Finite Group Databasebeta

← C3×#10592538 #13162879

Subgroups — C3×#10592538 #13162879

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3743C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3×#64081C211yesmaximal, characteristic
4C33×C4⋊C8C211yesmaximal, characteristic, complemented
5C3×#63641C211yesmaximal, characteristic, complemented
6C3×#63641C211yesmaximal, characteristic, complemented
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592538C311yesmaximal, characteristic, direct product factor
9C3×#4839—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C3×#3752C411yescharacteristic
15C3×#3752C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×#3750C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#3743—22no—
22C3×C8×C3⋊D3—22no—
23C2×#3743C2×C311yescharacteristic
24C2×C8×C3⋊D3C2×C311yescharacteristic
25#64081C2×C311yescharacteristic
26C2×C3×ΓL(1,𝔽25)—34no—
27C2×C3×D3×C8—34no—
28C3×#872—34no—
29C32×C4⋊C8—24no—
30C32×C4⋊C8D314yescomplemented
31C32×C4⋊C8C2×C311yescharacteristic, complemented
32#63641C2×C311yescharacteristic, complemented
33#63641C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×Dic3⋊C8—34no—
36C3×Dic3⋊C8—34no—
37C3×C4⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4×C3⋊D3D412yes—
46C3×C4×C3⋊D3Q812yes—
47C33×C8—21no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×C3⋊(C3⋊C8)—21no—
51C3×#424—91no—
52#4839—34no—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×C3×C4×D3—34no—
55C2×C32×C8—24no—
56C2×C32×C8—24no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C42—24no—
62C32×C42C2×D314yes—
63C32×C42C22×C311yescharacteristic
64#3752C3×C411yescharacteristic
65#3752C3×C411yescharacteristic
66#3750C3×C411yescharacteristic
67#3750C3×C411yescharacteristic
68C4×C3⋊Dic3C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71#3743—22no—
72C8×C3⋊D3—22no—
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—34no—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8—34no—
80C3×ΓL(1,𝔽25)—68no—
81C3×D3×C8—68no—
82C22×C33C22×C411yescharacteristic
83C2×C3×C3⋊D3—21no—
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3C4⋊C412yes—
86C33×C4—21no—
87C33×C4—21no—
88C33×C4C2×D411yescharacteristic
89C33×C4C2×Q811yescharacteristic
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3C4⋊C412yes—
93C2×C3×M16—91no—
94C22×C3×C8—91no—
95C3×#117—91no—
96C2×ΓL(1,𝔽25)—34no—
97C2×D3×C8—34no—
98#872—34no—
99C3×C4⋊C8—24no—
100C3×C4⋊C8—91no—
101C3×C4⋊C8—91no—
102C3×C4⋊C8—91no—
103C3×C4⋊C8C3×D314yescomplemented
104C3×C4⋊C8C3⋊D311yescharacteristic, complemented
105Dic3⋊C8—34no—
106Dic3⋊C8—34no—
107C4⋊(C3⋊C8)—34no—
108C22×C3⋊D3C2×C3×C411yescharacteristic
109C22×C3×D3—34no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4C4×D314yes—
114C2×C32×C4C4×D314yes—
115C2×C32×C4C2×C3×C411yescharacteristic
116C2×C32×C4C2×C3×C411yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3C2×C3×C411yescharacteristic
120C2×C3⋊Dic3C2×C3×C411yescharacteristic
121C2×C3⋊Dic3C2×C3×C411yescharacteristic
122C4×C3⋊D3C3×D412yes—
123C4×C3⋊D3C3×Q812yes—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C3×C4×D3—38no—
128C3×C4×D3—38no—
129C32×C8—21no—
130C32×C8—21no—
131C32×C8—24no—
132C32×C8—24no—
133C32×C8—44no—
134C32×C8—44no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—64no—
138C3×C3⋊C8—64no—
139#424—91noSylow
140C2×C33C2×C4⋊C411yescharacteristic
141C2×C33#13111yescharacteristic
142C2×C33#13111yescharacteristic
143C3×C3⋊D3—22no—
144C22×C3×C4—91no—
145C2×C4×D3—34no—
146C3×M16—182no—
147C2×C3×C8—24no—
148C2×C3×C8—24no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C2×C3×C8—182no—
152C2×C3×C8C2×C3×D314yes—
153C2×C3×C8C2×C3×D314yes—
154C2×C3×C8C2×C3⋊D311yescharacteristic
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—91no—
159C3×C22⋊C4—91no—
160C3×C42—24no—
161C3×C42—91no—
162C3×C42C2×C3×D314yes—
163C3×C42C2×C3⋊D311yescharacteristic
164(C2×D3)⋊C4—34no—
165(C2×D3)⋊C4—34no—
166Dic3⋊C4—34no—
167Dic3⋊C4—34no—
168C4×Dic3—34no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8—34no—
171ΓL(1,𝔽25)—68no—
172D3×C8—68no—
173C22×C32—24no—
174C22×C32C2×C4×D314yes—
175C22×C32C22×C3×C411yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3C3×C4⋊C412yes—
179C2×C3×D3—38no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C32×C4—28no—
183C32×C4—24no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—44no—
190C32×C4D3×D414yes—
191C32×C4D3×Q814yes—
192C32×C4C2×C3×D411yescharacteristic
193C32×C4C2×C3×Q811yescharacteristic
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3C3×C4⋊C412yes—
197C3×Dic3—38no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C2×M16—91no—
201C22×C8—91no—
202#117—91no—
203C4⋊C8—91no—
204C4⋊C8—91no—
205C4⋊C8—91no—
206C4⋊C8C3×C3⋊D311yescharacteristic, complemented
207C33#42411yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—34no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4C3×C4×D314yes—
219C2×C3×C4C3×C4×D314yes—
220C2×C3×C4C4×C3⋊D311yescharacteristic
221C2×C3×C4C4×C3⋊D311yescharacteristic
222C2×C3×C4C22×C3×D314yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—34no—
225C2×Dic3—34no—
226C2×Dic3—34no—
227C4×D3—38no—
228C4×D3—38no—
229C3×C8—24no—
230C3×C8—24no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—44no—
234C3×C8—44no—
235C3×C8—181no—
236C3×C8—181no—
237C3⋊C8—64no—
238C3⋊C8—64no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32D3×C4⋊C414yes—
243C2×C32#90114yes—
244C2×C32#90714yes—
245C2×C32C2×C3×C4⋊C411yescharacteristic
246C2×C32C3×#13111yescharacteristic
247C2×C32C3×#13111yescharacteristic
248C3⋊D3—22no—
249C3×D3—68no—
250C22×C4—91no—
251M16—182no—
252C2×C8—91no—
253C2×C8—91no—
254C2×C8—182no—
255C2×C8C2×C3×C3⋊D311yescharacteristic
256C2×C8C2×C3×C3⋊D311yescharacteristic
257C4⋊C4—91no—
258C4⋊C4—91no—
259C22⋊C4—91no—
260C22⋊C4—91no—
261C42—91no—
262C42C2×C3×C3⋊D311yescharacteristic
263C22×C3—24no—
264C22×C3—92no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3C2×C3×C4×D314yes—
268C22×C3C2×C4×C3⋊D311yescharacteristic
269C2×D3—38no—
270C2×D3—64no—
271C2×D3—64no—
272C3×C4—28no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—44no—
280C3×C4—92no—
281C3×C4—181no—
282C3×C4—181no—
283C3×C4C3×D3×D414yes—
284C3×C4C3×D3×Q814yes—
285C3×C4D4×C3⋊D311yescharacteristic
286C3×C4Q8×C3⋊D311yescharacteristic
287Dic3—38no—
288Dic3—64no—
289Dic3—64no—
290C32—24no—
291C32#483914yescomplemented
292C32C3×#42411yescharacteristic, complemented
293C23—91no—
294C2×C4—91no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4C3×C4×C3⋊D311yescharacteristic
300C2×C4C3×C4×C3⋊D311yescharacteristic
301C2×C4C22×C3×C3⋊D311yescharacteristic
302C8—21no—
303C8—21no—
304C8—181no—
305C8—181no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—182no—
310C2×C3C3×D3×C4⋊C414yes—
311C2×C3C3×#90114yes—
312C2×C3C3×#90714yes—
313C2×C3C4⋊C4×C3⋊D311yescharacteristic
314C2×C3#6411011yescharacteristic
315C2×C3#6411611yescharacteristic
316D3—68no—
317C22—92no—
318C22—181no—
319C22—181no—
320C22C2×C3×C4×C3⋊D311yescharacteristic
321C4—21no—
322C4—21no—
323C4—92no—
324C4—181no—
325C4—181no—
326C4C3×D4×C3⋊D311yescharacteristic
327C4C3×Q8×C3⋊D311yescharacteristic
328C3—24no—
329C3C3×#483914yescomplemented
330C3#1059253811yescharacteristic, direct product factor
331C2—182no—
332C2C3×C4⋊C4×C3⋊D311yescharacteristic
333C2C3×#6411011yescharacteristic
334C2C3×#6411611yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊C8C2
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C8C3⋊D3
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C33×C4⋊C8C2semidirect
5. C3×#63641C2semidirect
6. C3×#63641C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592538C3direct (N × H)
30. C32×C4⋊C8D3semidirect
31. C32×C4⋊C8C2×C3semidirect
32. #63641C2×C3semidirect
33. #63641C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
103. C3×C4⋊C8C3×D3semidirect
104. C3×C4⋊C8C3⋊D3semidirect
206. C4⋊C8C3×C3⋊D3semidirect
207. C33#424semidirect
291. C32#4839semidirect
292. C32C3×#424semidirect
329. C3C3×#4839semidirect
330. C3#10592538direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C2×C3×C8×C3⋊D32
C3×#640812
C33×C4⋊C82
C3×#636412
C3×C4⋊(C3⋊(C3⋊C8))2
#105925383
C3×#48393