Finite Group Databasebeta

← C3×#10592562 #13162901

Subgroups — C3×#10592562 #13162901

Order 1728.

828 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64104C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C33×#106C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592562C311yesmaximal, characteristic, direct product factor
9C3×#4863—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C33×C4⋊C4C2211yescharacteristic
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64104C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
23C3×#895—34no—
24C3×D3×C4⋊C4—34no—
25C32×#106—24no—
26C32×#106D314yescomplemented
27C32×#106C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×#821—34no—
33C3×(C2×D3)⋊C8—34no—
34C3×#809—34no—
35C3×#808—34no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—42no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—42no—
49C33×C8—21no—
50C3×C3⋊(C3⋊C8)—21no—
51C3×#481—91no—
52#4863—34no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×D12—34no—
58C2×C3×D12—64no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—64no—
61C2×C32×C8—24no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C22×C311yescharacteristic
64C32×C4⋊C4—24no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C22×C311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70#3752—21no—
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72#3750—21no—
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×(C2×D3)⋊C4—64no—
75C3×C4⋊Dic3—34no—
76C3×Dic3⋊C4—64no—
77C2×C3×C3⋊C8—34no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3SD1612yes—
85C33×C4—41no—
86C33×C4—41no—
87C33×C4#4111yescharacteristic
88C33×C4#4111yescharacteristic
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3—41no—
91C3×#121—91no—
92C2×C3×C4⋊C4—91no—
93#895—34no—
94D3×C4⋊C4—34no—
95C3×#106—24no—
96C3×#106—91no—
97C3×#106C3×D314yescomplemented
98C3×#106C3⋊D311yescharacteristic, complemented
99C3×#102—91no—
100C3×#102—91no—
101C3×C22⋊C8—91no—
102#821—34no—
103(C2×D3)⋊C8—34no—
104#809—34no—
105#808—34no—
106C22×C3⋊D3—21no—
107C22×C3⋊D3C3×D411yescharacteristic
108C22×C3×D3—34no—
109C22×C3×D3—64no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—44no—
116C2×C32×C4—44no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3C3×D411yescharacteristic
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—42no—
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C4×C3⋊D3—42no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×D12—64no—
130C3×D12—64no—
131C3×D12—128no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C3×C4×D3—128no—
135C32×C8—28no—
136C32×C8—24no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—64no—
140#481—91noSylow
141C2×C33#12311yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C3×C3⋊D3—22no—
145C3×C3⋊D3—81no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—91no—
149C22×C3×C4—181no—
150C2×D12—34no—
151C2×D12—64no—
152C2×C4×D3—34no—
153C2×C4×D3—64no—
154C2×C3×C8—24no—
155C2×C3×C8—91no—
156C2×C3×C8C2×C3×D314yes—
157C2×C3×C8C2×C3⋊D311yescharacteristic
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×C3×D314yes—
163C3×C4⋊C4C2×C3×D314yes—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C4⋊C4C2×C3⋊D311yescharacteristic
166C3×C22⋊C4—181no—
167(C2×D3)⋊C4—64no—
168C4⋊Dic3—34no—
169Dic3⋊C4—64no—
170C2×C3⋊C8—34no—
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32C2×C3×D411yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—41no—
178C2×C3⋊D3—41no—
179C2×C3⋊D3C3×SD1612yes—
180C2×C3×D3—38no—
181C2×C3×D3—64no—
182C2×C3×D3—64no—
183C2×C3×D3—124no—
184C2×C3×D3—124no—
185C2×C3×D3—124no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—44no—
191C32×C4—44no—
192C32×C4—84no—
193C32×C4—84no—
194C32×C4#23514yes—
195C32×C4#23914yes—
196C32×C4C3×#4111yescharacteristic
197C32×C4C3×#4111yescharacteristic
198C3⋊Dic3—21no—
199C3⋊Dic3—41no—
200C3×Dic3—64no—
201C3×Dic3—124no—
202#121—91no—
203C2×C4⋊C4—91no—
204#106—91no—
205#106C3×C3⋊D311yescharacteristic, complemented
206#102—91no—
207#102—91no—
208C22⋊C8—91no—
209C33#48111yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C23×C3—181no—
212C22×D3—34no—
213C22×D3—64no—
214C3×D4—181no—
215C3×D4—181no—
216C3×D4—362no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—24no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—362no—
229C2×C3×C4C22×C3×D314yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—34no—
232C2×Dic3—64no—
233D12—64no—
234D12—64no—
235D12—128no—
236C4×D3—64no—
237C4×D3—64no—
238C4×D3—128no—
239C3×C8—28no—
240C3×C8—24no—
241C3×C8—21no—
242C3×C8—181no—
243C3⋊C8—64no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32#89414yes—
248C2×C32D3×SD1614yes—
249C2×C32#91414yes—
250C2×C32C3×#12311yescharacteristic
251C2×C32C2×C3×SD1611yescharacteristic
252C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
253C3⋊D3—22no—
254C3⋊D3—81no—
255C3×D3—68no—
256C3×D3—244no—
257C2×D4—91no—
258C2×D4—181no—
259C22×C4—91no—
260C22×C4—181no—
261C2×C8—91no—
262C2×C8C2×C3×C3⋊D311yescharacteristic
263C4⋊C4—91no—
264C4⋊C4—181no—
265C4⋊C4C2×C3×C3⋊D311yescharacteristic
266C4⋊C4C2×C3×C3⋊D311yescharacteristic
267C22⋊C4—181no—
268C22×C3—24no—
269C22×C3—92no—
270C22×C3—181no—
271C22×C3—181no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3C3×D3×D414yes—
276C22×C3D4×C3⋊D311yescharacteristic
277C2×D3—38no—
278C2×D3—64no—
279C2×D3—64no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—124no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—44no—
286C3×C4—44no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—84no—
290C3×C4—84no—
291C3×C4—181no—
292C3×C4—361no—
293C3×C4C3×#23514yes—
294C3×C4C3×#23914yes—
295C3×C4#382811yescharacteristic
296C3×C4#383211yescharacteristic
297Dic3—64no—
298Dic3—124no—
299C32—24no—
300C32#486314yescomplemented
301C32C3×#48111yescharacteristic, complemented
302C23—91no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—362no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—91no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—362no—
314C2×C4C22×C3×C3⋊D311yescharacteristic
315C8—21no—
316C8—181no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—182no—
321C2×C3—721no—
322C2×C3C3×#89414yes—
323C2×C3C3×D3×SD1614yes—
324C2×C3C3×#91414yes—
325C2×C3#6410311yescharacteristic
326C2×C3SD16×C3⋊D311yescharacteristic
327C2×C3#6412311yescharacteristic
328D3—68no—
329D3—244no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22C3×D4×C3⋊D311yescharacteristic
337C4—41no—
338C4—41no—
339C4—181no—
340C4—361no—
341C4C3×#382811yescharacteristic
342C4C3×#383211yescharacteristic
343C3—24no—
344C3C3×#486314yescomplemented
345C3#1059256211yescharacteristic, direct product factor
346C2—182no—
347C2—721no—
348C2C3×#6410311yescharacteristic
349C2C3×SD16×C3⋊D311yescharacteristic
350C2C3×#6412311yescharacteristic
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×#481
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
2. C3×C4⋊C4×C3⋊D3C2semidirect
3. C33×#106C2semidirect
4. C3×#63648C2semidirect
5. C3×#63647C2semidirect
6. C3×#63636C2semidirect
7. C3×#63635C2semidirect
8. #10592562C3direct (N × H)
22. C4⋊C4×C3⋊D3C2×C3semidirect
26. C32×#106D3semidirect
27. C32×#106C2×C3semidirect
28. #63648C2×C3semidirect
29. #63647C2×C3semidirect
30. #63636C2×C3semidirect
31. #63635C2×C3semidirect
97. C3×#106C3×D3semidirect
98. C3×#106C3⋊D3semidirect
205. #106C3×C3⋊D3semidirect
209. C33#481semidirect
300. C32#4863semidirect
301. C32C3×#481semidirect
344. C3C3×#4863semidirect
345. C3#10592562direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641042
C3×C4⋊C4×C3⋊D32
C33×#1062
C3×#636482
C3×#636472
C3×#636362
C3×#636352
#105925623
C3×#48633