Finite Group Databasebeta

← C3×#10592849 #13163185

Subgroups — C3×#10592849 #13163185

Order 1728.

1144 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×D8C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592849C311yesmaximal, characteristic, direct product factor
9C3×#5150—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×#638—21no—
14C22×C3×C3⋊Dic3—21no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C33×D8—22no—
17C2×C33×C8C2211yescharacteristic
18C3×#3757—21no—
19C3×#3753—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×D8—24no—
24C2×C32×D8D314yescomplemented
25C2×C32×D8C2×C311yescharacteristic, complemented
26#64149C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×#3753C2×C311yescharacteristic, complemented
29C3×#940—34no—
30C3×D4×Dic3—34no—
31C2×C3×#213—34no—
32#63659C2×C311yescharacteristic, complemented
33#63648C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#832—34no—
36C3×#821—34no—
37C3×Dic3⋊C8—34no—
38C23×C33—21no—
39C23×C33—21no—
40C22×C3×C3⋊D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C33×D4—21no—
44C33×D4D412yescomplemented
45C2×C33×C4C2311yescharacteristic
46C3×#638—42no—
47C3×#638—42no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—42no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×#459—91no—
57#5150—34no—
58C2×C32×D4—24no—
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C22×C311yescharacteristic
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—21no—
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C3×#67—64no—
69C2×C3×#67—64no—
70C22×C3×Dic3—64no—
71C2×C3×D12—34no—
72C32×D8—216no—
73C32×D8—28no—
74C32×D8—22no—
75C2×C32×C8—24no—
76C2×C32×C8C2×D314yes—
77C2×C32×C8C22×C311yescharacteristic
78#3757—21no—
79#3753—22no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4×C3⋊Dic3C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×#217—64no—
84C3×#213—68no—
85C3×C4⋊Dic3—34no—
86C3×C4×Dic3—34no—
87C2×C3×C3⋊C8—34no—
88C22×C33—22no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33—41no—
93C22×C33—41no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C3×C3⋊Dic3—21no—
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3D812yescomplemented
103C2×C3×D8—24no—
104C2×C3×D8—91no—
105C2×C3×D8C3×D314yescomplemented
106C2×C3×D8C3⋊D311yescharacteristic, complemented
107C3×#127—91no—
108C3×C4×D4—91no—
109#940—34no—
110D4×Dic3—34no—
111C2×#213—34no—
112C3×C4⋊C8—91no—
113C3×#102—91no—
114C3×#102—91no—
115#832—34no—
116#821—34no—
117Dic3⋊C8—34no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—44no—
123C23×C32—44no—
124C22×C3⋊D3—21no—
125C22×C3×D3—64no—
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—28no—
132C32×D4—24no—
133C32×D4—24no—
134C32×D4—44no—
135C32×D4—44no—
136C32×D4#6718yescomplemented
137C32×D4C3×D412yescomplemented
138C2×C32×C4—24no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C23×C311yescharacteristic
141#638—42no—
142#638—42no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—42no—
145C2×C3⋊Dic3C3×D411yescharacteristic
146C2×C3⋊Dic3C3×D411yescharacteristic
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)—21no—
149C3×#67—128no—
150C3×#67—128no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—128no—
155C3×D12—64no—
156C3×D12—64no—
157C32×C8—28no—
158C32×C8—24no—
159C32×C8—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—64no—
162#459—91noSylow
163C2×C33—42no—
164C2×C33—81no—
165C2×C33#12111yescharacteristic
166C2×C33C2×D811yescharacteristic
167C2×C33AGL(1,ℤ/8)11yescharacteristic
168C3×C3⋊D3—81no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C2×C3×D4—181no—
174C2×C3×D4C2×C3×D314yes—
175C2×C3×D4C2×C3×D314yes—
176C2×C3×D4C2×C3⋊D311yescharacteristic
177C2×C3×D4C2×C3⋊D311yescharacteristic
178C22×C3×C4—181no—
179C2×#67—64no—
180C2×#67—64no—
181C22×Dic3—64no—
182C2×D12—34no—
183C3×D8—216no—
184C3×D8—28no—
185C3×D8—22no—
186C3×D8—182no—
187C2×C3×C8—24no—
188C2×C3×C8—91no—
189C2×C3×C8C2×C3×D314yes—
190C2×C3×C8C2×C3⋊D311yescharacteristic
191C3×C4⋊C4—91no—
192C3×C22⋊C4—181no—
193C3×C42—91no—
194#217—64no—
195#213—68no—
196C4⋊Dic3—34no—
197C4×Dic3—34no—
198C2×C3⋊C8—34no—
199C22×C32—22no—
200C22×C32—24no—
201C22×C32—28no—
202C22×C32—48no—
203C22×C32—48no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—44no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—44no—
211C22×C32—44no—
212C22×C32—44no—
213C22×C32—44no—
214C22×C32—84no—
215C22×C32—84no—
216C22×C32—84no—
217C22×C32—84no—
218C22×C32D3×D414yes—
219C22×C32C2×C3×D411yescharacteristic
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C2×C3×D3—124no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4#23714yes—
229C32×C4C2×#6714yes—
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C3×#4111yescharacteristic
232C3⋊Dic3—21no—
233C3⋊Dic3—41no—
234C3⋊Dic3C3×D812yescomplemented
235C3×Dic3—38no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C2×D8—91no—
239C2×D8C3×C3⋊D311yescharacteristic, complemented
240#127—91no—
241C4×D4—91no—
242C4⋊C8—91no—
243#102—91no—
244#102—91no—
245C33#45911yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—24no—
249C23×C3—21no—
250C23×C3—44no—
251C23×C3—44no—
252C23×C3—181no—
253C22×D3—64no—
254C3×D4—28no—
255C3×D4—24no—
256C3×D4—24no—
257C3×D4—24no—
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—28no—
262C3×D4—44no—
263C3×D4—44no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—362no—
267C3×D4—362no—
268C3×D4C3×#6718yescomplemented
269C3×D4#63812yescomplemented
270C2×C3×C4—24no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—181no—
274C2×C3×C4—362no—
275C2×C3×C4C22×C3×D314yes—
276C2×C3×C4C22×C3⋊D311yescharacteristic
277#67—128no—
278#67—128no—
279C2×Dic3—34no—
280C2×Dic3—34no—
281C2×Dic3—64no—
282C2×Dic3—128no—
283D12—64no—
284D12—64no—
285C3×C8—28no—
286C3×C8—24no—
287C3×C8—21no—
288C3×C8—181no—
289C3⋊C8—64no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—42no—
294C2×C32—48no—
295C2×C32—88no—
296C2×C32—88no—
297C2×C32—84no—
298C2×C32—81no—
299C2×C32D3×D814yes—
300C2×C32#91114yes—
301C2×C32#93914yes—
302C2×C32C3×#12111yescharacteristic
303C2×C32C2×C3×D811yescharacteristic
304C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
305C3⋊D3—81no—
306C3×D3—244no—
307C2×D4—91no—
308C2×D4—181no—
309C2×D4—181no—
310C2×D4C2×C3×C3⋊D311yescharacteristic
311C2×D4C2×C3×C3⋊D311yescharacteristic
312C22×C4—181no—
313D8—22no—
314D8—182no—
315C2×C8—91no—
316C2×C8C2×C3×C3⋊D311yescharacteristic
317C4⋊C4—91no—
318C22⋊C4—181no—
319C42—91no—
320C22×C3—28no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—48no—
324C22×C3—48no—
325C22×C3—44no—
326C22×C3—44no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—44no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—84no—
336C22×C3—84no—
337C22×C3—84no—
338C22×C3—84no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3C3×D3×D414yes—
343C22×C3D4×C3⋊D311yescharacteristic
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—124no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—92no—
350C3×C4—181no—
351C3×C4—361no—
352C3×C4C3×#23714yes—
353C3×C4C2×C3×#6714yes—
354C3×C4#383011yescharacteristic
355C3×C4C2×#63811yescharacteristic
356Dic3—38no—
357Dic3—64no—
358Dic3—124no—
359C32—24no—
360C32#515014yescomplemented
361C32C3×#45911yescharacteristic, complemented
362C23—21no—
363C23—21no—
364C23—181no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—181no—
369D4—181no—
370D4—362no—
371D4—362no—
372D4C3×#63812yescomplemented
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—181no—
376C2×C4—362no—
377C2×C4C22×C3×C3⋊D311yescharacteristic
378C8—21no—
379C8—181no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—42no—
384C2×C3—48no—
385C2×C3—88no—
386C2×C3—88no—
387C2×C3—81no—
388C2×C3—84no—
389C2×C3—721no—
390C2×C3C3×D3×D814yes—
391C2×C3C3×#91114yes—
392C2×C3C3×#93914yes—
393C2×C3D8×C3⋊D311yescharacteristic
394C2×C3#6412011yescharacteristic
395C2×C3#6414811yescharacteristic
396D3—244no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—361no—
404C22—361no—
405C22—361no—
406C22C3×D4×C3⋊D311yescharacteristic
407C4—92no—
408C4—181no—
409C4—361no—
410C4C3×#383011yescharacteristic
411C4C2×C3×#63811yescharacteristic
412C3—24no—
413C3C3×#515014yescomplemented
414C3#1059284911yescharacteristic, direct product factor
415C2—42no—
416C2—81no—
417C2—721no—
418C2C3×D8×C3⋊D311yescharacteristic
419C2C3×#6412011yescharacteristic
420C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×D8C2
socleC22×C33C2×D4
hypercenterC2×C3×D8C3⋊D3
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

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

NH ≅ G/Ntype
1. C2×C33×D8C2semidirect
2. C3×#64149C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. C2×C3×#3753C2semidirect
5. C3×#63659C2semidirect
6. C3×#63648C2semidirect
7. C3×#63641C2semidirect
8. #10592849C3direct (N × H)
24. C2×C32×D8D3semidirect
25. C2×C32×D8C2×C3semidirect
26. #64149C2×C3semidirect
27. D4×C3⋊Dic3C2×C3semidirect
28. C2×#3753C2×C3semidirect
32. #63659C2×C3semidirect
33. #63648C2×C3semidirect
34. #63641C2×C3semidirect
44. C33×D4D4semidirect
102. C3×C3⋊Dic3D8semidirect
105. C2×C3×D8C3×D3semidirect
106. C2×C3×D8C3⋊D3semidirect
136. C32×D4#67semidirect
137. C32×D4C3×D4semidirect
234. C3⋊Dic3C3×D8semidirect
239. C2×D8C3×C3⋊D3semidirect
245. C33#459semidirect
268. C3×D4C3×#67semidirect
269. C3×D4#638semidirect
360. C32#5150semidirect
361. C32C3×#459semidirect
372. D4C3×#638semidirect
413. C3C3×#5150semidirect
414. C3#10592849direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×D82
C3×#641492
C3×D4×C3⋊Dic32
C2×C3×#37532
C3×#636592
C3×#636482
C3×#636412
#105928493
C3×#51503