Finite Group Databasebeta

← C3×#10592509 #13162855

Subgroups — C3×#10592509 #13162855

Order 1728.

1140 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592509C311yesmaximal, characteristic, direct product factor
9C3×#4810—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832—22no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)—21no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C3×#3755—22no—
22C3×#3752—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3745—22no—
25C2×#3832C2×C311yescharacteristic, complemented
26C2×C3×#239—34no—
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#215—34no—
31C2×C3×D24—34no—
32C3×#895—34no—
33C32×#103—24no—
34C32×#103D314yescomplemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#809—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—24no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—42no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C3×C4×C3⋊D3—24no—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#449—91no—
61#4810—34no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C22×C311yescharacteristic
65#3832—22no—
66#3832—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C3×#239—68no—
74C3×#239—68no—
75C2×C3×D12—34no—
76C2×C3×D12—34no—
77C2×C3×D12—64no—
78C2×C3×D12—64no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—64no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yes—
83C2×C32×C8C22×C311yescharacteristic
84C32×C4⋊C4—24no—
85C32×C4⋊C4C2×D314yes—
86C32×C4⋊C4C22×C311yescharacteristic
87#3755—22no—
88#3752—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90#3745—22no—
91C3×#215—68no—
92C3×(C2×D3)⋊C4—64no—
93C2×C3×C3⋊C8—34no—
94C3×D24—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C33×C4—22no—
106C33×C4—41no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—21no—
110C2×C3×#41—91no—
111C2×#239—34no—
112C2×C3×SD16—91no—
113C2×C3×D8—91no—
114C3×#121—91no—
115C2×#215—34no—
116C2×D24—34no—
117#895—34no—
118C3×#103—24no—
119C3×#103C3×D314yescomplemented
120C3×#103C3⋊D311yescharacteristic, complemented
121C3×#102—91no—
122C3×C22⋊C8—91no—
123(C2×D3)⋊C8—34no—
124#809—34no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3C3×D411yescharacteristic
128C22×C3×D3—34no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C32×Q8—21no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—44no—
135C32×Q8D1218yescomplemented
136C32×Q8C3×D412yescomplemented
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—44no—
143C2×C32×C4—44no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C23×C311yescharacteristic
146C2×C3⋊Dic3C3×D411yescharacteristic
147C4⋊(C3⋊D3)—24no—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—42no—
152C4⋊(C3⋊D3)C3×D412yescomplemented
153C4×C3⋊D3—24no—
154C4×C3⋊D3—21no—
155C4×C3⋊D3—21no—
156C2×C3×Dic3—34no—
157C3×D12—38no—
158C3×D12—616no—
159C3×D12—64no—
160C3×D12—64no—
161C3×D12—64no—
162C3×D12—128no—
163C3×C4×D3—616no—
164C3×C4×D3—64no—
165C3×C4×D3—64no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—64no—
171#449—91noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3×C3⋊D3—41no—
176C3×C3⋊D3—42no—
177C3×C3⋊D3—81no—
178C3×#41—182no—
179C3×#41—182no—
180C2×C3×Q8—24no—
181C2×C3×Q8C2×C3×D314yes—
182C2×C3×Q8C2×C3⋊D311yescharacteristic
183C2×C3×D4—91no—
184C2×C3×D4—91no—
185C2×C3×D4—181no—
186C2×C3×D4—181no—
187C22×C3×C4—91no—
188C22×C3×C4—181no—
189#239—68no—
190#239—68no—
191C2×D12—34no—
192C2×D12—34no—
193C2×D12—64no—
194C2×D12—64no—
195C2×C4×D3—34no—
196C2×C4×D3—64no—
197C3×SD16—182no—
198C3×D8—182no—
199C2×C3×C8—24no—
200C2×C3×C8—91no—
201C2×C3×C8C2×C3×D314yes—
202C2×C3×C8C2×C3⋊D311yescharacteristic
203C3×C4⋊C4—24no—
204C3×C4⋊C4C2×C3×D314yes—
205C3×C4⋊C4C2×C3⋊D311yescharacteristic
206C3×C22⋊C4—181no—
207#215—68no—
208(C2×D3)⋊C4—64no—
209C2×C3⋊C8—34no—
210D24—68no—
211C22×C32—24no—
212C22×C32D3×D414yes—
213C22×C32C2×C3×D411yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—22no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—68no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C2×C3×D3—124no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—22no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—28no—
236C32×C4—48no—
237C32×C4—41no—
238C32×C4—44no—
239C32×C4—84no—
240C32×C4C2×D1214yes—
241C32×C4D3×D414yes—
242C32×C4C2×C3×D411yescharacteristic
243C32×C4C2×C3×D411yescharacteristic
244C3⋊Dic3—21no—
245C3×Dic3—64no—
246C2×#41—91no—
247C2×SD16—91no—
248C2×D8—91no—
249#121—91no—
250#103C3×C3⋊D311yescharacteristic, complemented
251#102—91no—
252C22⋊C8—91no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—34no—
258C22×D3—64no—
259C22×D3—64no—
260C3×Q8—24no—
261C3×Q8—21no—
262C3×Q8—28no—
263C3×Q8—44no—
264C3×Q8C3×D1218yescomplemented
265C3×Q8C4⋊(C3⋊D3)12yescomplemented
266C3×D4—92no—
267C3×D4—184no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—362no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—91no—
280C2×C3×C4—184no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4C22×C3×D314yes—
284C2×C3×C4C22×C3⋊D311yescharacteristic
285C2×Dic3—34no—
286D12—38no—
287D12—616no—
288D12—64no—
289D12—64no—
290D12—64no—
291D12—128no—
292C4×D3—616no—
293C4×D3—64no—
294C4×D3—64no—
295C3×C8—24no—
296C3×C8—21no—
297C3×C8—44no—
298C3×C8—181no—
299C3⋊C8—64no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32#88214yes—
304C2×C32#91414yes—
305C2×C32#91914yes—
306C2×C32C3×#12011yescharacteristic
307C2×C32C3×#13511yescharacteristic
308C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
309C3⋊D3—41no—
310C3⋊D3—42no—
311C3⋊D3—81no—
312C3×D3—124no—
313C3×D3—128no—
314C3×D3—244no—
315#41—182no—
316#41—182no—
317C2×Q8C2×C3×C3⋊D311yescharacteristic
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—181no—
322C22×C4—91no—
323C22×C4—181no—
324SD16—182no—
325D8—182no—
326C2×C8—91no—
327C2×C8C2×C3×C3⋊D311yescharacteristic
328C4⋊C4C2×C3×C3⋊D311yescharacteristic
329C22⋊C4—181no—
330C22×C3—24no—
331C22×C3—182no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3C3×D3×D414yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—68no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—28no—
352C3×C4—22no—
353C3×C4—24no—
354C3×C4—24no—
355C3×C4—48no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—84no—
359C3×C4—181no—
360C3×C4C2×C3×D1214yes—
361C3×C4C3×D3×D414yes—
362C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
363C3×C4D4×C3⋊D311yescharacteristic
364Dic3—64no—
365C32—24no—
366C32#481014yescomplemented
367C32C3×#44911yescharacteristic, complemented
368C23—91no—
369C23—181no—
370C23—181no—
371Q8—21no—
372Q8C3×C4⋊(C3⋊D3)12yescomplemented
373D4—92no—
374D4—184no—
375D4—181no—
376D4—181no—
377D4—181no—
378D4—362no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—91no—
382C2×C4—184no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4C22×C3×C3⋊D311yescharacteristic
386C8—21no—
387C8—181no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—361no—
392C2×C3—362no—
393C2×C3—721no—
394C2×C3C3×#88214yes—
395C2×C3C3×#91414yes—
396C2×C3C3×#91914yes—
397C2×C3#6409111yescharacteristic
398C2×C3#6412311yescharacteristic
399C2×C3#6412811yescharacteristic
400D3—124no—
401D3—128no—
402D3—244no—
403C22—182no—
404C22—181no—
405C22—181no—
406C22—181no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22C3×D4×C3⋊D311yescharacteristic
413C4—22no—
414C4—41no—
415C4—181no—
416C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
417C4C3×D4×C3⋊D311yescharacteristic
418C3—24no—
419C3C3×#481014yescomplemented
420C3#1059250911yescharacteristic, direct product factor
421C2—361no—
422C2—362no—
423C2—721no—
424C2C3×#6409111yescharacteristic
425C2C3×#6412311yescharacteristic
426C2C3×#6412811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C3×#3745C2semidirect
4. C3×#64104C2semidirect
5. C33×#103C2semidirect
6. C3×#63647C2semidirect
7. C3×#63636C2semidirect
8. #10592509C3direct (N × H)
25. C2×#3832C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
44. C33×Q8D4semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
119. C3×#103C3×D3semidirect
120. C3×#103C3⋊D3semidirect
135. C32×Q8D12semidirect
136. C32×Q8C3×D4semidirect
152. C4⋊(C3⋊D3)C3×D4semidirect
250. #103C3×C3⋊D3semidirect
253. C33#449semidirect
264. C3×Q8C3×D12semidirect
265. C3×Q8C4⋊(C3⋊D3)semidirect
366. C32#4810semidirect
367. C32C3×#449semidirect
372. Q8C3×C4⋊(C3⋊D3)semidirect
419. C3C3×#4810semidirect
420. C3#10592509direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×#37552
C2×C3×#37452
C3×#641042
C33×#1032
C3×#636472
C3×#636362
#105925093
C3×#48103