Finite Group Databasebeta

← C3×#10592501 #13162839

Subgroups — C3×#10592501 #13162839

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic
3C3×#64101C211yesmaximal, characteristic
4C33×#103C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592501C311yesmaximal, characteristic, direct product factor
9C3×#4802—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3C414yescomplemented
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3743—22no—
24C2×Q8×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×Q8—34no—
26C2×#3743C2×C311yescharacteristic
27#64101C2×C311yescharacteristic
28C2×C3×ΓL(1,𝔽25)—34no—
29C3×#892—34no—
30C32×#103—24no—
31C32×#103D314yescomplemented
32C32×#103C2×C311yescharacteristic, complemented
33#63662C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×#835—34no—
37C3×#816—34no—
38C3×#810—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yes—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—24no—
49C3×C4×C3⋊D3C2×C412yes—
50C3×C4×C3⋊D3D412yes—
51C3×#634—21no—
52C3×#634—24no—
53C3×#634C2×C412yes—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×#419—91no—
57#4802—34no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C22×C311yescharacteristic
61Q8×C3⋊D3—22no—
62Q8×C3⋊D3C3×C414yescomplemented
63C2×C4×C3⋊D3—21no—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×#634—21no—
66C2×#634C22×C311yescharacteristic
67C3×D3×Q8—316no—
68C3×D3×Q8—68no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—64no—
71C2×C3×Dic6—34no—
72C2×C3×Dic6—64no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C22×C311yescharacteristic
76C32×C4⋊C4—24no—
77C32×C4⋊C4C2×D314yes—
78C32×C4⋊C4C22×C311yescharacteristic
79#3752—21no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4×C3⋊Dic3—21no—
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83#3743—22no—
84C3×(C2×D3)⋊C4—64no—
85C3×C4⋊Dic3—34no—
86C3×C4×Dic3—64no—
87C2×C3×C3⋊C8—34no—
88C3×ΓL(1,𝔽25)—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3C22⋊C412yes—
93C33×C4—22no—
94C33×C4—41no—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—22no—
98C3×C3⋊Dic3—41no—
99C3×C3⋊Dic3C22⋊C412yes—
100C22×C3×Q8—91no—
101C2×D3×Q8—34no—
102C2×C3×M16—91no—
103C3×#117—91no—
104C2×ΓL(1,𝔽25)—34no—
105#892—34no—
106C3×#103—24no—
107C3×#103—91no—
108C3×#103—91no—
109C3×#103—91no—
110C3×#103C3×D314yescomplemented
111C3×#103C3⋊D311yescharacteristic, complemented
112#835—34no—
113#816—34no—
114#810—34no—
115C22×C3⋊D3C3×D411yescharacteristic
116C22×C3×D3—34no—
117C32×Q8—21no—
118C32×Q8—28no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8C4×D318yes—
122C32×Q8C2×C3×C412yes—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4—44no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3C3×D411yescharacteristic
135C4×C3⋊D3—24no—
136C4×C3⋊D3C2×C3×C412yes—
137C4×C3⋊D3C3×D412yes—
138#634—21no—
139#634—24no—
140#634C2×C3×C412yes—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C3×C4×D3—38no—
145C3×C4×D3—38no—
146C3×C4×D3—616no—
147C3×Dic6—38no—
148C3×Dic6—616no—
149C3×Dic6—64no—
150C32×C8—21no—
151C32×C8—24no—
152C32×C8—44no—
153C3⋊(C3⋊C8)—21no—
154C3×C3⋊C8—64no—
155#419—91noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33#13711yescharacteristic
158C2×C33#13711yescharacteristic
159C3×C3⋊D3—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8—94no—
162C2×C3×Q8—91no—
163C2×C3×Q8—181no—
164C2×C3×Q8—182no—
165C2×C3×Q8C2×C3×D314yes—
166C2×C3×Q8C2×C3⋊D311yescharacteristic
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169D3×Q8—316no—
170D3×Q8—68no—
171C2×C4×D3—34no—
172C2×C4×D3—64no—
173C2×Dic6—34no—
174C2×Dic6—64no—
175C3×M16—182no—
176C2×C3×C8—24no—
177C2×C3×C8—91no—
178C2×C3×C8C2×C3×D314yes—
179C2×C3×C8C2×C3⋊D311yescharacteristic
180C3×C4⋊C4—24no—
181C3×C4⋊C4—91no—
182C3×C4⋊C4C2×C3×D314yes—
183C3×C4⋊C4C2×C3⋊D311yescharacteristic
184C3×C22⋊C4—181no—
185C3×C42—181no—
186(C2×D3)⋊C4—64no—
187C4⋊Dic3—34no—
188C4×Dic3—64no—
189C2×C3⋊C8—34no—
190ΓL(1,𝔽25)—68no—
191C22×C32—24no—
192C22×C32D3×D414yes—
193C22×C32C2×C3×D411yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3⋊D3—21no—
196C2×C3⋊D3C3×C22⋊C412yes—
197C2×C3×D3—38no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C32×C4—22no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—28no—
204C32×C4—48no—
205C32×C4—41no—
206C32×C4—44no—
207C32×C4—84no—
208C32×C4C2×C4×D314yes—
209C32×C4D3×D414yes—
210C32×C4C22×C3×C411yescharacteristic
211C32×C4C2×C3×D411yescharacteristic
212C3⋊Dic3—22no—
213C3⋊Dic3—41no—
214C3⋊Dic3C3×C22⋊C412yes—
215C3×Dic3—38no—
216C3×Dic3—68no—
217C3×Dic3—124no—
218C22×Q8—91no—
219C2×M16—91no—
220#117—91no—
221#103—91no—
222#103—91no—
223#103—91no—
224#103C3×C3⋊D311yescharacteristic, complemented
225C33#41911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—34no—
228C3×Q8—24no—
229C3×Q8—21no—
230C3×Q8—28no—
231C3×Q8—44no—
232C3×Q8—92no—
233C3×Q8—184no—
234C3×Q8—181no—
235C3×Q8C3×C4×D318yes—
236C3×Q8C4×C3⋊D312yes—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4—91no—
245C2×C3×C4—92no—
246C2×C3×C4—92no—
247C2×C3×C4—184no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252C2×Dic3—34no—
253C2×Dic3—64no—
254C2×Dic3—64no—
255C4×D3—38no—
256C4×D3—38no—
257C4×D3—616no—
258Dic6—38no—
259Dic6—616no—
260Dic6—64no—
261C3×C8—24no—
262C3×C8—21no—
263C3×C8—44no—
264C3×C8—181no—
265C3⋊C8—64no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32D3×C22⋊C414yes—
270C2×C32#91514yes—
271C2×C32#91814yes—
272C2×C32C2×C3×C22⋊C411yescharacteristic
273C2×C32C3×#13711yescharacteristic
274C2×C32C3×#13711yescharacteristic
275C3⋊D3—22no—
276C3×D3—68no—
277C2×Q8—94no—
278C2×Q8—91no—
279C2×Q8—181no—
280C2×Q8—182no—
281C2×Q8C2×C3×C3⋊D311yescharacteristic
282C22×C4—91no—
283C22×C4—181no—
284M16—182no—
285C2×C8—91no—
286C2×C8C2×C3×C3⋊D311yescharacteristic
287C4⋊C4—91no—
288C4⋊C4C2×C3×C3⋊D311yescharacteristic
289C22⋊C4—181no—
290C42—181no—
291C22×C3—24no—
292C22×C3—92no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3C3×D3×D414yes—
296C22×C3D4×C3⋊D311yescharacteristic
297C2×D3—38no—
298C2×D3—64no—
299C2×D3—64no—
300C3×C4—28no—
301C3×C4—22no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—44no—
306C3×C4—41no—
307C3×C4—84no—
308C3×C4—92no—
309C3×C4—182no—
310C3×C4—361no—
311C3×C4C2×C3×C4×D314yes—
312C3×C4C3×D3×D414yes—
313C3×C4C2×C4×C3⋊D311yescharacteristic
314C3×C4D4×C3⋊D311yescharacteristic
315Dic3—38no—
316Dic3—68no—
317Dic3—124no—
318C32—24no—
319C32#480214yescomplemented
320C32C3×#41911yescharacteristic, complemented
321C23—91no—
322Q8—21no—
323Q8—92no—
324Q8—184no—
325Q8—181no—
326Q8C3×C4×C3⋊D312yes—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—184no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4C22×C3×C3⋊D311yescharacteristic
336C8—21no—
337C8—181no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—182no—
342C2×C3C3×D3×C22⋊C414yes—
343C2×C3C3×#91514yes—
344C2×C3C3×#91814yes—
345C2×C3C22⋊C4×C3⋊D311yescharacteristic
346C2×C3#6412411yescharacteristic
347C2×C3#6412711yescharacteristic
348D3—68no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22C3×D4×C3⋊D311yescharacteristic
353C4—22no—
354C4—41no—
355C4—92no—
356C4—182no—
357C4—361no—
358C4C2×C3×C4×C3⋊D311yescharacteristic
359C4C3×D4×C3⋊D311yescharacteristic
360C3—24no—
361C3C3×#480214yescomplemented
362C3#1059250111yescharacteristic, direct product factor
363C2—182no—
364C2C3×C22⋊C4×C3⋊D311yescharacteristic
365C2C3×#6412411yescharacteristic
366C2C3×#6412711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#419
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
4. C33×#103C2semidirect
5. C3×#63662C2semidirect
6. C3×#63643C2semidirect
7. C3×#63637C2semidirect
8. #10592501C3direct (N × H)
12. C3×Q8×C3⋊D3C4semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
62. Q8×C3⋊D3C3×C4semidirect
110. C3×#103C3×D3semidirect
111. C3×#103C3⋊D3semidirect
224. #103C3×C3⋊D3semidirect
225. C33#419semidirect
319. C32#4802semidirect
320. C32C3×#419semidirect
361. C3C3×#4802semidirect
362. C3#10592501direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C3×#37432
C3×#641012
C33×#1032
C3×#636622
C3×#636432
C3×#636372
#105925013
C3×#48023