Finite Group Databasebeta

← C3×#10592753 #13163098

Subgroups — C3×#10592753 #13163098

Order 1728.

976 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#124C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic
4C3×C4×#634C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63659C211yesmaximal, characteristic
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592753C311yesmaximal, characteristic, direct product factor
9C3×#5054—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×C22⋊C4—21no—
14C33×C22⋊C4—21no—
15C33×C42C2211yescharacteristic
16C3×#3756—22no—
17C3×#3754—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C3×C4×C3⋊Dic3—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#124—24no—
24C32×#124D314yes—
25C32×#124C2×C311yescharacteristic
26C2×#3756C2×C311yescharacteristic, complemented
27C2×#3754C2×C311yescharacteristic
28C4×#634C2×C311yescharacteristic, complemented
29C2×C3×#216—34no—
30C2×C3×#214—34no—
31C3×C4×Dic6—34no—
32#63662C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#832—34no—
37C3×C4⋊(C3⋊C8)—34no—
38C23×C33—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4—21no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634D412yes—
51C3×C3⋊(C3⋊C8)—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×#464—91no—
54#5054—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C22×C311yescharacteristic
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×Dic6—34no—
63C32×C22⋊C4—28no—
64C32×C22⋊C4—28no—
65C32×C22⋊C4—24no—
66C32×C22⋊C4—24no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4—21no—
69C32×C42—24no—
70C32×C42C2×D314yes—
71C32×C42C22×C311yescharacteristic
72#3756—22no—
73#3754—22no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75#3750—21no—
76C4×C3⋊Dic3—21no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#216—68no—
80C3×#214—68no—
81C3×C4⋊Dic3—34no—
82C3×Dic3⋊C4—64no—
83C3×C4×Dic3—64no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C33×C4—21no—
91C33×C4—21no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—22no—
96C3×C3⋊Dic3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—24no—
100C3×#124C3×D314yes—
101C3×#124C3⋊D311yescharacteristic
102C3×C4×Q8—91no—
103C2×#216—34no—
104C2×#214—34no—
105C4×Dic6—34no—
106C3×C4⋊C8—91no—
107C3×#103—91no—
108C3×#102—91no—
109#835—34no—
110#832—34no—
111C4⋊(C3⋊C8)—34no—
112C23×C32—28no—
113C23×C32—24no—
114C23×C32—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—24no—
119C32×Q8—44no—
120C32×Q8—44no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4—44no—
127C2×C32×C4—28no—
128C2×C32×C4—24no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4#6714yes—
134C2×C32×C4#6714yes—
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D314yes—
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141#634—21no—
142#634C3×D412yes—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C3×Dic6—38no—
146C3×Dic6—64no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—64no—
150C3×C3⋊C8—64no—
151#464—91noSylow
152C2×C33—81no—
153C2×C33#12111yescharacteristic
154C2×C33#13511yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—24no—
161C2×C3×D4C2×C3×D314yes—
162C2×C3×D4C2×C3⋊D311yescharacteristic
163C2×Dic6—34no—
164C3×Q16—182no—
165C3×SD16—182no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C22⋊C4—28no—
171C3×C22⋊C4—28no—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C42—24no—
177C3×C42—181no—
178C3×C42C2×C3×D314yes—
179C3×C42C2×C3⋊D311yescharacteristic
180#216—68no—
181#214—68no—
182C4⋊Dic3—34no—
183Dic3⋊C4—64no—
184C4×Dic3—64no—
185C2×C3⋊C8—34no—
186C2×C3⋊C8—34no—
187C22×C32—24no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32—48no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32C2×#6714yes—
198C22×C32C2×C3×D411yescharacteristic
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—44no—
206C32×C4—44no—
207C32×C4—48no—
208C32×C4—44no—
209C32×C4—41no—
210C32×C4D3×D414yes—
211C32×C4#23714yes—
212C32×C4C2×C3×D411yescharacteristic
213C32×C4C3×#4111yescharacteristic
214C3⋊Dic3—22no—
215C3⋊Dic3—41no—
216C3×Dic3—68no—
217C3×Dic3—124no—
218C2×Q16—91no—
219C2×SD16—91no—
220#124C3×C3⋊D311yescharacteristic
221C4×Q8—91no—
222C4⋊C8—91no—
223#103—91no—
224#102—91no—
225C33#46411yescharacteristic, complemented, Sylow
226C23×C3—28no—
227C23×C3—24no—
228C23×C3—21no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—44no—
242C3×D4—44no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C3×#6714yes—
252C2×C3×C4C3×#6714yes—
253C2×C3×C4#63811yescharacteristic
254C2×C3×C4#63811yescharacteristic
255C2×C3×C4C22×C3×D314yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×Dic3—64no—
258C2×Dic3—64no—
259Dic6—38no—
260Dic6—64no—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—64no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—88no—
269C2×C32—84no—
270C2×C32—81no—
271C2×C32#93814yes—
272C2×C32#95014yes—
273C2×C32#95114yes—
274C2×C32C3×#12111yescharacteristic
275C2×C32C3×#13511yescharacteristic
276C2×C32C3×#13711yescharacteristic
277C2×Q8—91no—
278C2×Q8C2×C3×C3⋊D311yescharacteristic
279C2×D4C2×C3×C3⋊D311yescharacteristic
280Q16—182no—
281SD16—182no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—181no—
289C42C2×C3×C3⋊D311yescharacteristic
290C22×C3—24no—
291C22×C3—48no—
292C22×C3—48no—
293C22×C3—48no—
294C22×C3—44no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C2×C3×#6714yes—
301C22×C3C2×#63811yescharacteristic
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—48no—
311C3×C4—44no—
312C3×C4—41no—
313C3×C4—182no—
314C3×C4—361no—
315C3×C4C3×D3×D414yes—
316C3×C4C3×#23714yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—68no—
320Dic3—124no—
321C32—24no—
322C32#505414yescomplemented
323C32C3×#46411yescharacteristic, complemented
324C23—21no—
325Q8—21no—
326Q8—21no—
327Q8—92no—
328Q8—181no—
329D4—21no—
330D4—21no—
331C2×C4—21no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4C3×#63811yescharacteristic
335C2×C4C3×#63811yescharacteristic
336C2×C4C22×C3×C3⋊D311yescharacteristic
337C8—181no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—88no—
343C2×C3—84no—
344C2×C3—81no—
345C2×C3C3×#93814yes—
346C2×C3C3×#95014yes—
347C2×C3C3×#95114yes—
348C2×C3#6414711yescharacteristic
349C2×C3#6415911yescharacteristic
350C2×C3#6416011yescharacteristic
351C22—41no—
352C22—41no—
353C22—41no—
354C22C2×C3×#63811yescharacteristic
355C4—21no—
356C4—21no—
357C4—41no—
358C4—182no—
359C4—361no—
360C4C3×D4×C3⋊D311yescharacteristic
361C4C3×#383011yescharacteristic
362C3—24no—
363C3C3×#505414yescomplemented
364C3#1059275311yescharacteristic, direct product factor
365C2—81no—
366C2C3×#6414711yescharacteristic
367C2C3×#6415911yescharacteristic
368C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#124C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC3×#124C3⋊D3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
4. C3×C4×#634C2semidirect
5. C3×#63662C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592753C3direct (N × H)
26. C2×#3756C2×C3semidirect
28. C4×#634C2×C3semidirect
32. #63662C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
225. C33#464semidirect
322. C32#5054semidirect
323. C32C3×#464semidirect
363. C3C3×#5054semidirect
364. C3#10592753direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1242
C2×C3×#37562
C2×C3×#37542
C3×C4×#6342
C3×#636622
C3×#636592
C3×C4⋊(C3⋊(C3⋊C8))2
#105927533
C3×#50543