Finite Group Databasebeta

← C3×#10592718 #13163057

Subgroups — C3×#10592718 #13163057

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C2×C3×#3754C212yesmaximal
3C3×#64078C211yesmaximal, characteristic, complemented
4C3×#63637C212yesmaximal, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10592718C311yesmaximal, characteristic, direct product factor
7C3×#5019—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C33×C4—21no—
10C2×C3×#634C2212yes—
11C33×C4⋊C4C2211yescharacteristic
12C33×C22⋊C4—21no—
13C33×C42C2211yescharacteristic
14C3×#3754—24no—
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C32×C4×D4—24no—
19C32×C4×D4D314yes—
20C32×C4×D4C2×C311yescharacteristic
21C2×#3754C2×C312yes—
22#64078C2×C311yescharacteristic, complemented
23C2×C3×#214—38no—
24C3×#869—34no—
25#63637C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#810—38no—
28C3×C4⋊(C3⋊C8)—34no—
29C23×C33—21no—
30C33×D4—21no—
31C33×D4D412yes—
32C2×C33×C4—21no—
33C2×C33×C4—42no—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C3×#634—22no—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)—22no—
41C3×#461—91no—
42#5019—34no—
43C2×C32×D4—24no—
44C2×C32×D4C2×D314yes—
45C2×C32×D4C22×C311yescharacteristic
46C22×C32×C4—28no—
47C22×C32×C4—24no—
48C22×C32×C4—21no—
49C2×#634C22×C312yes—
50C2×C3×Dic6—38no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4C2×D314yes—
53C32×C4⋊C4C22×C311yescharacteristic
54C32×C22⋊C4—28no—
55C32×C22⋊C4—24no—
56C32×C22⋊C4—21no—
57C32×C42—24no—
58C32×C42C2×D314yes—
59C32×C42C22×C311yescharacteristic
60#3754—24no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)—21no—
63C2×C3⋊(C3⋊C8)C22×C312yes—
64C3×#214—616no—
65C3×C4⋊Dic3—64no—
66C3×C4⋊Dic3—64no—
67C2×C3×C3⋊C8—38no—
68C22×C33—22no—
69C22×C33—41no—
70C22×C33—41no—
71C22×C33C2×D411yescharacteristic
72C33×C4—21no—
73C33×C4—41no—
74C33×C4SD1612yescomplemented
75C33×C4C2×D411yescharacteristic
76C33×C4#4111yescharacteristic
77C3×C3⋊Dic3—42no—
78C2×C3×SD16—92no—
79C3×C4⋊Q8—91no—
80C3×C4×D4—24no—
81C3×C4×D4C3×D314yes—
82C3×C4×D4C3⋊D311yescharacteristic
83C2×#214—38no—
84#869—34no—
85C3×C4⋊C8—91no—
86C3×#103—92no—
87#810—38no—
88C4⋊(C3⋊C8)—34no—
89C23×C32—28no—
90C23×C32—24no—
91C23×C32—21no—
92C32×D4—28no—
93C32×D4—24no—
94C32×D4—21no—
95C32×D4—28no—
96C32×D4D1218yes—
97C32×D4C3×D412yes—
98C2×C32×C4—28no—
99C2×C32×C4—24no—
100C2×C32×C4—21no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—416no—
105C2×C32×C4—48no—
106C2×C32×C4—42no—
107C2×C32×C4#6714yes—
108C2×C32×C4#6714yes—
109C2×C32×C4C3×D411yescharacteristic
110C2×C32×C4C3×D411yescharacteristic
111C2×C32×C4C22×D314yes—
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—22no—
114#634—22no—
115#634—22no—
116C2×C3×Dic3—68no—
117C3×Dic6—68no—
118C3×Dic6—68no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—68no—
121#461—91noSylow
122C2×C33—42no—
123C2×C33#12111yescharacteristic
124C2×C33C2×SD1611yescharacteristic
125C2×C33#13711yescharacteristic
126C2×C3×Q8—92no—
127C2×C3×D4—24no—
128C2×C3×D4C2×C3×D314yes—
129C2×C3×D4C2×C3⋊D311yescharacteristic
130C22×C3×C4—28no—
131C22×C3×C4—24no—
132C22×C3×C4—21no—
133C2×Dic6—38no—
134C3×SD16—184no—
135C2×C3×C8—92no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4C2×C3×D314yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C22⋊C4—28no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—21no—
144C3×C42—24no—
145C3×C42C2×C3×D314yes—
146C3×C42C2×C3⋊D311yescharacteristic
147#214—616no—
148C4⋊Dic3—64no—
149C4⋊Dic3—64no—
150C2×C3⋊C8—38no—
151C22×C32—216no—
152C22×C32—28no—
153C22×C32—22no—
154C22×C32—24no—
155C22×C32—48no—
156C22×C32—48no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32C2×#6714yes—
162C22×C32C2×C3×D411yescharacteristic
163C32×C4—21no—
164C32×C4—28no—
165C32×C4—24no—
166C32×C4—24no—
167C32×C4—24no—
168C32×C4—44no—
169C32×C4—48no—
170C32×C4—44no—
171C32×C4—41no—
172C32×C4#21418yescomplemented
173C32×C4C3×SD1612yescomplemented
174C32×C4C2×D1214yes—
175C32×C4#23514yes—
176C32×C4C2×C3×D411yescharacteristic
177C32×C4C3×#4111yescharacteristic
178C3⋊Dic3—42no—
179C3×Dic3—128no—
180C2×SD16—92no—
181C4⋊Q8—91no—
182C4×D4C3×C3⋊D311yescharacteristic
183C4⋊C8—91no—
184#103—92no—
185C33#46111yescharacteristic, complemented, Sylow
186C23×C3—28no—
187C23×C3—24no—
188C23×C3—21no—
189C3×Q8—182no—
190C3×Q8—182no—
191C3×D4—28no—
192C3×D4—24no—
193C3×D4—21no—
194C3×D4—28no—
195C3×D4C3×D1218yes—
196C3×D4C4⋊(C3⋊D3)12yes—
197C2×C3×C4—28no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—416no—
204C2×C3×C4—48no—
205C2×C3×C4—42no—
206C2×C3×C4—182no—
207C2×C3×C4C3×#6714yes—
208C2×C3×C4C3×#6714yes—
209C2×C3×C4#63811yescharacteristic
210C2×C3×C4#63811yescharacteristic
211C2×C3×C4C22×C3×D314yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—68no—
214Dic6—68no—
215Dic6—68no—
216C3×C8—182no—
217C3⋊C8—68no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—416no—
222C2×C32—48no—
223C2×C32—42no—
224C2×C32#93014yes—
225C2×C32C2×#21414yes—
226C2×C32#95114yes—
227C2×C32C3×#12111yescharacteristic
228C2×C32C2×C3×SD1611yescharacteristic
229C2×C32C3×#13711yescharacteristic
230C2×Q8—92no—
231C2×D4C2×C3×C3⋊D311yescharacteristic
232C22×C4—21no—
233SD16—184no—
234C2×C8—92no—
235C4⋊C4—181no—
236C4⋊C4—181no—
237C4⋊C4C2×C3×C3⋊D311yescharacteristic
238C22⋊C4—21no—
239C42C2×C3×C3⋊D311yescharacteristic
240C22×C3—216no—
241C22×C3—28no—
242C22×C3—22no—
243C22×C3—24no—
244C22×C3—48no—
245C22×C3—48no—
246C22×C3—44no—
247C22×C3—44no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3C2×C3×#6714yes—
251C22×C3C2×#63811yescharacteristic
252C3×C4—24no—
253C3×C4—21no—
254C3×C4—28no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—44no—
258C3×C4—48no—
259C3×C4—44no—
260C3×C4—41no—
261C3×C4—362no—
262C3×C4C3×#21418yescomplemented
263C3×C4#375412yescomplemented
264C3×C4C2×C3×D1214yes—
265C3×C4C3×#23514yes—
266C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
267C3×C4#382811yescharacteristic
268Dic3—128no—
269C32—24no—
270C32#501914yescomplemented
271C32C3×#46111yescharacteristic, complemented
272C23—21no—
273Q8—182no—
274Q8—182no—
275D4—21no—
276D4C3×C4⋊(C3⋊D3)12yes—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—182no—
280C2×C4C3×#63811yescharacteristic
281C2×C4C3×#63811yescharacteristic
282C2×C4C22×C3×C3⋊D311yescharacteristic
283C8—182no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—416no—
288C2×C3—48no—
289C2×C3—42no—
290C2×C3C3×#93014yes—
291C2×C3C2×C3×#21414yes—
292C2×C3C3×#95114yes—
293C2×C3#6413911yescharacteristic
294C2×C3C2×#375411yescharacteristic
295C2×C3#6416011yescharacteristic
296C22—22no—
297C22—41no—
298C22—41no—
299C22C2×C3×#63811yescharacteristic
300C4—21no—
301C4—41no—
302C4—362no—
303C4C3×#375412yescomplemented
304C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
305C4C3×#382811yescharacteristic
306C3—24no—
307C3C3×#501914yescomplemented
308C3#1059271811yescharacteristic, direct product factor
309C2—42no—
310C2C3×#6413911yescharacteristic
311C2C2×C3×#375411yescharacteristic
312C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×#64078C2semidirect
4. C3×#63637C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10592718C3direct (N × H)
22. #64078C2×C3semidirect
25. #63637C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
74. C33×C4SD16semidirect
172. C32×C4#214semidirect
173. C32×C4C3×SD16semidirect
185. C33#461semidirect
262. C3×C4C3×#214semidirect
263. C3×C4#3754semidirect
270. C32#5019semidirect
271. C32C3×#461semidirect
303. C4C3×#3754semidirect
307. C3C3×#5019semidirect
308. C3#10592718direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C2×C3×#37542
C3×#640782
C3×#636372
C3×C4⋊(C3⋊(C3⋊C8))2
#105927183
C3×#50193