Finite Group Databasebeta

← C3×#10592665 #13163023

Subgroups — C3×#10592665 #13163023

Order 1728.

1340 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
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4C3×#63637C212yesmaximal, complemented
5C3×#63636C212yesmaximal, complemented
6#10592665C311yesmaximal, characteristic, direct product factor
7C3×#4966—34nomaximal
8C22×C33×C4—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C3×#3828—22no—
12C3×#3828C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C2211yescharacteristic
16C33×C4⋊C4—21no—
17C33×C4⋊C4C2212yes—
18C3×#3748—22no—
19C2×C3×C3⋊(C3⋊C8)C2212yes—
20C2×#3828C2×C311yescharacteristic
21C2×C3×#235—34no—
22C2×C32×C4⋊C4—24no—
23C2×C32×C4⋊C4D314yescomplemented
24C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
25C2×#3748C2×C311yescharacteristic, complemented
26C2×C3×#208—34no—
27#63637C2×C312yescomplemented
28#63636C2×C312yescomplemented
29C3×#810—38no—
30C3×#809—38no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4—22no—
34C2×C33×C4—42no—
35C2×C33×C4C2×C412yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4C2311yescharacteristic
39C3×#638—24no—
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)C2×C412yescomplemented
43C3×C4×C3⋊D3—24no—
44C3×#634—21no—
45C3×#634C2×C412yescomplemented
46C3×C3⋊(C3⋊C8)—22no—
47C3×#417—91no—
48#4966—34no—
49C22×C32×C4—24no—
50C22×C32×C4—28no—
51C22×C32×C4—24no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C22×C311yescharacteristic
55C2×#638—21no—
56#3828—22no—
57#3828C3×C414yescomplemented
58C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
59C2×C4×C3⋊D3—21no—
60C2×#634C22×C311yescharacteristic
61C2×C3×#67—64no—
62C3×#235—316no—
63C3×#235—68no—
64C2×C3×D12—34no—
65C2×C3×C4×D3—64no—
66C2×C3×Dic6—34no—
67C32×C4⋊C4—28no—
68C32×C4⋊C4—28no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2×D318yes—
72C32×C4⋊C4C22×C312yes—
73#3748—22no—
74C2×C3⋊(C3⋊C8)C22×C312yes—
75C3×#208—68no—
76C2×C3×C3⋊C8—38no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33C22⋊C412yes—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—42no—
85C33×C4C22⋊C412yes—
86C33×C4C22×C411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C3×C3⋊Dic3—22no—
89C2×C3×#41—91no—
90C2×#235—34no—
91C2×C3×M16—91no—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4C3×D314yescomplemented
94C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
95C2×#208—34no—
96C3×#103—92no—
97C3×#102—92no—
98#810—38no—
99#809—38no—
100C23×C32—24no—
101C23×C32#6714yes—
102C23×C32C3×D411yescharacteristic
103C22×C3⋊D3—21no—
104C22×C3×D3—64no—
105C2×C32×C4—216no—
106C2×C32×C4—28no—
107C2×C32×C4—22no—
108C2×C32×C4—28no—
109C2×C32×C4—28no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—416no—
113C2×C32×C4—48no—
114C2×C32×C4—42no—
115C2×C32×C4C4×D318yes—
116C2×C32×C4D1218yes—
117C2×C32×C4#6714yes—
118C2×C32×C4C2×C3×C412yes—
119C2×C32×C4C3×D412yes—
120C2×C32×C4C3×D411yescharacteristic
121C2×C32×C4C22×D314yes—
122C2×C32×C4C23×C311yescharacteristic
123#638—24no—
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C2×C3×C412yescomplemented
127C4×C3⋊D3—24no—
128#634—21no—
129#634C2×C3×C412yescomplemented
130C3×#67—616no—
131C2×C3×Dic3—64no—
132C3×D12—38no—
133C3×D12—64no—
134C3×C4×D3—616no—
135C3×Dic6—38no—
136C3×Dic6—64no—
137C3⋊(C3⋊C8)—22no—
138C3×C3⋊C8—68no—
139#417—91noSylow
140C2×C33—22no—
141C2×C33C2×C22⋊C411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3×C3⋊D3—42no—
145C3×#41—94no—
146C3×#41—182no—
147C2×C3×Q8—91no—
148C2×C3×D4—91no—
149C2×C3×D4—181no—
150C22×C3×C4—28no—
151C22×C3×C4—24no—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—181no—
155C22×C3×C4C2×C3×D314yes—
156C22×C3×C4C2×C3⋊D311yescharacteristic
157C2×#67—64no—
158#235—316no—
159#235—68no—
160C2×D12—34no—
161C2×C4×D3—64no—
162C2×Dic6—34no—
163C3×M16—182no—
164C2×C3×C8—92no—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4C2×C3×D318yes—
170C3×C4⋊C4C2×C3⋊D312yes—
171#208—68no—
172C2×C3⋊C8—38no—
173C22×C32—28no—
174C22×C32—28no—
175C22×C32—24no—
176C22×C32—24no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—28no—
180C22×C32—24no—
181C22×C32(C2×D3)⋊C418yes—
182C22×C32C3×C22⋊C412yes—
183C22×C32C2×#6714yes—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3⋊D3—22no—
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3×D3—68no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C32×C4—28no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—416no—
195C32×C4—48no—
196C32×C4—42no—
197C32×C4(C2×D3)⋊C418yes—
198C32×C4C3×C22⋊C412yes—
199C32×C4C2×C4×D314yes—
200C32×C4C2×D1214yes—
201C32×C4C22×C3×C411yescharacteristic
202C32×C4C2×C3×D411yescharacteristic
203C3⋊Dic3—22no—
204C3×Dic3—68no—
205C2×#41—91no—
206C2×M16—91no—
207C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
208#103—92no—
209#102—92no—
210C33#41711yescharacteristic, complemented, Sylow
211C23×C3—24no—
212C23×C3—181no—
213C23×C3C3×#6714yes—
214C23×C3#63811yescharacteristic
215C22×D3—64no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×D4—92no—
219C3×D4—184no—
220C3×D4—181no—
221C2×C3×C4—216no—
222C2×C3×C4—28no—
223C2×C3×C4—22no—
224C2×C3×C4—28no—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—416no—
229C2×C3×C4—48no—
230C2×C3×C4—42no—
231C2×C3×C4—181no—
232C2×C3×C4—184no—
233C2×C3×C4C3×C4×D318yes—
234C2×C3×C4C3×D1218yes—
235C2×C3×C4C3×#6714yes—
236C2×C3×C4C4×C3⋊D312yes—
237C2×C3×C4C4⋊(C3⋊D3)12yes—
238C2×C3×C4#63811yescharacteristic
239C2×C3×C4C22×C3×D314yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241#67—616no—
242C2×Dic3—64no—
243D12—38no—
244D12—64no—
245C4×D3—616no—
246Dic6—38no—
247Dic6—64no—
248C3×C8—182no—
249C3⋊C8—68no—
250C2×C32—216no—
251C2×C32—28no—
252C2×C32—22no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32C2×(C2×D3)⋊C414yes—
257C2×C32#93214yes—
258C2×C32#94214yes—
259C2×C32C2×C3×C22⋊C411yescharacteristic
260C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
261C2×C32C3×#13711yescharacteristic
262C3⋊D3—42no—
263C3×D3—128no—
264#41—94no—
265#41—182no—
266C2×Q8—91no—
267C2×D4—91no—
268C2×D4—181no—
269C22×C4—21no—
270C22×C4—181no—
271C22×C4C2×C3×C3⋊D311yescharacteristic
272M16—182no—
273C2×C8—92no—
274C4⋊C4—21no—
275C4⋊C4C2×C3×C3⋊D312yes—
276C22×C3—28no—
277C22×C3—28no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—28no—
283C22×C3—24no—
284C22×C3—182no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3C3×(C2×D3)⋊C418yes—
288C22×C3#375212yes—
289C22×C3C2×C3×#6714yes—
290C22×C3C2×#63811yescharacteristic
291C2×D3—68no—
292C2×D3—124no—
293C2×D3—124no—
294C3×C4—28no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—416no—
298C3×C4—48no—
299C3×C4—42no—
300C3×C4—182no—
301C3×C4C3×(C2×D3)⋊C418yes—
302C3×C4#375212yes—
303C3×C4C2×C3×C4×D314yes—
304C3×C4C2×C3×D1214yes—
305C3×C4C2×C4×C3⋊D311yescharacteristic
306C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
307Dic3—68no—
308C32—24no—
309C32#496614yescomplemented
310C32C3×#41711yescharacteristic, complemented
311C23—181no—
312C23C3×#63811yescharacteristic
313Q8—92no—
314Q8—181no—
315D4—92no—
316D4—184no—
317D4—181no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—181no—
321C2×C4—184no—
322C2×C4C3×C4×C3⋊D312yes—
323C2×C4C3×C4⋊(C3⋊D3)12yes—
324C2×C4C3×#63811yescharacteristic
325C2×C4C22×C3×C3⋊D311yescharacteristic
326C8—182no—
327C2×C3—216no—
328C2×C3—28no—
329C2×C3—22no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—362no—
334C2×C3C2×C3×(C2×D3)⋊C414yes—
335C2×C3C3×#93214yes—
336C2×C3C3×#94214yes—
337C2×C3C2×#375211yescharacteristic
338C2×C3#6414111yescharacteristic
339C2×C3#6415111yescharacteristic
340D3—128no—
341C22—21no—
342C22—21no—
343C22—182no—
344C22—361no—
345C22—361no—
346C22C3×#375212yes—
347C22C2×C3×#63811yescharacteristic
348C4—42no—
349C4—182no—
350C4C3×#375212yes—
351C4C2×C3×C4×C3⋊D311yescharacteristic
352C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
353C3—24no—
354C3C3×#496614yescomplemented
355C3#1059266511yescharacteristic, direct product factor
356C2—22no—
357C2—362no—
358C2C2×C3×#375211yescharacteristic
359C2C3×#6414111yescharacteristic
360C2C3×#6415111yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C33×C4⋊C4C2semidirect
3. C2×C3×#3748C2semidirect
4. C3×#63637C2semidirect
5. C3×#63636C2semidirect
6. #10592665C3direct (N × H)
12. C3×#3828C4semidirect
23. C2×C32×C4⋊C4D3semidirect
24. C2×C32×C4⋊C4C2×C3semidirect
25. C2×#3748C2×C3semidirect
27. #63637C2×C3semidirect
28. #63636C2×C3semidirect
42. C3×C4⋊(C3⋊D3)C2×C4semidirect
45. C3×#634C2×C4semidirect
57. #3828C3×C4semidirect
93. C2×C3×C4⋊C4C3×D3semidirect
94. C2×C3×C4⋊C4C3⋊D3semidirect
126. C4⋊(C3⋊D3)C2×C3×C4semidirect
129. #634C2×C3×C4semidirect
207. C2×C4⋊C4C3×C3⋊D3semidirect
210. C33#417semidirect
309. C32#4966semidirect
310. C32C3×#417semidirect
354. C3C3×#4966semidirect
355. C3#10592665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C33×C4⋊C42
C2×C3×#37482
C3×#636372
C3×#636362
#105926653
C3×#49663