Finite Group Databasebeta

← C3×#10592704 #13163044

Subgroups — C3×#10592704 #13163044

Order 1728.

1228 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C33×#117C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4C3×#63632C214yesmaximal, complemented
5#10592704C311yesmaximal, characteristic, direct product factor
6C3×#5005—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638—21no—
9C3×#3828—21no—
10C3×#3828—21no—
11C3×#3828C412yescomplemented
12C3×#3828C2212yes—
13C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C411yescharacteristic, complemented
16C33×C4⋊C4—21no—
17C33×C22⋊C4—21no—
18C33×C42C2212yescomplemented
19C3×#3748—21no—
20C3×#3748C2212yescomplemented
21C2×C3×C3⋊(C3⋊C8)—21no—
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—34no—
24C32×#117—24no—
25C32×#117D314yescomplemented
26C32×#117C2×C311yescharacteristic, complemented
27C2×#3748C2×C311yescharacteristic, complemented
28C2×C3×#208—34no—
29#63632C2×C314yescomplemented
30C3×#805—316no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4—22no—
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4C2×C411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C3×#638—22no—
41C3×#638—22no—
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yes—
45C3×C4×C3⋊D3—22no—
46C3×C4×C3⋊D3—22no—
47C3×#634—21no—
48C3×#634C2×C412yes—
49C3×C3⋊(C3⋊C8)—22no—
50C3×#421—91no—
51#5005—34no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C22×C311yescharacteristic
55C2×#638—21no—
56#3828—21no—
57#3828—21no—
58#3828C3×C412yescomplemented
59#3828C22×C312yes—
60C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
61C2×C4×C3⋊D3—21no—
62C2×#634C3×C411yescharacteristic, complemented
63C2×C3×#67—64no—
64C3×#235—38no—
65C3×#235—38no—
66C3×#235—64no—
67C3×#235—64no—
68C2×C3×D12—34no—
69C2×C3×C4×D3—64no—
70C2×C3×Dic6—34no—
71C32×C4⋊C4—28no—
72C32×C4⋊C4—24no—
73C32×C4⋊C4—21no—
74C32×C22⋊C4—28no—
75C32×C22⋊C4—24no—
76C32×C22⋊C4—21no—
77C32×C42—28no—
78C32×C42C2×D318yescomplemented
79C32×C42C22×C312yescomplemented
80#3748—21no—
81#3748C22×C312yescomplemented
82C2×C3⋊(C3⋊C8)—21no—
83C3×#208—38no—
84C3×#208—64no—
85C2×C3×C3⋊C8—64no—
86C22×C33—41no—
87C22×C33C22⋊C411yescharacteristic
88C22×C33C22⋊C411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—42no—
92C33×C4—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—22no—
98C2×C3×#41—91no—
99C2×#235—34no—
100C2×C3×M16—91no—
101C3×#117—24no—
102C3×#117C3×D314yescomplemented
103C3×#117C3⋊D311yescharacteristic, complemented
104C2×#208—34no—
105C3×#104—94no—
106#805—316no—
107C23×C32—24no—
108C23×C32#6714yes—
109C23×C32C3×D411yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C2×C32×C4—216no—
113C2×C32×C4—28no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4C4×D314yes—
122C2×C32×C4C4×D314yes—
123C2×C32×C4D1214yes—
124C2×C32×C4D1214yes—
125C2×C32×C4#6714yes—
126C2×C32×C4C2×C3×C411yescharacteristic
127C2×C32×C4C2×C3×C411yescharacteristic
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C3×D411yescharacteristic
130C2×C32×C4C3×D411yescharacteristic
131C2×C32×C4C22×D314yes—
132C2×C32×C4C23×C311yescharacteristic
133#638—22no—
134#638—22no—
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)C2×C3×C412yes—
138C4×C3⋊D3—22no—
139C4×C3⋊D3—22no—
140#634—21no—
141#634C2×C3×C412yes—
142C3×#67—68no—
143C3×#67—68no—
144C2×C3×Dic3—64no—
145C3×D12—38no—
146C3×D12—64no—
147C3×C4×D3—68no—
148C3×C4×D3—68no—
149C3×Dic6—38no—
150C3×Dic6—64no—
151C3⋊(C3⋊C8)—22no—
152C3×C3⋊C8—68no—
153#421—91noSylow
154C2×C33—21no—
155C2×C33—21no—
156C2×C33—21no—
157C2×C33C2×C22⋊C411yescharacteristic
158C3×C3⋊D3—42no—
159C3×#41—92no—
160C3×#41—92no—
161C3×#41—181no—
162C3×#41—181no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—24no—
167C22×C3×C4—181no—
168C22×C3×C4C2×C3×D314yes—
169C22×C3×C4C2×C3⋊D311yescharacteristic
170C2×#67—64no—
171#235—38no—
172#235—38no—
173#235—64no—
174#235—64no—
175C2×D12—34no—
176C2×C4×D3—64no—
177C2×Dic6—34no—
178C3×M16—92no—
179C3×M16—181no—
180C2×C3×C8—181no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—21no—
187C3×C42—28no—
188C3×C42C2×C3×D318yescomplemented
189C3×C42C2×C3⋊D312yescomplemented
190#208—38no—
191#208—64no—
192C2×C3⋊C8—64no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—44no—
197C22×C32—41no—
198C22×C32—48no—
199C22×C32(C2×D3)⋊C414yes—
200C22×C32(C2×D3)⋊C414yes—
201C22×C32C3×C22⋊C411yescharacteristic
202C22×C32C3×C22⋊C411yescharacteristic
203C22×C32C2×#6714yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3—42no—
207C2×C3×D3—68no—
208C2×C3×D3—128no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—416no—
214C32×C4—48no—
215C32×C4—42no—
216C32×C4(C2×D3)⋊C414yes—
217C32×C4(C2×D3)⋊C414yes—
218C32×C4C3×C22⋊C411yescharacteristic
219C32×C4C3×C22⋊C411yescharacteristic
220C32×C4C2×C4×D314yes—
221C32×C4C2×D1214yes—
222C32×C4C22×C3×C411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C3⋊Dic3—22no—
225C3×Dic3—68no—
226C2×#41—91no—
227C2×M16—91no—
228#117C3×C3⋊D311yescharacteristic, complemented
229#104—94no—
230C33#42111yescharacteristic, complemented, Sylow
231C23×C3—24no—
232C23×C3—181no—
233C23×C3C3×#6714yes—
234C23×C3#63811yescharacteristic
235C22×D3—64no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×D4—92no—
239C3×D4—182no—
240C3×D4—182no—
241C3×D4—181no—
242C2×C3×C4—216no—
243C2×C3×C4—28no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—181no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×C3×C4C3×C4×D314yes—
255C2×C3×C4C3×C4×D314yes—
256C2×C3×C4C3×D1214yes—
257C2×C3×C4C3×D1214yes—
258C2×C3×C4C3×#6714yes—
259C2×C3×C4C4×C3⋊D311yescharacteristic
260C2×C3×C4C4×C3⋊D311yescharacteristic
261C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
262C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
263C2×C3×C4#63811yescharacteristic
264C2×C3×C4C22×C3×D314yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—68no—
267#67—68no—
268C2×Dic3—64no—
269D12—38no—
270D12—64no—
271C4×D3—68no—
272C4×D3—68no—
273Dic6—38no—
274Dic6—64no—
275C3×C8—182no—
276C3⋊C8—68no—
277C2×C32—28no—
278C2×C32—24no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—44no—
286C2×C32—44no—
287C2×C32C2×(C2×D3)⋊C414yes—
288C2×C32C2×C3×C22⋊C411yescharacteristic
289C3⋊D3—42no—
290C3×D3—128no—
291#41—92no—
292#41—92no—
293#41—181no—
294#41—181no—
295C2×Q8—91no—
296C2×D4—91no—
297C2×D4—181no—
298C22×C4—181no—
299C22×C4C2×C3×C3⋊D311yescharacteristic
300M16—92no—
301M16—181no—
302C2×C8—181no—
303C4⋊C4—21no—
304C22⋊C4—21no—
305C42C2×C3×C3⋊D312yescomplemented
306C22×C3—24no—
307C22×C3—24no—
308C22×C3—24no—
309C22×C3—44no—
310C22×C3—41no—
311C22×C3—48no—
312C22×C3—182no—
313C22×C3—362no—
314C22×C3C3×(C2×D3)⋊C414yes—
315C22×C3C3×(C2×D3)⋊C414yes—
316C22×C3#375211yescharacteristic
317C22×C3#375211yescharacteristic
318C22×C3C2×C3×#6714yes—
319C22×C3C2×#63811yescharacteristic
320C2×D3—68no—
321C2×D3—128no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—416no—
327C3×C4—48no—
328C3×C4—42no—
329C3×C4—182no—
330C3×C4C3×(C2×D3)⋊C414yes—
331C3×C4C3×(C2×D3)⋊C414yes—
332C3×C4#375211yescharacteristic
333C3×C4#375211yescharacteristic
334C3×C4C2×C3×C4×D314yes—
335C3×C4C2×C3×D1214yes—
336C3×C4C2×C4×C3⋊D311yescharacteristic
337C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
338Dic3—68no—
339C32—24no—
340C32#500514yescomplemented
341C32C3×#42111yescharacteristic, complemented
342C23—181no—
343C23C3×#63811yescharacteristic
344Q8—92no—
345Q8—181no—
346D4—92no—
347D4—182no—
348D4—182no—
349D4—181no—
350C2×C4—22no—
351C2×C4—181no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4C3×C4×C3⋊D311yescharacteristic
355C2×C4C3×C4×C3⋊D311yescharacteristic
356C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
357C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
358C2×C4C3×#63811yescharacteristic
359C2×C4C22×C3×C3⋊D311yescharacteristic
360C8—182no—
361C2×C3—28no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—24no—
369C2×C3—44no—
370C2×C3—44no—
371C2×C3—362no—
372C2×C3C2×C3×(C2×D3)⋊C414yes—
373C2×C3C2×#375211yescharacteristic
374D3—128no—
375C22—41no—
376C22—182no—
377C22—362no—
378C22C3×#375211yescharacteristic
379C22C3×#375211yescharacteristic
380C22C2×C3×#63811yescharacteristic
381C4—42no—
382C4—182no—
383C4C3×#375211yescharacteristic
384C4C3×#375211yescharacteristic
385C4C2×C3×C4×C3⋊D311yescharacteristic
386C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
387C3—24no—
388C3C3×#500514yescomplemented
389C3#1059270411yescharacteristic, direct product factor
390C2—21no—
391C2—21no—
392C2—21no—
393C2—362no—
394C2C2×C3×#375211yescharacteristic
subgroupG/H
centerC3×C4C2×C4⋊(C3⋊D3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#117C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#117C3⋊D3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C33×#117C2semidirect
3. C2×C3×#3748C2semidirect
4. C3×#63632C2semidirect
5. #10592704C3direct (N × H)
11. C3×#3828C4semidirect
13. C2×C3×C4⋊(C3⋊D3)C4semidirect
15. C2×C3×#634C4semidirect
18. C33×C42C22semidirect
20. C3×#3748C22semidirect
25. C32×#117D3semidirect
26. C32×#117C2×C3semidirect
27. C2×#3748C2×C3semidirect
29. #63632C2×C3semidirect
58. #3828C3×C4semidirect
60. C2×C4⋊(C3⋊D3)C3×C4semidirect
62. C2×#634C3×C4semidirect
78. C32×C42C2×D3semidirect
79. C32×C42C22×C3semidirect
81. #3748C22×C3semidirect
102. C3×#117C3×D3semidirect
103. C3×#117C3⋊D3semidirect
188. C3×C42C2×C3×D3semidirect
189. C3×C42C2×C3⋊D3semidirect
228. #117C3×C3⋊D3semidirect
230. C33#421semidirect
305. C42C2×C3×C3⋊D3semidirect
340. C32#5005semidirect
341. C32C3×#421semidirect
388. C3C3×#5005semidirect
389. C3#10592704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C33×#1172
C2×C3×#37482
C3×#636322
#105927043
C3×#50053