Finite Group Databasebeta

← C3×#10592524 #13162863

Subgroups — C3×#10592524 #13162863

Order 1728.

1088 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic
2C3×#64160C211yesmaximal, characteristic, complemented
3C3×#64118C211yesmaximal, characteristic, complemented
4C3×#64084C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6C3×#63650C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8#10592524C311yesmaximal, characteristic, direct product factor
9C3×#4825—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×D4×C3⋊D3—22no—
13C3×D4×C3⋊D3—21no—
14C3×#3828—21no—
15C3×#3828C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)—21no—
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C33×M16C2211yescharacteristic, complemented
20C33×C42C2211yescharacteristic, complemented
21C3×#3756—21no—
22C3×#3754—21no—
23C3×#3752—22no—
24C3×#3748C2211yescharacteristic, complemented
25C3×#3746—21no—
26C3×#3744—21no—
27#64366C2×C311yescharacteristic
28C3×#1009—34no—
29#64160C2×C311yescharacteristic, complemented
30#64118C2×C311yescharacteristic, complemented
31#64084C2×C311yescharacteristic, complemented
32C3×#951—34no—
33C3×#909—34no—
34C3×#875—34no—
35C32×#104—24no—
36C32×#104D314yescomplemented
37C32×#104C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#823—34no—
41C3×#805—34no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D412yes—
44C33×Q8D411yescharacteristic
45C33×D4—21no—
46C33×D4D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—22no—
51C3×#638—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—22no—
58C3×C4×C3⋊D3—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#455—91no—
65#4825—34no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C22×C311yescharacteristic
69#3832—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—21no—
72#3828—21no—
73#3828C22×C311yescharacteristic
74C2×C4⋊(C3⋊D3)—21no—
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic, complemented
77C3×#239—64no—
78C3×D3×D4—68no—
79C3×D3×D4—64no—
80C3×#235—34no—
81C3×#235—64no—
82C2×C3×D12—34no—
83C2×C3×D12—64no—
84C2×C3×Dic6—34no—
85C32×M16—24no—
86C32×M16C2×D314yescomplemented
87C32×M16C22×C311yescharacteristic, complemented
88C32×C42—24no—
89C32×C42C2×D314yescomplemented
90C32×C42C22×C311yescharacteristic, complemented
91#3756—21no—
92#3754—21no—
93#3752—22no—
94#3748C22×C311yescharacteristic, complemented
95#3746—21no—
96#3744—21no—
97C3×#216—64no—
98C3×#214—64no—
99C3×(C2×D3)⋊C4—68no—
100C3×#208—34no—
101C3×Dic12—64no—
102C3×#204—64no—
103C22×C33—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—42no—
108C2×C3×C3⋊D3—82no—
109C33×C4—21no—
110C33×C4—41no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—41no—
115C3×H(5,𝔽2)—91no—
116#1009—34no—
117C3×#137—91no—
118C3×#137—91no—
119C3×#124—91no—
120#951—34no—
121#909—34no—
122#875—34no—
123C3×#104—24no—
124C3×#104—91no—
125C3×#104C3×D314yescomplemented
126C3×#104C3⋊D311yescharacteristic, complemented
127C3×#100—91no—
128#823—34no—
129#805—34no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3C3×D412yes—
132C22×C3×D3—38no—
133C22×C3×D3—68no—
134C32×Q8—24no—
135C32×Q8D1214yes—
136C32×Q8C3×D411yescharacteristic
137C32×D4—21no—
138C32×D4—24no—
139C32×D4—24no—
140C32×D4—44no—
141C32×D4D1214yes—
142C32×D4C3×D411yescharacteristic
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—24no—
146C2×C32×C4—24no—
147C2×C32×C4—24no—
148C2×C32×C4—44no—
149C2×C32×C4—44no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C23×C311yescharacteristic
152#638—22no—
153#638—21no—
154C2×C3⋊Dic3—21no—
155C4⋊(C3⋊D3)—22no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)C3×D411yescharacteristic
159C4×C3⋊D3—22no—
160C4×C3⋊D3—21no—
161#634—21no—
162#634—21no—
163#634C3×D411yescharacteristic
164C3×#67—68no—
165C3×#67—64no—
166C2×C3×Dic3—64no—
167C3×D12—34no—
168C3×D12—68no—
169C3×D12—64no—
170C3×D12—64no—
171C3×C4×D3—68no—
172C3×C4×D3—64no—
173C3×Dic6—34no—
174C3×Dic6—64no—
175C3×Dic6—64no—
176C32×C8—21no—
177C32×C8—24no—
178C32×C8—44no—
179C3⋊(C3⋊C8)—21no—
180C3×C3⋊C8—64no—
181#455—91noSylow
182C2×C33—21no—
183C2×C33—41no—
184C2×C33#12011yescharacteristic
185C3×C3⋊D3—42no—
186C3×C3⋊D3—41no—
187C3×#41—24no—
188C3×#41—91no—
189C3×#41—181no—
190C3×#41—181no—
191C3×#41C2×C3×D314yes—
192C3×#41C2×C3⋊D311yescharacteristic
193C2×C3×Q8—91no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C2×C3×D4—182no—
197C2×C3×D4—181no—
198#239—64no—
199D3×D4—68no—
200D3×D4—64no—
201#235—34no—
202#235—64no—
203C2×D12—34no—
204C2×D12—64no—
205C2×Dic6—34no—
206C3×Q16—181no—
207C3×Q16—181no—
208C3×SD16—181no—
209C3×SD16—181no—
210C3×M16—24no—
211C3×M16—91no—
212C3×M16C2×C3×D314yescomplemented
213C3×M16C2×C3⋊D311yescharacteristic, complemented
214C3×C22⋊C4—182no—
215C3×C42—24no—
216C3×C42C2×C3×D314yescomplemented
217C3×C42C2×C3⋊D311yescharacteristic, complemented
218#216—64no—
219#214—64no—
220(C2×D3)⋊C4—68no—
221#208—34no—
222Dic12—64no—
223#204—64no—
224C22×C32—21no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—44no—
228C22×C32D3×D414yes—
229C22×C32C2×C3×D411yescharacteristic
230C2×C3⋊D3—22no—
231C2×C3⋊D3—21no—
232C2×C3⋊D3—42no—
233C2×C3⋊D3—82no—
234C2×C3×D3—64no—
235C2×C3×D3—68no—
236C2×C3×D3—128no—
237C2×C3×D3—248no—
238C32×C4—21no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—44no—
243C32×C4—41no—
244C32×C4—44no—
245C32×C4—48no—
246C32×C4C2×D1214yes—
247C32×C4D3×D414yes—
248C32×C4C2×C3×D411yescharacteristic
249C32×C4C2×C3×D411yescharacteristic
250C3⋊Dic3—21no—
251C3⋊Dic3—41no—
252C3×Dic3—64no—
253C3×Dic3—124no—
254H(5,𝔽2)—91no—
255#137—91no—
256#137—91no—
257#124—91no—
258#104—91no—
259#104C3×C3⋊D311yescharacteristic, complemented
260#100—91no—
261C33#45511yescharacteristic, complemented, Sylow
262C23×C3—92no—
263C23×C3—182no—
264C22×D3—38no—
265C22×D3—68no—
266C3×Q8—24no—
267C3×Q8—91no—
268C3×Q8—181no—
269C3×Q8—181no—
270C3×Q8C3×D1214yes—
271C3×Q8C4⋊(C3⋊D3)11yescharacteristic
272C3×D4—24no—
273C3×D4—21no—
274C3×D4—24no—
275C3×D4—44no—
276C3×D4—91no—
277C3×D4—182no—
278C3×D4—182no—
279C3×D4—181no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4C3×D1214yes—
283C3×D4C4⋊(C3⋊D3)11yescharacteristic
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C22×C3×D314yes—
295C2×C3×C4C22×C3⋊D311yescharacteristic
296#67—68no—
297#67—64no—
298C2×Dic3—64no—
299D12—34no—
300D12—68no—
301D12—64no—
302D12—64no—
303C4×D3—68no—
304C4×D3—64no—
305Dic6—34no—
306Dic6—64no—
307Dic6—64no—
308C3×C8—24no—
309C3×C8—21no—
310C3×C8—44no—
311C3×C8—181no—
312C3⋊C8—64no—
313C2×C32—21no—
314C2×C32—24no—
315C2×C32—24no—
316C2×C32—44no—
317C2×C32—41no—
318C2×C32—44no—
319C2×C32—84no—
320C2×C32#88214yes—
321C2×C32C3×#12011yescharacteristic
322C3⋊D3—42no—
323C3⋊D3—41no—
324C3×D3—128no—
325C3×D3—124no—
326#41—91no—
327#41—181no—
328#41—181no—
329#41C2×C3×C3⋊D311yescharacteristic
330C2×Q8—91no—
331C2×D4—91no—
332C2×D4—181no—
333C2×D4—182no—
334C2×D4—181no—
335Q16—181no—
336Q16—181no—
337SD16—181no—
338SD16—181no—
339M16—91no—
340M16C2×C3×C3⋊D311yescharacteristic, complemented
341C22⋊C4—182no—
342C42C2×C3×C3⋊D311yescharacteristic, complemented
343C22×C3—24no—
344C22×C3—21no—
345C22×C3—24no—
346C22×C3—44no—
347C22×C3—181no—
348C22×C3—182no—
349C22×C3—362no—
350C22×C3—722no—
351C22×C3C3×D3×D414yes—
352C22×C3D4×C3⋊D311yescharacteristic
353C2×D3—64no—
354C2×D3—68no—
355C2×D3—128no—
356C2×D3—248no—
357C3×C4—24no—
358C3×C4—21no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—44no—
362C3×C4—41no—
363C3×C4—44no—
364C3×C4—48no—
365C3×C4—181no—
366C3×C4—361no—
367C3×C4C2×C3×D1214yes—
368C3×C4C3×D3×D414yes—
369C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
370C3×C4D4×C3⋊D311yescharacteristic
371Dic3—64no—
372Dic3—124no—
373C32—24no—
374C32#482514yescomplemented
375C32C3×#45511yescharacteristic, complemented
376C23—92no—
377C23—182no—
378Q8—91no—
379Q8—181no—
380Q8—181no—
381Q8C3×C4⋊(C3⋊D3)11yescharacteristic
382D4—21no—
383D4—91no—
384D4—182no—
385D4—182no—
386D4—181no—
387D4—181no—
388D4—181no—
389D4C3×C4⋊(C3⋊D3)11yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—182no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4C22×C3×C3⋊D311yescharacteristic
396C8—21no—
397C8—181no—
398C2×C3—21no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—44no—
402C2×C3—41no—
403C2×C3—44no—
404C2×C3—84no—
405C2×C3—362no—
406C2×C3—361no—
407C2×C3C3×#88214yes—
408C2×C3#6409111yescharacteristic
409D3—128no—
410D3—124no—
411C22—21no—
412C22—181no—
413C22—182no—
414C22—362no—
415C22—722no—
416C22C3×D4×C3⋊D311yescharacteristic
417C4—21no—
418C4—41no—
419C4—181no—
420C4—361no—
421C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
422C4C3×D4×C3⋊D311yescharacteristic
423C3—24no—
424C3C3×#482514yescomplemented
425C3#1059252411yescharacteristic, direct product factor
426C2—21no—
427C2—41no—
428C2—362no—
429C2—361no—
430C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#104C2
socleC2×C33#120
prime radicalC3×#64366C2
hypercenterC3×#104C3⋊D3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×#64160C2semidirect
3. C3×#64118C2semidirect
4. C3×#64084C2semidirect
5. C33×#104C2semidirect
6. C3×#63650C2semidirect
7. C3×#63632C2semidirect
8. #10592524C3direct (N × H)
18. C2×C3×#634C22semidirect
19. C33×M16C22semidirect
20. C33×C42C22semidirect
24. C3×#3748C22semidirect
29. #64160C2×C3semidirect
30. #64118C2×C3semidirect
31. #64084C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63650C2×C3semidirect
39. #63632C2×C3semidirect
76. C2×#634C22×C3semidirect
86. C32×M16C2×D3semidirect
87. C32×M16C22×C3semidirect
89. C32×C42C2×D3semidirect
90. C32×C42C22×C3semidirect
94. #3748C22×C3semidirect
125. C3×#104C3×D3semidirect
126. C3×#104C3⋊D3semidirect
212. C3×M16C2×C3×D3semidirect
213. C3×M16C2×C3⋊D3semidirect
216. C3×C42C2×C3×D3semidirect
217. C3×C42C2×C3⋊D3semidirect
259. #104C3×C3⋊D3semidirect
261. C33#455semidirect
340. M16C2×C3×C3⋊D3semidirect
342. C42C2×C3×C3⋊D3semidirect
374. C32#4825semidirect
375. C32C3×#455semidirect
424. C3C3×#4825semidirect
425. C3#10592524direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643662
C3×#641602
C3×#641182
C3×#640842
C33×#1042
C3×#636502
C3×#636322
#105925243
C3×#48253