Finite Group Databasebeta

← C2×#63445⋊C3 #13152380

Subgroups — C2×#63445⋊C3 #13152380

Order 1728.

1092 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#611⋊C3C211yesmaximal, characteristic
2C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4C2×#3676⋊C3C211yesmaximal, characteristic, complemented
5C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
6C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
7C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#63445⋊C3C218yesmaximal, direct product factor
9C2×C3×#883—31nomaximal
10C2×#63445C311yesmaximal, characteristic, complemented
11C24×3−1+2C2211yescharacteristic
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic
15C2×#611⋊C3—24no—
16C2×#611⋊C3C2214yes—
17C22×3−1+2⋊C4C2211yescharacteristic
18C22×3−1+2⋊C4C2211yescharacteristic
19C22×3−1+2⋊C4C2211yescharacteristic
20C2×C4×AGL(1,ℤ/9)—24no—
21C2×C4×AGL(1,ℤ/9)C2214yescomplemented
22C22⋊C4×3−1+2C2214yescomplemented
23#3676⋊C3C2214yescomplemented
24(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
25C4⋊(3−1+2⋊C4)C2214yescomplemented
26(3−1+2⋊C4)⋊C4C2214yescomplemented
27C22×C3×#67—31no—
28C22×C3×C4×D3—31no—
29C2×C32×C22⋊C4D311yescharacteristic
30C2×C3×#217—31no—
31C2×C3×(C2×D3)⋊C4—31no—
32C2×C3×C4⋊Dic3—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#883—38no—
35C22×#611C2×C311yescharacteristic
36C22×C4×D9C2×C311yescharacteristic, complemented
37C2×C9×C22⋊C4—21no—
38C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
39C2×#3676C2×C311yescharacteristic, complemented
40C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
41C2×C4⋊Dic9C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63445C2×C318yescomplemented
44C23×3−1+2—22no—
45C23×3−1+2—22no—
46C23×3−1+2—22no—
47C23×3−1+2C2312yes—
48C23×3−1+2C2311yescharacteristic
49C22×AGL(1,ℤ/9)—22no—
50C22×AGL(1,ℤ/9)—22no—
51C22×AGL(1,ℤ/9)D414yes—
52C22×AGL(1,ℤ/9)C2312yes—
53C2×C4×3−1+2—22no—
54C2×C4×3−1+2—22no—
55C2×C4×3−1+2C2312yes—
56C2×C4×3−1+2C2312yes—
57#611⋊C3—28no—
58C2×3−1+2⋊C4—22no—
59C2×3−1+2⋊C4—22no—
60C2×3−1+2⋊C4—22no—
61C2×3−1+2⋊C4C2312yes—
62C2×3−1+2⋊C4C2312yes—
63C2×3−1+2⋊C4C2312yes—
64C4×AGL(1,ℤ/9)—28no—
65C2×C3×#123—91no—
66C2×#883—31no—
67C24×C32C2×D311yescharacteristic
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C2×C3×#67—34no—
72C2×C3×#67—64no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—64no—
78C24×C9—21no—
79C24×C9C22×C311yescharacteristic
80C23×D9C22×C311yescharacteristic
81C32×C22⋊C4C2×D314yes—
82C3×#217—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×C4⋊Dic3—34no—
85C3×Dic3⋊C4—34no—
86C22×C4×C9—21no—
87C22×C4×C9—21no—
88C22×C4×C9C22×C311yescharacteristic
89C22×C4×C9C22×C311yescharacteristic
90C2×#611—24no—
91C2×#611C22×C314yes—
92C22×Dic9C22×C311yescharacteristic
93C22×Dic9C22×C311yescharacteristic
94C22×Dic9C22×C311yescharacteristic
95C2×C4×D9—24no—
96C2×C4×D9C22×C314yescomplemented
97C9×C22⋊C4—24no—
98C9×C22⋊C4C22×C314yescomplemented
99#3676C22×C314yescomplemented
100(C2×D9)⋊C4C22×C314yescomplemented
101C4⋊Dic9C22×C314yescomplemented
102Dic9⋊C4C22×C314yescomplemented
103C22×3−1+2—22no—
104C22×3−1+2—22no—
105C22×3−1+2—22no—
106C22×3−1+2—44no—
107C22×3−1+2C2×D412yes—
108C22×3−1+2#4112yes—
109C22×3−1+2#4112yes—
110C22×3−1+2C2411yescharacteristic
111C2×AGL(1,ℤ/9)—28no—
112C2×AGL(1,ℤ/9)—22no—
113C2×AGL(1,ℤ/9)—22no—
114C2×AGL(1,ℤ/9)C2×D414yes—
115C4×3−1+2—22no—
116C4×3−1+2—22no—
1173−1+2⋊C4—22no—
1183−1+2⋊C4—22no—
1193−1+2⋊C4—22no—
120C22×C3×D4—91no—
121C23×C3×C4—91no—
122C22×#67—31no—
123C22×C4×D3—31no—
124C3×#123—98no—
125C2×C3×C4⋊C4—91no—
126C2×C3×C4⋊C4—91no—
127C2×C3×C22⋊C4—31no—
128C2×C3×C22⋊C4—91no—
129C2×C3×C22⋊C4—91no—
130C2×C3×C22⋊C4C3×D311yescharacteristic
131C2×#217—31no—
132C2×(C2×D3)⋊C4—31no—
133C2×C4⋊Dic3—31no—
134C2×Dic3⋊C4—31no—
135#883—38no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32C22×D312yes—
140C23×C32C22×D311yescharacteristic
141C22×C3×D3—34no—
142C22×C3×D3—32no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D312yes—
149C3×#67—68no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C3×C4×D3—68no—
157C23×C9—24no—
158C23×C9—22no—
159C23×C9—22no—
160C23×C9—22no—
161C23×C9—21no—
162C23×C9—22no—
163C23×C9—42no—
164C23×C9—42no—
165C23×C9C23×C312yes—
166C23×C9C23×C311yescharacteristic
167C22×D9—22no—
168C22×D9—22no—
169C22×D9C3×D414yes—
170C22×D9C23×C312yes—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—22no—
175C2×C4×C9—42no—
176C2×C4×C9—42no—
177C2×C4×C9C23×C312yes—
178C2×C4×C9C23×C312yes—
179#611—28no—
180C2×Dic9—22no—
181C2×Dic9—22no—
182C2×Dic9—22no—
183C2×Dic9C23×C312yes—
184C2×Dic9C23×C312yes—
185C2×Dic9C23×C312yes—
186C4×D9—28no—
187C2×#123—91noSylow
188C2×3−1+2—42no—
189C2×3−1+2#12314yescomplemented
190C2×3−1+2C22×D411yescharacteristic
191C2×3−1+2C2×#4111yescharacteristic
192C2×3−1+2C2×#4111yescharacteristic
193AGL(1,ℤ/9)—24no—
194C24×C3—31no—
195C24×C3—91no—
196C24×C3C2×C3×D311yescharacteristic
197C23×D3—31no—
198C2×C3×D4—94no—
199C2×C3×D4—184no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—94no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4—184no—
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C2×#67—34no—
210C2×#67—64no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C2×C4×D3—34no—
215C2×C4×D3—64no—
216C3×C4⋊C4—94no—
217C3×C4⋊C4—94no—
218C3×C22⋊C4—34no—
219C3×C22⋊C4—94no—
220C3×C22⋊C4—94no—
221C3×C22⋊C4C2×C3×D314yes—
222#217—34no—
223(C2×D3)⋊C4—34no—
224C4⋊Dic3—34no—
225Dic3⋊C4—34no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—44no—
230C22×C32#23512yes—
231C22×C32D3×D412yes—
232C22×C32#23712yes—
233C22×C32C23×D311yescharacteristic
234C2×C3×D3—34no—
235C2×C3×D3—68no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C32×C4—22no—
239C32×C4—22no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C22×C9—24no—
244C22×C9—22no—
245C22×C9—22no—
246C22×C9—22no—
247C22×C9—21no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9—22no—
251C22×C9—42no—
252C22×C9—42no—
253C22×C9—48no—
254C22×C9—44no—
255C22×C9C2×C3×D412yes—
256C22×C9C3×#4112yes—
257C22×C9C3×#4112yes—
258C22×C9C24×C311yescharacteristic
259C2×D9—28no—
260C2×D9—22no—
261C2×D9—22no—
262C2×D9C2×C3×D414yes—
263C4×C9—22no—
264C4×C9—22no—
265C4×C9—42no—
266C4×C9—42no—
267Dic9—22no—
268Dic9—22no—
269Dic9—22no—
270C22×D4—91no—
271C23×C4—91no—
272#123—98no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4—91no—
275C2×C22⋊C4—91no—
276C2×C22⋊C4—91no—
277C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
2783−1+2C2×#12311yescharacteristic, complemented, Sylow
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—32no—
283C23×C3—31no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—92no—
288C23×C3—94no—
289C23×C3—182no—
290C23×C3—182no—
291C23×C3C22×C3×D312yes—
292C23×C3C22×C3×D311yescharacteristic
293C22×D3—32no—
294C22×D3—34no—
295C22×D3—62no—
296C22×D3—62no—
297C3×D4—188no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—188no—
308C2×C3×C4—182no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4C22×C3×D312yes—
312C2×C3×C4C22×C3×D312yes—
313#67—68no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C4×D3—68no—
321C2×C32—42no—
322C2×C32#88314yes—
323C2×C32C2×#23511yescharacteristic
324C2×C32C2×D3×D411yescharacteristic
325C2×C32C2×#23711yescharacteristic
326C3×D3—64no—
327C2×C9—24no—
328C2×C9—21no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—44no—
332C2×C9—42no—
333C2×C9C3×#12314yescomplemented
334C2×C9C22×C3×D411yescharacteristic
335C2×C9C2×C3×#4111yescharacteristic
336C2×C9C2×C3×#4111yescharacteristic
337D9—24no—
338C24—91no—
339C24C2×AGL(1,ℤ/9)11yescharacteristic
340C2×D4—94no—
341C2×D4—184no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—94no—
346C22×C4—184no—
347C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
348C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
349C4⋊C4—94no—
350C4⋊C4—94no—
351C22⋊C4—94no—
352C22⋊C4—94no—
353C22⋊C4C2×AGL(1,ℤ/9)14yescomplemented
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—31no—
358C22×C3—32no—
359C22×C3—32no—
360C22×C3—32no—
361C22×C3—44no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—62no—
365C22×C3—94no—
366C22×C3—124no—
367C22×C3—182no—
368C22×C3—182no—
369C22×C3—188no—
370C22×C3C3×#23512yes—
371C22×C3C3×D3×D412yes—
372C22×C3C3×#23712yes—
373C22×C3C23×C3×D311yescharacteristic
374C2×D3—34no—
375C2×D3—68no—
376C2×D3—62no—
377C2×D3—62no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—62no—
381C3×C4—62no—
382C3×C4—182no—
383C3×C4—182no—
384C3×C4—182no—
385Dic3—62no—
386Dic3—62no—
387Dic3—62no—
388C32C2×#88311yescharacteristic
389C9—21no—
390C9C2×C3×#12311yescharacteristic, complemented
391C23—22no—
392C23—22no—
393C23—22no—
394C23—92no—
395C23—94no—
396C23—182no—
397C23—182no—
398C23C22×AGL(1,ℤ/9)12yes—
399C23C22×AGL(1,ℤ/9)11yescharacteristic
400D4—188no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4—92no—
406C2×C4—188no—
407C2×C4—182no—
408C2×C4—182no—
409C2×C4—182no—
410C2×C4C22×AGL(1,ℤ/9)12yes—
411C2×C4C22×AGL(1,ℤ/9)12yes—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—34no—
416C2×C3—42no—
417C2×C3—122no—
418C2×C3—184no—
419C2×C3C3×#88314yes—
420C2×C3C2×C3×#23511yescharacteristic
421C2×C3C2×C3×D3×D411yescharacteristic
422C2×C3C2×C3×#23711yescharacteristic
423D3—64no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—44no—
428C22—94no—
429C22—182no—
430C22—182no—
431C22—188no—
432C22#3697⋊C312yes—
433C22D4×AGL(1,ℤ/9)12yes—
434C22#3699⋊C312yes—
435C22C23×AGL(1,ℤ/9)11yescharacteristic
436C4—22no—
437C4—22no—
438C4—182no—
439C4—182no—
440C4—182no—
441C3—31no—
442C3C2×C3×#88311yescharacteristic
443C2—42no—
444C2—184no—
445C2#63445⋊C314yesdirect product factor
446C2C2×#3697⋊C311yescharacteristic
447C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
448C2C2×#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 radicalC22×#611⋊C3C2
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#123
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×AGL(1,ℤ/9)C2semidirect
3. C2×C22⋊C4×3−1+2C2semidirect
4. C2×#3676⋊C3C2semidirect
5. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
6. C2×C4⋊(3−1+2⋊C4)C2semidirect
7. C2×(3−1+2⋊C4)⋊C4C2semidirect
8. #63445⋊C3C2direct (N × H)
10. C2×#63445C3semidirect
21. C2×C4×AGL(1,ℤ/9)C22semidirect
22. C22⋊C4×3−1+2C22semidirect
23. #3676⋊C3C22semidirect
24. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
25. C4⋊(3−1+2⋊C4)C22semidirect
26. (3−1+2⋊C4)⋊C4C22semidirect
36. C22×C4×D9C2×C3semidirect
38. C2×C9×C22⋊C4C2×C3semidirect
39. C2×#3676C2×C3semidirect
40. C2×(C2×D9)⋊C4C2×C3semidirect
41. C2×C4⋊Dic9C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63445C2×C3semidirect
96. C2×C4×D9C22×C3semidirect
98. C9×C22⋊C4C22×C3semidirect
99. #3676C22×C3semidirect
100. (C2×D9)⋊C4C22×C3semidirect
101. C4⋊Dic9C22×C3semidirect
102. Dic9⋊C4C22×C3semidirect
189. C2×3−1+2#123semidirect
277. C2×C22⋊C4AGL(1,ℤ/9)semidirect
278. 3−1+2C2×#123semidirect
333. C2×C9C3×#123semidirect
353. C22⋊C4C2×AGL(1,ℤ/9)semidirect
390. C9C2×C3×#123semidirect
445. C2#63445⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#611⋊C32
C22×C4×AGL(1,ℤ/9)2
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
#63445⋊C32
C2×C3×#8833
C2×#634453