Finite Group Databasebeta

← C3×Dic3×#208 #13158082

Subgroups — C3×Dic3×#208 #13158082

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C211yesmaximal, characteristic
2C2×C32×C4×Dic3C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic
4C3×#63554C212yesmaximal, complemented
5C3×Dic3×C3⋊C8C212yesmaximal, complemented
6C3×Dic3×M16—31nomaximal
7C3×C4×#208—31nomaximal
8Dic3×#208C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3C412yes—
11C3×#3748C414yescomplemented
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C412yes—
14C32×C4×Dic3C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C32×#208C414yescomplemented
17C2×C32×C3⋊C8C2212yes—
18C2×C32×M16—31no—
19C2×C32×C42—31no—
20C2×#3748C2×C311yescharacteristic
21C2×C3×C4×Dic3—21no—
22C2×C3×C4×Dic3D311yescharacteristic
23C2×C3×C4×Dic3C2×C311yescharacteristic
24C2×C3×#208—21no—
25C2×C3×#208—31no—
26C2×C3×#208—31no—
27C2×C3×#208—61no—
28C2×C3×#208D311yescharacteristic
29C2×C3×#208C2×C311yescharacteristic
30C3×#815—32no—
31C3×C8×Dic3—32no—
32C3×#803—32no—
33C3×C4×C3⋊C8—32no—
34#63554C2×C312yescomplemented
35Dic3×C3⋊C8C2×C312yescomplemented
36C23×C33C2×C411yescharacteristic
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4C2×C412yes—
40C2×C33×C4C2311yescharacteristic
41C2×C32×Dic3—24no—
42C2×C32×Dic3C2×C412yes—
43C2×C32×Dic3C2×C412yes—
44C3×C3⋊(C3⋊C8)C2×C414yescomplemented
45C32×C3⋊C8C2×C414yescomplemented
46C3×C4×M16—91no—
47Dic3×M16—31no—
48C4×#208—31no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—32no—
53C22×C32×C4—41no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C3×Dic3—22no—
58C22×C3×Dic3Dic312yes—
59C22×C3×Dic3C3×C412yes—
60C32×M16—34no—
61C2×C32×C8—32no—
62C32×C42—32no—
63C32×C42—31no—
64C32×C42—31no—
65#3748C3×C414yescomplemented
66C2×C3⋊(C3⋊C8)C22×C312yes—
67C3×C4×Dic3—22no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3—21no—
70C3×C4×Dic3Dic312yes—
71C3×C4×Dic3C3×C412yes—
72C3×C4×Dic3C2×D311yescharacteristic
73C3×C4×Dic3C2×D311yescharacteristic
74C3×C4×Dic3C22×C311yescharacteristic
75C3×C4×Dic3C22×C311yescharacteristic
76C3×#208—24no—
77C3×#208—34no—
78C3×#208—34no—
79C3×#208—64no—
80C3×#208Dic314yesdirect product factor
81C3×#208C3×C414yescomplemented
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—62no—
86C2×C3×C3⋊C8C2×D312yes—
87C2×C3×C3⋊C8C22×C312yes—
88C22×C33—22no—
89C22×C33C4212yes—
90C22×C33C22×C411yescharacteristic
91C33×C4C4212yes—
92C33×C4C22×C411yescharacteristic
93C33×C4C22×C411yescharacteristic
94C32×Dic3—22no—
95C32×Dic3M1614yescomplemented
96C2×C3×M16—31no—
97C2×C3×M16—31no—
98C2×C3×M16—61no—
99C2×C3×M16—91no—
100C2×C3×C42—31no—
101C2×C3×C42—31no—
102C2×C3×C42—61no—
103C2×C4×Dic3C3×D311yescharacteristic
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—61no—
107C2×#208C3×D311yescharacteristic
108C3×#97—92no—
109C3×C4×C8—92no—
110#815—32no—
111C8×Dic3—32no—
112#803—32no—
113C4×C3⋊C8—32no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C23×C32C4×D311yescharacteristic
119C23×C32C2×Dic311yescharacteristic
120C23×C32C2×C3×C411yescharacteristic
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—32no—
134C2×C32×C4—32no—
135C2×C32×C4—41no—
136C2×C32×C4—42no—
137C2×C32×C4—42no—
138C2×C32×C4—41no—
139C2×C32×C4—64no—
140C2×C32×C4C4×D311yescharacteristic
141C2×C32×C4C4×D312yes—
142C2×C32×C4C4×D312yes—
143C2×C32×C4C4×D312yes—
144C2×C32×C4C2×Dic311yescharacteristic
145C2×C32×C4C2×Dic312yes—
146C2×C32×C4C2×C3×C412yes—
147C2×C32×C4C2×C3×C411yescharacteristic
148C2×C32×C4C2×C3×C412yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C3×Dic3—28no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3C2×Dic312yes—
158C2×C3×Dic3C2×Dic312yes—
159C2×C3×Dic3C2×C3×C412yes—
160C2×C3×Dic3C2×C3×C412yes—
161C32×C8—34no—
162C3⋊(C3⋊C8)C2×C3×C414yescomplemented
163C3×C3⋊C8—24no—
164C3×C3⋊C8—34no—
165C3×C3⋊C8—34no—
166C3×C3⋊C8—64no—
167C3×C3⋊C8C2×Dic314yescomplemented
168C3×C3⋊C8C2×C3×C414yescomplemented
169C4×M16—91noSylow
170C2×C33—22no—
171C2×C33C2×C4211yescharacteristic
172C2×C33C2×M1612yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—41no—
179C22×C3×C4—62no—
180C22×C3×C4D3211yescharacteristic
181C22×C3×C4C2×C3×D311yescharacteristic
182C22×C3×C4C2×C3×D311yescharacteristic
183C22×Dic3C3×Dic312yes—
184C3×M16—34no—
185C3×M16—34no—
186C3×M16—64no—
187C3×M16—94no—
188C2×C3×C8—32no—
189C2×C3×C8—32no—
190C2×C3×C8—62no—
191C2×C3×C8—92no—
192C3×C42—32no—
193C3×C42—32no—
194C3×C42—31no—
195C3×C42—31no—
196C3×C42—31no—
197C3×C42—31no—
198C3×C42—62no—
199C3×C42—61no—
200C3×C42—61no—
201C4×Dic3C3×Dic312yes—
202C4×Dic3C2×C3×D311yescharacteristic
203C4×Dic3C2×C3×D311yescharacteristic
204#208—34no—
205#208—34no—
206#208—64no—
207#208C3×Dic314yesdirect product factor
208C2×C3⋊C8—32no—
209C2×C3⋊C8—32no—
210C2×C3⋊C8—62no—
211C2×C3⋊C8C2×C3×D312yes—
212C22×C32—22no—
213C22×C32—24no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—42no—
225C22×C32—42no—
226C22×C32—44no—
227C22×C32C4×Dic312yes—
228C22×C32C4×Dic312yes—
229C22×C32C3×C4212yes—
230C22×C32C2×C4×D311yescharacteristic
231C22×C32C22×Dic311yescharacteristic
232C22×C32C22×C3×C411yescharacteristic
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—34no—
243C32×C4—42no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—62no—
247C32×C4C4×Dic312yes—
248C32×C4C4×Dic312yes—
249C32×C4C3×C4212yes—
250C32×C4C2×C4×D311yescharacteristic
251C32×C4C2×C4×D311yescharacteristic
252C32×C4C2×C4×D311yescharacteristic
253C32×C4C22×Dic311yescharacteristic
254C32×C4C22×C3×C411yescharacteristic
255C32×C4C22×C3×C411yescharacteristic
256C3×Dic3—24no—
257C3×Dic3—22no—
258C3×Dic3—22no—
259C3×Dic3—24no—
260C3×Dic3#20814yesdirect product factor
261C3×Dic3C3×M1614yescomplemented
262C2×M16—31no—
263C2×M16—91no—
264C2×C42—31no—
265#97—92no—
266C4×C8—92no—
267C33C4×M1611yescharacteristic, complemented, Sylow
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—41no—
272C23×C3D3×Dic311yescharacteristic
273C23×C3C3×C4×D311yescharacteristic
274C23×C3C2×C3×Dic311yescharacteristic
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—64no—
298C2×C3×C4—64no—
299C2×C3×C4—68no—
300C2×C3×C4D3×Dic311yescharacteristic
301C2×C3×C4D3×Dic312yes—
302C2×C3×C4#62412yes—
303C2×C3×C4C3×C4×D311yescharacteristic
304C2×C3×C4C3×C4×D312yes—
305C2×C3×C4C3×C4×D312yes—
306C2×C3×C4C3×C4×D312yes—
307C2×C3×C4C2×C3×Dic311yescharacteristic
308C2×C3×C4C2×C3×Dic312yes—
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C22×C3×D311yescharacteristic
311C2×C3×C4C22×C3×D311yescharacteristic
312C2×Dic3—24no—
313C2×Dic3C2×C3×Dic312yes—
314C2×Dic3C2×C3×Dic312yes—
315C3×C8—34no—
316C3×C8—34no—
317C3×C8—64no—
318C3×C8—94no—
319C3⋊C8—34no—
320C3⋊C8—34no—
321C3⋊C8—64no—
322C3⋊C8C2×C3×Dic314yescomplemented
323C2×C32—22no—
324C2×C32—24no—
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—22no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32—41no—
334C2×C32—42no—
335C2×C32—42no—
336C2×C32—42no—
337C2×C32—44no—
338C2×C32D3×M1612yes—
339C2×C32C2×#20812yes—
340C2×C32C2×C4×Dic311yescharacteristic
341C2×C32C2×C4×Dic311yescharacteristic
342C2×C32C2×C3×C4211yescharacteristic
343C2×C32C2×C3×M1612yes—
344C22×C4—32no—
345C22×C4C3×D3211yescharacteristic
346M16—34no—
347M16—94no—
348C2×C8—32no—
349C2×C8—92no—
350C42—32no—
351C42—31no—
352C42—31no—
353C22×C3—24no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—44no—
366C22×C3—41no—
367C22×C3—42no—
368C22×C3Dic3212yes—
369C22×C3C3×C4×Dic312yes—
370C22×C3C3×C4×Dic312yes—
371C22×C3C2×D3×Dic311yescharacteristic
372C22×C3C2×C3×C4×D311yescharacteristic
373C22×C3C22×C3×Dic311yescharacteristic
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—34no—
384C3×C4—34no—
385C3×C4—41no—
386C3×C4—41no—
387C3×C4—42no—
388C3×C4—64no—
389C3×C4—64no—
390C3×C4—62no—
391C3×C4—62no—
392C3×C4Dic3212yes—
393C3×C4C3×C4×Dic312yes—
394C3×C4C3×C4×Dic312yes—
395C3×C4C2×D3×Dic311yescharacteristic
396C3×C4C2×#62411yescharacteristic
397C3×C4C2×C3×C4×D311yescharacteristic
398C3×C4C2×C3×C4×D311yescharacteristic
399C3×C4C2×C3×C4×D311yescharacteristic
400C3×C4C22×C3×Dic311yescharacteristic
401Dic3—22no—
402Dic3C3×#20814yesdirect product factor
403C32—21no—
404C32—21no—
405C32—21no—
406C32—41no—
407C32C4×#20811yescharacteristic, complemented
408C32Dic3×M1611yescharacteristic, complemented
409C32C3×C4×M1611yescharacteristic, complemented
410C23C3×D3×Dic311yescharacteristic
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—64no—
414C2×C4C3×D3×Dic312yes—
415C2×C4C3×D3×Dic311yescharacteristic
416C2×C4C3×#62412yes—
417C2×C4C2×C3×D3211yescharacteristic
418C8—34no—
419C8—94no—
420C2×C3—24no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—44no—
433C2×C3—41no—
434C2×C3—42no—
435C2×C3D3×#20812yes—
436C2×C3C2×Dic3211yescharacteristic
437C2×C3C3×D3×M1612yes—
438C2×C3C2×C3×#20812yes—
439C2×C3C2×C3×C4×Dic311yescharacteristic
440C2×C3C2×C3×C4×Dic311yescharacteristic
441C22—22no—
442C22C3×Dic3212yes—
443C22C2×C3×D3×Dic311yescharacteristic
444C4—34no—
445C4—62no—
446C4C3×Dic3212yes—
447C4C2×C3×D3×Dic311yescharacteristic
448C4C2×C3×#62411yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3Dic3×#20811yescharacteristic, direct product factor
454C3C3×C4×#20811yescharacteristic, complemented
455C3C3×Dic3×M1611yescharacteristic, complemented
456C2—22no—
457C2C3×D3×#20812yes—
458C2C2×C3×Dic3211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×C4×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63554C2semidirect
5. C3×Dic3×C3⋊C8C2semidirect
8. Dic3×#208C3direct (N × H)
11. C3×#3748C4semidirect
16. C32×#208C4semidirect
34. #63554C2×C3semidirect
35. Dic3×C3⋊C8C2×C3semidirect
44. C3×C3⋊(C3⋊C8)C2×C4semidirect
45. C32×C3⋊C8C2×C4semidirect
65. #3748C3×C4semidirect
80. C3×#208Dic3direct (N × H)
81. C3×#208C3×C4semidirect
95. C32×Dic3M16semidirect
162. C3⋊(C3⋊C8)C2×C3×C4semidirect
167. C3×C3⋊C8C2×Dic3semidirect
168. C3×C3⋊C8C2×C3×C4semidirect
207. #208C3×Dic3direct (N × H)
260. C3×Dic3#208direct (N × H)
261. C3×Dic3C3×M16semidirect
267. C33C4×M16semidirect
322. C3⋊C8C2×C3×Dic3semidirect
402. Dic3C3×#208direct (N × H)
407. C32C4×#208semidirect
408. C32Dic3×M16semidirect
409. C32C3×C4×M16semidirect
453. C3Dic3×#208direct (N × H)
454. C3C3×C4×#208semidirect
455. C3C3×Dic3×M16semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C2×C32×C4×Dic32
C2×C32×#2082
C3×#635542
C3×Dic3×C3⋊C82
C3×Dic3×M163
C3×C4×#2083
Dic3×#2083