Finite Group Databasebeta

← C3×Dic3×#67 #11723297

Subgroups — C3×Dic3×#67 #11723297

Order 864.

550 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#67C211yesmaximal, characteristic
2C22×C32×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×D3×Dic3C211yesmaximal, characteristic, complemented
4C3×#3757C211yesmaximal, characteristic, complemented
5C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic32C211yesmaximal, characteristic, complemented
8C3×D4×Dic3—31nomaximal
9C3×C4×#67—31nomaximal
10Dic3×#67C311yesmaximal, characteristic, direct product factor
11C23×C33C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C32×#67C414yescomplemented
16C2×C32×Dic3—22no—
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19C2×C32×Dic3C2211yescharacteristic
20C3×D3×Dic3—22no—
21C2×C32×D4—31no—
22C22×C32×C4—31no—
23C2×C3×#67—21no—
24C2×C3×#67D311yescharacteristic
25C2×C3×#67C2×C311yescharacteristic
26C22×C3×Dic3—21no—
27C22×C3×Dic3—31no—
28C22×C3×Dic3D311yescharacteristic, complemented
29C22×C3×Dic3C2×C311yescharacteristic, complemented
30C2×C3×C4×D3—31no—
31C2×D3×Dic3C2×C311yescharacteristic, complemented
32#3757C2×C311yescharacteristic, complemented
33C3×#217—31no—
34C3×#217—31no—
35C3×#217—31no—
36C3×#217—61no—
37C3×(C2×D3)⋊C4—31no—
38C3×C4⋊Dic3—31no—
39C3×Dic3⋊C4—31no—
40C3×C4×Dic3—31no—
41C3×C4×Dic3—31no—
42Dic3⋊Dic3C2×C311yescharacteristic, complemented
43(C2×D3)⋊Dic3C2×C311yescharacteristic, complemented
44Dic32C2×C311yescharacteristic, complemented
45C22×C33—21no—
46C22×C33—21no—
47C22×C33C2×C412yescomplemented
48C22×C33C2311yescharacteristic
49C2×C32×D3—21no—
50C2×C32×D3—21no—
51C2×C32×D3C2×C412yescomplemented
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—21no—
54C32×Dic3—21no—
55C32×Dic3C2×C412yescomplemented
56C32×Dic3D412yescomplemented
57C3×C4×D4—91no—
58D4×Dic3—31no—
59C4×#67—31no—
60C23×C32—21no—
61C23×C32—21no—
62C23×C32—21no—
63C23×C32—31no—
64C23×C32—41no—
65C23×C32C2×D311yescharacteristic
66C23×C32C2×D311yescharacteristic
67C23×C32C22×C311yescharacteristic
68C22×C3×D3—21no—
69C22×C3×D3C2×D311yescharacteristic
70C22×C3×D3C22×C311yescharacteristic
71C32×D4—34no—
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—62no—
76C2×C3⋊Dic3C22×C311yescharacteristic
77C2×C3⋊Dic3C22×C311yescharacteristic
78C3×#67—24no—
79C3×#67Dic314yesdirect product factor
80C3×#67C3×C414yescomplemented
81C2×C3×Dic3—24no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—21no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—31no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—31no—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3C2×D311yescharacteristic
95C2×C3×Dic3C2×D311yescharacteristic
96C2×C3×Dic3C2×D311yescharacteristic
97C2×C3×Dic3C22×C311yescharacteristic
98C2×C3×Dic3C22×C311yescharacteristic
99C2×C3×Dic3C22×C311yescharacteristic
100C3×C4×D3—62no—
101D3×Dic3—22no—
102C2×C33—22no—
103C2×C33C22×C411yescharacteristic
104C2×C33C2×D411yescharacteristic
105C2×C33#4111yescharacteristic
106C32×D3—22no—
107C2×C3×D4—31no—
108C2×C3×D4—31no—
109C2×C3×D4—61no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—61no—
113C22×C3×C4—91no—
114C2×#67C3×D311yescharacteristic
115C22×Dic3—31no—
116C22×Dic3C3×D311yescharacteristic, complemented
117C2×C4×D3—31no—
118C3×C4⋊C4—91no—
119C3×C22⋊C4—91no—
120C3×C22⋊C4—91no—
121C3×C42—91no—
122#217—31no—
123#217—31no—
124#217—31no—
125#217—61no—
126(C2×D3)⋊C4—31no—
127C4⋊Dic3—31no—
128Dic3⋊C4—31no—
129C4×Dic3—31no—
130C4×Dic3—31no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—32no—
146C22×C32—42no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—41no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—61no—
155C22×C32—61no—
156C22×C32C4×D312yescomplemented
157C22×C32C2×Dic312yescomplemented
158C22×C32C2×C3×C412yescomplemented
159C22×C32C22×D311yescharacteristic
160C22×C32C22×D311yescharacteristic
161C22×C32C23×C311yescharacteristic
162C2×C3×D3—21no—
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3—21no—
166C2×C3×D3—22no—
167C2×C3×D3—41no—
168C2×C3×D3—41no—
169C2×C3×D3C2×Dic312yescomplemented
170C2×C3×D3C2×C3×C412yescomplemented
171C32×C4—32no—
172C32×C4—32no—
173C32×C4—61no—
174C3⋊Dic3—21no—
175C3⋊Dic3—21no—
176C3×Dic3—22no—
177C3×Dic3—21no—
178C3×Dic3—21no—
179C3×Dic3—22no—
180C3×Dic3—22no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3C2×Dic312yescomplemented
188C3×Dic3#6712yesdirect product factor
189C3×Dic3C2×C3×C412yescomplemented
190C3×Dic3C3×D412yescomplemented
191C4×D4—91noSylow
192C33C4×D411yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—41no—
199C23×C3—61no—
200C23×C3D3211yescharacteristic
201C23×C3C2×C3×D311yescharacteristic
202C23×C3C2×C3×D311yescharacteristic
203C22×D3C2×C3×D311yescharacteristic
204C3×D4—34no—
205C3×D4—34no—
206C3×D4—64no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—64no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—182no—
222#67C3×Dic314yesdirect product factor
223C2×Dic3—22no—
224C2×Dic3—31no—
225C2×Dic3—31no—
226C2×Dic3—31no—
227C2×Dic3—31no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—62no—
231C2×Dic3C2×C3×D311yescharacteristic
232C2×Dic3C2×C3×D311yescharacteristic
233C2×Dic3C2×C3×D311yescharacteristic
234C4×D3—62no—
235C2×C32—22no—
236C2×C32—24no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32—42no—
253C2×C32—44no—
254C2×C32—62no—
255C2×C32C2×C4×D311yescharacteristic
256C2×C32#23511yescharacteristic
257C2×C32D3×D411yescharacteristic
258C2×C32#23711yescharacteristic
259C2×C32C22×Dic311yescharacteristic
260C2×C32C2×#6711yescharacteristic
261C2×C32C22×C3×C411yescharacteristic
262C2×C32C2×C3×D411yescharacteristic
263C2×C32C3×#4111yescharacteristic
264C3×D3—22no—
265C3×D3—22no—
266C3×D3—42no—
267C2×D4—31no—
268C22×C4—31no—
269C22×C4—91no—
270C4⋊C4—91no—
271C22⋊C4—91no—
272C22⋊C4—91no—
273C42—91no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—32no—
289C22×C3—32no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3D3×Dic312yescomplemented
306C22×C3C3×C4×D312yescomplemented
307C22×C3C2×C3×Dic312yescomplemented
308C22×C3C2×D3211yescharacteristic
309C22×C3C22×C3×D311yescharacteristic
310C22×C3C22×C3×D311yescharacteristic
311C2×D3—21no—
312C2×D3—21no—
313C2×D3C2×C3×Dic312yescomplemented
314C3×C4—32no—
315C3×C4—32no—
316C3×C4—32no—
317C3×C4—32no—
318C3×C4—62no—
319C3×C4—62no—
320C3×C4—62no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—181no—
324C3×C4—181no—
325Dic3—21no—
326Dic3—61no—
327Dic3—61no—
328Dic3—61no—
329Dic3—61no—
330Dic3—121no—
331Dic3—121no—
332Dic3C2×C3×Dic312yescomplemented
333Dic3C3×#6712yesdirect product factor
334C32—21no—
335C32—21no—
336C32—21no—
337C32—41no—
338C32C4×#6711yescharacteristic, complemented
339C32D4×Dic311yescharacteristic, complemented
340C32C3×C4×D411yescharacteristic, complemented
341C23—31no—
342C23C3×D3211yescharacteristic
343D4—34no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—62no—
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—182no—
351C2×C3—24no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—42no—
365C2×C3—42no—
366C2×C3—44no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—62no—
371C2×C3—62no—
372C2×C3—122no—
373C2×C3C2×D3×Dic311yescharacteristic
374C2×C3#380411yescharacteristic
375C2×C3D3×#6711yescharacteristic
376C2×C3C2×C3×C4×D311yescharacteristic
377C2×C3C3×#23511yescharacteristic
378C2×C3C3×D3×D411yescharacteristic
379C2×C3C3×#23711yescharacteristic
380C2×C3C22×C3×Dic311yescharacteristic
381C2×C3C2×C3×#6711yescharacteristic
382D3—22no—
383C22—21no—
384C22—21no—
385C22—32no—
386C22—61no—
387C22—61no—
388C22C3×D3×Dic312yescomplemented
389C22C2×C3×D3211yescharacteristic
390C4—32no—
391C4—32no—
392C4—61no—
393C4—181no—
394C4—181no—
395C3—21no—
396C3—21no—
397C3—21no—
398C3—41no—
399C3Dic3×#6711yescharacteristic, direct product factor
400C3C3×C4×#6711yescharacteristic, complemented
401C3C3×D4×Dic311yescharacteristic, complemented
402C2—22no—
403C2—62no—
404C2C2×C3×D3×Dic311yescharacteristic
405C2C3×#380411yescharacteristic
406C2C3×D3×#6711yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutatorC2×C32C22×C3×C4
FrattiniC22C2×C3×D32
FittingC23×C33C22
socleC22×C33C23
prime radicalC2×C32×#67C2
cosocleC22×C32C23×C3
hypercenterC23×C3D32
hypocenterC32C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C32×Dic3C2semidirect
3. C2×C3×D3×Dic3C2semidirect
4. C3×#3757C2semidirect
5. C3×Dic3⋊Dic3C2semidirect
6. C3×(C2×D3)⋊Dic3C2semidirect
7. C3×Dic32C2semidirect
10. Dic3×#67C3direct (N × H)
15. C32×#67C4semidirect
28. C22×C3×Dic3D3semidirect
29. C22×C3×Dic3C2×C3semidirect
31. C2×D3×Dic3C2×C3semidirect
32. #3757C2×C3semidirect
42. Dic3⋊Dic3C2×C3semidirect
43. (C2×D3)⋊Dic3C2×C3semidirect
44. Dic32C2×C3semidirect
47. C22×C33C2×C4semidirect
51. C2×C32×D3C2×C4semidirect
55. C32×Dic3C2×C4semidirect
56. C32×Dic3D4semidirect
79. C3×#67Dic3direct (N × H)
80. C3×#67C3×C4semidirect
116. C22×Dic3C3×D3semidirect
156. C22×C32C4×D3semidirect
157. C22×C32C2×Dic3semidirect
158. C22×C32C2×C3×C4semidirect
169. C2×C3×D3C2×Dic3semidirect
170. C2×C3×D3C2×C3×C4semidirect
187. C3×Dic3C2×Dic3semidirect
188. C3×Dic3#67direct (N × H)
189. C3×Dic3C2×C3×C4semidirect
190. C3×Dic3C3×D4semidirect
192. C33C4×D4semidirect
222. #67C3×Dic3direct (N × H)
305. C22×C3D3×Dic3semidirect
306. C22×C3C3×C4×D3semidirect
307. C22×C3C2×C3×Dic3semidirect
313. C2×D3C2×C3×Dic3semidirect
332. Dic3C2×C3×Dic3semidirect
333. Dic3C3×#67direct (N × H)
338. C32C4×#67semidirect
339. C32D4×Dic3semidirect
340. C32C3×C4×D4semidirect
388. C22C3×D3×Dic3semidirect
399. C3Dic3×#67direct (N × H)
400. C3C3×C4×#67semidirect
401. C3C3×D4×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#672
C22×C32×Dic32
C2×C3×D3×Dic32
C3×#37572
C3×Dic3⋊Dic32
C3×(C2×D3)⋊Dic32
C3×Dic322
C3×D4×Dic33
C3×C4×#673
Dic3×#673