Finite Group Databasebeta

← C3×#10592484 #13162822

Subgroups — C3×#10592484 #13162822

Order 1728.

992 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic
3C3×#64102C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C3×#63642C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592484C311yesmaximal, characteristic, direct product factor
9C3×#4785—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3×C3⋊Dic3—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3757—21no—
17C3×#3753C414yescomplemented
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22D4×C3⋊Dic3C2×C311yescharacteristic, complemented
23C2×#3753C2×C311yescharacteristic
24#64102C2×C311yescharacteristic, complemented
25C3×D4×Dic3—34no—
26C2×C3×#213—34no—
27C3×#893—34no—
28C32×#102—24no—
29C32×#102D314yescomplemented
30C32×#102C2×C311yescharacteristic, complemented
31#63648C2×C311yescharacteristic, complemented
32#63642C2×C311yescharacteristic, complemented
33#63635C2×C311yescharacteristic, complemented
34C3×#821—34no—
35C3×#815—34no—
36C3×#808—34no—
37C23×C33—21no—
38C22×C3×C3⋊D3—21no—
39C33×D4—21no—
40C33×D4C2×C412yescomplemented
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—42no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yescomplemented
49C3×C4×C3⋊D3—42no—
50C33×C8—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52C3×#442—91no—
53#4785—34no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C22×C311yescharacteristic
57C22×C3⋊Dic3—21no—
58C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
59C2×C4×C3⋊D3—21no—
60C22×C3×Dic3—64no—
61C2×C3×D12—34no—
62C2×C3×C4×D3—64no—
63C2×C32×C8—24no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C22×C311yescharacteristic
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C22×C311yescharacteristic
69#3757—21no—
70#3753C3×C414yescomplemented
71#3752—21no—
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73C4×C3⋊Dic3C22×C311yescharacteristic
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C3×#217—64no—
76C3×#213—316no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—34no—
79C3×C4×Dic3—34no—
80C2×C3×C3⋊C8—34no—
81C22×C33—22no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—41no—
94C2×C3×D8—91no—
95C3×C4×D4—91no—
96C3×C4×D4—91no—
97D4×Dic3—34no—
98C2×#213—34no—
99#893—34no—
100C3×#106—91no—
101C3×#102—24no—
102C3×#102—91no—
103C3×#102C3×D314yescomplemented
104C3×#102C3⋊D311yescharacteristic, complemented
105C3×#97—91no—
106#821—34no—
107#815—34no—
108#808—34no—
109C23×C32—21no—
110C23×C32—24no—
111C23×C32—44no—
112C22×C3⋊D3—21no—
113C22×C3×D3—64no—
114C32×D4—21no—
115C32×D4—28no—
116C32×D4—24no—
117C32×D4—44no—
118C32×D4C4×D318yescomplemented
119C32×D4C2×C3×C412yescomplemented
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—44no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—42no—
128C2×C3⋊Dic3C3×D411yescharacteristic
129C2×C3⋊Dic3C3×D411yescharacteristic
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C2×C3×C412yescomplemented
132C4×C3⋊D3—42no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—128no—
137C3×D12—38no—
138C3×D12—64no—
139C3×C4×D3—128no—
140C32×C8—21no—
141C32×C8—24no—
142C32×C8—44no—
143C3⋊(C3⋊C8)C2×C3×C412yescomplemented
144C3×C3⋊C8—38no—
145#442—91noSylow
146C2×C33—42no—
147C2×C33C4×D411yescharacteristic
148C2×C33AGL(1,ℤ/8)11yescharacteristic
149C2×C33AGL(1,ℤ/8)11yescharacteristic
150C3×C3⋊D3—42no—
151C2×C3×D4—24no—
152C2×C3×D4—91no—
153C2×C3×D4C2×C3×D314yes—
154C2×C3×D4C2×C3⋊D311yescharacteristic
155C22×C3×C4—181no—
156C22×C3×C4—181no—
157C22×Dic3—64no—
158C2×D12—34no—
159C2×C4×D3—64no—
160C3×D8—94no—
161C2×C3×C8—24no—
162C2×C3×C8—91no—
163C2×C3×C8C2×C3×D314yes—
164C2×C3×C8C2×C3⋊D311yescharacteristic
165C3×C4⋊C4—24no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4C2×C3×D314yes—
168C3×C4⋊C4C2×C3⋊D311yescharacteristic
169C3×C22⋊C4—181no—
170C3×C22⋊C4—181no—
171C3×C42—91no—
172#217—64no—
173#213—316no—
174(C2×D3)⋊C4—64no—
175C4⋊Dic3—34no—
176C4×Dic3—34no—
177C2×C3⋊C8—34no—
178C22×C32—22no—
179C22×C32—24no—
180C22×C32—28no—
181C22×C32—48no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—44no—
185C22×C32—44no—
186C22×C32—84no—
187C22×C32—84no—
188C22×C32D3×D414yes—
189C22×C32C2×C3×D411yescharacteristic
190C2×C3⋊D3—22no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—68no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—41no—
199C32×C4—44no—
200C32×C4—84no—
201C32×C4C2×C4×D314yes—
202C32×C4#23714yes—
203C32×C4C22×C3×C411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—41no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—124no—
211C2×D8—91no—
212C4×D4—91no—
213C4×D4—91no—
214#106—91no—
215#102—91no—
216#102C3×C3⋊D311yescharacteristic, complemented
217#97—91no—
218C33#44211yescharacteristic, complemented, Sylow
219C23×C3—24no—
220C23×C3—21no—
221C23×C3—44no—
222C23×C3—181no—
223C22×D3—64no—
224C3×D4—24no—
225C3×D4—21no—
226C3×D4—28no—
227C3×D4—44no—
228C3×D4—92no—
229C3×D4—181no—
230C3×D4C3×C4×D318yescomplemented
231C3×D4C4×C3⋊D312yescomplemented
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—24no—
235C2×C3×C4—44no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—362no—
240C2×C3×C4—362no—
241C2×C3×C4C22×C3×D314yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—34no—
244C2×Dic3—34no—
245C2×Dic3—64no—
246C2×Dic3—128no—
247D12—38no—
248D12—64no—
249C4×D3—128no—
250C3×C8—24no—
251C3×C8—21no—
252C3×C8—44no—
253C3×C8—92no—
254C3⋊C8—38no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—42no—
259C2×C32—48no—
260C2×C32—88no—
261C2×C32#67⋊C414yes—
262C2×C32#91114yes—
263C2×C32#91414yes—
264C2×C32C3×C4×D411yescharacteristic
265C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
266C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
267C3⋊D3—42no—
268C3×D3—128no—
269C2×D4—91no—
270C2×D4C2×C3×C3⋊D311yescharacteristic
271C22×C4—181no—
272C22×C4—181no—
273D8—94no—
274C2×C8—91no—
275C2×C8C2×C3×C3⋊D311yescharacteristic
276C4⋊C4—91no—
277C4⋊C4C2×C3×C3⋊D311yescharacteristic
278C22⋊C4—181no—
279C22⋊C4—181no—
280C42—91no—
281C22×C3—28no—
282C22×C3—22no—
283C22×C3—24no—
284C22×C3—48no—
285C22×C3—44no—
286C22×C3—44no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—84no—
290C22×C3—84no—
291C22×C3—182no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3C3×D3×D414yes—
295C22×C3D4×C3⋊D311yescharacteristic
296C2×D3—68no—
297C2×D3—124no—
298C2×D3—124no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—44no—
302C3×C4—41no—
303C3×C4—84no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4C2×C3×C4×D314yes—
308C3×C4C3×#23714yes—
309C3×C4C2×C4×C3⋊D311yescharacteristic
310C3×C4#383011yescharacteristic
311Dic3—64no—
312Dic3—64no—
313Dic3—124no—
314C32—24no—
315C32#478514yescomplemented
316C32C3×#44211yescharacteristic, complemented
317C23—21no—
318C23—181no—
319D4—21no—
320D4—92no—
321D4—181no—
322D4C3×C4×C3⋊D312yescomplemented
323C2×C4—21no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—362no—
328C2×C4—362no—
329C2×C4C22×C3×C3⋊D311yescharacteristic
330C8—21no—
331C8—92no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—48no—
336C2×C3—42no—
337C2×C3—88no—
338C2×C3—362no—
339C2×C3C3×#67⋊C414yes—
340C2×C3C3×#91114yes—
341C2×C3C3×#91414yes—
342C2×C3#6409011yescharacteristic
343C2×C3#6412011yescharacteristic
344C2×C3#6412311yescharacteristic
345D3—128no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—182no—
350C22—361no—
351C22—361no—
352C22C3×D4×C3⋊D311yescharacteristic
353C4—41no—
354C4—181no—
355C4—181no—
356C4—361no—
357C4C2×C3×C4×C3⋊D311yescharacteristic
358C4C3×#383011yescharacteristic
359C3—24no—
360C3C3×#478514yescomplemented
361C3#1059248411yescharacteristic, direct product factor
362C2—42no—
363C2—362no—
364C2C3×#6409011yescharacteristic
365C2C3×#6412011yescharacteristic
366C2C3×#6412311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#442
LCSG ▷ 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
1. C3×D4×C3⋊Dic3C2semidirect
3. C3×#64102C2semidirect
4. C33×#102C2semidirect
5. C3×#63648C2semidirect
6. C3×#63642C2semidirect
7. C3×#63635C2semidirect
8. #10592484C3direct (N × H)
17. C3×#3753C4semidirect
22. D4×C3⋊Dic3C2×C3semidirect
24. #64102C2×C3semidirect
29. C32×#102D3semidirect
30. C32×#102C2×C3semidirect
31. #63648C2×C3semidirect
32. #63642C2×C3semidirect
33. #63635C2×C3semidirect
40. C33×D4C2×C4semidirect
48. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
70. #3753C3×C4semidirect
103. C3×#102C3×D3semidirect
104. C3×#102C3⋊D3semidirect
118. C32×D4C4×D3semidirect
119. C32×D4C2×C3×C4semidirect
131. C4⋊(C3⋊D3)C2×C3×C4semidirect
143. C3⋊(C3⋊C8)C2×C3×C4semidirect
216. #102C3×C3⋊D3semidirect
218. C33#442semidirect
230. C3×D4C3×C4×D3semidirect
231. C3×D4C4×C3⋊D3semidirect
315. C32#4785semidirect
316. C32C3×#442semidirect
322. D4C3×C4×C3⋊D3semidirect
360. C3C3×#4785semidirect
361. C3#10592484direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C3×#37532
C3×#641022
C33×#1022
C3×#636482
C3×#636422
C3×#636352
#105924843
C3×#47853