213C2×C3×C4D3×Dic318yes—
214C2×C3×C4#62514yes—
215C2×C3×C4#62618yes—
216C2×C3×C4C4×C3⋊D312yes—
217C2×C3×C4C4⋊(C3⋊D3)12yes—
218C2×C3×C4#63811yescharacteristic
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221#67—616no—
222C2×Dic3—22no—
223C2×Dic3—64no—
224D12—38no—
225D12—64no—
226C4×D3—616no—
227Dic6—38no—
228Dic6—64no—
229C3⋊C8—1816no—
230C3⋊C8—188no—
231C3⋊C8—182no—
232C2×C32—216no—
233C2×C32—22no—
234C2×C32—28no—
235C2×C32—28no—
236C2×C32—24no—
237C2×C32C2×(C2×D3)⋊C414yes—
238C2×C32#95018yes—
239C2×C32#95012yes—
240C2×C32C2×#21711yescharacteristic
241C3⋊D3—42no—
242C3×D3—128no—
243#41—94no—
244#41—182no—
245C2×Q8—91no—
246C2×D4—91no—
247C2×D4—181no—
248C22×C4—181no—
249C22×C4D3×C3⋊D311yescharacteristic
250C2×C8—272no—
251C2×C8—542no—
252C4⋊C4—32no—
253C22⋊C4—61no—
254C42—61no—
255C22×C3—216no—
256C22×C3—28no—
257C22×C3—22no—
258C22×C3—24no—
259C22×C3—28no—
260C22×C3—182no—
261C22×C3—362no—
262C22×C3(C2×D3)⋊Dic318yes—
263C22×C3#375212yes—
264C22×C3C2×#62514yes—
265C22×C3C2×#63811yescharacteristic
266C2×D3—68no—
267C2×D3—128no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—28no—
271C3×C4—128no—
272C3×C4—182no—
273C3×C4(C2×D3)⋊Dic318yes—
274C3×C4#375212yes—
275C3×C4C2×D3×Dic314yes—
276C3×C4C2×#62614yes—
277C3×C4C2×C4×C3⋊D311yescharacteristic
278C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
279Dic3—42no—
280Dic3—68no—
281C32—24no—
282C32#500214yescomplemented
283C32#523311yescharacteristic, complemented
284C23—181no—
285C23#629011yescharacteristic
286Q8—92no—
287Q8—181no—
288D4—92no—
289D4—184no—
290D4—181no—
291C2×C4—62no—
292C2×C4—181no—
293C2×C4—184no—
294C2×C4Dic3×C3⋊D312yes—
295C2×C4#629011yescharacteristic
296C2×C4#629212yes—
297C2×C4C2×D3×C3⋊D311yescharacteristic
298C8—542no—
299C2×C3—216no—
300C2×C3—28no—
301C2×C3—22no—
302C2×C3—24no—
303C2×C3—28no—
304C2×C3—362no—
305C2×C3#6382218yes—
306C2×C3C2×(C2×D3)⋊Dic314yes—
307C2×C3C2×#375211yescharacteristic
308C2×C3#6415912yes—
309D3—128no—
310C22—22no—
311C22—182no—
312C22—362no—
313C22#8880312yes—
314C22C2×#629011yescharacteristic
315C4—122no—
316C4—182no—
317C4#8880312yes—
318C4C2×Dic3×C3⋊D311yescharacteristic
319C4C2×#629211yescharacteristic
320C3—24no—
321C3#1059108314yescomplemented
322C3#1059270111yescharacteristic, complemented
323C2—22no—
324C2—362no—
325C2#1172336312yes—
326C2C2×#8880311yescharacteristic
subgroupG/H
centerC2×C4#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊(C3⋊(C3⋊C8))C2semidirect
3. C32×#926C2semidirect
4. #11722039C2semidirect
5. #11722033C2semidirect
11. C3×#3828C4semidirect
26. C3×#926D3semidirect
53. #3828Dic3semidirect
88. #926C3⋊D3semidirect
195. C33#416semidirect
282. C32#5002semidirect
283. C32#5233semidirect
321. C3#10591083semidirect
322. C3#10592701semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C22×C3⋊(C3⋊(C3⋊C8))2
C32×#9262
#117220392
#117220332
#105927013
#105910833