Finite Group Databasebeta

← C3×#10592563 #13162902

Subgroups — C3×#10592563 #13162902

Order 1728.

888 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3744C211yesmaximal, characteristic, complemented
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#64106C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic
5C33×#106C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592563C311yesmaximal, characteristic, direct product factor
9C3×#4864—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C33×C4⋊C4C2211yescharacteristic
17C3×#3752—21no—
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#3744—22no—
22C3×C8×C3⋊D3—22no—
23C2×#3744C2×C311yescharacteristic, complemented
24C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
25#64106C2×C311yescharacteristic, complemented
26#64104C2×C311yescharacteristic
27C2×C3×#204—34no—
28C2×C3×D3×C8—34no—
29C3×#897—34no—
30C3×#895—34no—
31C32×#106—24no—
32C32×#106D314yescomplemented
33C32×#106C2×C311yescharacteristic, complemented
34#63637C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#810—34no—
37C3×#809—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—42no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C33×C8D412yescomplemented
53C3×C3⋊(C3⋊C8)—21no—
54C3×#465—91no—
55#4864—34no—
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×#634C22×C311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×D12—64no—
62C2×C3×C4×D3—34no—
63C2×C3×Dic6—34no—
64C2×C32×C8—24no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C22×C311yescharacteristic
67C32×C4⋊C4—24no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4C2×D314yes—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C22×C311yescharacteristic
72C32×C4⋊C4C22×C311yescharacteristic
73#3752—21no—
74#3752—21no—
75C4⋊(C3⋊Dic3)—21no—
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77#3744—22no—
78C8×C3⋊D3—22no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×C4⋊Dic3—64no—
82C2×C3×C3⋊C8—34no—
83C3×#204—68no—
84C3×D3×C8—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3SD1612yescomplemented
92C33×C4—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C2×C3×SD16—91no—
99C22×C3×C8—91no—
100C3×C22⋊Q8—91no—
101C3×#121—91no—
102C2×#204—34no—
103C2×D3×C8—34no—
104#897—34no—
105#895—34no—
106C3×#106—24no—
107C3×#106C3×D314yescomplemented
108C3×#106C3⋊D311yescharacteristic, complemented
109C3×#103—91no—
110C3×#102—91no—
111#810—34no—
112#809—34no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3C3×D411yescharacteristic
115C22×C3×D3—34no—
116C22×C3×D3—64no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—44no—
123C2×C32×C4—44no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3C3×D411yescharacteristic
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—42no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×D12—64no—
138C3×D12—64no—
139C3×D12—128no—
140C3×C4×D3—64no—
141C3×C4×D3—64no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C32×C8—28no—
145C32×C8D1218yescomplemented
146C32×C8C3×D412yescomplemented
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—64no—
149#465—91noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13511yescharacteristic
153C3×C3⋊D3—22no—
154C3×C3⋊D3—81no—
155C2×C3×Q8—91no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—91no—
159C2×D12—34no—
160C2×D12—64no—
161C2×C4×D3—34no—
162C2×Dic6—34no—
163C3×SD16—182no—
164C2×C3×C8—24no—
165C2×C3×C8—91no—
166C2×C3×C8—182no—
167C2×C3×C8C2×C3×D314yes—
168C2×C3×C8C2×C3⋊D311yescharacteristic
169C3×C4⋊C4—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4C2×C3×D314yes—
173C3×C4⋊C4C2×C3×D314yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C3×C4⋊C4C2×C3⋊D311yescharacteristic
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178(C2×D3)⋊C4—64no—
179(C2×D3)⋊C4—64no—
180C4⋊Dic3—64no—
181C2×C3⋊C8—34no—
182#204—68no—
183D3×C8—68no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32C2×C3×D411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3C3×SD1612yescomplemented
193C2×C3×D3—38no—
194C2×C3×D3—64no—
195C2×C3×D3—64no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4—44no—
204C32×C4—44no—
205C32×C4—84no—
206C32×C4—84no—
207C32×C4C2×D1214yes—
208C32×C4#23914yes—
209C32×C4C2×C3×D411yescharacteristic
210C32×C4C3×#4111yescharacteristic
211C3⋊Dic3—21no—
212C3⋊Dic3—41no—
213C3×Dic3—64no—
214C3×Dic3—124no—
215C2×SD16—91no—
216C22×C8—91no—
217C22⋊Q8—91no—
218#121—91no—
219#106C3×C3⋊D311yescharacteristic, complemented
220#103—91no—
221#102—91no—
222C33#46511yescharacteristic, complemented, Sylow
223C23×C3—91no—
224C23×C3—181no—
225C22×D3—34no—
226C22×D3—64no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—181no—
230C3×D4—181no—
231C3×D4—362no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—24no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C22×C3×D314yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—34no—
246C2×Dic3—64no—
247D12—64no—
248D12—64no—
249D12—128no—
250C4×D3—64no—
251C4×D3—64no—
252Dic6—64no—
253Dic6—64no—
254C3×C8—28no—
255C3×C8—181no—
256C3×C8C3×D1218yescomplemented
257C3×C8C4⋊(C3⋊D3)12yescomplemented
258C3⋊C8—64no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32#89514yes—
263C2×C32D3×SD1614yes—
264C2×C32#91614yes—
265C2×C32C3×#12111yescharacteristic
266C2×C32C2×C3×SD1611yescharacteristic
267C2×C32C3×#13511yescharacteristic
268C3⋊D3—22no—
269C3⋊D3—81no—
270C3×D3—68no—
271C3×D3—244no—
272C2×Q8—91no—
273C2×D4—91no—
274C2×D4—181no—
275C22×C4—91no—
276SD16—182no—
277C2×C8—91no—
278C2×C8—182no—
279C2×C8C2×C3×C3⋊D311yescharacteristic
280C4⋊C4—181no—
281C4⋊C4C2×C3×C3⋊D311yescharacteristic
282C4⋊C4C2×C3×C3⋊D311yescharacteristic
283C22⋊C4—181no—
284C22⋊C4—181no—
285C22×C3—24no—
286C22×C3—92no—
287C22×C3—181no—
288C22×C3—181no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3C3×D3×D414yes—
293C22×C3D4×C3⋊D311yescharacteristic
294C2×D3—38no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—124no—
298C2×D3—124no—
299C2×D3—124no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—44no—
303C3×C4—44no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—84no—
307C3×C4—84no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4C2×C3×D1214yes—
311C3×C4C3×#23914yes—
312C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
313C3×C4#383211yescharacteristic
314Dic3—64no—
315Dic3—124no—
316C32—24no—
317C32#486414yescomplemented
318C32C3×#46511yescharacteristic, complemented
319C23—91no—
320C23—181no—
321Q8—181no—
322Q8—181no—
323D4—181no—
324D4—181no—
325D4—362no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—91no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4C22×C3×C3⋊D311yescharacteristic
333C8—181no—
334C8C3×C4⋊(C3⋊D3)12yescomplemented
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3—182no—
339C2×C3—721no—
340C2×C3C3×#89514yes—
341C2×C3C3×D3×SD1614yes—
342C2×C3C3×#91614yes—
343C2×C3#6410411yescharacteristic
344C2×C3SD16×C3⋊D311yescharacteristic
345C2×C3#6412511yescharacteristic
346D3—68no—
347D3—244no—
348C22—92no—
349C22—181no—
350C22—181no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22C3×D4×C3⋊D311yescharacteristic
355C4—41no—
356C4—41no—
357C4—181no—
358C4—361no—
359C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
360C4C3×#383211yescharacteristic
361C3—24no—
362C3C3×#486414yescomplemented
363C3#1059256311yescharacteristic, direct product factor
364C2—182no—
365C2—721no—
366C2C3×#6410411yescharacteristic
367C2C3×SD16×C3⋊D311yescharacteristic
368C2C3×#6412511yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
2. C2×C3×C8×C3⋊D3C2semidirect
3. C3×#64106C2semidirect
5. C33×#106C2semidirect
6. C3×#63637C2semidirect
7. C3×#63636C2semidirect
8. #10592563C3direct (N × H)
23. C2×#3744C2×C3semidirect
24. C2×C8×C3⋊D3C2×C3semidirect
25. #64106C2×C3semidirect
32. C32×#106D3semidirect
33. C32×#106C2×C3semidirect
34. #63637C2×C3semidirect
35. #63636C2×C3semidirect
52. C33×C8D4semidirect
91. C2×C3×C3⋊D3SD16semidirect
107. C3×#106C3×D3semidirect
108. C3×#106C3⋊D3semidirect
145. C32×C8D12semidirect
146. C32×C8C3×D4semidirect
192. C2×C3⋊D3C3×SD16semidirect
219. #106C3×C3⋊D3semidirect
222. C33#465semidirect
256. C3×C8C3×D12semidirect
257. C3×C8C4⋊(C3⋊D3)semidirect
317. C32#4864semidirect
318. C32C3×#465semidirect
334. C8C3×C4⋊(C3⋊D3)semidirect
362. C3C3×#4864semidirect
363. C3#10592563direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37442
C2×C3×C8×C3⋊D32
C3×#641062
C3×#641042
C33×#1062
C3×#636372
C3×#636362
#105925633
C3×#48643