Finite Group Databasebeta

← C3×#10592476 #13162814

Subgroups — C3×#10592476 #13162814

Order 1728.

884 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C33×#102C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63644C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592476C311yesmaximal, characteristic, direct product factor
9C3×#4777—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4×C3⋊D3—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64147C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
23C3×#938—34no—
24C3×D3×C4⋊C4—34no—
25C32×#102—24no—
26C32×#102D314yescomplemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63644C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×#832—34no—
33C3×(C2×D3)⋊C8—34no—
34C3×#817—34no—
35C3×#808—34no—
36C23×C33—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—42no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
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#4777—34no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C2×C32×D4C22×C311yescharacteristic
56C2×#638—21no—
57C2×C4×C3⋊D3—21no—
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×C3×#67—64no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—64no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C22×C311yescharacteristic
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C22×C311yescharacteristic
68#3757—21no—
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71#3750—21no—
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×#217—64no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×Dic3⋊C4—64no—
77C2×C3×C3⋊C8—34no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33—41no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—21no—
84C2×C3×C3⋊D3SD1612yes—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—41no—
90C3×C3⋊Dic3—41no—
91C3×#121—91no—
92C2×C3×C4⋊C4—91no—
93#938—34no—
94D3×C4⋊C4—34no—
95C3×#106—91no—
96C3×#106—91no—
97C3×#102—24no—
98C3×#102—91no—
99C3×#102C3×D314yescomplemented
100C3×#102C3⋊D311yescharacteristic, complemented
101C3×C22⋊C8—91no—
102#832—34no—
103(C2×D3)⋊C8—34no—
104#817—34no—
105#808—34no—
106C23×C32—21no—
107C23×C32—24no—
108C23×C32—44no—
109C22×C3⋊D3C3×D411yescharacteristic
110C22×C3×D3—34no—
111C32×D4—28no—
112C32×D4—24no—
113C32×D4—21no—
114C32×D4—21no—
115C32×D4—24no—
116C32×D4—44no—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—44no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C23×C311yescharacteristic
123#638—42no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3C3×D411yescharacteristic
127C4×C3⋊D3—21no—
128C4×C3⋊D3—21no—
129C4×C3⋊D3—42no—
130C3×#67—128no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—64no—
134C3×C4×D3—64no—
135C3×C4×D3—64no—
136C3×C4×D3—128no—
137C32×C8—21no—
138C32×C8—24no—
139C32×C8—44no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—64no—
142#481—91noSylow
143C2×C33—81no—
144C2×C33#12311yescharacteristic
145C2×C33C2×SD1611yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C3×C3⋊D3—22no—
148C2×C3×D4—24no—
149C2×C3×D4—181no—
150C2×C3×D4C2×C3×D314yes—
151C2×C3×D4C2×C3⋊D311yescharacteristic
152C22×C3×C4—91no—
153C22×C3×C4—181no—
154C2×#67—64no—
155C2×C4×D3—34no—
156C2×C4×D3—64no—
157C2×C3×C8—24no—
158C2×C3×C8—91no—
159C2×C3×C8C2×C3×D314yes—
160C2×C3×C8C2×C3⋊D311yescharacteristic
161C3×C4⋊C4—24no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4C2×C3×D314yes—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167C3×C22⋊C4—181no—
168#217—64no—
169C4⋊Dic3—34no—
170C4⋊Dic3—34no—
171Dic3⋊C4—64no—
172C2×C3⋊C8—34no—
173C22×C32—24no—
174C22×C32—48no—
175C22×C32—44no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32—84no—
182C22×C32—84no—
183C22×C32D3×D414yes—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3C3×SD1612yes—
188C2×C3×D3—38no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—41no—
194C32×C4—44no—
195C32×C4—84no—
196C32×C4#23514yes—
197C32×C4#23714yes—
198C32×C4C3×#4111yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3⋊Dic3—21no—
201C3⋊Dic3—41no—
202C3⋊Dic3—41no—
203C3×Dic3—64no—
204C3×Dic3—124no—
205C3×Dic3—124no—
206#121—91no—
207C2×C4⋊C4—91no—
208#106—91no—
209#106—91no—
210#102—91no—
211#102C3×C3⋊D311yescharacteristic, complemented
212C22⋊C8—91no—
213C33#48111yescharacteristic, complemented, Sylow
214C23×C3—24no—
215C23×C3—21no—
216C23×C3—44no—
217C23×C3—91no—
218C22×D3—34no—
219C3×D4—28no—
220C3×D4—24no—
221C3×D4—24no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—44no—
225C3×D4—362no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—44no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—362no—
236C2×C3×C4C22×C3×D314yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—128no—
239C2×Dic3—34no—
240C2×Dic3—64no—
241C2×Dic3—64no—
242C4×D3—64no—
243C4×D3—64no—
244C4×D3—128no—
245C3×C8—24no—
246C3×C8—21no—
247C3×C8—44no—
248C3×C8—181no—
249C3⋊C8—64no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—88no—
254C2×C32—84no—
255C2×C32—81no—
256C2×C32#88314yes—
257C2×C32#91114yes—
258C2×C32D3×SD1614yes—
259C2×C32C3×#12311yescharacteristic
260C2×C32C2×C3×SD1611yescharacteristic
261C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
262C3⋊D3—22no—
263C3×D3—68no—
264C2×D4—181no—
265C2×D4C2×C3×C3⋊D311yescharacteristic
266C22×C4—91no—
267C22×C4—181no—
268C2×C8—91no—
269C2×C8C2×C3×C3⋊D311yescharacteristic
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—181no—
273C4⋊C4C2×C3×C3⋊D311yescharacteristic
274C22⋊C4—181no—
275C22×C3—24no—
276C22×C3—48no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—84no—
284C22×C3—84no—
285C22×C3—92no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3C3×D3×D414yes—
289C22×C3D4×C3⋊D311yescharacteristic
290C2×D3—38no—
291C2×D3—64no—
292C2×D3—64no—
293C3×C4—24no—
294C3×C4—24no—
295C3×C4—44no—
296C3×C4—41no—
297C3×C4—84no—
298C3×C4—181no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4C3×#23514yes—
302C3×C4C3×#23714yes—
303C3×C4#382811yescharacteristic
304C3×C4#383011yescharacteristic
305Dic3—64no—
306Dic3—124no—
307Dic3—124no—
308C32—24no—
309C32#477714yescomplemented
310C32C3×#48111yescharacteristic, complemented
311C23—21no—
312C23—91no—
313D4—21no—
314D4—21no—
315D4—362no—
316C2×C4—21no—
317C2×C4—91no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—362no—
323C2×C4C22×C3×C3⋊D311yescharacteristic
324C8—21no—
325C8—181no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—88no—
330C2×C3—84no—
331C2×C3—81no—
332C2×C3—182no—
333C2×C3C3×#88314yes—
334C2×C3C3×#91114yes—
335C2×C3C3×D3×SD1614yes—
336C2×C3#6409211yescharacteristic
337C2×C3#6412011yescharacteristic
338C2×C3SD16×C3⋊D311yescharacteristic
339D3—68no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—92no—
344C22—181no—
345C22—181no—
346C22C3×D4×C3⋊D311yescharacteristic
347C4—41no—
348C4—181no—
349C4—361no—
350C4—361no—
351C4C3×#382811yescharacteristic
352C4C3×#383011yescharacteristic
353C3—24no—
354C3C3×#477714yescomplemented
355C3#1059247611yescharacteristic, direct product factor
356C2—81no—
357C2—182no—
358C2C3×#6409211yescharacteristic
359C2C3×#6412011yescharacteristic
360C2C3×SD16×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#481
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊C4×C3⋊D3C2semidirect
3. C33×#102C2semidirect
4. C3×#63659C2semidirect
5. C3×#63647C2semidirect
6. C3×#63644C2semidirect
7. C3×#63635C2semidirect
8. #10592476C3direct (N × H)
22. C4⋊C4×C3⋊D3C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63647C2×C3semidirect
30. #63644C2×C3semidirect
31. #63635C2×C3semidirect
99. C3×#102C3×D3semidirect
100. C3×#102C3⋊D3semidirect
211. #102C3×C3⋊D3semidirect
213. C33#481semidirect
309. C32#4777semidirect
310. C32C3×#481semidirect
354. C3C3×#4777semidirect
355. C3#10592476direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641472
C3×C4⋊C4×C3⋊D32
C33×#1022
C3×#636592
C3×#636472
C3×#636442
C3×#636352
#105924763
C3×#47773