Finite Group Databasebeta

← C2×C3×#63456 #13152403

Subgroups — C2×C3×#63456 #13152403

Order 1728.

1118 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D36C211yesmaximal, characteristic
2C22×C3×C4×D9C211yesmaximal, characteristic, complemented
3C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
8C3×#63456C218yesmaximal, direct product factor
9C2×C3×#894—31nomaximal
10C2×#63456C311yesmaximal, characteristic, direct product factor
11C23×C3×D9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C3×D36—24no—
19C2×C3×D36C2214yes—
20C2×C3×C4×D9—24no—
21C2×C3×C4×D9C2214yescomplemented
22C3×C9×C4⋊C4C2214yescomplemented
23C3×(C2×D9)⋊C4C2214yescomplemented
24C3×(C2×D9)⋊C4C2214yescomplemented
25C3×(C2×D9)⋊C4C2214yescomplemented
26C3×Dic9⋊C4C2214yescomplemented
27C22×C3×D12—31no—
28C22×C3×C4×D3—31no—
29C2×C32×C4⋊C4D311yescharacteristic
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×(C2×D3)⋊C4—31no—
32C2×C3×(C2×D3)⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#894—38no—
35C22×D36C2×C311yescharacteristic
36C22×C4×D9C2×C311yescharacteristic, complemented
37C2×C9×C4⋊C4—21no—
38C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
39C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
40C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
41C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63456C2×C318yesdirect product factor
44C23×C3×C9C2311yescharacteristic
45C22×C3×D9—22no—
46C22×C3×D9—22no—
47C22×C3×D9—22no—
48C22×C3×D9—22no—
49C22×C3×D9—22no—
50C22×C3×D9D414yes—
51C22×C3×D9C2312yes—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9—22no—
55C2×C3×C4×C9—22no—
56C2×C3×C4×C9C2312yes—
57C2×C3×C4×C9C2312yes—
58C2×C3×C4×C9C2312yes—
59C2×C3×Dic9—22no—
60C2×C3×Dic9—22no—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63C3×D36—28no—
64C3×C4×D9—28no—
65C2×C3×#123—91no—
66C2×#894—31no—
67C23×C3×D3—31no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C2×C3×D12—34no—
75C2×C3×D12—64no—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—64no—
78C23×D9C22×C311yescharacteristic
79C23×D9C22×C311yescharacteristic
80C32×C4⋊C4C2×D314yes—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×Dic3⋊C4—34no—
85C22×C4×C9—21no—
86C22×C4×C9—21no—
87C22×C4×C9—21no—
88C22×C4×C9C22×C311yescharacteristic
89C22×C4×C9C22×C311yescharacteristic
90C22×C4×C9C22×C311yescharacteristic
91C22×Dic9C22×C311yescharacteristic
92C22×Dic9C22×C311yescharacteristic
93C2×D36—24no—
94C2×D36C22×C314yes—
95C2×C4×D9—24no—
96C2×C4×D9C22×C314yescomplemented
97C9×C4⋊C4—24no—
98C9×C4⋊C4C22×C314yescomplemented
99(C2×D9)⋊C4C22×C314yescomplemented
100(C2×D9)⋊C4C22×C314yescomplemented
101(C2×D9)⋊C4C22×C314yescomplemented
102Dic9⋊C4C22×C314yescomplemented
103C22×C3×C9C2×D412yes—
104C22×C3×C9#4112yes—
105C22×C3×C9#4112yes—
106C22×C3×C9C2411yescharacteristic
107C2×C3×D9—28no—
108C2×C3×D9—22no—
109C2×C3×D9—22no—
110C2×C3×D9—22no—
111C2×C3×D9—22no—
112C2×C3×D9—22no—
113C2×C3×D9—44no—
114C2×C3×D9C2×D414yes—
115C3×C4×C9—22no—
116C3×C4×C9—22no—
117C3×C4×C9—22no—
118C3×Dic9—22no—
119C3×Dic9—22no—
120C22×C3×D4—91no—
121C23×C3×C4—91no—
122C22×D12—31no—
123C22×C4×D3—31no—
124C3×#123—98no—
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4—91no—
127C2×C3×C4⋊C4D911yescharacteristic, complemented
128C2×C3×C4⋊C4C3×D311yescharacteristic
129C2×C3×C22⋊C4—91no—
130C2×C3×C22⋊C4—91no—
131C2×C3×C22⋊C4—91no—
132C2×(C2×D3)⋊C4—31no—
133C2×(C2×D3)⋊C4—31no—
134C2×(C2×D3)⋊C4—31no—
135C2×Dic3⋊C4—31no—
136#894—38no—
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—34no—
139C22×C3×D3—32no—
140C22×C3×D3—32no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D312yes—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C3×D12—68no—
157C3×C4×D3—68no—
158C23×C9—21no—
159C23×C9C23×C311yescharacteristic
160C22×D9—22no—
161C22×D9—22no—
162C22×D9—22no—
163C22×D9—22no—
164C22×D9—22no—
165C22×D9C3×D414yes—
166C22×D9C23×C312yes—
167C22×D9C23×C312yes—
168C2×C4×C9—24no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—22no—
175C2×C4×C9—42no—
176C2×C4×C9—42no—
177C2×C4×C9C23×C312yes—
178C2×C4×C9C23×C312yes—
179C2×C4×C9C23×C312yes—
180C2×Dic9—22no—
181C2×Dic9—22no—
182C2×Dic9C23×C312yes—
183C2×Dic9C23×C312yes—
184D36—28no—
185C4×D9—28no—
186C2×#123—91noSylow
187C2×C3×C9#12314yescomplemented
188C2×C3×C9C22×D411yescharacteristic
189C2×C3×C9C2×#4111yescharacteristic
190C2×C3×C9C2×#4111yescharacteristic
191C3×D9—24no—
192C3×D9—42no—
193C24×C3—91no—
194C24×C3—91no—
195C23×D3—31no—
196C23×D3—31no—
197C2×C3×D4—94no—
198C2×C3×D4—184no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—94no—
205C22×C3×C4—184no—
206C22×C3×C4C2×D911yescharacteristic
207C22×C3×C4C2×D911yescharacteristic
208C22×C3×C4C2×D911yescharacteristic
209C22×C3×C4C2×C3×D311yescharacteristic
210C22×C3×C4C2×C3×D311yescharacteristic
211C22×C3×C4C2×C3×D311yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C2×D12—34no—
215C2×D12—64no—
216C2×C4×D3—34no—
217C2×C4×D3—64no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—94no—
220C3×C4⋊C4C2×D914yescomplemented
221C3×C4⋊C4C2×C3×D314yes—
222C3×C22⋊C4—94no—
223C3×C22⋊C4—94no—
224C3×C22⋊C4—94no—
225(C2×D3)⋊C4—34no—
226(C2×D3)⋊C4—34no—
227(C2×D3)⋊C4—34no—
228Dic3⋊C4—34no—
229C22×C32#23512yes—
230C22×C32D3×D412yes—
231C22×C32#23912yes—
232C22×C32C23×D311yescharacteristic
233C2×C3×D3—34no—
234C2×C3×D3—68no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—124no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—22no—
244C3×Dic3—62no—
245C3×Dic3—62no—
246C22×C9—21no—
247C22×C9—22no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9C2×C3×D412yes—
251C22×C9C3×#4112yes—
252C22×C9C3×#4112yes—
253C22×C9C24×C311yescharacteristic
254C2×D9—28no—
255C2×D9—22no—
256C2×D9—22no—
257C2×D9—22no—
258C2×D9—22no—
259C2×D9—22no—
260C2×D9—44no—
261C2×D9C2×C3×D414yes—
262C4×C9—24no—
263C4×C9—22no—
264C4×C9—22no—
265C4×C9—22no—
266C4×C9—42no—
267C4×C9—42no—
268Dic9—22no—
269Dic9—22no—
270C22×D4—91no—
271C23×C4—91no—
272#123—98no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4C3×D911yescharacteristic, complemented
275C2×C22⋊C4—91no—
276C2×C22⋊C4—91no—
277C2×C22⋊C4—91no—
278C3×C9C2×#12311yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3—94no—
281C23×C3—92no—
282C23×C3—92no—
283C23×C3—182no—
284C23×C3—182no—
285C23×C3—182no—
286C23×C3—182no—
287C23×C3—182no—
288C23×C3C22×D911yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C22×D3—32no—
291C22×D3—34no—
292C22×D3—32no—
293C22×D3—62no—
294C22×D3—62no—
295C22×D3—62no—
296C22×D3—62no—
297C22×D3—62no—
298C3×D4—188no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—92no—
312C2×C3×C4—92no—
313C2×C3×C4—188no—
314C2×C3×C4—182no—
315C2×C3×C4—182no—
316C2×C3×C4C22×D912yes—
317C2×C3×C4C22×D912yes—
318C2×C3×C4C22×D912yes—
319C2×C3×C4C22×C3×D312yes—
320C2×C3×C4C22×C3×D312yes—
321C2×C3×C4C22×C3×D312yes—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326D12—68no—
327C4×D3—68no—
328C2×C32#89414yes—
329C2×C32C2×#23511yescharacteristic
330C2×C32C2×D3×D411yescharacteristic
331C2×C32C2×#23911yescharacteristic
332C3×D3—64no—
333C3×D3—122no—
334C2×C9—24no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—21no—
338C2×C9C3×#12314yescomplemented
339C2×C9C22×C3×D411yescharacteristic
340C2×C9C2×C3×#4111yescharacteristic
341C2×C9C2×C3×#4111yescharacteristic
342D9—24no—
343D9—42no—
344C24—91no—
345C24—91no—
346C2×D4—94no—
347C2×D4—184no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—94no—
351C22×C4—184no—
352C22×C4C2×C3×D911yescharacteristic
353C22×C4C2×C3×D911yescharacteristic
354C22×C4C2×C3×D911yescharacteristic
355C4⋊C4—94no—
356C4⋊C4C2×C3×D914yescomplemented
357C22⋊C4—94no—
358C22⋊C4—94no—
359C22⋊C4—94no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—94no—
365C22×C3—182no—
366C22×C3—188no—
367C22×C3—182no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3—364no—
372C22×C3#369712yes—
373C22×C3D4×D912yes—
374C22×C3#370112yes—
375C22×C3C23×D911yescharacteristic
376C22×C3C3×#23512yes—
377C22×C3C3×D3×D412yes—
378C22×C3C3×#23912yes—
379C22×C3C23×C3×D311yescharacteristic
380C2×D3—34no—
381C2×D3—68no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—124no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—42no—
396C3×C4—42no—
397C3×C4—182no—
398C3×C4—182no—
399Dic3—62no—
400Dic3—62no—
401C32C2×#89411yescharacteristic
402C9—21no—
403C9C2×C3×#12311yescharacteristic, complemented
404C23—92no—
405C23—94no—
406C23—92no—
407C23—182no—
408C23—182no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23C22×C3×D911yescharacteristic
413D4—188no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—92no—
418C2×C4—92no—
419C2×C4—188no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4C22×C3×D912yes—
423C2×C4C22×C3×D912yes—
424C2×C4C22×C3×D912yes—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—24no—
429C2×C3—184no—
430C2×C3—362no—
431C2×C3#6345614yesdirect product factor
432C2×C3C2×#369711yescharacteristic
433C2×C3C2×D4×D911yescharacteristic
434C2×C3C2×#370111yescharacteristic
435C2×C3C3×#89414yes—
436C2×C3C2×C3×#23511yescharacteristic
437C2×C3C2×C3×D3×D411yescharacteristic
438C2×C3C2×C3×#23911yescharacteristic
439D3—64no—
440D3—122no—
441C22—94no—
442C22—182no—
443C22—182no—
444C22—188no—
445C22—182no—
446C22—182no—
447C22—182no—
448C22—364no—
449C22C3×#369712yes—
450C22C3×D4×D912yes—
451C22C3×#370112yes—
452C22C23×C3×D911yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—182no—
457C4—182no—
458C3—21no—
459C3C2×#6345611yescharacteristic, direct product factor
460C3C2×C3×#89411yescharacteristic
461C2—184no—
462C2—362no—
463C2C3×#6345614yesdirect product factor
464C2C2×C3×#369711yescharacteristic
465C2C2×C3×D4×D911yescharacteristic
466C2C2×C3×#370111yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC22×C3×D36C2
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×#123
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×D9C2semidirect
3. C2×C3×C9×C4⋊C4C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×C3×Dic9⋊C4C2semidirect
8. C3×#63456C2direct (N × H)
10. C2×#63456C3direct (N × H)
21. C2×C3×C4×D9C22semidirect
22. C3×C9×C4⋊C4C22semidirect
23. C3×(C2×D9)⋊C4C22semidirect
24. C3×(C2×D9)⋊C4C22semidirect
25. C3×(C2×D9)⋊C4C22semidirect
26. C3×Dic9⋊C4C22semidirect
36. C22×C4×D9C2×C3semidirect
38. C2×C9×C4⋊C4C2×C3semidirect
39. C2×(C2×D9)⋊C4C2×C3semidirect
40. C2×(C2×D9)⋊C4C2×C3semidirect
41. C2×(C2×D9)⋊C4C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63456C2×C3direct (N × H)
96. C2×C4×D9C22×C3semidirect
98. C9×C4⋊C4C22×C3semidirect
99. (C2×D9)⋊C4C22×C3semidirect
100. (C2×D9)⋊C4C22×C3semidirect
101. (C2×D9)⋊C4C22×C3semidirect
102. Dic9⋊C4C22×C3semidirect
127. C2×C3×C4⋊C4D9semidirect
187. C2×C3×C9#123semidirect
220. C3×C4⋊C4C2×D9semidirect
274. C2×C4⋊C4C3×D9semidirect
278. C3×C9C2×#123semidirect
338. C2×C9C3×#123semidirect
356. C4⋊C4C2×C3×D9semidirect
403. C9C2×C3×#123semidirect
431. C2×C3#63456direct (N × H)
459. C3C2×#63456direct (N × H)
463. C2C3×#63456direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D362
C22×C3×C4×D92
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×Dic9⋊C42
C3×#634562
C2×C3×#8943
C2×#634563