Finite Group Databasebeta

← C3×#10592871 #13163214

Subgroups — C3×#10592871 #13163214

Order 1728.

1244 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63647C211yesmaximal, characteristic, complemented
8#10592871C311yesmaximal, characteristic, direct product factor
9C3×#5172—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×#3832—22no—
14C3×#3832—22no—
15C2×C3×C4⋊(C3⋊D3)—21no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C33×SD16—22no—
20C2×C33×C8C2211yescharacteristic
21C3×#3757—21no—
22C3×#3753—22no—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3832C2×C311yescharacteristic, complemented
26C2×C3×#239—34no—
27C2×C32×SD16—24no—
28C2×C32×SD16D314yescomplemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64147C2×C311yescharacteristic, complemented
31C2×#3753C2×C311yescharacteristic, complemented
32C3×#938—34no—
33C2×C3×#213—34no—
34#63662C2×C311yescharacteristic, complemented
35#63648C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#835—34no—
38C3×#821—34no—
39C3×(C2×D3)⋊C8—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C33×D4—21no—
46C33×D4—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C3×#638—42no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—24no—
53C3×C4⋊(C3⋊D3)—21no—
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#5172—34no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C22×C311yescharacteristic
65C2×C32×D4—24no—
66C2×C32×D4C2×D314yes—
67C2×C32×D4C22×C311yescharacteristic
68C2×#638—21no—
69#3832—22no—
70#3832—22no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
73C2×C4×C3⋊D3—21no—
74C2×C4×C3⋊D3C22×C311yescharacteristic
75C2×C3×#67—64no—
76C3×#239—68no—
77C3×#239—68no—
78C2×C3×D12—34no—
79C2×C3×D12—64no—
80C2×C3×C4×D3—34no—
81C2×C3×C4×D3—64no—
82C32×SD16—216no—
83C32×SD16—28no—
84C32×SD16—22no—
85C2×C32×C8—24no—
86C2×C32×C8C2×D314yes—
87C2×C32×C8C22×C311yescharacteristic
88#3757—21no—
89#3753—22no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—64no—
93C3×#213—68no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8—34no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C33×C4—22no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—21no—
110C3×C3⋊Dic3—41no—
111C2×C3×#41—91no—
112C2×#239—34no—
113C2×C3×SD16—24no—
114C2×C3×SD16C3×D314yescomplemented
115C2×C3×SD16C3⋊D311yescharacteristic, complemented
116C2×C3×D8—91no—
117C3×#121—91no—
118#938—34no—
119C2×#213—34no—
120C3×#103—91no—
121C3×#102—91no—
122C3×C22⋊C8—91no—
123#835—34no—
124#821—34no—
125(C2×D3)⋊C8—34no—
126C23×C32—21no—
127C23×C32—24no—
128C23×C32—44no—
129C22×C3⋊D3—21no—
130C22×C3⋊D3C3×D411yescharacteristic
131C22×C3×D3—34no—
132C22×C3×D3—64no—
133C32×Q8—21no—
134C32×Q8—28no—
135C32×Q8—24no—
136C32×Q8—44no—
137C32×Q8#6718yescomplemented
138C32×Q8C3×D412yescomplemented
139C32×D4—28no—
140C32×D4—24no—
141C32×D4—21no—
142C32×D4—21no—
143C32×D4—24no—
144C32×D4—44no—
145C2×C32×C4—21no—
146C2×C32×C4—24no—
147C2×C32×C4—24no—
148C2×C32×C4—44no—
149C2×C32×C4C22×D314yes—
150C2×C32×C4C23×C311yescharacteristic
151#638—42no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3C3×D411yescharacteristic
154C4⋊(C3⋊D3)—24no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)C3×D412yescomplemented
157C4×C3⋊D3—24no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C3×#67—128no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—64no—
163C3×D12—38no—
164C3×D12—616no—
165C3×D12—64no—
166C3×C4×D3—616no—
167C3×C4×D3—64no—
168C3×C4×D3—64no—
169C32×C8—28no—
170C32×C8—24no—
171C32×C8—21no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—64no—
174#449—91noSylow
175C2×C33—81no—
176C2×C33#12011yescharacteristic
177C2×C33#13511yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3×C3⋊D3—41no—
180C3×C3⋊D3—42no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—24no—
184C2×C3×Q8C2×C3×D314yes—
185C2×C3×Q8C2×C3⋊D311yescharacteristic
186C2×C3×D4—24no—
187C2×C3×D4—91no—
188C2×C3×D4—181no—
189C2×C3×D4—181no—
190C2×C3×D4C2×C3×D314yes—
191C2×C3×D4C2×C3⋊D311yescharacteristic
192C22×C3×C4—91no—
193C22×C3×C4—181no—
194C2×#67—64no—
195#239—68no—
196#239—68no—
197C2×D12—34no—
198C2×D12—64no—
199C2×C4×D3—34no—
200C2×C4×D3—64no—
201C3×SD16—216no—
202C3×SD16—28no—
203C3×SD16—22no—
204C3×D8—182no—
205C2×C3×C8—24no—
206C2×C3×C8—91no—
207C2×C3×C8C2×C3×D314yes—
208C2×C3×C8C2×C3⋊D311yescharacteristic
209C3×C4⋊C4—91no—
210C3×C22⋊C4—181no—
211#217—64no—
212#213—68no—
213C4⋊Dic3—34no—
214C2×C3⋊C8—34no—
215C22×C32—24no—
216C22×C32—48no—
217C22×C32—44no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—84no—
224C22×C32—84no—
225C22×C32D3×D414yes—
226C22×C32C2×C3×D411yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—22no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3—21no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C2×C3×D3—68no—
236C2×C3×D3—64no—
237C2×C3×D3—124no—
238C2×C3×D3—124no—
239C32×C4—22no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—28no—
243C32×C4—48no—
244C32×C4D3×D414yes—
245C32×C4C2×#6714yes—
246C32×C4C2×C3×D411yescharacteristic
247C32×C4C2×C3×D411yescharacteristic
248C3⋊Dic3—21no—
249C3⋊Dic3—41no—
250C3×Dic3—64no—
251C3×Dic3—124no—
252C2×#41—91no—
253C2×SD16C3×C3⋊D311yescharacteristic, complemented
254C2×D8—91no—
255#121—91no—
256#103—91no—
257#102—91no—
258C22⋊C8—91no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—24no—
261C23×C3—21no—
262C23×C3—44no—
263C23×C3—91no—
264C23×C3—181no—
265C22×D3—34no—
266C22×D3—64no—
267C3×Q8—24no—
268C3×Q8—21no—
269C3×Q8—28no—
270C3×Q8—44no—
271C3×Q8C3×#6718yescomplemented
272C3×Q8#63812yescomplemented
273C3×D4—28no—
274C3×D4—24no—
275C3×D4—24no—
276C3×D4—21no—
277C3×D4—21no—
278C3×D4—44no—
279C3×D4—92no—
280C3×D4—184no—
281C3×D4—181no—
282C3×D4—362no—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—44no—
287C2×C3×C4—91no—
288C2×C3×C4—184no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4C22×C3×D314yes—
293C2×C3×C4C22×C3⋊D311yescharacteristic
294#67—128no—
295C2×Dic3—34no—
296C2×Dic3—64no—
297D12—38no—
298D12—616no—
299D12—64no—
300C4×D3—616no—
301C4×D3—64no—
302C4×D3—64no—
303C3×C8—28no—
304C3×C8—24no—
305C3×C8—21no—
306C3×C8—181no—
307C3⋊C8—64no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—88no—
312C2×C32—84no—
313C2×C32—81no—
314C2×C32#91414yes—
315C2×C32#91614yes—
316C2×C32#93714yes—
317C2×C32C3×#12011yescharacteristic
318C2×C32C3×#13511yescharacteristic
319C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
320C3⋊D3—41no—
321C3⋊D3—42no—
322C3×D3—124no—
323C3×D3—128no—
324#41—182no—
325#41—182no—
326C2×Q8C2×C3×C3⋊D311yescharacteristic
327C2×D4—91no—
328C2×D4—181no—
329C2×D4—181no—
330C2×D4C2×C3×C3⋊D311yescharacteristic
331C22×C4—91no—
332C22×C4—181no—
333SD16—22no—
334D8—182no—
335C2×C8—91no—
336C2×C8C2×C3×C3⋊D311yescharacteristic
337C4⋊C4—91no—
338C22⋊C4—181no—
339C22×C3—24no—
340C22×C3—48no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3—44no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—84no—
348C22×C3—84no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—182no—
352C22×C3—181no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3C3×D3×D414yes—
356C22×C3D4×C3⋊D311yescharacteristic
357C2×D3—64no—
358C2×D3—64no—
359C2×D3—68no—
360C2×D3—64no—
361C2×D3—124no—
362C2×D3—124no—
363C3×C4—28no—
364C3×C4—22no—
365C3×C4—24no—
366C3×C4—24no—
367C3×C4—48no—
368C3×C4—181no—
369C3×C4—361no—
370C3×C4C3×D3×D414yes—
371C3×C4C2×C3×#6714yes—
372C3×C4D4×C3⋊D311yescharacteristic
373C3×C4C2×#63811yescharacteristic
374Dic3—64no—
375Dic3—124no—
376C32—24no—
377C32#517214yescomplemented
378C32C3×#44911yescharacteristic, complemented
379C23—21no—
380C23—91no—
381C23—181no—
382Q8—21no—
383Q8C3×#63812yescomplemented
384D4—21no—
385D4—21no—
386D4—92no—
387D4—184no—
388D4—181no—
389D4—362no—
390C2×C4—21no—
391C2×C4—91no—
392C2×C4—184no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4C22×C3×C3⋊D311yescharacteristic
397C8—21no—
398C8—181no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—88no—
403C2×C3—84no—
404C2×C3—81no—
405C2×C3—361no—
406C2×C3—362no—
407C2×C3C3×#91414yes—
408C2×C3C3×#91614yes—
409C2×C3C3×#93714yes—
410C2×C3#6412311yescharacteristic
411C2×C3#6412511yescharacteristic
412C2×C3#6414611yescharacteristic
413D3—124no—
414D3—128no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—181no—
419C22—181no—
420C22—182no—
421C22—181no—
422C22—361no—
423C22—361no—
424C22C3×D4×C3⋊D311yescharacteristic
425C4—22no—
426C4—181no—
427C4—361no—
428C4C3×D4×C3⋊D311yescharacteristic
429C4C2×C3×#63811yescharacteristic
430C3—24no—
431C3C3×#517214yescomplemented
432C3#1059287111yescharacteristic, direct product factor
433C2—81no—
434C2—361no—
435C2—362no—
436C2C3×#6412311yescharacteristic
437C2C3×#6412511yescharacteristic
438C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C33×SD16C2semidirect
3. C3×#64147C2semidirect
4. C2×C3×#3753C2semidirect
5. C3×#63662C2semidirect
6. C3×#63648C2semidirect
7. C3×#63647C2semidirect
8. #10592871C3direct (N × H)
25. C2×#3832C2×C3semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64147C2×C3semidirect
31. C2×#3753C2×C3semidirect
34. #63662C2×C3semidirect
35. #63648C2×C3semidirect
36. #63647C2×C3semidirect
44. C33×Q8D4semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
114. C2×C3×SD16C3×D3semidirect
115. C2×C3×SD16C3⋊D3semidirect
137. C32×Q8#67semidirect
138. C32×Q8C3×D4semidirect
156. C4⋊(C3⋊D3)C3×D4semidirect
253. C2×SD16C3×C3⋊D3semidirect
259. C33#449semidirect
271. C3×Q8C3×#67semidirect
272. C3×Q8#638semidirect
377. C32#5172semidirect
378. C32C3×#449semidirect
383. Q8C3×#638semidirect
431. C3C3×#5172semidirect
432. C3#10592871direct (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×C33×SD162
C3×#641472
C2×C3×#37532
C3×#636622
C3×#636482
C3×#636472
#105928713
C3×#51723