Finite Group Databasebeta

← C3×#4989 #10592086

Subgroups — C3×#4989 #10592086

Order 576.

514 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D3C211yesmaximal, characteristic
2C2×C32×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
4C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
5C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C3×C3⋊#95C211yesmaximal, characteristic, complemented
7C3×C3⋊#95C211yesmaximal, characteristic, complemented
8C3×#385—31nomaximal
9#4989C311yesmaximal, characteristic, direct product factor
10C23×C3×D3C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic
13C22×C32×C4C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C2×C3×C4×D3—22no—
18C2×C3×C4×D3—22no—
19C2×C3×C4×D3—22no—
20C32×C4⋊C4—22no—
21C3×(C2×D3)⋊C4—22no—
22C3×(C2×D3)⋊C4C414yescomplemented
23C3×Dic3⋊C4—22no—
24C23×C3×C4—31no—
25C22×C4×D3C2×C311yescharacteristic
26C2×C3×C4⋊C4—21no—
27C2×C3×C4⋊C4—31no—
28C2×C3×C4⋊C4D311yescharacteristic, complemented
29C2×C3×C4⋊C4C2×C311yescharacteristic, complemented
30C2×C3×C22⋊C4—31no—
31C2×C3×C22⋊C4—31no—
32C2×(C2×D3)⋊C4C2×C311yescharacteristic
33C2×(C2×D3)⋊C4C2×C311yescharacteristic
34C2×Dic3⋊C4C2×C311yescharacteristic, complemented
35C3×#95—31no—
36C3×#95—31no—
37C3⋊#95C2×C311yescharacteristic, complemented
38C3⋊#95C2×C311yescharacteristic, complemented
39C23×C32C2311yescharacteristic
40C22×C3×D3—21no—
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C22×C3×D3—21no—
44C22×C3×D3C2×C412yes—
45C22×C3×D3D412yes—
46C22×C3×D3Q812yes—
47C2×C32×C4—21no—
48C2×C32×C4—21no—
49C2×C32×C4—21no—
50C2×C32×C4—21no—
51C2×C32×C4—21no—
52C2×C32×C4—21no—
53C2×C32×C4—21no—
54C2×C32×C4C2×C412yescomplemented
55C2×C32×C4D412yes—
56C2×C3×Dic3—21no—
57C2×C3×Dic3—21no—
58C2×C3×Dic3—21no—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3—21no—
63C2×C3×Dic3C2×C412yescomplemented
64C2×C3×Dic3D412yes—
65C3×C4×D3—28no—
66#385—31noSylow
67C24×C3—31no—
68C23×D3C22×C311yescharacteristic
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—31no—
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4—62no—
76C22×C3×C4—62no—
77C22×C3×C4—62no—
78C22×C3×C4C2×D311yescharacteristic
79C22×C3×C4C2×D311yescharacteristic
80C22×C3×C4C2×D311yescharacteristic
81C22×C3×C4C22×C311yescharacteristic
82C22×C3×C4C22×C311yescharacteristic
83C22×C3×C4C22×C311yescharacteristic
84C22×Dic3C22×C311yescharacteristic
85C22×Dic3C22×C311yescharacteristic
86C22×Dic3C22×C311yescharacteristic
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C3×C4⋊C4—24no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—62no—
94C3×C22⋊C4—34no—
95C3×C22⋊C4—62no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4C3×C414yescomplemented
98Dic3⋊C4—22no—
99C22×C32C22×C411yescharacteristic
100C22×C32C2×D411yescharacteristic
101C22×C32C2×D411yescharacteristic
102C22×C32C2×D411yescharacteristic
103C22×C32C2×Q811yescharacteristic
104C22×C32#4111yescharacteristic
105C22×C32#4111yescharacteristic
106C2×C3×D3—22no—
107C2×C3×D3—22no—
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C2×C3×D3C4⋊C414yescomplemented
113C32×C4—22no—
114C32×C4—22no—
115C32×C4—41no—
116C3×Dic3—22no—
117C3×Dic3—22no—
118C3×Dic3—41no—
119C23×C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4C3×D311yescharacteristic, complemented
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124#95—31no—
125#95—31no—
126C23×C3—21no—
127C23×C3—32no—
128C23×C3—32no—
129C23×C3—32no—
130C23×C3—61no—
131C23×C3—61no—
132C23×C3—61no—
133C23×C3—61no—
134C23×C3C22×D311yescharacteristic
135C23×C3C23×C311yescharacteristic
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—21no—
139C22×D3—21no—
140C22×D3C2×C3×C412yes—
141C22×D3C3×D412yes—
142C22×D3C3×Q812yes—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—22no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—68no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4C4×D312yescomplemented
177C2×C3×C4#6712yes—
178C2×C3×C4C2×C3×C412yescomplemented
179C2×C3×C4C3×D412yes—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×C3×C412yescomplemented
188C2×Dic3C3×D412yes—
189C4×D3—28no—
190C2×C32C2×C4⋊C411yescharacteristic
191C2×C32C4×D411yescharacteristic
192C2×C32C4×D411yescharacteristic
193C2×C32#12011yescharacteristic
194C2×C32C22⋊Q811yescharacteristic
195C2×C32C22⋊Q811yescharacteristic
196C2×C32#12311yescharacteristic
197C3×D3—24no—
198C24—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4C2×C3×D311yescharacteristic
206C22×C4C2×C3×D311yescharacteristic
207C22×C4C2×C3×D311yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—62no—
210C22⋊C4—34no—
211C22⋊C4—62no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—34no—
220C22×C3—62no—
221C22×C3—62no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3—62no—
225C22×C3—62no—
226C22×C3C2×C4×D311yescharacteristic
227C22×C3#23511yescharacteristic
228C22×C3D3×D411yescharacteristic
229C22×C3D3×D411yescharacteristic
230C22×C3D3×Q811yescharacteristic
231C22×C3#23911yescharacteristic
232C22×C3C2×#6711yescharacteristic
233C22×C3C22×C3×C411yescharacteristic
234C22×C3C2×C3×D411yescharacteristic
235C22×C3C2×C3×D411yescharacteristic
236C22×C3C2×C3×D411yescharacteristic
237C22×C3C2×C3×Q811yescharacteristic
238C22×C3C3×#4111yescharacteristic
239C22×C3C3×#4111yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—22no—
246C2×D3C3×C4⋊C414yescomplemented
247C3×C4—24no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—42no—
253C3×C4—42no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—121no—
259Dic3—22no—
260Dic3—22no—
261Dic3—41no—
262C32#38511yescharacteristic, complemented, Sylow
263C23—32no—
264C23—32no—
265C23—32no—
266C23—61no—
267C23—61no—
268C23—61no—
269C23—61no—
270C23C22×C3×D311yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—68no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4C3×C4×D312yescomplemented
289C2×C4C3×#6712yes—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—64no—
298C2×C3D3×C4⋊C411yescharacteristic
299C2×C3#89311yescharacteristic
300C2×C3#89411yescharacteristic
301C2×C3#89611yescharacteristic
302C2×C3C4×#6711yescharacteristic
303C2×C3#93711yescharacteristic
304C2×C3#94611yescharacteristic
305C2×C3C2×C3×C4⋊C411yescharacteristic
306C2×C3C3×C4×D411yescharacteristic
307C2×C3C3×C4×D411yescharacteristic
308C2×C3C3×#12011yescharacteristic
309C2×C3C3×C22⋊Q811yescharacteristic
310C2×C3C3×C22⋊Q811yescharacteristic
311C2×C3C3×#12311yescharacteristic
312D3—24no—
313C22—34no—
314C22—62no—
315C22—62no—
316C22—62no—
317C22—62no—
318C22—62no—
319C22—62no—
320C22C2×C3×C4×D311yescharacteristic
321C22C3×#23511yescharacteristic
322C22C3×D3×D411yescharacteristic
323C22C3×D3×D411yescharacteristic
324C22C3×D3×Q811yescharacteristic
325C22C3×#23911yescharacteristic
326C22C2×C3×#6711yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—62no—
331C4—62no—
332C4—121no—
333C3—21no—
334C3#498911yescharacteristic, direct product factor
335C3C3×#38511yescharacteristic, complemented
336C2—64no—
337C2C3×D3×C4⋊C411yescharacteristic
338C2C3×#89311yescharacteristic
339C2C3×#89411yescharacteristic
340C2C3×#89611yescharacteristic
341C2C3×C4×#6711yescharacteristic
342C2C3×#93711yescharacteristic
343C2C3×#94611yescharacteristic
subgroupG/H
centerC23×C3C22×D3
commutatorC22×C3C22×C3×C4
FrattiniC23C22×C3×D3
FittingC2×C32×C4⋊C4C2
socleC23×C32C23
prime radicalC23×C3×D3C22
cosocleC23×C3C23×C3
hypercenterC2×C3×C4⋊C4D3
hypocenterC3C3×#385
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4⋊C4C2semidirect
5. C2×C3×Dic3⋊C4C2semidirect
6. C3×C3⋊#95C2semidirect
7. C3×C3⋊#95C2semidirect
9. #4989C3direct (N × H)
22. C3×(C2×D3)⋊C4C4semidirect
28. C2×C3×C4⋊C4D3semidirect
29. C2×C3×C4⋊C4C2×C3semidirect
34. C2×Dic3⋊C4C2×C3semidirect
37. C3⋊#95C2×C3semidirect
38. C3⋊#95C2×C3semidirect
54. C2×C32×C4C2×C4semidirect
63. C2×C3×Dic3C2×C4semidirect
97. (C2×D3)⋊C4C3×C4semidirect
112. C2×C3×D3C4⋊C4semidirect
121. C2×C4⋊C4C3×D3semidirect
176. C2×C3×C4C4×D3semidirect
178. C2×C3×C4C2×C3×C4semidirect
187. C2×Dic3C2×C3×C4semidirect
246. C2×D3C3×C4⋊C4semidirect
262. C32#385semidirect
288. C2×C4C3×C4×D3semidirect
334. C3#4989direct (N × H)
335. C3C3×#385semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D32
C2×C32×C4⋊C42
C2×C3×(C2×D3)⋊C42
C2×C3×Dic3⋊C42
C3×C3⋊#952
C3×#3853
#49893