Finite Group Databasebeta

← C3×#10592543 #13162882

Subgroups — C3×#10592543 #13162882

Order 1728.

896 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3745C211yesmaximal, characteristic
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3C3×#64084C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
5C33×C4⋊C8C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592543C311yesmaximal, characteristic, direct product factor
9C3×#4844—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C3×#634C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C2×C33×C8C2211yescharacteristic
16C33×C42C2211yescharacteristic
17C3×#3752—21no—
18C3×#3752—21no—
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3745—22no—
22C3×#3744—22no—
23C2×#3745C2×C311yescharacteristic
24C2×#3744C2×C311yescharacteristic, complemented
25#64084C2×C311yescharacteristic, complemented
26C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
27C2×C3×D24—34no—
28C2×C3×#204—34no—
29C3×#875—34no—
30C3×C4×D12—34no—
31C32×C4⋊C8—24no—
32C32×C4⋊C8D314yescomplemented
33C32×C4⋊C8C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63643C2×C311yescharacteristic, complemented
36C3×#821—34no—
37C3×#816—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)D412yes—
49C3×C4×C3⋊D3—42no—
50C3×#634—21no—
51C3×#634—21no—
52C33×C8—21no—
53C33×C8—21no—
54C3×#463—91no—
55#4844—34no—
56C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634C22×C311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×D12—34no—
62C2×C3×C4×D3—64no—
63C2×C3×Dic6—34no—
64C2×C32×C8—24no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C22×C311yescharacteristic
69C2×C32×C8C22×C311yescharacteristic
70C32×C42—24no—
71C32×C42C2×D314yes—
72C32×C42C22×C311yescharacteristic
73#3752—21no—
74#3752—21no—
75#3752—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77#3745—22no—
78#3744—22no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×C4⋊Dic3—34no—
83C3×D24—68no—
84C3×#204—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4—21no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—41no—
97C3×C3⋊Dic3—41no—
98C2×C3×SD16—91no—
99C2×C3×D8—91no—
100C3×#124—91no—
101C3×C4×D4—91no—
102C2×D24—34no—
103C2×#204—34no—
104#875—34no—
105C4×D12—34no—
106C3×C4⋊C8—24no—
107C3×C4⋊C8C3×D314yescomplemented
108C3×C4⋊C8C3⋊D311yescharacteristic, complemented
109C3×#103—91no—
110C3×#102—91no—
111#821—34no—
112#816—34no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3—64no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4D1214yes—
121C2×C32×C4D1214yes—
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C3×D411yescharacteristic
124C2×C32×C4C22×D314yes—
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C3×D412yes—
132C4×C3⋊D3—42no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—64no—
137C3×D12—38no—
138C3×D12—64no—
139C3×D12—64no—
140C3×D12—64no—
141C3×C4×D3—128no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C32×C8—21no—
145C32×C8—21no—
146C32×C8—24no—
147C32×C8—24no—
148C32×C8—44no—
149C32×C8—44no—
150#463—91noSylow
151C2×C33#12111yescharacteristic
152C2×C33#13511yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3×C3⋊D3—42no—
155C3×C3⋊D3—81no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C2×D12—34no—
161C2×D12—34no—
162C2×C4×D3—64no—
163C2×Dic6—34no—
164C3×SD16—182no—
165C3×D8—182no—
166C2×C3×C8—24no—
167C2×C3×C8—24no—
168C2×C3×C8C2×C3×D314yes—
169C2×C3×C8C2×C3×D314yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C2×C3×C8C2×C3⋊D311yescharacteristic
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174C3×C22⋊C4—181no—
175C3×C22⋊C4—181no—
176C3×C42—24no—
177C3×C42C2×C3×D314yes—
178C3×C42C2×C3⋊D311yescharacteristic
179(C2×D3)⋊C4—64no—
180(C2×D3)⋊C4—64no—
181(C2×D3)⋊C4—64no—
182C4⋊Dic3—34no—
183D24—68no—
184#204—68no—
185C22×C32—24no—
186C22×C32C2×D1214yes—
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—22no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—68no—
195C2×C3×D3—124no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C32×C4—28no—
201C32×C4—24no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—44no—
208C32×C4D3×D414yes—
209C32×C4#23914yes—
210C32×C4C2×C3×D411yescharacteristic
211C32×C4C3×#4111yescharacteristic
212C3⋊Dic3—41no—
213C3⋊Dic3—41no—
214C3×Dic3—124no—
215C3×Dic3—124no—
216C2×SD16—91no—
217C2×D8—91no—
218#124—91no—
219C4×D4—91no—
220C4⋊C8C3×C3⋊D311yescharacteristic, complemented
221#103—91no—
222#102—91no—
223C33#46311yescharacteristic, complemented, Sylow
224C23×C3—181no—
225C23×C3—181no—
226C22×D3—64no—
227C22×D3—64no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×D4—92no—
231C3×D4—181no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—362no—
240C2×C3×C4C3×D1214yes—
241C2×C3×C4C3×D1214yes—
242C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
243C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
244C2×C3×C4C22×C3×D314yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×Dic3—64no—
247C2×Dic3—64no—
248D12—38no—
249D12—64no—
250D12—64no—
251D12—64no—
252C4×D3—128no—
253Dic6—64no—
254Dic6—64no—
255C3×C8—24no—
256C3×C8—24no—
257C3×C8—21no—
258C3×C8—21no—
259C3×C8—44no—
260C3×C8—44no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32#89514yes—
265C2×C32#90414yes—
266C2×C32#90814yes—
267C2×C32C3×#12111yescharacteristic
268C2×C32C3×#13511yescharacteristic
269C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
270C3⋊D3—42no—
271C3⋊D3—81no—
272C3×D3—128no—
273C3×D3—244no—
274C2×Q8—91no—
275C2×D4—91no—
276C2×D4—91no—
277C22×C4—181no—
278SD16—182no—
279D8—182no—
280C2×C8C2×C3×C3⋊D311yescharacteristic
281C2×C8C2×C3×C3⋊D311yescharacteristic
282C4⋊C4—91no—
283C22⋊C4—181no—
284C22⋊C4—181no—
285C22⋊C4—181no—
286C42C2×C3×C3⋊D311yescharacteristic
287C22×C3—24no—
288C22×C3—182no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3C2×C3×D1214yes—
295C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
296C2×D3—68no—
297C2×D3—124no—
298C2×D3—124no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—124no—
302C3×C4—28no—
303C3×C4—24no—
304C3×C4—24no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—24no—
308C3×C4—24no—
309C3×C4—44no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4C3×D3×D414yes—
313C3×C4C3×#23914yes—
314C3×C4D4×C3⋊D311yescharacteristic
315C3×C4#383211yescharacteristic
316Dic3—124no—
317Dic3—124no—
318C32—24no—
319C32#484414yescomplemented
320C32C3×#46311yescharacteristic, complemented
321C23—181no—
322C23—181no—
323Q8—181no—
324Q8—181no—
325D4—92no—
326D4—181no—
327D4—181no—
328D4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—362no—
332C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
333C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
334C2×C4C22×C3×C3⋊D311yescharacteristic
335C8—21no—
336C8—21no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—362no—
341C2×C3—721no—
342C2×C3C3×#89514yes—
343C2×C3C3×#90414yes—
344C2×C3C3×#90814yes—
345C2×C3#6410411yescharacteristic
346C2×C3#6411311yescharacteristic
347C2×C3#6411711yescharacteristic
348D3—128no—
349D3—244no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
357C4—21no—
358C4—21no—
359C4—361no—
360C4—361no—
361C4C3×D4×C3⋊D311yescharacteristic
362C4C3×#383211yescharacteristic
363C3—24no—
364C3C3×#484414yescomplemented
365C3#1059254311yescharacteristic, direct product factor
366C2—362no—
367C2—721no—
368C2C3×#6410411yescharacteristic
369C2C3×#6411311yescharacteristic
370C2C3×#6411711yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊C8C2
socleC22×C33C2×D4
prime radicalC2×C3×#3745C2
hypercenterC3×C4⋊C8C3⋊D3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3744C2semidirect
3. C3×#64084C2semidirect
4. C3×C4×C4⋊(C3⋊D3)C2semidirect
5. C33×C4⋊C8C2semidirect
6. C3×#63648C2semidirect
7. C3×#63643C2semidirect
8. #10592543C3direct (N × H)
24. C2×#3744C2×C3semidirect
25. #64084C2×C3semidirect
26. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. C32×C4⋊C8D3semidirect
33. C32×C4⋊C8C2×C3semidirect
34. #63648C2×C3semidirect
35. #63643C2×C3semidirect
107. C3×C4⋊C8C3×D3semidirect
108. C3×C4⋊C8C3⋊D3semidirect
220. C4⋊C8C3×C3⋊D3semidirect
223. C33#463semidirect
319. C32#4844semidirect
320. C32C3×#463semidirect
364. C3C3×#4844semidirect
365. C3#10592543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37452
C2×C3×#37442
C3×#640842
C3×C4×C4⋊(C3⋊D3)2
C33×C4⋊C82
C3×#636482
C3×#636432
#105925433
C3×#48443