Finite Group Databasebeta

← C2×#63451⋊C3 #13152450

Subgroups — C2×#63451⋊C3 #13152450

Order 1728.

724 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×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
3C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
5C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
7C2×C4×3−1+2⋊C4C211yesmaximal, characteristic
8#63451⋊C3C218yesmaximal, direct product factor
9C2×C3×#889—31nomaximal
10C2×#63451C311yesmaximal, characteristic, complemented
11C22×C4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C22×3−1+2⋊C4C2211yescharacteristic
18C4⋊C4×3−1+2C2214yes—
19C4⋊(3−1+2⋊C4)C2214yes—
20(3−1+2⋊C4)⋊C4C2214yes—
21(3−1+2⋊C4)⋊C4C2214yes—
22(3−1+2⋊C4)⋊C4C2214yes—
23(3−1+2⋊C4)⋊C4C2214yes—
24C4×3−1+2⋊C4C2214yes—
25C2×C32×C4⋊C4D311yescharacteristic
26C2×C3×C4⋊Dic3—31no—
27C2×C3×Dic3⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×C4×Dic3—31no—
32C3×#889—38no—
33C2×C9×C4⋊C4—21no—
34C2×C9×C4⋊C4C2×C311yescharacteristic
35C2×C4⋊Dic9C2×C311yescharacteristic
36C2×Dic9⋊C4C2×C311yescharacteristic
37C2×Dic9⋊C4C2×C311yescharacteristic
38C2×Dic9⋊C4C2×C311yescharacteristic
39C2×Dic9⋊C4C2×C311yescharacteristic
40C2×C4×Dic9C2×C311yescharacteristic
41#63451C2×C318yescomplemented
42C23×3−1+2C2311yescharacteristic
43C2×C4×3−1+2—22no—
44C2×C4×3−1+2—22no—
45C2×C4×3−1+2—22no—
46C2×C4×3−1+2C2312yes—
47C2×C4×3−1+2C2312yes—
48C2×C4×3−1+2C2312yes—
49C2×3−1+2⋊C4—22no—
50C2×3−1+2⋊C4—22no—
51C2×3−1+2⋊C4—22no—
52C2×3−1+2⋊C4Q814yes—
53C2×3−1+2⋊C4C2312yes—
54C2×3−1+2⋊C4C2312yes—
55C2×3−1+2⋊C4C2312yes—
56C2×3−1+2⋊C4C2312yes—
57C2×C3×#125—91no—
58C2×#889—31no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C3×Dic3—31no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C32×C4⋊C4C2×D314yes—
67C3×C4⋊Dic3—34no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—34no—
70C3×Dic3⋊C4—34no—
71C3×Dic3⋊C4—34no—
72C3×C4×Dic3—34no—
73C22×C4×C9—21no—
74C22×C4×C9—21no—
75C22×C4×C9—21no—
76C22×C4×C9C22×C311yescharacteristic
77C22×C4×C9C22×C311yescharacteristic
78C22×C4×C9C22×C311yescharacteristic
79C22×Dic9C22×C311yescharacteristic
80C22×Dic9C22×C311yescharacteristic
81C22×Dic9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C9×C4⋊C4—24no—
84C9×C4⋊C4C22×C314yes—
85C4⋊Dic9C22×C314yes—
86Dic9⋊C4C22×C314yes—
87Dic9⋊C4C22×C314yes—
88Dic9⋊C4C22×C314yes—
89Dic9⋊C4C22×C314yes—
90C4×Dic9C22×C314yes—
91C22×3−1+2C2×Q812yes—
92C22×3−1+2#4112yes—
93C22×3−1+2#4112yes—
94C22×3−1+2C2411yescharacteristic
95C4×3−1+2—22no—
96C4×3−1+2—22no—
97C4×3−1+2—22no—
983−1+2⋊C4—22no—
993−1+2⋊C4—22no—
1003−1+2⋊C4—22no—
1013−1+2⋊C4C2×Q814yes—
102C3×#125—98no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—91no—
105C2×C3×C4⋊C4—91no—
106C2×C3×C4⋊C4—91no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C4⋊C4C3×D311yescharacteristic
110C2×C3×C42—91no—
111C2×C4⋊Dic3—31no—
112C2×Dic3⋊C4—31no—
113C2×Dic3⋊C4—31no—
114C2×Dic3⋊C4—31no—
115C2×Dic3⋊C4—31no—
116C2×C4×Dic3—31no—
117#889—38no—
118C23×C32C22×D311yescharacteristic
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D312yes—
124C2×C32×C4C22×D312yes—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C23×C9—21no—
134C23×C9C23×C311yescharacteristic
135C2×C4×C9—24no—
136C2×C4×C9—22no—
137C2×C4×C9—22no—
138C2×C4×C9—22no—
139C2×C4×C9—22no—
140C2×C4×C9—22no—
141C2×C4×C9—22no—
142C2×C4×C9—42no—
143C2×C4×C9—42no—
144C2×C4×C9C23×C312yes—
145C2×C4×C9C23×C312yes—
146C2×C4×C9C23×C312yes—
147C2×Dic9—22no—
148C2×Dic9—22no—
149C2×Dic9—22no—
150C2×Dic9C3×Q814yes—
151C2×Dic9C23×C312yes—
152C2×Dic9C23×C312yes—
153C2×Dic9C23×C312yes—
154C2×Dic9C23×C312yes—
155C2×#125—91noSylow
156C2×3−1+2#12514yescomplemented
157C2×3−1+2C22×Q811yescharacteristic
158C2×3−1+2C2×#4111yescharacteristic
159C2×3−1+2C2×#4111yescharacteristic
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—91no—
166C22×C3×C4—91no—
167C22×C3×C4C2×C3×D311yescharacteristic
168C22×C3×C4C2×C3×D311yescharacteristic
169C22×C3×C4C2×C3×D311yescharacteristic
170C22×Dic3—31no—
171C22×Dic3—31no—
172C22×Dic3—31no—
173C22×Dic3—31no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—94no—
176C3×C4⋊C4—94no—
177C3×C4⋊C4—94no—
178C3×C4⋊C4—94no—
179C3×C4⋊C4—94no—
180C3×C4⋊C4C2×C3×D314yes—
181C3×C42—94no—
182C4⋊Dic3—34no—
183Dic3⋊C4—34no—
184Dic3⋊C4—34no—
185Dic3⋊C4—34no—
186Dic3⋊C4—34no—
187C4×Dic3—34no—
188C22×C32#23512yes—
189C22×C32#23712yes—
190C22×C32D3×Q812yes—
191C22×C32C23×D311yescharacteristic
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—22no—
195C3×Dic3—34no—
196C3×Dic3—62no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9—22no—
202C22×C9—22no—
203C22×C9C2×C3×Q812yes—
204C22×C9C3×#4112yes—
205C22×C9C3×#4112yes—
206C22×C9C24×C311yescharacteristic
207C4×C9—24no—
208C4×C9—22no—
209C4×C9—22no—
210C4×C9—22no—
211C4×C9—42no—
212C4×C9—42no—
213Dic9—22no—
214Dic9—22no—
215Dic9—22no—
216Dic9C2×C3×Q814yes—
217#125—98no—
218C2×C4⋊C4—91no—
219C2×C4⋊C4—91no—
220C2×C4⋊C4—91no—
221C2×C4⋊C4—91no—
222C2×C4⋊C4—91no—
223C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
224C2×C42—91no—
2253−1+2C2×#12511yescharacteristic, complemented, Sylow
226C23×C3—31no—
227C23×C3C22×C3×D311yescharacteristic
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—92no—
238C2×C3×C4—94no—
239C2×C3×C4—92no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—182no—
245C2×C3×C4C22×C3×D312yes—
246C2×C3×C4C22×C3×D312yes—
247C2×C3×C4C22×C3×D312yes—
248C2×Dic3—32no—
249C2×Dic3—34no—
250C2×Dic3—32no—
251C2×Dic3—32no—
252C2×Dic3—32no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×C32#88914yes—
257C2×C32C2×#23511yescharacteristic
258C2×C32C2×#23711yescharacteristic
259C2×C32C2×D3×Q811yescharacteristic
260C2×C9—24no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—21no—
264C2×C9C3×#12514yescomplemented
265C2×C9C22×C3×Q811yescharacteristic
266C2×C9C2×C3×#4111yescharacteristic
267C2×C9C2×C3×#4111yescharacteristic
268C22×C4—91no—
269C22×C4—91no—
270C22×C4—91no—
271C22×C4—91no—
272C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
273C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
274C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
275C4⋊C4—94no—
276C4⋊C4—94no—
277C4⋊C4—94no—
278C4⋊C4—94no—
279C4⋊C4—94no—
280C4⋊C4C2×AGL(1,ℤ/9)14yes—
281C42—94no—
282C22×C3—31no—
283C22×C3—32no—
284C22×C3—32no—
285C22×C3—32no—
286C22×C3C3×#23512yes—
287C22×C3C3×#23712yes—
288C22×C3C3×D3×Q812yes—
289C22×C3C23×C3×D311yescharacteristic
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—62no—
294C3×C4—62no—
295C3×C4—62no—
296C3×C4—94no—
297C3×C4—182no—
298C3×C4—182no—
299C3×C4—182no—
300Dic3—34no—
301Dic3—62no—
302Dic3—62no—
303Dic3—62no—
304C32C2×#88911yescharacteristic
305C9—21no—
306C9C2×C3×#12511yescharacteristic, complemented
307C23C22×AGL(1,ℤ/9)11yescharacteristic
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—92no—
312C2×C4—94no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—92no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4C22×AGL(1,ℤ/9)12yes—
320C2×C4C22×AGL(1,ℤ/9)12yes—
321C2×C4C22×AGL(1,ℤ/9)12yes—
322C2×C3—34no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3C3×#88914yes—
327C2×C3C2×C3×#23511yescharacteristic
328C2×C3C2×C3×#23711yescharacteristic
329C2×C3C2×C3×D3×Q811yescharacteristic
330C22#3697⋊C312yes—
331C22#3699⋊C312yes—
332C22Q8×AGL(1,ℤ/9)12yes—
333C22C23×AGL(1,ℤ/9)11yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—94no—
338C4—182no—
339C4—182no—
340C4—182no—
341C3—31no—
342C3C2×C3×#88911yescharacteristic
343C2#63451⋊C314yesdirect product factor
344C2C2×#3697⋊C311yescharacteristic
345C2C2×#3699⋊C311yescharacteristic
346C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#125
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #63451⋊C3C2direct (N × H)
10. C2×#63451C3semidirect
41. #63451C2×C3semidirect
156. C2×3−1+2#125semidirect
225. 3−1+2C2×#125semidirect
264. C2×C9C3×#125semidirect
306. C9C2×C3×#125semidirect
343. C2#63451⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
#63451⋊C32
C2×C3×#8893
C2×#634513