Finite Group Databasebeta

← C2×#63458⋊C3 #13152448

Subgroups — C2×#63458⋊C3 #13152448

Order 1728.

1012 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C22×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
5C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
7C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#63458⋊C3C218yesmaximal, direct product factor
9C2×C3×#896—31nomaximal
10C2×#63458C311yesmaximal, characteristic, complemented
11C23×AGL(1,ℤ/9)C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C22×3−1+2⋊C4C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)—24no—
19C2×C4×AGL(1,ℤ/9)C2214yes—
20C2×3−1+2⋊Q8—24no—
21C2×3−1+2⋊Q8C2214yescomplemented
22C4⋊C4×3−1+2C2214yescomplemented
23(C2×AGL(1,ℤ/9))⋊C4C2214yes—
24(C2×AGL(1,ℤ/9))⋊C4C2214yes—
25(3−1+2⋊C4)⋊C4C2214yescomplemented
26(3−1+2⋊C4)⋊C4C2214yescomplemented
27C22×C3×C4×D3—31no—
28C22×C3×Dic6—31no—
29C2×C32×C4⋊C4D311yescharacteristic
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×(C2×D3)⋊C4—31no—
32C2×C3×Dic3⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#896—38no—
35C22×C4×D9C2×C311yescharacteristic
36C22×Dic18C2×C311yescharacteristic, complemented
37C2×C9×C4⋊C4—21no—
38C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
39C2×(C2×D9)⋊C4C2×C311yescharacteristic
40C2×(C2×D9)⋊C4C2×C311yescharacteristic
41C2×Dic9⋊C4C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63458C2×C318yescomplemented
44C23×3−1+2C2311yescharacteristic
45C22×AGL(1,ℤ/9)—22no—
46C22×AGL(1,ℤ/9)—22no—
47C22×AGL(1,ℤ/9)Q814yescomplemented
48C22×AGL(1,ℤ/9)C2312yes—
49C2×C4×3−1+2—22no—
50C2×C4×3−1+2—22no—
51C2×C4×3−1+2—22no—
52C2×C4×3−1+2C2312yes—
53C2×C4×3−1+2C2312yes—
54C2×C4×3−1+2C2312yes—
55C2×3−1+2⋊C4—22no—
56C2×3−1+2⋊C4—22no—
57C2×3−1+2⋊C4D414yes—
58C2×3−1+2⋊C4C2312yes—
59C2×3−1+2⋊C4C2312yes—
60C2×3−1+2⋊C4C2312yes—
61C4×AGL(1,ℤ/9)—28no—
623−1+2⋊Q8—28no—
63C2×C3×C22⋊Q8—91no—
64C2×#896—31no—
65C23×C3×D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—64no—
74C2×C3×Dic6—34no—
75C2×C3×Dic6—64no—
76C23×D9C22×C311yescharacteristic
77C32×C4⋊C4C2×D314yes—
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×Dic3⋊C4—34no—
81C3×Dic3⋊C4—34no—
82C22×C4×C9—21no—
83C22×C4×C9—21no—
84C22×C4×C9—21no—
85C22×C4×C9C22×C311yescharacteristic
86C22×C4×C9C22×C311yescharacteristic
87C22×C4×C9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C22×Dic9C22×C311yescharacteristic
90C22×Dic9C22×C311yescharacteristic
91C2×C4×D9—24no—
92C2×C4×D9C22×C314yes—
93C2×Dic18—24no—
94C2×Dic18C22×C314yescomplemented
95C9×C4⋊C4—24no—
96C9×C4⋊C4C22×C314yescomplemented
97(C2×D9)⋊C4C22×C314yes—
98(C2×D9)⋊C4C22×C314yes—
99Dic9⋊C4C22×C314yescomplemented
100Dic9⋊C4C22×C314yescomplemented
101C22×3−1+2C2×D412yes—
102C22×3−1+2C2×Q812yes—
103C22×3−1+2#4112yes—
104C22×3−1+2C2411yescharacteristic
105C2×AGL(1,ℤ/9)—28no—
106C2×AGL(1,ℤ/9)—22no—
107C2×AGL(1,ℤ/9)—22no—
108C2×AGL(1,ℤ/9)C2×Q814yescomplemented
109C4×3−1+2—22no—
110C4×3−1+2—22no—
111C4×3−1+2—22no—
1123−1+2⋊C4—22no—
1133−1+2⋊C4—22no—
1143−1+2⋊C4C2×D414yes—
115C22×C3×Q8—91no—
116C23×C3×C4—91no—
117C22×C4×D3—31no—
118C22×Dic6—31no—
119C3×C22⋊Q8—98no—
120C2×C3×C4⋊C4—31no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C4⋊C4C3×D311yescharacteristic
124C2×C3×C22⋊C4—91no—
125C2×C3×C22⋊C4—91no—
126C2×(C2×D3)⋊C4—31no—
127C2×(C2×D3)⋊C4—31no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—31no—
130#896—38no—
131C23×C32C22×D311yescharacteristic
132C22×C3×D3—34no—
133C22×C3×D3—32no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4C22×D312yes—
140C2×C32×C4C22×D312yes—
141C2×C32×C4C22×D312yes—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C3×C4×D3—68no—
149C3×Dic6—68no—
150C23×C9—21no—
151C23×C9C23×C311yescharacteristic
152C22×D9—22no—
153C22×D9—22no—
154C22×D9C3×Q814yescomplemented
155C22×D9C23×C312yes—
156C2×C4×C9—24no—
157C2×C4×C9—22no—
158C2×C4×C9—22no—
159C2×C4×C9—22no—
160C2×C4×C9—22no—
161C2×C4×C9—22no—
162C2×C4×C9—22no—
163C2×C4×C9—42no—
164C2×C4×C9—42no—
165C2×C4×C9C23×C312yes—
166C2×C4×C9C23×C312yes—
167C2×C4×C9C23×C312yes—
168C2×Dic9—22no—
169C2×Dic9—22no—
170C2×Dic9C3×D414yes—
171C2×Dic9C23×C312yes—
172C2×Dic9C23×C312yes—
173C2×Dic9C23×C312yes—
174C4×D9—28no—
175Dic18—28no—
176C2×C22⋊Q8—91noSylow
177C2×3−1+2C22⋊Q814yescomplemented
178C2×3−1+2C22×D411yescharacteristic
179C2×3−1+2C22×Q811yescharacteristic
180C2×3−1+2C2×#4111yescharacteristic
181AGL(1,ℤ/9)—24no—
182C24×C3—91no—
183C23×D3—31no—
184C2×C3×Q8—94no—
185C2×C3×Q8—184no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—94no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—184no—
194C22×C3×C4C2×C3×D311yescharacteristic
195C22×C3×C4C2×C3×D311yescharacteristic
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—31no—
200C2×C4×D3—34no—
201C2×C4×D3—64no—
202C2×Dic6—34no—
203C2×Dic6—64no—
204C3×C4⋊C4—34no—
205C3×C4⋊C4—94no—
206C3×C4⋊C4—94no—
207C3×C4⋊C4C2×C3×D314yes—
208C3×C22⋊C4—94no—
209C3×C22⋊C4—94no—
210(C2×D3)⋊C4—34no—
211(C2×D3)⋊C4—34no—
212Dic3⋊C4—34no—
213Dic3⋊C4—34no—
214C22×C32#23512yes—
215C22×C32D3×D412yes—
216C22×C32D3×Q812yes—
217C22×C32C23×D311yescharacteristic
218C2×C3×D3—34no—
219C2×C3×D3—68no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4—22no—
225C3×Dic3—34no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C22×C9—21no—
229C22×C9—22no—
230C22×C9—22no—
231C22×C9—22no—
232C22×C9C2×C3×D412yes—
233C22×C9C2×C3×Q812yes—
234C22×C9C3×#4112yes—
235C22×C9C24×C311yescharacteristic
236C2×D9—28no—
237C2×D9—22no—
238C2×D9—22no—
239C2×D9C2×C3×Q814yescomplemented
240C4×C9—24no—
241C4×C9—22no—
242C4×C9—22no—
243C4×C9—22no—
244C4×C9—42no—
245C4×C9—42no—
246Dic9—22no—
247Dic9—22no—
248Dic9C2×C3×D414yes—
249C22×Q8—91no—
250C23×C4—91no—
251C22⋊Q8—98no—
252C2×C4⋊C4—91no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
255C2×C22⋊C4—91no—
256C2×C22⋊C4—91no—
2573−1+2C2×C22⋊Q811yescharacteristic, complemented, Sylow
258C23×C3—31no—
259C23×C3—92no—
260C23×C3—94no—
261C23×C3—182no—
262C23×C3—182no—
263C23×C3C22×C3×D311yescharacteristic
264C22×D3—32no—
265C22×D3—34no—
266C22×D3—62no—
267C22×D3—62no—
268C3×Q8—188no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—32no—
273C2×C3×C4—32no—
274C2×C3×C4—32no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—92no—
279C2×C3×C4—94no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—188no—
283C2×C3×C4—182no—
284C2×C3×C4—182no—
285C2×C3×C4C22×C3×D312yes—
286C2×C3×C4C22×C3×D312yes—
287C2×C3×C4C22×C3×D312yes—
288C2×Dic3—32no—
289C2×Dic3—34no—
290C2×Dic3—32no—
291C2×Dic3—32no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C4×D3—68no—
295Dic6—68no—
296C2×C32#89614yes—
297C2×C32C2×#23511yescharacteristic
298C2×C32C2×D3×D411yescharacteristic
299C2×C32C2×D3×Q811yescharacteristic
300C3×D3—64no—
301C2×C9—24no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9C3×C22⋊Q814yescomplemented
306C2×C9C22×C3×D411yescharacteristic
307C2×C9C22×C3×Q811yescharacteristic
308C2×C9C2×C3×#4111yescharacteristic
309D9—24no—
310C24—91no—
311C2×Q8—94no—
312C2×Q8—184no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—91no—
316C22×C4—94no—
317C22×C4—184no—
318C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
319C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
320C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
321C4⋊C4—94no—
322C4⋊C4—94no—
323C4⋊C4C2×AGL(1,ℤ/9)14yescomplemented
324C22⋊C4—94no—
325C22⋊C4—94no—
326C22×C3—31no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—32no—
330C22×C3—94no—
331C22×C3—182no—
332C22×C3—188no—
333C22×C3—182no—
334C22×C3C3×#23512yes—
335C22×C3C3×D3×D412yes—
336C22×C3C3×D3×Q812yes—
337C22×C3C23×C3×D311yescharacteristic
338C2×D3—34no—
339C2×D3—68no—
340C2×D3—62no—
341C2×D3—62no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—62no—
348C3×C4—94no—
349C3×C4—182no—
350C3×C4—182no—
351Dic3—34no—
352Dic3—62no—
353Dic3—62no—
354C32C2×#89611yescharacteristic
355C9—21no—
356C9C2×C3×C22⋊Q811yescharacteristic, complemented
357C23—92no—
358C23—94no—
359C23—182no—
360C23—182no—
361C23C22×AGL(1,ℤ/9)11yescharacteristic
362Q8—188no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—92no—
367C2×C4—94no—
368C2×C4—92no—
369C2×C4—92no—
370C2×C4—188no—
371C2×C4—182no—
372C2×C4—182no—
373C2×C4C22×AGL(1,ℤ/9)12yes—
374C2×C4C22×AGL(1,ℤ/9)12yes—
375C2×C4C22×AGL(1,ℤ/9)12yes—
376C2×C3—34no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—184no—
381C2×C3C3×#89614yes—
382C2×C3C2×C3×#23511yescharacteristic
383C2×C3C2×C3×D3×D411yescharacteristic
384C2×C3C2×C3×D3×Q811yescharacteristic
385D3—64no—
386C22—94no—
387C22—188no—
388C22—182no—
389C22—182no—
390C22#3697⋊C312yes—
391C22D4×AGL(1,ℤ/9)12yes—
392C22Q8×AGL(1,ℤ/9)12yes—
393C22C23×AGL(1,ℤ/9)11yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—94no—
398C4—182no—
399C4—182no—
400C3—31no—
401C3C2×C3×#89611yescharacteristic
402C2—184no—
403C2#63458⋊C314yesdirect product factor
404C2C2×#3697⋊C311yescharacteristic
405C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
406C2C2×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×AGL(1,ℤ/9)C22
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×3−1+2⋊Q8C2semidirect
3. C2×C4⋊C4×3−1+2C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. C2×(3−1+2⋊C4)⋊C4C2semidirect
8. #63458⋊C3C2direct (N × H)
10. C2×#63458C3semidirect
21. C2×3−1+2⋊Q8C22semidirect
22. C4⋊C4×3−1+2C22semidirect
25. (3−1+2⋊C4)⋊C4C22semidirect
26. (3−1+2⋊C4)⋊C4C22semidirect
36. C22×Dic18C2×C3semidirect
38. C2×C9×C4⋊C4C2×C3semidirect
41. C2×Dic9⋊C4C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63458C2×C3semidirect
47. C22×AGL(1,ℤ/9)Q8semidirect
94. C2×Dic18C22×C3semidirect
96. C9×C4⋊C4C22×C3semidirect
99. Dic9⋊C4C22×C3semidirect
100. Dic9⋊C4C22×C3semidirect
108. C2×AGL(1,ℤ/9)C2×Q8semidirect
154. C22×D9C3×Q8semidirect
177. C2×3−1+2C22⋊Q8semidirect
239. C2×D9C2×C3×Q8semidirect
254. C2×C4⋊C4AGL(1,ℤ/9)semidirect
257. 3−1+2C2×C22⋊Q8semidirect
305. C2×C9C3×C22⋊Q8semidirect
323. C4⋊C4C2×AGL(1,ℤ/9)semidirect
356. C9C2×C3×C22⋊Q8semidirect
403. C2#63458⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×AGL(1,ℤ/9)2
C22×3−1+2⋊Q82
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
#63458⋊C32
C2×C3×#8963
C2×#634583