Finite Group Databasebeta

← C3×#63877 #11723202

Subgroups — C3×#63877 #11723202

Order 864.

410 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic3C211yesmaximal, characteristic
2C3×#3750C211yesmaximal, characteristic, complemented
3C32×(C2×D3)⋊C4C211yesmaximal, characteristic
4C32×C4×Dic3C211yesmaximal, characteristic, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic
7C3×Dic32C211yesmaximal, characteristic, complemented
8C3×#877—31nomaximal
9C3×#872—31nomaximal
10#63877C311yesmaximal, characteristic, direct product factor
11C22×C32×D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C3×D3×Dic3C414yescomplemented
19C22×C3×Dic3—31no—
20C2×C3×C4×D3—31no—
21C2×D3×Dic3C2×C311yescharacteristic
22C32×C22⋊C4—31no—
23C32×C42—31no—
24#3750C2×C311yescharacteristic, complemented
25C3×#217—31no—
26C3×(C2×D3)⋊C4—21no—
27C3×(C2×D3)⋊C4—31no—
28C3×(C2×D3)⋊C4D311yescharacteristic
29C3×(C2×D3)⋊C4C2×C311yescharacteristic
30C3×Dic3⋊C4—31no—
31C3×Dic3⋊C4—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C3×Dic3⋊C4—61no—
35C3×C4×Dic3—21no—
36C3×C4×Dic3—31no—
37C3×C4×Dic3—31no—
38C3×C4×Dic3D311yescharacteristic, complemented
39C3×C4×Dic3C2×C311yescharacteristic, complemented
40#3724C2×C311yescharacteristic, complemented
41(C2×D3)⋊Dic3C2×C311yescharacteristic
42Dic32C2×C311yescharacteristic, complemented
43C22×C33C2311yescharacteristic
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C2×C32×D3C2×C412yes—
47C33×C4—21no—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3C2×C412yes—
50C32×Dic3—21no—
51C32×Dic3—21no—
52C32×Dic3C2×C412yes—
53C3×#117—91no—
54#877—31no—
55#872—31no—
56C23×C32—31no—
57C22×C3×D3—21no—
58C22×C3×D3C2×D311yescharacteristic
59C22×C3×D3C22×C311yescharacteristic
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—41no—
67C2×C32×C4C2×D311yescharacteristic
68C2×C32×C4C2×D311yescharacteristic
69C2×C32×C4C22×C311yescharacteristic
70C2×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊Dic3C22×C311yescharacteristic
72C2×C3×Dic3—21no—
73C2×C3×Dic3—21no—
74C2×C3×Dic3—21no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—34no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C22×C311yescharacteristic
86C2×C3×Dic3C22×C311yescharacteristic
87C2×C3×Dic3C22×C311yescharacteristic
88C3×C4×D3—34no—
89D3×Dic3C3×C414yescomplemented
90C2×C33C22×C411yescharacteristic
91C2×C33#4111yescharacteristic
92C2×C33#4111yescharacteristic
93C32×D3—22no—
94C22×C3×C4—91no—
95C22×Dic3—31no—
96C2×C4×D3—31no—
97C3×C4⋊C4—91no—
98C3×C4⋊C4—91no—
99C3×C22⋊C4—31no—
100C3×C22⋊C4—31no—
101C3×C22⋊C4—61no—
102C3×C22⋊C4—91no—
103C3×C42—31no—
104C3×C42—31no—
105C3×C42—61no—
106C3×C42—91no—
107#217—31no—
108(C2×D3)⋊C4—31no—
109(C2×D3)⋊C4C3×D311yescharacteristic
110Dic3⋊C4—31no—
111Dic3⋊C4—31no—
112Dic3⋊C4—31no—
113Dic3⋊C4—31no—
114Dic3⋊C4—61no—
115C4×Dic3—31no—
116C4×Dic3—31no—
117C4×Dic3C3×D311yescharacteristic, complemented
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—21no—
121C22×C32—32no—
122C22×C32—41no—
123C22×C32—61no—
124C22×C32—61no—
125C22×C32C22×D311yescharacteristic
126C22×C32C22×D311yescharacteristic
127C22×C32C23×C311yescharacteristic
128C2×C3×D3—21no—
129C2×C3×D3—21no—
130C2×C3×D3—21no—
131C2×C3×D3—21no—
132C2×C3×D3—22no—
133C2×C3×D3—41no—
134C2×C3×D3—41no—
135C2×C3×D3C4×D312yes—
136C2×C3×D3C2×C3×C412yes—
137C32×C4—21no—
138C32×C4—22no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—32no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—42no—
145C32×C4—61no—
146C32×C4—61no—
147C3⋊Dic3—21no—
148C3⋊Dic3C2×C3×C412yes—
149C3×Dic3—22no—
150C3×Dic3—21no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—22no—
155C3×Dic3—32no—
156C3×Dic3—32no—
157C3×Dic3—41no—
158C3×Dic3—62no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—121no—
162C3×Dic3C4×D312yes—
163C3×Dic3C2×C3×C412yes—
164#117—91noSylow
165C33#11711yescharacteristic, complemented, Sylow
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—61no—
169C22×D3C2×C3×D311yescharacteristic
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—41no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—94no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4D3211yescharacteristic
187C2×C3×C4C2×C3×D311yescharacteristic
188C2×C3×C4C2×C3×D311yescharacteristic
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—34no—
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3C2×C3×D311yescharacteristic
197C2×Dic3C2×C3×D311yescharacteristic
198C2×Dic3C2×C3×D311yescharacteristic
199C4×D3—34no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—62no—
213C2×C32C2×C4×D311yescharacteristic
214C2×C32C2×C4×D311yescharacteristic
215C2×C32#23511yescharacteristic
216C2×C32#23511yescharacteristic
217C2×C32#23711yescharacteristic
218C2×C32#23711yescharacteristic
219C2×C32C22×C3×C411yescharacteristic
220C2×C32C3×#4111yescharacteristic
221C2×C32C3×#4111yescharacteristic
222C3×D3—22no—
223C3×D3—22no—
224C3×D3—42no—
225C22×C4—91no—
226C4⋊C4—91no—
227C4⋊C4—91no—
228C22⋊C4—31no—
229C22⋊C4—91no—
230C42—31no—
231C42—91no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—41no—
238C22×C3—62no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3C2×D3211yescharacteristic
246C22×C3C22×C3×D311yescharacteristic
247C22×C3C22×C3×D311yescharacteristic
248C2×D3—21no—
249C2×D3—21no—
250C2×D3C3×C4×D312yes—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—42no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—92no—
267C3×C4—121no—
268C3×C4—181no—
269Dic3—21no—
270Dic3—21no—
271Dic3—32no—
272Dic3—32no—
273Dic3—62no—
274Dic3—61no—
275Dic3—61no—
276Dic3—121no—
277Dic3C3×C4×D312yes—
278C32—21no—
279C32—21no—
280C32—21no—
281C32—41no—
282C32#87211yescharacteristic, complemented
283C32#87711yescharacteristic, complemented
284C32C3×#11711yescharacteristic, complemented
285C23—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—94no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4C3×D3211yescharacteristic
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—62no—
306C2×C3—62no—
307C2×C3—122no—
308C2×C3#379511yescharacteristic
309C2×C3C4×D3211yescharacteristic
310C2×C3#380411yescharacteristic
311C2×C3C2×C3×C4×D311yescharacteristic
312C2×C3C2×C3×C4×D311yescharacteristic
313C2×C3C3×#23511yescharacteristic
314C2×C3C3×#23511yescharacteristic
315C2×C3C3×#23711yescharacteristic
316C2×C3C3×#23711yescharacteristic
317D3—22no—
318C22—32no—
319C22—61no—
320C22—61no—
321C22C2×C3×D3211yescharacteristic
322C4—21no—
323C4—32no—
324C4—61no—
325C4—61no—
326C4—92no—
327C4—181no—
328C3—21no—
329C3—21no—
330C3—21no—
331C3—41no—
332C3#6387711yescharacteristic, direct product factor
333C3C3×#87211yescharacteristic, complemented
334C3C3×#87711yescharacteristic, complemented
335C2—62no—
336C2C3×#379511yescharacteristic
337C2C3×C4×D3211yescharacteristic
338C2C3×#380411yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutatorC2×C32C22×C3×C4
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC22×C32×D3C22
cosocleC22×C32C23×C3
hypercenterC2×C3×C4D32
hypocenterC32C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#3750C2semidirect
4. C32×C4×Dic3C2semidirect
5. C3×#3724C2semidirect
7. C3×Dic32C2semidirect
10. #63877C3direct (N × H)
18. C3×D3×Dic3C4semidirect
24. #3750C2×C3semidirect
38. C3×C4×Dic3D3semidirect
39. C3×C4×Dic3C2×C3semidirect
40. #3724C2×C3semidirect
42. Dic32C2×C3semidirect
89. D3×Dic3C3×C4semidirect
117. C4×Dic3C3×D3semidirect
165. C33#117semidirect
282. C32#872semidirect
283. C32#877semidirect
284. C32C3×#117semidirect
332. C3#63877direct (N × H)
333. C3C3×#872semidirect
334. C3C3×#877semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic32
C3×#37502
C32×(C2×D3)⋊C42
C32×C4×Dic32
C3×#37242
C3×(C2×D3)⋊Dic32
C3×Dic322
C3×#8773
C3×#8723
#638773