Finite Group Databasebeta

← C3×#10596192 #13179473

Subgroups — C3×#10596192 #13179473

Order 1728.

1472 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C212yesmaximal
3C3×#64153C212yesmaximal
4C3×#64099C212yesmaximal
5C3×#64098C214yesmaximal
6C3×#64096C212yesmaximal
7C3×C4×#634C211yesmaximal, characteristic
8C3×C4×#634C211yesmaximal, characteristic
9#10596192C311yesmaximal, characteristic, direct product factor
10C3×#5721—34nomaximal
11C2×C33×Q8C2212yes—
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×C4⋊C4C2212yes—
15C33×C4⋊C4C2212yes—
16C33×C42C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2214yes—
20C3×#3750C2212yes—
21C3×#3750C2212yes—
22C3×#3750C2214yes—
23C3×#3750C2214yes—
24C3×C4×C3⋊Dic3C2212yes—
25C3×C4×C3⋊Dic3C2212yes—
26C3×C4×C3⋊Dic3C2214yes—
27C32×C4⋊Q8—24no—
28C32×C4⋊Q8D314yes—
29C32×C4⋊Q8C2×C311yescharacteristic
30Q8×C3⋊Dic3C2×C312yes—
31#64153C2×C312yes—
32#64099C2×C312yes—
33#64098C2×C314yes—
34#64096C2×C312yes—
35C4×#634C2×C311yescharacteristic
36C4×#634C2×C311yescharacteristic
37C3×Q8×Dic3—38no—
38C3×#944—38no—
39C3×#890—38no—
40C3×#889—316no—
41C3×#887—38no—
42C3×C4×Dic6—34no—
43C3×C4×Dic6—34no—
44C33×Q8—24no—
45C2×C33×C4C2314yes—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3C2312yes—
50C2×C3×C3⋊Dic3C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C3×#634—22no—
54C3×#634Q814yescomplemented
55C3×#557—91no—
56#5721—34no—
57C2×C32×Q8—28no—
58C2×C32×Q8C2×D318yes—
59C2×C32×Q8C22×C312yes—
60C2×#634C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×Dic6—34no—
63C2×C3×Dic6—34no—
64C32×C4⋊C4—28no—
65C32×C4⋊C4—28no—
66C32×C4⋊C4C2×D318yes—
67C32×C4⋊C4C2×D318yes—
68C32×C4⋊C4C22×C312yes—
69C32×C4⋊C4C22×C312yes—
70C32×C42—24no—
71C32×C42C2×D314yes—
72C32×C42C22×C311yescharacteristic
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75C4⋊(C3⋊Dic3)C22×C314yes—
76#3750C22×C312yes—
77#3750C22×C312yes—
78#3750C22×C314yes—
79#3750C22×C314yes—
80C4×C3⋊Dic3C22×C312yes—
81C4×C3⋊Dic3C22×C312yes—
82C4×C3⋊Dic3C22×C314yes—
83C3×C4⋊Dic3—316no—
84C3×C4⋊Dic3—34no—
85C3×C4⋊Dic3—34no—
86C3×Dic3⋊C4—316no—
87C3×Dic3⋊C4—38no—
88C3×Dic3⋊C4—38no—
89C3×Dic3⋊C4—316no—
90C3×C4×Dic3—316no—
91C3×C4×Dic3—38no—
92C3×C4×Dic3—38no—
93C22×C33C2411yescharacteristic
94C33×C4—24no—
95C33×C4—21no—
96C33×C4C2×Q812yes—
97C33×C4#4112yes—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3C2×Q814yes—
102C3×C4⋊Q8—24no—
103C3×C4⋊Q8—92no—
104C3×C4⋊Q8C3×D314yes—
105C3×C4⋊Q8C3⋊D311yescharacteristic
106C3×#125—94no—
107C3×#125—92no—
108C3×C4×Q8—92no—
109C3×C4×Q8—92no—
110C3×C4×Q8—91no—
111C3×C4×Q8—91no—
112Q8×Dic3—38no—
113#944—38no—
114#890—38no—
115#889—316no—
116#887—38no—
117C4×Dic6—34no—
118C4×Dic6—34no—
119C32×Q8—24no—
120C32×Q8—216no—
121C32×Q8—416no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—216no—
126C2×C32×C4C22×D3116yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C23×C314yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C32×C4C23×C311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3C23×C312yes—
135C2×C3⋊Dic3C23×C312yes—
136C2×C3⋊Dic3C23×C312yes—
137C2×C3⋊Dic3C23×C312yes—
138#634—22no—
139#634C3×Q814yescomplemented
140C2×C3×Dic3—38no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3—38no—
143C2×C3×Dic3—38no—
144C3×Dic6—316no—
145C3×Dic6—68no—
146#557—91noSylow
147C2×C33C22×Q811yescharacteristic
148C2×C33C2×#4111yescharacteristic
149C2×C332−1+411yescharacteristic
150C2×C3×Q8—28no—
151C2×C3×Q8—91no—
152C2×C3×Q8—91no—
153C2×C3×Q8C2×C3×D318yes—
154C2×C3×Q8C2×C3⋊D312yes—
155C2×Dic6—34no—
156C2×Dic6—34no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—92no—
160C3×C4⋊C4—92no—
161C3×C4⋊C4—94no—
162C3×C4⋊C4—94no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—94no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4C2×C3×D318yes—
167C3×C4⋊C4C2×C3×D318yes—
168C3×C4⋊C4C2×C3⋊D312yes—
169C3×C4⋊C4C2×C3⋊D312yes—
170C3×C42—24no—
171C3×C42—94no—
172C3×C42—92no—
173C3×C42—92no—
174C3×C42C2×C3×D314yes—
175C3×C42C2×C3⋊D311yescharacteristic
176C4⋊Dic3—34no—
177C4⋊Dic3—34no—
178C4⋊Dic3—316no—
179Dic3⋊C4—38no—
180Dic3⋊C4—38no—
181Dic3⋊C4—316no—
182Dic3⋊C4—316no—
183C4×Dic3—38no—
184C4×Dic3—38no—
185C4×Dic3—316no—
186C22×C32—24no—
187C22×C32C23×D314yes—
188C22×C32C24×C311yescharacteristic
189C32×C4—28no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4—21no—
193C32×C4—28no—
194C32×C4—28no—
195C32×C4—216no—
196C32×C4—416no—
197C32×C4#23718yes—
198C32×C4D3×Q818yes—
199C32×C4C2×C3×Q812yes—
200C32×C4C3×#4112yes—
201C3⋊Dic3—22no—
202C3⋊Dic3—22no—
203C3⋊Dic3—22no—
204C3⋊Dic3C2×C3×Q814yes—
205C3×Dic3—316no—
206C3×Dic3—68no—
207C3×Dic3—68no—
208C3×Dic3—68no—
209C4⋊Q8—92no—
210C4⋊Q8C3×C3⋊D311yescharacteristic
211#125—94no—
212#125—92no—
213C4×Q8—92no—
214C4×Q8—92no—
215C4×Q8—91no—
216C4×Q8—91no—
217C33#55711yescharacteristic, complemented, Sylow
218C3×Q8—216no—
219C3×Q8—24no—
220C3×Q8—416no—
221C3×Q8—94no—
222C3×Q8—182no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—216no—
227C2×C3×C4—92no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—92no—
231C2×C3×C4C22×C3×D3116yes—
232C2×C3×C4C22×C3×D314yes—
233C2×C3×C4C22×C3×D314yes—
234C2×C3×C4C22×C3×D314yes—
235C2×C3×C4C22×C3⋊D314yes—
236C2×C3×C4C22×C3⋊D311yescharacteristic
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—38no—
240C2×Dic3—38no—
241C2×Dic3—38no—
242C2×Dic3—38no—
243Dic6—316no—
244Dic6—68no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32C2×#23714yes—
249C2×C32C2×D3×Q814yes—
250C2×C32#100714yes—
251C2×C32C22×C3×Q811yescharacteristic
252C2×C32C2×C3×#4111yescharacteristic
253C2×C32C3×2−1+411yescharacteristic
254C2×Q8—91no—
255C2×Q8—91no—
256C2×Q8C2×C3×C3⋊D312yes—
257C4⋊C4—91no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—94no—
261C4⋊C4—91no—
262C4⋊C4—94no—
263C4⋊C4—94no—
264C4⋊C4C2×C3×C3⋊D312yes—
265C4⋊C4C2×C3×C3⋊D312yes—
266C42—94no—
267C42—92no—
268C42—92no—
269C42C2×C3×C3⋊D311yescharacteristic
270C22×C3—24no—
271C22×C3C23×C3×D314yes—
272C22×C3C23×C3⋊D311yescharacteristic
273C3×C4—28no—
274C3×C4—216no—
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—21no—
278C3×C4—28no—
279C3×C4—28no—
280C3×C4—416no—
281C3×C4—94no—
282C3×C4—182no—
283C3×C4—182no—
284C3×C4—182no—
285C3×C4C3×#23718yes—
286C3×C4C3×D3×Q818yes—
287C3×C4#383012yes—
288C3×C4Q8×C3⋊D312yes—
289Dic3—316no—
290Dic3—68no—
291Dic3—68no—
292Dic3—68no—
293C32—24no—
294C32#572114yescomplemented
295C32C3×#55711yescharacteristic, complemented
296Q8—24no—
297Q8—94no—
298Q8—182no—
299C2×C4—92no—
300C2×C4—92no—
301C2×C4—92no—
302C2×C4—92no—
303C2×C4C22×C3×C3⋊D314yes—
304C2×C4C22×C3×C3⋊D311yescharacteristic
305C2×C4C22×C3×C3⋊D311yescharacteristic
306C2×C4C22×C3×C3⋊D311yescharacteristic
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3C2×C3×#23714yes—
311C2×C3C2×C3×D3×Q814yes—
312C2×C3C3×#100714yes—
313C2×C3C2×#383011yescharacteristic
314C2×C3C2×Q8×C3⋊D311yescharacteristic
315C2×C3#6436411yescharacteristic
316C22C23×C3×C3⋊D311yescharacteristic
317C4—24no—
318C4—21no—
319C4—94no—
320C4—182no—
321C4—182no—
322C4—182no—
323C4C3×#383012yes—
324C4C3×Q8×C3⋊D312yes—
325C3—24no—
326C3C3×#572114yescomplemented
327C3#1059619211yescharacteristic, direct product factor
328C2C2×C3×#383011yescharacteristic
329C2C2×C3×Q8×C3⋊D311yescharacteristic
330C2C3×#6436411yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socle, prime radicalC22×C33C24
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#557
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
9. #10596192C3direct (N × H)
54. C3×#634Q8semidirect
139. #634C3×Q8semidirect
217. C33#557semidirect
294. C32#5721semidirect
295. C32C3×#557semidirect
326. C3C3×#5721semidirect
327. C3#10596192direct (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
C3×Q8×C3⋊Dic32
C3×#641532
C3×#640992
C3×#640982
C3×#640962
C3×C4×#6342
#105961923
C3×#57213