Finite Group Databasebeta

← C3×#10592727 #13163066

Subgroups — C3×#10592727 #13163066

Order 1728.

952 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic
4C3×#64084C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592727C311yesmaximal, characteristic, direct product factor
9C3×#5028—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×C4⋊C4—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C42—21no—
16C33×C42C2211yescharacteristic
17C3×#3756—22no—
18C3×#3755—22no—
19C3×#3752—21no—
20C3×#3752—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×C4×Q8—24no—
24C32×C4×Q8D314yescomplemented
25C32×C4×Q8C2×C311yescharacteristic, complemented
26C2×#3756C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic
28#64084C2×C311yescharacteristic
29C2×C3×#216—34no—
30C2×C3×#215—34no—
31C3×#875—34no—
32#63637C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#810—34no—
36C3×#809—34no—
37C3×C4⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3—21no—
39C33×Q8—21no—
40C33×Q8D412yes—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4D411yescharacteristic
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×#634—21no—
50C3×#634—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×#464—91no—
54#5028—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C22×C311yescharacteristic
58C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
59C2×#634C22×C311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×Dic6—34no—
62C32×C4⋊C4—28no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C22×C311yescharacteristic
68C32×C42—28no—
69C32×C42—24no—
70C32×C42—24no—
71C32×C42—21no—
72C32×C42C2×D314yes—
73C32×C42C22×C311yescharacteristic
74#3756—22no—
75#3755—22no—
76#3752—21no—
77#3752—21no—
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#216—68no—
81C3×#215—68no—
82C3×(C2×D3)⋊C4—64no—
83C3×(C2×D3)⋊C4—64no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C33×C4—22no—
91C33×C4—21no—
92C33×C4—21no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—91no—
100C3×C4×Q8—24no—
101C3×C4×Q8C3×D314yescomplemented
102C3×C4×Q8C3⋊D311yescharacteristic, complemented
103C2×#216—34no—
104C2×#215—34no—
105#875—34no—
106C3×C4⋊C8—91no—
107C3×#103—91no—
108C3×#102—91no—
109#810—34no—
110#809—34no—
111C4⋊(C3⋊C8)—34no—
112C22×C3⋊D3—21no—
113C22×C3×D3—64no—
114C32×Q8—28no—
115C32×Q8—24no—
116C32×Q8—21no—
117C32×Q8—28no—
118C32×Q8D1218yes—
119C32×Q8C3×D412yes—
120C2×C32×C4—28no—
121C2×C32×C4—24no—
122C2×C32×C4—21no—
123C2×C32×C4—28no—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4#6714yes—
130C2×C32×C4#6714yes—
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C22×D314yes—
134C2×C32×C4C23×C311yescharacteristic
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—64no—
141C3×D12—64no—
142C3×D12—64no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—64no—
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—91no—
156C2×C3×Q8C2×C3×D314yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—91no—
159C2×D12—34no—
160C2×Dic6—34no—
161C3×Q16—182no—
162C3×SD16—182no—
163C2×C3×C8—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4C2×C3×D314yes—
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171C3×C22⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C42—28no—
174C3×C42—24no—
175C3×C42—21no—
176C3×C42—24no—
177C3×C42C2×C3×D314yes—
178C3×C42C2×C3⋊D311yescharacteristic
179#216—68no—
180#215—68no—
181(C2×D3)⋊C4—64no—
182(C2×D3)⋊C4—64no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8—34no—
185C22×C32—24no—
186C22×C32C2×#6714yes—
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C32×C4—28no—
195C32×C4—216no—
196C32×C4—28no—
197C32×C4—24no—
198C32×C4—22no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—44no—
205C32×C4—48no—
206C32×C4—44no—
207C32×C4—41no—
208C32×C4C2×D1214yes—
209C32×C4#23514yes—
210C32×C4C2×C3×D411yescharacteristic
211C32×C4C3×#4111yescharacteristic
212C3⋊Dic3—41no—
213C3×Dic3—124no—
214C2×Q16—91no—
215C2×SD16—91no—
216#124—91no—
217C4×Q8C3×C3⋊D311yescharacteristic, complemented
218C4⋊C8—91no—
219#103—91no—
220#102—91no—
221C33#46411yescharacteristic, complemented, Sylow
222C23×C3—181no—
223C22×D3—64no—
224C3×Q8—28no—
225C3×Q8—24no—
226C3×Q8—21no—
227C3×Q8—28no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8C3×D1218yes—
231C3×Q8C4⋊(C3⋊D3)12yes—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—181no—
244C2×C3×C4C3×#6714yes—
245C2×C3×C4C3×#6714yes—
246C2×C3×C4#63811yescharacteristic
247C2×C3×C4#63811yescharacteristic
248C2×C3×C4C22×C3×D314yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250C2×Dic3—64no—
251D12—64no—
252D12—64no—
253Dic6—64no—
254Dic6—64no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—64no—
258C3⋊C8—64no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32#93014yes—
263C2×C32#94214yes—
264C2×C32#95014yes—
265C2×C32C3×#12111yescharacteristic
266C2×C32C3×#13511yescharacteristic
267C2×C32C3×#13711yescharacteristic
268C3⋊D3—81no—
269C3×D3—244no—
270C2×Q8—91no—
271C2×Q8C2×C3×C3⋊D311yescharacteristic
272C2×D4—91no—
273Q16—182no—
274SD16—182no—
275C2×C8—91no—
276C2×C8—91no—
277C4⋊C4—21no—
278C4⋊C4C2×C3×C3⋊D311yescharacteristic
279C22⋊C4—181no—
280C22⋊C4—181no—
281C42—21no—
282C42C2×C3×C3⋊D311yescharacteristic
283C22×C3—24no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3C2×C3×#6714yes—
288C22×C3C2×#63811yescharacteristic
289C2×D3—124no—
290C2×D3—124no—
291C2×D3—124no—
292C3×C4—216no—
293C3×C4—28no—
294C3×C4—28no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—44no—
303C3×C4—48no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—361no—
307C3×C4C2×C3×D1214yes—
308C3×C4C3×#23514yes—
309C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
310C3×C4#382811yescharacteristic
311Dic3—124no—
312C32—24no—
313C32#502814yescomplemented
314C32C3×#46411yescharacteristic, complemented
315C23—181no—
316Q8—21no—
317Q8—181no—
318Q8—181no—
319Q8C3×C4⋊(C3⋊D3)12yes—
320D4—181no—
321D4—181no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—181no—
325C2×C4C3×#63811yescharacteristic
326C2×C4C3×#63811yescharacteristic
327C2×C4C22×C3×C3⋊D311yescharacteristic
328C8—181no—
329C8—181no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—721no—
334C2×C3C3×#93014yes—
335C2×C3C3×#94214yes—
336C2×C3C3×#95014yes—
337C2×C3#6413911yescharacteristic
338C2×C3#6415111yescharacteristic
339C2×C3#6415911yescharacteristic
340D3—244no—
341C22—361no—
342C22—361no—
343C22—361no—
344C22C2×C3×#63811yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—361no—
350C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
351C4C3×#382811yescharacteristic
352C3—24no—
353C3C3×#502814yescomplemented
354C3#1059272711yescharacteristic, direct product factor
355C2—721no—
356C2C3×#6413911yescharacteristic
357C2C3×#6415111yescharacteristic
358C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4×Q8C2semidirect
2. C2×C3×#3756C2semidirect
5. C3×#63637C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592727C3direct (N × H)
24. C32×C4×Q8D3semidirect
25. C32×C4×Q8C2×C3semidirect
26. C2×#3756C2×C3semidirect
32. #63637C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
101. C3×C4×Q8C3×D3semidirect
102. C3×C4×Q8C3⋊D3semidirect
217. C4×Q8C3×C3⋊D3semidirect
221. C33#464semidirect
313. C32#5028semidirect
314. C32C3×#464semidirect
353. C3C3×#5028semidirect
354. C3#10592727direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×Q82
C2×C3×#37562
C2×C3×#37552
C3×#640842
C3×#636372
C3×#636362
C3×C4⋊(C3⋊(C3⋊C8))2
#105927273
C3×#50283