Finite Group Databasebeta

← C3×#10592862 #13163201

Subgroups — C3×#10592862 #13163201

Order 1728.

1040 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×SD16C211yesmaximal, characteristic, complemented
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4C3×#64149C211yesmaximal, characteristic
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592862C311yesmaximal, characteristic, direct product factor
9C3×#5163—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×#638—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C33×SD16—22no—
16C2×C33×C8C2211yescharacteristic
17C3×#3755—22no—
18C3×C4⋊(C3⋊Dic3)—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3—21no—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×SD16—24no—
24C2×C32×SD16D314yescomplemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28#64149C2×C311yescharacteristic
29C3×Q8×Dic3—34no—
30C2×C3×#215—34no—
31C3×#940—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—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8D412yescomplemented
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C3×#638—42no—
47C3×#638—42no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C33×C8—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C3×#460—91no—
56#5163—34no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—24no—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—21no—
64C2×#638—21no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×#67—64no—
67C2×C3×#67—64no—
68C2×C3×D12—34no—
69C32×SD16—216no—
70C32×SD16—28no—
71C32×SD16—22no—
72C2×C32×C8—24no—
73C2×C32×C8C2×D314yes—
74C2×C32×C8C22×C311yescharacteristic
75#3755—22no—
76C4⋊(C3⋊Dic3)—21no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C4×C3⋊Dic3—21no—
79C4×C3⋊Dic3C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×#215—68no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—64no—
84C3×C4×Dic3—34no—
85C3×C4×Dic3—64no—
86C2×C3×C3⋊C8—34no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C33×C4—22no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3—41no—
99C3×C3⋊Dic3SD1612yescomplemented
100C2×C3×SD16—24no—
101C2×C3×SD16—91no—
102C2×C3×SD16C3×D314yescomplemented
103C2×C3×SD16C3⋊D311yescharacteristic, complemented
104C3×#127—91no—
105C3×C4×Q8—91no—
106Q8×Dic3—34no—
107C2×#215—34no—
108#940—34no—
109C3×C4⋊C8—91no—
110C3×#102—91no—
111C3×#102—91no—
112#832—34no—
113#821—34no—
114Dic3⋊C8—34no—
115C23×C32—21no—
116C23×C32—24no—
117C23×C32—44no—
118C22×C3⋊D3—21no—
119C22×C3×D3—64no—
120C32×Q8—21no—
121C32×Q8—28no—
122C32×Q8—24no—
123C32×Q8—44no—
124C32×Q8#6718yescomplemented
125C32×Q8C3×D412yescomplemented
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—24no—
131C32×D4—44no—
132C2×C32×C4—21no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—44no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C23×C311yescharacteristic
138#638—42no—
139#638—42no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3C3×D411yescharacteristic
142C2×C3⋊Dic3C3×D411yescharacteristic
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C3×#67—128no—
146C3×#67—128no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—64no—
150C3×D12—64no—
151C3×D12—64no—
152C32×C8—28no—
153C32×C8—24no—
154C32×C8—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—64no—
157#460—91noSylow
158C2×C33—81no—
159C2×C33#12111yescharacteristic
160C2×C33C2×SD1611yescharacteristic
161C2×C33AGL(1,ℤ/8)11yescharacteristic
162C3×C3⋊D3—81no—
163C2×C3×Q8—24no—
164C2×C3×Q8C2×C3×D314yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166C2×C3×D4—24no—
167C2×C3×D4—91no—
168C2×C3×D4—181no—
169C2×C3×D4—181no—
170C2×C3×D4C2×C3×D314yes—
171C2×C3×D4C2×C3⋊D311yescharacteristic
172C2×#67—64no—
173C2×#67—64no—
174C2×D12—34no—
175C3×SD16—216no—
176C3×SD16—28no—
177C3×SD16—22no—
178C3×SD16—182no—
179C2×C3×C8—24no—
180C2×C3×C8—91no—
181C2×C3×C8C2×C3×D314yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—91no—
184C3×C4⋊C4—181no—
185C3×C42—91no—
186C3×C42—181no—
187#215—68no—
188C4⋊Dic3—34no—
189C4⋊Dic3—64no—
190C4×Dic3—34no—
191C4×Dic3—64no—
192C2×C3⋊C8—34no—
193C22×C32—24no—
194C22×C32—48no—
195C22×C32—44no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32—84no—
202C22×C32—84no—
203C22×C32D3×D414yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—48no—
216C32×C4#23714yes—
217C32×C4C2×#6714yes—
218C32×C4C2×C3×D411yescharacteristic
219C32×C4C3×#4111yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3—41no—
222C3⋊Dic3C3×SD1612yescomplemented
223C3×Dic3—38no—
224C3×Dic3—64no—
225C3×Dic3—124no—
226C2×SD16—91no—
227C2×SD16C3×C3⋊D311yescharacteristic, complemented
228#127—91no—
229C4×Q8—91no—
230C4⋊C8—91no—
231#102—91no—
232#102—91no—
233C33#46011yescharacteristic, complemented, Sylow
234C23×C3—24no—
235C23×C3—21no—
236C23×C3—44no—
237C23×C3—181no—
238C22×D3—64no—
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—44no—
243C3×Q8C3×#6718yescomplemented
244C3×Q8#63812yescomplemented
245C3×D4—28no—
246C3×D4—24no—
247C3×D4—24no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—44no—
251C3×D4—181no—
252C3×D4—181no—
253C3×D4—362no—
254C3×D4—362no—
255C2×C3×C4—24no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—44no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4C22×C3×D314yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264#67—128no—
265#67—128no—
266C2×Dic3—34no—
267C2×Dic3—34no—
268C2×Dic3—64no—
269D12—64no—
270D12—64no—
271C3×C8—28no—
272C3×C8—24no—
273C3×C8—21no—
274C3×C8—181no—
275C3⋊C8—64no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—88no—
280C2×C32—84no—
281C2×C32—81no—
282C2×C32D3×SD1614yes—
283C2×C32#91414yes—
284C2×C32#93914yes—
285C2×C32C3×#12111yescharacteristic
286C2×C32C2×C3×SD1611yescharacteristic
287C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
288C3⋊D3—81no—
289C3×D3—244no—
290C2×Q8C2×C3×C3⋊D311yescharacteristic
291C2×D4—91no—
292C2×D4—181no—
293C2×D4—181no—
294C2×D4C2×C3×C3⋊D311yescharacteristic
295SD16—22no—
296SD16—182no—
297C2×C8—91no—
298C2×C8C2×C3×C3⋊D311yescharacteristic
299C4⋊C4—91no—
300C4⋊C4—181no—
301C42—91no—
302C42—181no—
303C22×C3—24no—
304C22×C3—48no—
305C22×C3—44no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—84no—
312C22×C3—84no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3C3×D3×D414yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—124no—
319C2×D3—124no—
320C2×D3—124no—
321C3×C4—28no—
322C3×C4—22no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—48no—
326C3×C4—92no—
327C3×C4—181no—
328C3×C4—361no—
329C3×C4C3×#23714yes—
330C3×C4C2×C3×#6714yes—
331C3×C4#383011yescharacteristic
332C3×C4C2×#63811yescharacteristic
333Dic3—38no—
334Dic3—64no—
335Dic3—124no—
336C32—24no—
337C32#516314yescomplemented
338C32C3×#46011yescharacteristic, complemented
339C23—21no—
340C23—181no—
341Q8—21no—
342Q8C3×#63812yescomplemented
343D4—21no—
344D4—21no—
345D4—181no—
346D4—181no—
347D4—362no—
348D4—362no—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4C22×C3×C3⋊D311yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—88no—
360C2×C3—84no—
361C2×C3—81no—
362C2×C3—721no—
363C2×C3C3×D3×SD1614yes—
364C2×C3C3×#91414yes—
365C2×C3C3×#93914yes—
366C2×C3SD16×C3⋊D311yescharacteristic
367C2×C3#6412311yescharacteristic
368C2×C3#6414811yescharacteristic
369D3—244no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—361no—
374C22—361no—
375C22—361no—
376C22C3×D4×C3⋊D311yescharacteristic
377C4—22no—
378C4—92no—
379C4—181no—
380C4—361no—
381C4C3×#383011yescharacteristic
382C4C2×C3×#63811yescharacteristic
383C3—24no—
384C3C3×#516314yescomplemented
385C3#1059286211yescharacteristic, direct product factor
386C2—81no—
387C2—721no—
388C2C3×SD16×C3⋊D311yescharacteristic
389C2C3×#6412311yescharacteristic
390C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#460
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×C33×SD16C2semidirect
2. C3×Q8×C3⋊Dic3C2semidirect
3. C2×C3×#3755C2semidirect
5. C3×#63659C2semidirect
6. C3×#63648C2semidirect
7. C3×#63641C2semidirect
8. #10592862C3direct (N × H)
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. Q8×C3⋊Dic3C2×C3semidirect
27. C2×#3755C2×C3semidirect
32. #63659C2×C3semidirect
33. #63648C2×C3semidirect
34. #63641C2×C3semidirect
41. C33×Q8D4semidirect
99. C3×C3⋊Dic3SD16semidirect
102. C2×C3×SD16C3×D3semidirect
103. C2×C3×SD16C3⋊D3semidirect
124. C32×Q8#67semidirect
125. C32×Q8C3×D4semidirect
222. C3⋊Dic3C3×SD16semidirect
227. C2×SD16C3×C3⋊D3semidirect
233. C33#460semidirect
243. C3×Q8C3×#67semidirect
244. C3×Q8#638semidirect
337. C32#5163semidirect
338. C32C3×#460semidirect
342. Q8C3×#638semidirect
384. C3C3×#5163semidirect
385. C3#10592862direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×SD162
C3×Q8×C3⋊Dic32
C2×C3×#37552
C3×#641492
C3×#636592
C3×#636482
C3×#636412
#105928623
C3×#51633