Finite Group Databasebeta

← C3×#10592937 #13163278

Subgroups — C3×#10592937 #13163278

Order 1728.

1628 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4C3×#64139C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63659C211yesmaximal, characteristic, complemented
7C3×#63657C211yesmaximal, characteristic, complemented
8#10592937C311yesmaximal, characteristic, direct product factor
9C3×#5238—34nomaximal
10C33×#41—22no—
11C33×#41—22no—
12C2×C33×Q8C2211yescharacteristic
13C2×C33×D4—21no—
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4—21no—
16C22×C33×C4C2211yescharacteristic
17C2×C3×#638—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C3×#3755—22no—
20C3×#3753—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×C32×#41—24no—
26C2×C32×#41D314yescomplemented
27C2×C32×#41C2×C311yescharacteristic, complemented
28C2×#3755C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic, complemented
30#64139C2×C311yescharacteristic, complemented
31C2×C3×#215—34no—
32C2×C3×#213—34no—
33C3×#930—34no—
34#63662C2×C311yescharacteristic, complemented
35#63659C2×C311yescharacteristic, complemented
36#63657C2×C311yescharacteristic, complemented
37C3×#835—34no—
38C3×#832—34no—
39C3×#830—34no—
40C23×C33—21no—
41C23×C33D411yescharacteristic
42C22×C3×C3⋊D3—21no—
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C33×D4—24no—
46C33×D4—21no—
47C33×D4D412yescomplemented
48C2×C33×C4—24no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4—21no—
52C2×C33×C4D411yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C3×#638—42no—
55C2×C3×C3⋊Dic3—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#449—91no—
61#5238—34no—
62C32×#41—216no—
63C32×#41—216no—
64C32×#41—28no—
65C32×#41—22no—
66C32×#41—28no—
67C32×#41—22no—
68C2×C32×Q8—24no—
69C2×C32×Q8C2×D314yes—
70C2×C32×Q8C22×C311yescharacteristic
71C2×C32×D4—28no—
72C2×C32×D4—24no—
73C2×C32×D4—24no—
74C2×C32×D4—21no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C22×C311yescharacteristic
77C22×C32×C4—28no—
78C22×C32×C4—24no—
79C22×C32×C4—24no—
80C22×C32×C4—21no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C22×C311yescharacteristic
83C2×#638—21no—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C3×#67—64no—
86C2×C3×D12—34no—
87#3755—22no—
88#3753—22no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#215—68no—
94C3×#213—68no—
95C3×(C2×D3)⋊C4—64no—
96C3×C4⋊Dic3—34no—
97C2×C3×C3⋊C8—34no—
98C2×C3×C3⋊C8—34no—
99C22×C33—21no—
100C22×C33—22no—
101C22×C33—21no—
102C22×C33—21no—
103C22×C33—41no—
104C22×C33—41no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C33×C4—21no—
110C33×C4—22no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—41no—
114C2×C3×#41—24no—
115C2×C3×#41C3×D314yescomplemented
116C2×C3×#41C3⋊D311yescharacteristic, complemented
117C2×C3×SD16—91no—
118C2×C3×D8—91no—
119C3×#121—91no—
120C2×#215—34no—
121C2×#213—34no—
122#930—34no—
123C3×#103—91no—
124C3×#102—91no—
125C3×C22⋊C8—91no—
126#835—34no—
127#832—34no—
128#830—34no—
129C23×C32—28no—
130C23×C32—24no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32#6714yes—
134C23×C32C3×D411yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3×D3—64no—
137C32×Q8—28no—
138C32×Q8—24no—
139C32×Q8—21no—
140C32×Q8—28no—
141C32×Q8#6718yescomplemented
142C32×Q8C3×D412yescomplemented
143C32×D4—232no—
144C32×D4—216no—
145C32×D4—24no—
146C32×D4—28no—
147C32×D4—24no—
148C32×D4—21no—
149C32×D4—28no—
150C32×D4#6718yescomplemented
151C32×D4C3×D412yescomplemented
152C2×C32×C4—28no—
153C2×C32×C4—232no—
154C2×C32×C4—216no—
155C2×C32×C4—24no—
156C2×C32×C4—28no—
157C2×C32×C4—28no—
158C2×C32×C4—24no—
159C2×C32×C4—21no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—24no—
165C2×C32×C4—24no—
166C2×C32×C4#6714yes—
167C2×C32×C4C3×D411yescharacteristic
168C2×C32×C4C22×D314yes—
169C2×C32×C4C23×C311yescharacteristic
170#638—42no—
171C2×C3⋊Dic3—21no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)—21no—
174C3×#67—128no—
175C2×C3×Dic3—64no—
176C3×D12—64no—
177C3×D12—64no—
178C3⋊(C3⋊C8)—21no—
179C3⋊(C3⋊C8)—21no—
180C3×C3⋊C8—64no—
181C3×C3⋊C8—64no—
182#449—91noSylow
183C2×C33—41no—
184C2×C33—42no—
185C2×C33#12011yescharacteristic
186C2×C33#13511yescharacteristic
187C2×C33AGL(1,ℤ/8)11yescharacteristic
188C3×C3⋊D3—81no—
189C3×#41—216no—
190C3×#41—216no—
191C3×#41—28no—
192C3×#41—22no—
193C3×#41—28no—
194C3×#41—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8C2×C3×D314yes—
197C2×C3×Q8C2×C3⋊D311yescharacteristic
198C2×C3×D4—28no—
199C2×C3×D4—24no—
200C2×C3×D4—21no—
201C2×C3×D4—24no—
202C2×C3×D4—91no—
203C2×C3×D4—181no—
204C2×C3×D4C2×C3×D314yes—
205C2×C3×D4C2×C3⋊D311yescharacteristic
206C22×C3×C4—28no—
207C22×C3×C4—24no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4C2×C3×D314yes—
211C22×C3×C4C2×C3⋊D311yescharacteristic
212C2×#67—64no—
213C2×D12—34no—
214C3×SD16—182no—
215C3×D8—182no—
216C2×C3×C8—91no—
217C2×C3×C8—91no—
218C3×C4⋊C4—91no—
219C3×C22⋊C4—181no—
220#215—68no—
221#213—68no—
222(C2×D3)⋊C4—64no—
223C4⋊Dic3—34no—
224C2×C3⋊C8—34no—
225C2×C3⋊C8—34no—
226C22×C32—28no—
227C22×C32—216no—
228C22×C32—24no—
229C22×C32—28no—
230C22×C32—21no—
231C22×C32—22no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—24no—
237C22×C32—44no—
238C22×C32—44no—
239C22×C32—48no—
240C22×C32—48no—
241C22×C32—44no—
242C22×C32—44no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32C2×#6714yes—
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—41no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C32×C4—28no—
254C32×C4—216no—
255C32×C4—24no—
256C32×C4—28no—
257C32×C4—21no—
258C32×C4—22no—
259C32×C4—24no—
260C32×C4—24no—
261C32×C4C2×#6714yes—
262C32×C4C2×#6714yes—
263C32×C4C2×C3×D411yescharacteristic
264C32×C4C2×C3×D411yescharacteristic
265C3⋊Dic3—41no—
266C3×Dic3—124no—
267C2×#41C3×C3⋊D311yescharacteristic, complemented
268C2×SD16—91no—
269C2×D8—91no—
270#121—91no—
271#103—91no—
272#102—91no—
273C22⋊C8—91no—
274C33#44911yescharacteristic, complemented, Sylow
275C23×C3—28no—
276C23×C3—24no—
277C23×C3—21no—
278C23×C3—24no—
279C23×C3—181no—
280C23×C3C3×#6714yes—
281C23×C3#63811yescharacteristic
282C22×D3—64no—
283C3×Q8—28no—
284C3×Q8—24no—
285C3×Q8—21no—
286C3×Q8—28no—
287C3×Q8C3×#6718yescomplemented
288C3×Q8#63812yescomplemented
289C3×D4—232no—
290C3×D4—216no—
291C3×D4—24no—
292C3×D4—28no—
293C3×D4—24no—
294C3×D4—21no—
295C3×D4—28no—
296C3×D4—181no—
297C3×D4—181no—
298C3×D4—362no—
299C3×D4C3×#6718yescomplemented
300C3×D4#63812yescomplemented
301C2×C3×C4—28no—
302C2×C3×C4—232no—
303C2×C3×C4—216no—
304C2×C3×C4—24no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4—24no—
308C2×C3×C4—21no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—181no—
316C2×C3×C4C3×#6714yes—
317C2×C3×C4#63811yescharacteristic
318C2×C3×C4C22×C3×D314yes—
319C2×C3×C4C22×C3⋊D311yescharacteristic
320#67—128no—
321C2×Dic3—64no—
322D12—64no—
323D12—64no—
324C3×C8—181no—
325C3×C8—181no—
326C3⋊C8—64no—
327C3⋊C8—64no—
328C2×C32—24no—
329C2×C32—24no—
330C2×C32—24no—
331C2×C32—48no—
332C2×C32—416no—
333C2×C32—44no—
334C2×C32—48no—
335C2×C32—41no—
336C2×C32—42no—
337C2×C32#94914yes—
338C2×C32#95014yes—
339C2×C32#95314yes—
340C2×C32C3×#12011yescharacteristic
341C2×C32C3×#13511yescharacteristic
342C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
343C3⋊D3—81no—
344C3×D3—244no—
345#41—22no—
346#41—22no—
347C2×Q8C2×C3×C3⋊D311yescharacteristic
348C2×D4—21no—
349C2×D4—91no—
350C2×D4—181no—
351C2×D4C2×C3×C3⋊D311yescharacteristic
352C22×C4—21no—
353C22×C4C2×C3×C3⋊D311yescharacteristic
354SD16—182no—
355D8—182no—
356C2×C8—91no—
357C2×C8—91no—
358C4⋊C4—91no—
359C22⋊C4—181no—
360C22×C3—28no—
361C22×C3—216no—
362C22×C3—24no—
363C22×C3—24no—
364C22×C3—24no—
365C22×C3—28no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—24no—
371C22×C3—44no—
372C22×C3—44no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3C2×C3×#6714yes—
383C22×C3C2×#63811yescharacteristic
384C2×D3—124no—
385C2×D3—124no—
386C2×D3—124no—
387C3×C4—28no—
388C3×C4—216no—
389C3×C4—24no—
390C3×C4—28no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—24no—
394C3×C4—24no—
395C3×C4—361no—
396C3×C4C2×C3×#6714yes—
397C3×C4C2×C3×#6714yes—
398C3×C4C2×#63811yescharacteristic
399C3×C4C2×#63811yescharacteristic
400Dic3—124no—
401C32—24no—
402C32#523814yescomplemented
403C32C3×#44911yescharacteristic, complemented
404C23—21no—
405C23—181no—
406C23C3×#63811yescharacteristic
407Q8—21no—
408Q8C3×#63812yescomplemented
409D4—24no—
410D4—21no—
411D4—181no—
412D4—181no—
413D4—362no—
414D4C3×#63812yescomplemented
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—181no—
420C2×C4C3×#63811yescharacteristic
421C2×C4C22×C3×C3⋊D311yescharacteristic
422C8—181no—
423C8—181no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—48no—
428C2×C3—416no—
429C2×C3—44no—
430C2×C3—48no—
431C2×C3—41no—
432C2×C3—42no—
433C2×C3—721no—
434C2×C3C3×#94914yes—
435C2×C3C3×#95014yes—
436C2×C3C3×#95314yes—
437C2×C3#6415811yescharacteristic
438C2×C3#6415911yescharacteristic
439C2×C3#6416211yescharacteristic
440D3—244no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22C2×C3×#63811yescharacteristic
451C4—21no—
452C4—22no—
453C4—361no—
454C4C2×C3×#63811yescharacteristic
455C4C2×C3×#63811yescharacteristic
456C3—24no—
457C3C3×#523814yescomplemented
458C3#1059293711yescharacteristic, direct product factor
459C2—41no—
460C2—42no—
461C2—721no—
462C2C3×#6415811yescharacteristic
463C2C3×#6415911yescharacteristic
464C2C3×#6416211yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×#41C2
socleC22×C33C2×D4
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C3×#3753C2semidirect
4. C3×#64139C2semidirect
5. C3×#63662C2semidirect
6. C3×#63659C2semidirect
7. C3×#63657C2semidirect
8. #10592937C3direct (N × H)
26. C2×C32×#41D3semidirect
27. C2×C32×#41C2×C3semidirect
28. C2×#3755C2×C3semidirect
29. C2×#3753C2×C3semidirect
30. #64139C2×C3semidirect
34. #63662C2×C3semidirect
35. #63659C2×C3semidirect
36. #63657C2×C3semidirect
44. C33×Q8D4semidirect
47. C33×D4D4semidirect
115. C2×C3×#41C3×D3semidirect
116. C2×C3×#41C3⋊D3semidirect
141. C32×Q8#67semidirect
142. C32×Q8C3×D4semidirect
150. C32×D4#67semidirect
151. C32×D4C3×D4semidirect
267. C2×#41C3×C3⋊D3semidirect
274. C33#449semidirect
287. C3×Q8C3×#67semidirect
288. C3×Q8#638semidirect
299. C3×D4C3×#67semidirect
300. C3×D4#638semidirect
402. C32#5238semidirect
403. C32C3×#449semidirect
408. Q8C3×#638semidirect
414. D4C3×#638semidirect
457. C3C3×#5238semidirect
458. C3#10592937direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C2×C3×#37552
C2×C3×#37532
C3×#641392
C3×#636622
C3×#636592
C3×#636572
#105929373
C3×#52383