Finite Group Databasebeta

← C3×#10592544 #13162888

Subgroups — C3×#10592544 #13162888

Order 1728.

824 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic, complemented
2C2×C3×#3744C211yesmaximal, characteristic
3C3×#64084C211yesmaximal, characteristic
4C3×C4×#634C211yesmaximal, characteristic, complemented
5C33×C4⋊C8C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592544C311yesmaximal, characteristic, direct product factor
9C3×#4845—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C33×C42C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C3×C4×C3⋊Dic3—21no—
21C3×#3746—22no—
22C3×#3744—22no—
23C2×#3746C2×C311yescharacteristic, complemented
24C2×#3744C2×C311yescharacteristic
25#64084C2×C311yescharacteristic
26C4×#634C2×C311yescharacteristic, complemented
27C2×C3×Dic12—34no—
28C2×C3×#204—34no—
29C3×#875—34no—
30C3×C4×Dic6—34no—
31C32×C4⋊C8—24no—
32C32×C4⋊C8D314yescomplemented
33C32×C4⋊C8C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic
35#63643C2×C311yescharacteristic, complemented
36C3×#821—34no—
37C3×#816—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×#634—21no—
48C3×#634—21no—
49C3×#634—21no—
50C3×#634D412yes—
51C33×C8—21no—
52C33×C8—21no—
53C3×#464—91no—
54#4845—34no—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×#634C22×C311yescharacteristic
57C2×#634C22×C311yescharacteristic
58C2×C3×D12—34no—
59C2×C3×Dic6—34no—
60C2×C3×Dic6—34no—
61C2×C32×C8—24no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C22×C311yescharacteristic
66C2×C32×C8C22×C311yescharacteristic
67C32×C42—24no—
68C32×C42C2×D314yes—
69C32×C42C22×C311yescharacteristic
70#3752—21no—
71#3752—21no—
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73#3750—21no—
74C4×C3⋊Dic3—21no—
75#3746—22no—
76#3744—22no—
77C3×(C2×D3)⋊C4—64no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3—34no—
80C3×Dic3⋊C4—64no—
81C3×C4×Dic3—64no—
82C3×Dic12—68no—
83C3×#204—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—41no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4—21no—
89C33×C4—21no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—41no—
94C3×C3⋊Dic3—41no—
95C2×C3×Q16—91no—
96C2×C3×SD16—91no—
97C3×#124—91no—
98C3×C4×Q8—91no—
99C2×Dic12—34no—
100C2×#204—34no—
101#875—34no—
102C4×Dic6—34no—
103C3×C4⋊C8—24no—
104C3×C4⋊C8C3×D314yescomplemented
105C3×C4⋊C8C3⋊D311yescharacteristic, complemented
106C3×#103—91no—
107C3×#102—91no—
108#821—34no—
109#816—34no—
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4D1214yes—
116C2×C32×C4D1214yes—
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C3×D411yescharacteristic
119C2×C32×C4C22×D314yes—
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C4⋊(C3⋊D3)—21no—
125C4⋊(C3⋊D3)—21no—
126#634—21no—
127#634—21no—
128#634—21no—
129#634C3×D412yes—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C3×D12—64no—
134C3×D12—64no—
135C3×Dic6—38no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6—64no—
139C32×C8—21no—
140C32×C8—21no—
141C32×C8—24no—
142C32×C8—24no—
143C32×C8—44no—
144C32×C8—44no—
145#464—91noSylow
146C2×C33#12111yescharacteristic
147C2×C33#13511yescharacteristic
148C2×C33#13711yescharacteristic
149C3×C3⋊D3—81no—
150C2×C3×Q8—91no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×D12—34no—
154C2×Dic6—34no—
155C2×Dic6—34no—
156C3×Q16—182no—
157C3×SD16—182no—
158C2×C3×C8—24no—
159C2×C3×C8—24no—
160C2×C3×C8C2×C3×D314yes—
161C2×C3×C8C2×C3×D314yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C2×C3×C8C2×C3⋊D311yescharacteristic
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—24no—
169C3×C42—181no—
170C3×C42C2×C3×D314yes—
171C3×C42C2×C3⋊D311yescharacteristic
172(C2×D3)⋊C4—64no—
173(C2×D3)⋊C4—64no—
174C4⋊Dic3—34no—
175Dic3⋊C4—64no—
176C4×Dic3—64no—
177Dic12—68no—
178#204—68no—
179C22×C32—24no—
180C22×C32C2×D1214yes—
181C22×C32C2×C3×D411yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—124no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C32×C4—28no—
189C32×C4—24no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—44no—
196C32×C4D3×D414yes—
197C32×C4#23914yes—
198C32×C4C2×C3×D411yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3⋊Dic3—22no—
201C3⋊Dic3—41no—
202C3⋊Dic3—41no—
203C3×Dic3—68no—
204C3×Dic3—124no—
205C3×Dic3—124no—
206C2×Q16—91no—
207C2×SD16—91no—
208#124—91no—
209C4×Q8—91no—
210C4⋊C8C3×C3⋊D311yescharacteristic, complemented
211#103—91no—
212#102—91no—
213C33#46411yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×Q8—181no—
219C3×Q8—181no—
220C3×D4—181no—
221C3×D4—181no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4C3×D1214yes—
229C2×C3×C4C3×D1214yes—
230C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
231C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
232C2×C3×C4C22×C3×D314yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—64no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237D12—64no—
238D12—64no—
239Dic6—38no—
240Dic6—64no—
241Dic6—64no—
242Dic6—64no—
243C3×C8—24no—
244C3×C8—24no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—44no—
248C3×C8—44no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#89514yes—
253C2×C32#90414yes—
254C2×C32#90914yes—
255C2×C32C3×#12111yescharacteristic
256C2×C32C3×#13511yescharacteristic
257C2×C32C3×#13711yescharacteristic
258C3⋊D3—81no—
259C3×D3—244no—
260C2×Q8—91no—
261C2×Q8—91no—
262C2×D4—91no—
263Q16—182no—
264SD16—182no—
265C2×C8C2×C3×C3⋊D311yescharacteristic
266C2×C8C2×C3×C3⋊D311yescharacteristic
267C4⋊C4—91no—
268C4⋊C4—181no—
269C22⋊C4—181no—
270C22⋊C4—181no—
271C42—181no—
272C42C2×C3×C3⋊D311yescharacteristic
273C22×C3—24no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—361no—
277C22×C3C2×C3×D1214yes—
278C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C3×C4—28no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—44no—
290C3×C4—182no—
291C3×C4—361no—
292C3×C4—361no—
293C3×C4C3×D3×D414yes—
294C3×C4C3×#23914yes—
295C3×C4D4×C3⋊D311yescharacteristic
296C3×C4#383211yescharacteristic
297Dic3—68no—
298Dic3—124no—
299Dic3—124no—
300C32—24no—
301C32#484514yescomplemented
302C32C3×#46411yescharacteristic, complemented
303C23—181no—
304Q8—92no—
305Q8—181no—
306Q8—181no—
307Q8—181no—
308D4—181no—
309D4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
314C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
315C2×C4C22×C3×C3⋊D311yescharacteristic
316C8—21no—
317C8—21no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—721no—
322C2×C3C3×#89514yes—
323C2×C3C3×#90414yes—
324C2×C3C3×#90914yes—
325C2×C3#6410411yescharacteristic
326C2×C3#6411311yescharacteristic
327C2×C3#6411811yescharacteristic
328D3—244no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
333C4—21no—
334C4—21no—
335C4—182no—
336C4—361no—
337C4—361no—
338C4C3×D4×C3⋊D311yescharacteristic
339C4C3×#383211yescharacteristic
340C3—24no—
341C3C3×#484514yescomplemented
342C3#1059254411yescharacteristic, direct product factor
343C2—721no—
344C2C3×#6410411yescharacteristic
345C2C3×#6411311yescharacteristic
346C2C3×#6411811yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊C8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊C8C3⋊D3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
4. C3×C4×#634C2semidirect
5. C33×C4⋊C8C2semidirect
7. C3×#63643C2semidirect
8. #10592544C3direct (N × H)
23. C2×#3746C2×C3semidirect
26. C4×#634C2×C3semidirect
32. C32×C4⋊C8D3semidirect
33. C32×C4⋊C8C2×C3semidirect
35. #63643C2×C3semidirect
104. C3×C4⋊C8C3×D3semidirect
105. C3×C4⋊C8C3⋊D3semidirect
210. C4⋊C8C3×C3⋊D3semidirect
213. C33#464semidirect
301. C32#4845semidirect
302. C32C3×#464semidirect
341. C3C3×#4845semidirect
342. C3#10592544direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C2×C3×#37442
C3×#640842
C3×C4×#6342
C33×C4⋊C82
C3×#636482
C3×#636432
#105925443
C3×#48453