Finite Group Databasebeta

← C2×#63441⋊C3 #13152374

Subgroups — C2×#63441⋊C3 #13152374

Order 1728.

804 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
2C2×#3676⋊C3C211yesmaximal, characteristic
3C2×#3676⋊C3C211yesmaximal, characteristic
4C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
8#63441⋊C3C218yesmaximal, direct product factor
9C2×C3×#879—31nomaximal
10C2×#63441C311yesmaximal, characteristic, complemented
11C24×3−1+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
18C22⋊C4×3−1+2C2214yes—
19#3676⋊C3C2214yes—
20#3676⋊C3C2214yes—
21C4⋊(3−1+2⋊C4)C2214yescomplemented
22(3−1+2⋊C4)⋊C4C2214yescomplemented
23(3−1+2⋊C4)⋊C4C2214yescomplemented
24C4×3−1+2⋊C4C2214yescomplemented
25C2×C32×C22⋊C4D311yescharacteristic
26C2×C3×#217—31no—
27C2×C3×#217—31no—
28C2×C3×C4⋊Dic3—31no—
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×C4×Dic3—31no—
32C3×#879—38no—
33C2×C9×C22⋊C4—21no—
34C2×C9×C22⋊C4C2×C311yescharacteristic
35C2×#3676C2×C311yescharacteristic
36C2×#3676C2×C311yescharacteristic
37C2×C4⋊Dic9C2×C311yescharacteristic, complemented
38C2×Dic9⋊C4C2×C311yescharacteristic, complemented
39C2×Dic9⋊C4C2×C311yescharacteristic, complemented
40C2×C4×Dic9C2×C311yescharacteristic, complemented
41#63441C2×C318yescomplemented
42C23×3−1+2—22no—
43C23×3−1+2—22no—
44C23×3−1+2—22no—
45C23×3−1+2C2312yes—
46C23×3−1+2C2311yescharacteristic
47C2×C4×3−1+2—22no—
48C2×C4×3−1+2—22no—
49C2×C4×3−1+2C2312yes—
50C2×C4×3−1+2C2312yes—
51C2×3−1+2⋊C4—22no—
52C2×3−1+2⋊C4—22no—
53C2×3−1+2⋊C4—22no—
54C2×3−1+2⋊C4—22no—
55C2×3−1+2⋊C4C2312yes—
56C2×3−1+2⋊C4C2312yes—
57C2×3−1+2⋊C4C2312yes—
58C2×3−1+2⋊C4C2312yes—
59C2×C3×#126—91no—
60C2×#879—31no—
61C24×C32C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C24×C9—21no—
69C24×C9C22×C311yescharacteristic
70C32×C22⋊C4C2×D314yes—
71C3×#217—34no—
72C3×#217—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×C4×Dic3—34no—
77C22×C4×C9—21no—
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C22×C4×C9C22×C311yescharacteristic
81C22×Dic9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C9×C22⋊C4—24no—
86C9×C22⋊C4C22×C314yes—
87#3676C22×C314yes—
88#3676C22×C314yes—
89C4⋊Dic9C22×C314yescomplemented
90Dic9⋊C4C22×C314yescomplemented
91Dic9⋊C4C22×C314yescomplemented
92C4×Dic9C22×C314yescomplemented
93C22×3−1+2—22no—
94C22×3−1+2—22no—
95C22×3−1+2—22no—
96C22×3−1+2—44no—
97C22×3−1+2#4112yes—
98C22×3−1+2#4112yes—
99C22×3−1+2#4112yes—
100C22×3−1+2C2411yescharacteristic
101C4×3−1+2—22no—
102C4×3−1+2—22no—
1033−1+2⋊C4—22no—
1043−1+2⋊C4—22no—
1053−1+2⋊C4—22no—
1063−1+2⋊C4—22no—
107C3×#126—98no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—31no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4C3×D311yescharacteristic
115C2×C3×C42—91no—
116C2×#217—31no—
117C2×#217—31no—
118C2×C4⋊Dic3—31no—
119C2×Dic3⋊C4—31no—
120C2×Dic3⋊C4—31no—
121C2×C4×Dic3—31no—
122#879—38no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32C22×D312yes—
127C23×C32C22×D311yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4C22×D312yes—
131C2×C32×C4C22×D312yes—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C23×C9—24no—
141C23×C9—22no—
142C23×C9—22no—
143C23×C9—22no—
144C23×C9—21no—
145C23×C9—22no—
146C23×C9—42no—
147C23×C9—42no—
148C23×C9C23×C312yes—
149C23×C9C23×C311yescharacteristic
150C2×C4×C9—22no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—22no—
154C2×C4×C9—42no—
155C2×C4×C9—42no—
156C2×C4×C9C23×C312yes—
157C2×C4×C9C23×C312yes—
158C2×Dic9—22no—
159C2×Dic9—22no—
160C2×Dic9—22no—
161C2×Dic9—22no—
162C2×Dic9C23×C312yes—
163C2×Dic9C23×C312yes—
164C2×Dic9C23×C312yes—
165C2×Dic9C23×C312yes—
166C2×#126—91noSylow
167C2×3−1+2—42no—
168C2×3−1+2#12614yescomplemented
169C2×3−1+2C2×#4111yescharacteristic
170C2×3−1+2C2×#4111yescharacteristic
171C2×3−1+2C2×#4111yescharacteristic
172C24×C3—31no—
173C24×C3C2×C3×D311yescharacteristic
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4C2×C3×D311yescharacteristic
181C22×C3×C4C2×C3×D311yescharacteristic
182C22×Dic3—31no—
183C22×Dic3—31no—
184C22×Dic3—31no—
185C22×Dic3—31no—
186C3×C4⋊C4—94no—
187C3×C4⋊C4—94no—
188C3×C4⋊C4—94no—
189C3×C22⋊C4—34no—
190C3×C22⋊C4—94no—
191C3×C22⋊C4—94no—
192C3×C22⋊C4C2×C3×D314yes—
193C3×C42—94no—
194#217—34no—
195#217—34no—
196C4⋊Dic3—34no—
197Dic3⋊C4—34no—
198Dic3⋊C4—34no—
199C4×Dic3—34no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—44no—
204C22×C32#23512yes—
205C22×C32#23712yes—
206C22×C32#23712yes—
207C22×C32C23×D311yescharacteristic
208C32×C4—22no—
209C32×C4—22no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C22×C9—24no—
215C22×C9—22no—
216C22×C9—22no—
217C22×C9—22no—
218C22×C9—21no—
219C22×C9—22no—
220C22×C9—22no—
221C22×C9—22no—
222C22×C9—42no—
223C22×C9—42no—
224C22×C9—48no—
225C22×C9—44no—
226C22×C9C3×#4112yes—
227C22×C9C3×#4112yes—
228C22×C9C3×#4112yes—
229C22×C9C24×C311yescharacteristic
230C4×C9—22no—
231C4×C9—22no—
232C4×C9—42no—
233C4×C9—42no—
234Dic9—22no—
235Dic9—22no—
236Dic9—22no—
237Dic9—22no—
238#126—98no—
239C2×C4⋊C4—91no—
240C2×C4⋊C4—91no—
241C2×C4⋊C4—91no—
242C2×C22⋊C4—91no—
243C2×C22⋊C4—91no—
244C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
245C2×C42—91no—
2463−1+2C2×#12611yescharacteristic, complemented, Sylow
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—32no—
251C23×C3—31no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3—62no—
255C23×C3C22×C3×D312yes—
256C23×C3C22×C3×D311yescharacteristic
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—32no—
260C2×C3×C4—32no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—92no—
264C2×C3×C4—92no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—182no—
268C2×C3×C4—182no—
269C2×C3×C4—182no—
270C2×C3×C4—182no—
271C2×C3×C4C22×C3×D312yes—
272C2×C3×C4C22×C3×D312yes—
273C2×Dic3—32no—
274C2×Dic3—32no—
275C2×Dic3—32no—
276C2×Dic3—32no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×C32—42no—
282C2×C32#87914yes—
283C2×C32C2×#23511yescharacteristic
284C2×C32C2×#23711yescharacteristic
285C2×C32C2×#23711yescharacteristic
286C2×C9—24no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—44no—
291C2×C9—42no—
292C2×C9C3×#12614yescomplemented
293C2×C9C2×C3×#4111yescharacteristic
294C2×C9C2×C3×#4111yescharacteristic
295C2×C9C2×C3×#4111yescharacteristic
296C24C2×AGL(1,ℤ/9)11yescharacteristic
297C22×C4—91no—
298C22×C4—91no—
299C22×C4—91no—
300C22×C4—91no—
301C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
302C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
303C4⋊C4—94no—
304C4⋊C4—94no—
305C4⋊C4—94no—
306C22⋊C4—94no—
307C22⋊C4—94no—
308C22⋊C4C2×AGL(1,ℤ/9)14yes—
309C42—94no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—31no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—32no—
317C22×C3—44no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—62no—
321C22×C3—124no—
322C22×C3C3×#23512yes—
323C22×C3C3×#23712yes—
324C22×C3C3×#23712yes—
325C22×C3C23×C3×D311yescharacteristic
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—182no—
331C3×C4—182no—
332C3×C4—182no—
333C3×C4—182no—
334Dic3—62no—
335Dic3—62no—
336Dic3—62no—
337Dic3—62no—
338C32C2×#87911yescharacteristic
339C9—21no—
340C9C2×C3×#12611yescharacteristic, complemented
341C23—22no—
342C23—22no—
343C23—22no—
344C23C22×AGL(1,ℤ/9)12yes—
345C23C22×AGL(1,ℤ/9)11yescharacteristic
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4C22×AGL(1,ℤ/9)12yes—
357C2×C4C22×AGL(1,ℤ/9)12yes—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—34no—
362C2×C3—42no—
363C2×C3—122no—
364C2×C3C3×#87914yes—
365C2×C3C2×C3×#23511yescharacteristic
366C2×C3C2×C3×#23711yescharacteristic
367C2×C3C2×C3×#23711yescharacteristic
368C22—22no—
369C22—22no—
370C22—22no—
371C22—44no—
372C22#3697⋊C312yes—
373C22#3699⋊C312yes—
374C22#3699⋊C312yes—
375C22C23×AGL(1,ℤ/9)11yescharacteristic
376C4—22no—
377C4—22no—
378C4—182no—
379C4—182no—
380C4—182no—
381C4—182no—
382C3—31no—
383C3C2×C3×#87911yescharacteristic
384C2—42no—
385C2#63441⋊C314yesdirect product factor
386C2C2×#3697⋊C311yescharacteristic
387C2C2×#3699⋊C311yescharacteristic
388C2C2×#3699⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC24×C32C2×D3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#126
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×(3−1+2⋊C4)⋊C4C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. C2×C4×3−1+2⋊C4C2semidirect
8. #63441⋊C3C2direct (N × H)
10. C2×#63441C3semidirect
21. C4⋊(3−1+2⋊C4)C22semidirect
22. (3−1+2⋊C4)⋊C4C22semidirect
23. (3−1+2⋊C4)⋊C4C22semidirect
24. C4×3−1+2⋊C4C22semidirect
37. C2×C4⋊Dic9C2×C3semidirect
38. C2×Dic9⋊C4C2×C3semidirect
39. C2×Dic9⋊C4C2×C3semidirect
40. C2×C4×Dic9C2×C3semidirect
41. #63441C2×C3semidirect
89. C4⋊Dic9C22×C3semidirect
90. Dic9⋊C4C22×C3semidirect
91. Dic9⋊C4C22×C3semidirect
92. C4×Dic9C22×C3semidirect
168. C2×3−1+2#126semidirect
246. 3−1+2C2×#126semidirect
292. C2×C9C3×#126semidirect
340. C9C2×C3×#126semidirect
385. C2#63441⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
#63441⋊C32
C2×C3×#8793
C2×#634413