Finite Group Databasebeta

← C3×#10592433 #13162772

Subgroups — C3×#10592433 #13162772

Order 1728.

1172 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C33×C22⋊C8C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592433C311yesmaximal, characteristic, direct product factor
9C3×#4734—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×#638—21no—
13C3×#3828—22no—
14C3×#3828—22no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×#634C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C2×C33×C8C2211yescharacteristic
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3745—22no—
24C3×#3744—22no—
25C2×#3828C2×C311yescharacteristic, complemented
26C2×C3×#235—34no—
27#64139C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29C2×#3744C2×C311yescharacteristic, complemented
30C3×#930—34no—
31C2×C3×D24—34no—
32C2×C3×#204—34no—
33C32×C22⋊C8—24no—
34C32×C22⋊C8D314yescomplemented
35C32×C22⋊C8C2×C311yescharacteristic, complemented
36#63648C2×C311yescharacteristic, complemented
37#63643C2×C311yescharacteristic, complemented
38C3×#821—34no—
39C3×#816—34no—
40C23×C33D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C3×#638—24no—
48C3×#638—42no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C3×C4×C3⋊D3—24no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C33×C8—21no—
59C33×C8—21no—
60C3×#449—91no—
61#4734—34no—
62C22×C32×C4—24no—
63C22×C32×C4C2×D314yes—
64C22×C32×C4C22×C311yescharacteristic
65C2×#638—21no—
66C2×#638—21no—
67#3828—22no—
68#3828—22no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×#634C22×C311yescharacteristic
73C2×C3×#67—64no—
74C2×C3×#67—64no—
75C3×#235—68no—
76C3×#235—68no—
77C2×C3×D12—34no—
78C2×C3×D12—34no—
79C2×C3×C4×D3—64no—
80C2×C3×Dic6—34no—
81C2×C32×C8—24no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C22×C311yescharacteristic
86C2×C32×C8C22×C311yescharacteristic
87#3752—21no—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89#3745—22no—
90#3744—22no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×D24—68no—
94C3×#204—68no—
95C22×C33—21no—
96C22×C33—21no—
97C22×C33—21no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—22no—
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—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3—41no—
110C2×C3×#41—91no—
111C2×#235—34no—
112C2×C3×SD16—91no—
113C2×C3×D8—91no—
114C3×#121—91no—
115#930—34no—
116C2×D24—34no—
117C2×#204—34no—
118C3×#103—91no—
119C3×#102—91no—
120C3×C22⋊C8—24no—
121C3×C22⋊C8C3×D314yescomplemented
122C3×C22⋊C8C3⋊D311yescharacteristic, complemented
123#821—34no—
124#816—34no—
125C23×C32—24no—
126C23×C32D1214yes—
127C23×C32C3×D411yescharacteristic
128C22×C3⋊D3—21no—
129C22×C3⋊D3—21no—
130C22×C3×D3—64no—
131C22×C3×D3—64no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4—44no—
140C2×C32×C4D1214yes—
141C2×C32×C4C3×D411yescharacteristic
142C2×C32×C4C22×D314yes—
143C2×C32×C4C23×C311yescharacteristic
144#638—24no—
145#638—42no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)C3×D412yescomplemented
152C4×C3⋊D3—24no—
153#634—21no—
154#634C3×D412yescomplemented
155C3×#67—616no—
156C3×#67—128no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—64no—
159C3×D12—38no—
160C3×D12—64no—
161C3×D12—64no—
162C3×D12—64no—
163C3×C4×D3—616no—
164C3×Dic6—38no—
165C3×Dic6—64no—
166C32×C8—21no—
167C32×C8—21no—
168C32×C8—24no—
169C32×C8—24no—
170C32×C8—44no—
171C32×C8—44no—
172#449—91noSylow
173C2×C33—41no—
174C2×C33#12011yescharacteristic
175C2×C33#13511yescharacteristic
176C2×C33AGL(1,ℤ/8)11yescharacteristic
177C3×C3⋊D3—42no—
178C3×C3⋊D3—81no—
179C3×#41—182no—
180C3×#41—182no—
181C2×C3×Q8—91no—
182C2×C3×D4—91no—
183C2×C3×D4—91no—
184C2×C3×D4—181no—
185C2×C3×D4—181no—
186C22×C3×C4—24no—
187C22×C3×C4—181no—
188C22×C3×C4C2×C3×D314yes—
189C22×C3×C4C2×C3⋊D311yescharacteristic
190C2×#67—64no—
191C2×#67—64no—
192#235—68no—
193#235—68no—
194C2×D12—34no—
195C2×D12—34no—
196C2×C4×D3—64no—
197C2×Dic6—34no—
198C3×SD16—182no—
199C3×D8—182no—
200C2×C3×C8—24no—
201C2×C3×C8—24no—
202C2×C3×C8C2×C3×D314yes—
203C2×C3×C8C2×C3×D314yes—
204C2×C3×C8C2×C3⋊D311yescharacteristic
205C2×C3×C8C2×C3⋊D311yescharacteristic
206C3×C4⋊C4—91no—
207C3×C22⋊C4—181no—
208(C2×D3)⋊C4—64no—
209C4⋊Dic3—34no—
210D24—68no—
211#204—68no—
212C22×C32—28no—
213C22×C32—24no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—24no—
220C22×C32—44no—
221C22×C32—44no—
222C22×C32C2×D1214yes—
223C22×C32C2×C3×D411yescharacteristic
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3×D3—68no—
231C2×C3×D3—124no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236C32×C4—21no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—24no—
240C32×C4—44no—
241C32×C4D3×D414yes—
242C32×C4D3×D414yes—
243C32×C4C2×C3×D411yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C3⋊Dic3—22no—
246C3⋊Dic3—41no—
247C3×Dic3—68no—
248C3×Dic3—124no—
249C2×#41—91no—
250C2×SD16—91no—
251C2×D8—91no—
252#121—91no—
253#103—91no—
254#102—91no—
255C22⋊C8C3×C3⋊D311yescharacteristic, complemented
256C33#44911yescharacteristic, complemented, Sylow
257C23×C3—24no—
258C23×C3—181no—
259C23×C3—181no—
260C23×C3C3×D1214yes—
261C23×C3C4⋊(C3⋊D3)11yescharacteristic
262C22×D3—64no—
263C22×D3—64no—
264C3×Q8—92no—
265C3×Q8—181no—
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—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—184no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4C3×D1214yes—
284C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
285C2×C3×C4C22×C3×D314yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—616no—
288#67—128no—
289C2×Dic3—64no—
290C2×Dic3—64no—
291D12—38no—
292D12—64no—
293D12—64no—
294D12—64no—
295C4×D3—616no—
296Dic6—38no—
297Dic6—64no—
298C3×C8—24no—
299C3×C8—24no—
300C3×C8—21no—
301C3×C8—21no—
302C3×C8—44no—
303C3×C8—44no—
304C2×C32—24no—
305C2×C32—24no—
306C2×C32—24no—
307C2×C32—48no—
308C2×C32—44no—
309C2×C32—41no—
310C2×C32#88214yes—
311C2×C32#90414yes—
312C2×C32#90814yes—
313C2×C32C3×#12011yescharacteristic
314C2×C32C3×#13511yescharacteristic
315C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
316C3⋊D3—42no—
317C3⋊D3—81no—
318C3×D3—128no—
319C3×D3—244no—
320#41—182no—
321#41—182no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—181no—
326C2×D4—181no—
327C22×C4—181no—
328C22×C4C2×C3×C3⋊D311yescharacteristic
329SD16—182no—
330D8—182no—
331C2×C8C2×C3×C3⋊D311yescharacteristic
332C2×C8C2×C3×C3⋊D311yescharacteristic
333C4⋊C4—91no—
334C22⋊C4—181no—
335C22×C3—28no—
336C22×C3—24no—
337C22×C3—24no—
338C22×C3—24no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—24no—
343C22×C3—44no—
344C22×C3—44no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3C2×C3×D1214yes—
352C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
353C2×D3—68no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3—124no—
359C3×C4—24no—
360C3×C4—21no—
361C3×C4—24no—
362C3×C4—24no—
363C3×C4—44no—
364C3×C4—182no—
365C3×C4—361no—
366C3×C4C3×D3×D414yes—
367C3×C4C3×D3×D414yes—
368C3×C4D4×C3⋊D311yescharacteristic
369C3×C4D4×C3⋊D311yescharacteristic
370Dic3—68no—
371Dic3—124no—
372C32—24no—
373C32#473414yescomplemented
374C32C3×#44911yescharacteristic, complemented
375C23—181no—
376C23—181no—
377C23C3×C4⋊(C3⋊D3)11yescharacteristic
378Q8—92no—
379Q8—181no—
380D4—92no—
381D4—184no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—362no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—184no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
392C2×C4C22×C3×C3⋊D311yescharacteristic
393C8—21no—
394C8—21no—
395C2×C3—24no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—48no—
399C2×C3—44no—
400C2×C3—41no—
401C2×C3—362no—
402C2×C3—721no—
403C2×C3C3×#88214yes—
404C2×C3C3×#90414yes—
405C2×C3C3×#90814yes—
406C2×C3#6409111yescharacteristic
407C2×C3#6411311yescharacteristic
408C2×C3#6411711yescharacteristic
409D3—128no—
410D3—244no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—182no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22—361no—
419C22—361no—
420C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
421C4—21no—
422C4—182no—
423C4—361no—
424C4C3×D4×C3⋊D311yescharacteristic
425C4C3×D4×C3⋊D311yescharacteristic
426C3—24no—
427C3C3×#473414yescomplemented
428C3#1059243311yescharacteristic, direct product factor
429C2—41no—
430C2—362no—
431C2—721no—
432C2C3×#6409111yescharacteristic
433C2C3×#6411311yescharacteristic
434C2C3×#6411711yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C3×#64139C2semidirect
3. C2×C3×#3745C2semidirect
4. C2×C3×#3744C2semidirect
5. C33×C22⋊C8C2semidirect
6. C3×#63648C2semidirect
7. C3×#63643C2semidirect
8. #10592433C3direct (N × H)
25. C2×#3828C2×C3semidirect
27. #64139C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. C2×#3744C2×C3semidirect
34. C32×C22⋊C8D3semidirect
35. C32×C22⋊C8C2×C3semidirect
36. #63648C2×C3semidirect
37. #63643C2×C3semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
57. C3×#634D4semidirect
121. C3×C22⋊C8C3×D3semidirect
122. C3×C22⋊C8C3⋊D3semidirect
151. C4⋊(C3⋊D3)C3×D4semidirect
154. #634C3×D4semidirect
255. C22⋊C8C3×C3⋊D3semidirect
256. C33#449semidirect
373. C32#4734semidirect
374. C32C3×#449semidirect
427. C3C3×#4734semidirect
428. C3#10592433direct (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
C3×#641392
C2×C3×#37452
C2×C3×#37442
C33×C22⋊C82
C3×#636482
C3×#636432
#105924333
C3×#47343