Finite Group Databasebeta

← C3×#10592884 #13163220

Subgroups — C3×#10592884 #13163220

Order 1728.

912 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×Q16C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic
3C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592884C311yesmaximal, characteristic, direct product factor
9C3×#5185—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C33×Q16—22no—
14C2×C33×C8C2211yescharacteristic
15C3×#3755—22no—
16C3×#3752—21no—
17C3×#3752—21no—
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×Q16—24no—
24C2×C32×Q16D314yescomplemented
25C2×C32×Q16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic
27Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×#3755C2×C311yescharacteristic
29C3×#947—34no—
30C3×Q8×Dic3—34no—
31C2×C3×#215—34no—
32#63662C2×C311yescharacteristic, complemented
33#63648C2×C311yescharacteristic
34#63641C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#821—34no—
37C3×Dic3⋊C8—34no—
38C22×C3×C3⋊D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×#464—91no—
54#5185—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×Q8C22×C311yescharacteristic
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×C3×D12—34no—
63C32×Q16—216no—
64C32×Q16—28no—
65C32×Q16—22no—
66C2×C32×C8—24no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C22×C311yescharacteristic
69#3755—22no—
70#3752—21no—
71#3752—21no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74C4×C3⋊Dic3—21no—
75C4×C3⋊Dic3C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C3×#215—68no—
78C3×(C2×D3)⋊C4—64no—
79C3×(C2×D3)⋊C4—64no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—64no—
82C3×C4×Dic3—34no—
83C3×C4×Dic3—64no—
84C2×C3×C3⋊C8—34no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C33×C4—22no—
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—21no—
95C3×C3⋊Dic3—41no—
96C2×C3×Q16—24no—
97C2×C3×Q16C3×D314yescomplemented
98C2×C3×Q16C3⋊D311yescharacteristic, complemented
99C2×C3×SD16—91no—
100C3×#124—91no—
101C3×C4×Q8—91no—
102#947—34no—
103Q8×Dic3—34no—
104C2×#215—34no—
105C3×C4⋊C8—91no—
106C3×#103—91no—
107C3×#102—91no—
108#835—34no—
109#821—34no—
110Dic3⋊C8—34no—
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C32×Q8—21no—
114C32×Q8—28no—
115C32×Q8—24no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—28no—
119C32×Q8—24no—
120C32×Q8—24no—
121C32×Q8—44no—
122C32×Q8—44no—
123C32×Q8#6718yes—
124C32×Q8C3×D412yes—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—44no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D411yescharacteristic
136C2×C3⋊Dic3C3×D411yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—64no—
142C3×D12—64no—
143C3×D12—64no—
144C32×C8—28no—
145C32×C8—24no—
146C32×C8—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—64no—
149#464—91noSylow
150C2×C33#12111yescharacteristic
151C2×C33#13511yescharacteristic
152C2×C33#13711yescharacteristic
153C3×C3⋊D3—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—24no—
156C2×C3×Q8C2×C3×D314yes—
157C2×C3×Q8C2×C3×D314yes—
158C2×C3×Q8C2×C3⋊D311yescharacteristic
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—91no—
161C2×D12—34no—
162C3×Q16—216no—
163C3×Q16—28no—
164C3×Q16—22no—
165C3×SD16—182no—
166C2×C3×C8—24no—
167C2×C3×C8—91no—
168C2×C3×C8C2×C3×D314yes—
169C2×C3×C8C2×C3⋊D311yescharacteristic
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—91no—
175C3×C42—181no—
176#215—68no—
177(C2×D3)⋊C4—64no—
178(C2×D3)⋊C4—64no—
179C4⋊Dic3—34no—
180C4⋊Dic3—64no—
181C4×Dic3—34no—
182C4×Dic3—64no—
183C2×C3⋊C8—34no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32C2×C3×D411yescharacteristic
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C32×C4—22no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—28no—
197C32×C4—48no—
198C32×C4—48no—
199C32×C4—44no—
200C32×C4—41no—
201C32×C4#23714yes—
202C32×C4C2×#6714yes—
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—41no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—124no—
211C2×Q16C3×C3⋊D311yescharacteristic, complemented
212C2×SD16—91no—
213#124—91no—
214C4×Q8—91no—
215C4⋊C8—91no—
216#103—91no—
217#102—91no—
218C33#46411yescharacteristic, complemented, Sylow
219C23×C3—181no—
220C22×D3—64no—
221C3×Q8—24no—
222C3×Q8—21no—
223C3×Q8—28no—
224C3×Q8—24no—
225C3×Q8—24no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—28no—
229C3×Q8—44no—
230C3×Q8—44no—
231C3×Q8C3×#6718yes—
232C3×Q8#63812yes—
233C3×D4—181no—
234C3×D4—181no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4—44no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4C22×C3×D314yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×Dic3—34no—
248C2×Dic3—34no—
249C2×Dic3—64no—
250D12—64no—
251D12—64no—
252C3×C8—28no—
253C3×C8—24no—
254C3×C8—21no—
255C3×C8—181no—
256C3⋊C8—64no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32#91814yes—
261C2×C32#91914yes—
262C2×C32#93914yes—
263C2×C32C3×#12111yescharacteristic
264C2×C32C3×#13511yescharacteristic
265C2×C32C3×#13711yescharacteristic
266C3⋊D3—81no—
267C3×D3—244no—
268C2×Q8C2×C3×C3⋊D311yescharacteristic
269C2×Q8C2×C3×C3⋊D311yescharacteristic
270C2×D4—91no—
271Q16—22no—
272SD16—182no—
273C2×C8—91no—
274C2×C8C2×C3×C3⋊D311yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—181no—
277C22⋊C4—181no—
278C22⋊C4—181no—
279C42—91no—
280C42—181no—
281C22×C3—24no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3C3×D3×D414yes—
286C22×C3D4×C3⋊D311yescharacteristic
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—28no—
291C3×C4—22no—
292C3×C4—24no—
293C3×C4—24no—
294C3×C4—48no—
295C3×C4—48no—
296C3×C4—44no—
297C3×C4—41no—
298C3×C4—181no—
299C3×C4—181no—
300C3×C4—361no—
301C3×C4C3×#23714yes—
302C3×C4C2×C3×#6714yes—
303C3×C4#383011yescharacteristic
304C3×C4C2×#63811yescharacteristic
305Dic3—64no—
306Dic3—64no—
307Dic3—124no—
308C32—24no—
309C32#518514yescomplemented
310C32C3×#46411yescharacteristic, complemented
311C23—181no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8C3×#63812yes—
316D4—181no—
317D4—181no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—181no—
323C2×C4C22×C3×C3⋊D311yescharacteristic
324C8—21no—
325C8—181no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—721no—
330C2×C3C3×#91814yes—
331C2×C3C3×#91914yes—
332C2×C3C3×#93914yes—
333C2×C3#6412711yescharacteristic
334C2×C3#6412811yescharacteristic
335C2×C3#6414811yescharacteristic
336D3—244no—
337C22—361no—
338C22—361no—
339C22—361no—
340C22C3×D4×C3⋊D311yescharacteristic
341C4—22no—
342C4—41no—
343C4—181no—
344C4—181no—
345C4—361no—
346C4C3×#383011yescharacteristic
347C4C2×C3×#63811yescharacteristic
348C3—24no—
349C3C3×#518514yescomplemented
350C3#1059288411yescharacteristic, direct product factor
351C2—721no—
352C2C3×#6412711yescharacteristic
353C2C3×#6412811yescharacteristic
354C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×Q16C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C3×Q16C3⋊D3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C33×Q16C2semidirect
3. C3×Q8×C3⋊Dic3C2semidirect
5. C3×#63662C2semidirect
7. C3×#63641C2semidirect
8. #10592884C3direct (N × H)
24. C2×C32×Q16D3semidirect
25. C2×C32×Q16C2×C3semidirect
27. Q8×C3⋊Dic3C2×C3semidirect
32. #63662C2×C3semidirect
34. #63641C2×C3semidirect
97. C2×C3×Q16C3×D3semidirect
98. C2×C3×Q16C3⋊D3semidirect
211. C2×Q16C3×C3⋊D3semidirect
218. C33#464semidirect
309. C32#5185semidirect
310. C32C3×#464semidirect
349. C3C3×#5185semidirect
350. C3#10592884direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×Q162
C3×#641562
C3×Q8×C3⋊Dic32
C2×C3×#37552
C3×#636622
C3×#636482
C3×#636412
#105928843
C3×#51853