Finite Group Databasebeta

← C2×C3×#63364 #13142378

Subgroups — C2×C3×#63364 #13142378

Order 1728.

618 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×C3×C42×C9C211yesmaximal, characteristic, complemented
3C2×C3×#3667C211yesmaximal, characteristic, complemented
4C3×#63364C214yesmaximal, direct product factor
5C2×C3×#805—31nomaximal
6C2×#63364C311yesmaximal, characteristic, direct product factor
7C22×C3×C4×C9—21no—
8C22×C3×C4×C9C2211yescharacteristic
9C2×C3×#611—21no—
10C3×#3697—21no—
11C3×#3697—21no—
12C3×#3697C412yescomplemented
13C3×#3697C2212yes—
14C2×C3×D36C411yescharacteristic, complemented
15C2×C3×C4×D9—21no—
16C2×C3×Dic18C411yescharacteristic, complemented
17C3×C42×C9—21no—
18C3×C42×C9C2212yescomplemented
19C3×#3667—21no—
20C3×#3667C2212yescomplemented
21C2×C3×C9⋊C8—21no—
22C2×C3×#235—31no—
23C2×C32×C42D311yescharacteristic
24C2×C3×#208—31no—
25C2×#3697C2×C311yescharacteristic
26C3×#805—34no—
27C2×C42×C9—21no—
28C2×C42×C9C2×C311yescharacteristic, complemented
29C2×#3667C2×C311yescharacteristic, complemented
30#63364C2×C314yesdirect product factor
31C23×C3×C9D411yescharacteristic
32C22×C3×D9—21no—
33C2×C3×C4×C9—24no—
34C2×C3×C4×C9—22no—
35C2×C3×C4×C9C2×C411yescharacteristic
36C2×C3×C4×C9C2×C411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9D411yescharacteristic
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9C2311yescharacteristic
41C3×#611—22no—
42C3×#611—22no—
43C2×C3×Dic9—21no—
44C3×D36—21no—
45C3×D36C2×C412yescomplemented
46C3×C4×D9—22no—
47C3×C4×D9—22no—
48C3×Dic18—21no—
49C3×Dic18C2×C412yescomplemented
50C3×C9⋊C8—22no—
51C2×C3×#104—91no—
52C2×#805—31no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C2×C3×#67—61no—
56C3×#235—32no—
57C3×#235—32no—
58C3×#235—61no—
59C3×#235—61no—
60C2×C3×D12—31no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—31no—
63C32×C42—21no—
64C32×C42C2×D312yes—
65C3×#208—32no—
66C3×#208—61no—
67C2×C3×C3⋊C8—61no—
68C22×C4×C9—22no—
69C22×C4×C9—21no—
70C22×C4×C9—21no—
71C22×C4×C9C22×C311yescharacteristic
72C2×#611—21no—
73#3697—21no—
74#3697—21no—
75#3697C3×C412yescomplemented
76#3697C22×C312yes—
77C2×D36C3×C411yescharacteristic, complemented
78C2×C4×D9—21no—
79C2×Dic18C3×C411yescharacteristic, complemented
80C42×C9—22no—
81C42×C9—22no—
82C42×C9—21no—
83C42×C9C22×C312yescomplemented
84#3667—21no—
85#3667C22×C312yescomplemented
86C2×C9⋊C8—21no—
87C22×C3×C9—22no—
88C22×C3×C9C22⋊C411yescharacteristic
89C22×C3×C9C22⋊C411yescharacteristic
90C22×C3×C9C2×D411yescharacteristic
91C2×C3×D9—22no—
92C2×C3×D9—42no—
93C3×C4×C9—24no—
94C3×C4×C9C22⋊C411yescharacteristic
95C3×C4×C9C22⋊C411yescharacteristic
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3×Dic9—22no—
99C2×C3×#41—91no—
100C2×#235—31no—
101C2×C3×M16—91no—
102C2×C3×C42—21no—
103C2×C3×C42D911yescharacteristic, complemented
104C2×C3×C42C3×D311yescharacteristic
105C2×#208—31no—
106C3×#104—94no—
107#805—34no—
108C23×C32#6711yescharacteristic
109C22×C3×D3—61no—
110C2×C32×C4—24no—
111C2×C32×C4—22no—
112C2×C32×C4C4×D311yescharacteristic
113C2×C32×C4C4×D311yescharacteristic
114C2×C32×C4D1211yescharacteristic
115C2×C32×C4D1211yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C3×#67—62no—
119C3×#67—62no—
120C2×C3×Dic3—61no—
121C3×D12—32no—
122C3×D12—61no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C23×C9—21no—
128C23×C9C3×D411yescharacteristic
129C22×D9—21no—
130C3×C3⋊C8—62no—
131C2×C4×C9—24no—
132C2×C4×C9—28no—
133C2×C4×C9—24no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9C2×C3×C411yescharacteristic
142C2×C4×C9C2×C3×C411yescharacteristic
143C2×C4×C9C3×D411yescharacteristic
144C2×C4×C9C3×D411yescharacteristic
145C2×C4×C9C3×D411yescharacteristic
146C2×C4×C9C23×C311yescharacteristic
147#611—22no—
148#611—22no—
149C2×Dic9—21no—
150D36—21no—
151D36C2×C3×C412yescomplemented
152C4×D9—22no—
153C4×D9—22no—
154Dic18—21no—
155Dic18C2×C3×C412yescomplemented
156C9⋊C8—22no—
157C2×#104—91noSylow
158C2×C3×C9—21no—
159C2×C3×C9—21no—
160C2×C3×C9#10412yescomplemented
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C3×D9—42no—
163C3×#41—92no—
164C3×#41—92no—
165C3×#41—181no—
166C3×#41—181no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—181no—
175C22×C3×C4C2×D911yescharacteristic
176C22×C3×C4C2×C3×D311yescharacteristic
177C2×#67—61no—
178#235—32no—
179#235—32no—
180#235—61no—
181#235—61no—
182C2×D12—31no—
183C2×C4×D3—61no—
184C2×Dic6—31no—
185C3×M16—92no—
186C3×M16—181no—
187C2×C3×C8—181no—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—22no—
192C3×C42C2×D912yescomplemented
193C3×C42C2×C3×D312yes—
194#208—32no—
195#208—61no—
196C2×C3⋊C8—61no—
197C22×C32—22no—
198C22×C32(C2×D3)⋊C411yescharacteristic
199C22×C32(C2×D3)⋊C411yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C2×C3×D3—62no—
202C2×C3×D3—122no—
203C32×C4—24no—
204C32×C4(C2×D3)⋊C411yescharacteristic
205C32×C4(C2×D3)⋊C411yescharacteristic
206C32×C4C2×C4×D311yescharacteristic
207C32×C4C2×D1211yescharacteristic
208C3×Dic3—62no—
209C22×C9—24no—
210C22×C9—22no—
211C22×C9—21no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9C3×C22⋊C411yescharacteristic
215C22×C9C3×C22⋊C411yescharacteristic
216C22×C9C2×C3×D411yescharacteristic
217C2×D9—22no—
218C2×D9—42no—
219C4×C9—28no—
220C4×C9—24no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—21no—
224C4×C9—21no—
225C4×C9C3×C22⋊C411yescharacteristic
226C4×C9C3×C22⋊C411yescharacteristic
227C4×C9C22×C3×C411yescharacteristic
228C4×C9C2×C3×D411yescharacteristic
229Dic9—22no—
230C2×#41—91no—
231C2×M16—91no—
232C2×C42C3×D911yescharacteristic, complemented
233#104—94no—
234C3×C9C2×#10411yescharacteristic, complemented, Sylow
235C23×C3—21no—
236C23×C3—181no—
237C23×C3#61111yescharacteristic
238C23×C3C3×#6711yescharacteristic
239C22×D3—61no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—92no—
243C3×D4—182no—
244C3×D4—182no—
245C3×D4—181no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—28no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—181no—
259C2×C3×C4—182no—
260C2×C3×C4—182no—
261C2×C3×C4C4×D911yescharacteristic
262C2×C3×C4C4×D911yescharacteristic
263C2×C3×C4D3611yescharacteristic
264C2×C3×C4D3611yescharacteristic
265C2×C3×C4#61111yescharacteristic
266C2×C3×C4C22×D911yescharacteristic
267C2×C3×C4C3×C4×D311yescharacteristic
268C2×C3×C4C3×C4×D311yescharacteristic
269C2×C3×C4C3×D1211yescharacteristic
270C2×C3×C4C3×D1211yescharacteristic
271C2×C3×C4C3×#6711yescharacteristic
272C2×C3×C4C22×C3×D311yescharacteristic
273#67—62no—
274#67—62no—
275C2×Dic3—61no—
276D12—32no—
277D12—61no—
278C4×D3—62no—
279C4×D3—62no—
280Dic6—32no—
281Dic6—61no—
282C3×C8—182no—
283C3⋊C8—62no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#80512yes—
287C2×C32C2×(C2×D3)⋊C411yescharacteristic
288C3×D3—122no—
289C2×C9—22no—
290C2×C9—22no—
291C2×C9—21no—
292C2×C9—21no—
293C2×C9—21no—
294C2×C9—22no—
295C2×C9C3×#10412yescomplemented
296C2×C9C2×C3×C22⋊C411yescharacteristic
297D9—42no—
298#41—92no—
299#41—92no—
300#41—181no—
301#41—181no—
302C2×Q8—91no—
303C2×D4—91no—
304C2×D4—181no—
305C22×C4—21no—
306C22×C4—181no—
307C22×C4C2×C3×D911yescharacteristic
308M16—92no—
309M16—181no—
310C2×C8—181no—
311C42—21no—
312C42C2×C3×D912yescomplemented
313C22×C3—24no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—182no—
320C22×C3—362no—
321C22×C3(C2×D9)⋊C411yescharacteristic
322C22×C3(C2×D9)⋊C411yescharacteristic
323C22×C3C2×#61111yescharacteristic
324C22×C3C3×(C2×D3)⋊C411yescharacteristic
325C22×C3C3×(C2×D3)⋊C411yescharacteristic
326C22×C3C2×C3×#6711yescharacteristic
327C2×D3—62no—
328C2×D3—122no—
329C3×C4—24no—
330C3×C4—28no—
331C3×C4—24no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—182no—
337C3×C4(C2×D9)⋊C411yescharacteristic
338C3×C4(C2×D9)⋊C411yescharacteristic
339C3×C4C2×C4×D911yescharacteristic
340C3×C4C2×D3611yescharacteristic
341C3×C4C3×(C2×D3)⋊C411yescharacteristic
342C3×C4C3×(C2×D3)⋊C411yescharacteristic
343C3×C4C2×C3×C4×D311yescharacteristic
344C3×C4C2×C3×D1211yescharacteristic
345Dic3—62no—
346C32C2×#80511yescharacteristic
347C9—21no—
348C9C2×C3×#10411yescharacteristic, complemented
349C23—181no—
350C23C3×#61111yescharacteristic
351Q8—92no—
352Q8—181no—
353D4—92no—
354D4—182no—
355D4—182no—
356D4—181no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—181no—
360C2×C4—182no—
361C2×C4—182no—
362C2×C4C3×C4×D911yescharacteristic
363C2×C4C3×C4×D911yescharacteristic
364C2×C4C3×D3611yescharacteristic
365C2×C4C3×D3611yescharacteristic
366C2×C4C3×#61111yescharacteristic
367C2×C4C22×C3×D911yescharacteristic
368C8—182no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—362no—
378C2×C3#6336412yesdirect product factor
379C2×C3C2×(C2×D9)⋊C411yescharacteristic
380C2×C3C3×#80512yes—
381C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
382D3—122no—
383C22—22no—
384C22—182no—
385C22—362no—
386C22C3×(C2×D9)⋊C411yescharacteristic
387C22C3×(C2×D9)⋊C411yescharacteristic
388C22C2×C3×#61111yescharacteristic
389C4—24no—
390C4—182no—
391C4C3×(C2×D9)⋊C411yescharacteristic
392C4C3×(C2×D9)⋊C411yescharacteristic
393C4C2×C3×C4×D911yescharacteristic
394C4C2×C3×D3611yescharacteristic
395C3—21no—
396C3C2×#6336411yescharacteristic, direct product factor
397C3C2×C3×#80511yescharacteristic
398C2—21no—
399C2—21no—
400C2—362no—
401C2C3×#6336412yesdirect product factor
402C2C2×C3×(C2×D9)⋊C411yescharacteristic
subgroupG/H
centerC2×C3×C4D36
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C42×C9C2
socleC22×C32(C2×D3)⋊C4
prime radicalC2×C3×#3697C2
cosocleC2×C4×C9C23×C3
hypercenterC2×C3×C42D9
hypocenterC9C2×C3×#104
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×C42×C9C2semidirect
3. C2×C3×#3667C2semidirect
4. C3×#63364C2direct (N × H)
6. C2×#63364C3direct (N × H)
12. C3×#3697C4semidirect
14. C2×C3×D36C4semidirect
16. C2×C3×Dic18C4semidirect
18. C3×C42×C9C22semidirect
20. C3×#3667C22semidirect
28. C2×C42×C9C2×C3semidirect
29. C2×#3667C2×C3semidirect
30. #63364C2×C3direct (N × H)
45. C3×D36C2×C4semidirect
49. C3×Dic18C2×C4semidirect
75. #3697C3×C4semidirect
77. C2×D36C3×C4semidirect
79. C2×Dic18C3×C4semidirect
83. C42×C9C22×C3semidirect
85. #3667C22×C3semidirect
103. C2×C3×C42D9semidirect
151. D36C2×C3×C4semidirect
155. Dic18C2×C3×C4semidirect
160. C2×C3×C9#104semidirect
192. C3×C42C2×D9semidirect
232. C2×C42C3×D9semidirect
234. C3×C9C2×#104semidirect
295. C2×C9C3×#104semidirect
312. C42C2×C3×D9semidirect
348. C9C2×C3×#104semidirect
378. C2×C3#63364direct (N × H)
396. C3C2×#63364direct (N × H)
401. C2C3×#63364direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×C3×C42×C92
C2×C3×#36672
C3×#633642
C2×C3×#8053
C2×#633643