Finite Group Databasebeta

← C3×#10592897 #13163238

Subgroups — C3×#10592897 #13163238

Order 1728.

1280 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic, complemented
2C33×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C3×#64141C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63650C211yesmaximal, characteristic, complemented
8#10592897C311yesmaximal, characteristic, direct product factor
9C3×#5198—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C3×#638—22no—
13C3×#3832—21no—
14C3×D4×C3⋊D3—22no—
15C3×D4×C3⋊D3—21no—
16C3×#3828—21no—
17C3×#3828C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)—21no—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C33×SD16—21no—
21C33×D8—21no—
22C33×M16C2211yescharacteristic, complemented
23C3×#3754—21no—
24C3×#3753—21no—
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C3×#3748C2211yescharacteristic, complemented
27#64366C2×C311yescharacteristic, complemented
28C3×#1009—34no—
29C32×AGL(1,ℤ/8)—24no—
30C32×AGL(1,ℤ/8)D314yescomplemented
31C32×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
32#64149C2×C311yescharacteristic, complemented
33#64141C2×C311yescharacteristic, complemented
34C3×#940—34no—
35C3×#932—34no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39C3×#837—34no—
40C3×#825—34no—
41C3×#823—34no—
42C23×C33—21no—
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3D412yes—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4—21no—
48C33×D4—21no—
49C33×D4D411yescharacteristic, complemented
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—22no—
53C3×#638—21no—
54C3×#638—44no—
55C2×C3×C3⋊Dic3—21no—
56C3×C4⋊(C3⋊D3)—22no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
60C3×C4×C3⋊D3—22no—
61C3×C4×C3⋊D3—21no—
62C3×#634D411yescharacteristic, complemented
63C33×C8—21no—
64C3×C3⋊(C3⋊C8)—21no—
65C3×#453—91no—
66#5198—34no—
67C32×#41—24no—
68C32×#41C2×D314yes—
69C32×#41C22×C311yescharacteristic
70C2×C32×D4—24no—
71C2×C32×D4C2×D314yescomplemented
72C2×C32×D4C22×C311yescharacteristic, complemented
73C2×#638—22no—
74#3832—21no—
75D4×C3⋊D3—22no—
76D4×C3⋊D3—21no—
77#3828—21no—
78#3828C22×C311yescharacteristic
79C2×C4⋊(C3⋊D3)—21no—
80C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
81C2×C3×#67—68no—
82C3×#239—64no—
83C3×D3×D4—68no—
84C3×D3×D4—64no—
85C3×#235—34no—
86C3×#235—64no—
87C2×C3×D12—34no—
88C2×C3×D12—64no—
89C32×SD16—28no—
90C32×SD16—24no—
91C32×SD16—21no—
92C32×D8—28no—
93C32×D8—24no—
94C32×D8—21no—
95C32×M16—24no—
96C32×M16C2×D314yescomplemented
97C32×M16C22×C311yescharacteristic, complemented
98#3754—21no—
99#3753—21no—
100C4×C3⋊Dic3C22×C311yescharacteristic, complemented
101#3748C22×C311yescharacteristic, complemented
102C3×#214—64no—
103C3×#213—64no—
104C3×C4×Dic3—34no—
105C3×#208—34no—
106C22×C33—21no—
107C22×C33—42no—
108C22×C33—41no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3—42no—
113C2×C3×C3⋊D3—82no—
114C33×C4—21no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2×D411yescharacteristic
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—21no—
119C3×H(5,𝔽2)—91no—
120#1009—34no—
121C3×AGL(1,ℤ/8)—24no—
122C3×AGL(1,ℤ/8)—91no—
123C3×AGL(1,ℤ/8)C3×D314yescomplemented
124C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
125C3×#127—91no—
126#940—34no—
127#932—34no—
128C3×#104—91no—
129C3×#104—91no—
130C3×#100—91no—
131#837—34no—
132#825—34no—
133#823—34no—
134C23×C32—21no—
135C23×C32—24no—
136C23×C32—44no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3C3×D412yes—
139C22×C3×D3—38no—
140C22×C3×D3—68no—
141C32×Q8—24no—
142C32×Q8#6714yescomplemented
143C32×Q8C3×D411yescharacteristic, complemented
144C32×D4—21no—
145C32×D4—28no—
146C32×D4—24no—
147C32×D4—21no—
148C32×D4—21no—
149C32×D4—24no—
150C32×D4—24no—
151C32×D4—24no—
152C32×D4—44no—
153C32×D4—44no—
154C32×D4#6714yescomplemented
155C32×D4C3×D411yescharacteristic, complemented
156C2×C32×C4—21no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—44no—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C23×C311yescharacteristic
162#638—22no—
163#638—21no—
164#638—44no—
165C2×C3⋊Dic3—21no—
166C4⋊(C3⋊D3)—22no—
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
170C4×C3⋊D3—22no—
171C4×C3⋊D3—21no—
172#634C3×D411yescharacteristic, complemented
173C3×#67—68no—
174C3×#67—64no—
175C3×#67—1216no—
176C2×C3×Dic3—64no—
177C3×D12—34no—
178C3×D12—68no—
179C3×D12—64no—
180C3×D12—64no—
181C3×C4×D3—68no—
182C3×C4×D3—64no—
183C3×Dic6—34no—
184C32×C8—28no—
185C32×C8—24no—
186C32×C8—21no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—64no—
189#453—91noSylow
190C2×C33—21no—
191C2×C33—41no—
192C2×C33—81no—
193C2×C33#12011yescharacteristic
194C3×C3⋊D3—42no—
195C3×C3⋊D3—41no—
196C3×#41—24no—
197C3×#41—91no—
198C3×#41—181no—
199C3×#41—181no—
200C3×#41C2×C3×D314yes—
201C3×#41C2×C3⋊D311yescharacteristic
202C2×C3×D4—24no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4—182no—
206C2×C3×D4—181no—
207C2×C3×D4—182no—
208C2×C3×D4C2×C3×D314yescomplemented
209C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
210C2×#67—68no—
211#239—64no—
212D3×D4—68no—
213D3×D4—64no—
214#235—34no—
215#235—64no—
216C2×D12—34no—
217C2×D12—64no—
218C3×SD16—28no—
219C3×SD16—24no—
220C3×SD16—21no—
221C3×SD16—181no—
222C3×D8—28no—
223C3×D8—24no—
224C3×D8—21no—
225C3×D8—181no—
226C3×M16—24no—
227C3×M16—91no—
228C3×M16C2×C3×D314yescomplemented
229C3×M16C2×C3⋊D311yescharacteristic, complemented
230C3×C42—91no—
231#214—64no—
232#213—64no—
233C4×Dic3—34no—
234#208—34no—
235C22×C32—21no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—44no—
239C22×C32—42no—
240C22×C32—48no—
241C22×C32—44no—
242C22×C32—41no—
243C22×C32—48no—
244C22×C32—88no—
245C22×C32D3×D414yes—
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—21no—
249C2×C3⋊D3—42no—
250C2×C3⋊D3—82no—
251C2×C3×D3—64no—
252C2×C3×D3—68no—
253C2×C3×D3—128no—
254C2×C3×D3—248no—
255C32×C4—21no—
256C32×C4—24no—
257C32×C4—24no—
258C32×C4—24no—
259C32×C4—44no—
260C32×C4D3×D414yes—
261C32×C4C2×#6714yes—
262C32×C4C2×C3×D411yescharacteristic
263C32×C4C2×C3×D411yescharacteristic
264C3⋊Dic3—22no—
265C3⋊Dic3—21no—
266C3×Dic3—68no—
267C3×Dic3—64no—
268H(5,𝔽2)—91no—
269AGL(1,ℤ/8)—91no—
270AGL(1,ℤ/8)C3×C3⋊D311yescharacteristic, complemented
271#127—91no—
272#104—91no—
273#104—91no—
274#100—91no—
275C33#45311yescharacteristic, complemented, Sylow
276C23×C3—24no—
277C23×C3—21no—
278C23×C3—44no—
279C23×C3—92no—
280C23×C3—182no—
281C22×D3—38no—
282C22×D3—68no—
283C3×Q8—24no—
284C3×Q8—91no—
285C3×Q8C3×#6714yescomplemented
286C3×Q8#63811yescharacteristic, complemented
287C3×D4—24no—
288C3×D4—21no—
289C3×D4—28no—
290C3×D4—24no—
291C3×D4—24no—
292C3×D4—21no—
293C3×D4—21no—
294C3×D4—24no—
295C3×D4—44no—
296C3×D4—44no—
297C3×D4—91no—
298C3×D4—182no—
299C3×D4—182no—
300C3×D4—181no—
301C3×D4—181no—
302C3×D4—181no—
303C3×D4—364no—
304C3×D4C3×#6714yescomplemented
305C3×D4#63811yescharacteristic, complemented
306C2×C3×C4—24no—
307C2×C3×C4—21no—
308C2×C3×C4—24no—
309C2×C3×C4—44no—
310C2×C3×C4—182no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×C3×C4C22×C3×D314yes—
314C2×C3×C4C22×C3⋊D311yescharacteristic
315#67—68no—
316#67—64no—
317#67—1216no—
318C2×Dic3—64no—
319D12—34no—
320D12—68no—
321D12—64no—
322D12—64no—
323C4×D3—68no—
324C4×D3—64no—
325Dic6—34no—
326C3×C8—28no—
327C3×C8—24no—
328C3×C8—21no—
329C3×C8—181no—
330C3⋊C8—64no—
331C2×C32—21no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—44no—
335C2×C32—41no—
336C2×C32—44no—
337C2×C32—84no—
338C2×C32—88no—
339C2×C32—84no—
340C2×C32—81no—
341C2×C32#93714yes—
342C2×C32C3×#12011yescharacteristic
343C3⋊D3—42no—
344C3⋊D3—41no—
345C3×D3—128no—
346C3×D3—124no—
347#41—91no—
348#41—181no—
349#41—181no—
350#41C2×C3×C3⋊D311yescharacteristic
351C2×D4—91no—
352C2×D4—181no—
353C2×D4—182no—
354C2×D4—181no—
355C2×D4—182no—
356C2×D4C2×C3×C3⋊D311yescharacteristic, complemented
357SD16—21no—
358SD16—181no—
359D8—21no—
360D8—181no—
361M16—91no—
362M16C2×C3×C3⋊D311yescharacteristic, complemented
363C42—91no—
364C22×C3—24no—
365C22×C3—21no—
366C22×C3—24no—
367C22×C3—44no—
368C22×C3—48no—
369C22×C3—48no—
370C22×C3—42no—
371C22×C3—44no—
372C22×C3—41no—
373C22×C3—88no—
374C22×C3—181no—
375C22×C3—182no—
376C22×C3—362no—
377C22×C3—722no—
378C22×C3C3×D3×D414yes—
379C22×C3D4×C3⋊D311yescharacteristic
380C2×D3—64no—
381C2×D3—68no—
382C2×D3—128no—
383C2×D3—248no—
384C3×C4—24no—
385C3×C4—21no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—44no—
389C3×C4—182no—
390C3×C4—181no—
391C3×C4C3×D3×D414yes—
392C3×C4C2×C3×#6714yes—
393C3×C4D4×C3⋊D311yescharacteristic
394C3×C4C2×#63811yescharacteristic
395Dic3—68no—
396Dic3—64no—
397C32—24no—
398C32#519814yescomplemented
399C32C3×#45311yescharacteristic, complemented
400C23—21no—
401C23—92no—
402C23—182no—
403Q8—91no—
404Q8C3×#63811yescharacteristic, complemented
405D4—21no—
406D4—21no—
407D4—21no—
408D4—91no—
409D4—182no—
410D4—182no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—364no—
415D4C3×#63811yescharacteristic, complemented
416C2×C4—21no—
417C2×C4—182no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4C22×C3×C3⋊D311yescharacteristic
421C8—21no—
422C8—181no—
423C2×C3—24no—
424C2×C3—21no—
425C2×C3—24no—
426C2×C3—44no—
427C2×C3—44no—
428C2×C3—41no—
429C2×C3—84no—
430C2×C3—88no—
431C2×C3—84no—
432C2×C3—81no—
433C2×C3—362no—
434C2×C3—361no—
435C2×C3C3×#93714yes—
436C2×C3#6414611yescharacteristic
437D3—128no—
438D3—124no—
439C22—21no—
440C22—42no—
441C22—41no—
442C22—181no—
443C22—182no—
444C22—362no—
445C22—722no—
446C22C3×D4×C3⋊D311yescharacteristic
447C4—21no—
448C4—182no—
449C4—181no—
450C4C3×D4×C3⋊D311yescharacteristic
451C4C2×C3×#63811yescharacteristic
452C3—24no—
453C3C3×#519814yescomplemented
454C3#1059289711yescharacteristic, direct product factor
455C2—21no—
456C2—41no—
457C2—81no—
458C2—362no—
459C2—361no—
460C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33#120
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C3×#64366C2semidirect
2. C33×AGL(1,ℤ/8)C2semidirect
3. C3×#64149C2semidirect
4. C3×#64141C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63650C2semidirect
8. #10592897C3direct (N × H)
11. C2×C33×D4C22semidirect
22. C33×M16C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C3×#3748C22semidirect
27. #64366C2×C3semidirect
30. C32×AGL(1,ℤ/8)D3semidirect
31. C32×AGL(1,ℤ/8)C2×C3semidirect
32. #64149C2×C3semidirect
33. #64141C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63650C2×C3semidirect
45. C33×Q8D4semidirect
49. C33×D4D4semidirect
59. C3×C4⋊(C3⋊D3)D4semidirect
62. C3×#634D4semidirect
71. C2×C32×D4C2×D3semidirect
72. C2×C32×D4C22×C3semidirect
96. C32×M16C2×D3semidirect
97. C32×M16C22×C3semidirect
100. C4×C3⋊Dic3C22×C3semidirect
101. #3748C22×C3semidirect
123. C3×AGL(1,ℤ/8)C3×D3semidirect
124. C3×AGL(1,ℤ/8)C3⋊D3semidirect
142. C32×Q8#67semidirect
143. C32×Q8C3×D4semidirect
154. C32×D4#67semidirect
155. C32×D4C3×D4semidirect
169. C4⋊(C3⋊D3)C3×D4semidirect
172. #634C3×D4semidirect
208. C2×C3×D4C2×C3×D3semidirect
209. C2×C3×D4C2×C3⋊D3semidirect
228. C3×M16C2×C3×D3semidirect
229. C3×M16C2×C3⋊D3semidirect
270. AGL(1,ℤ/8)C3×C3⋊D3semidirect
275. C33#453semidirect
285. C3×Q8C3×#67semidirect
286. C3×Q8#638semidirect
304. C3×D4C3×#67semidirect
305. C3×D4#638semidirect
356. C2×D4C2×C3×C3⋊D3semidirect
362. M16C2×C3×C3⋊D3semidirect
398. C32#5198semidirect
399. C32C3×#453semidirect
404. Q8C3×#638semidirect
415. D4C3×#638semidirect
453. C3C3×#5198semidirect
454. C3#10592897direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643662
C33×AGL(1,ℤ/8)2
C3×#641492
C3×#641412
C3×#636642
C3×#636522
C3×#636502
#105928973
C3×#51983