Finite Group Databasebeta

← C2×#63456⋊C3 #13152457

Subgroups — C2×#63456⋊C3 #13152457

Order 1728.

1072 conjugacy classes of subgroups.

The subgroup lattice has 444 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×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
6C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
7C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#63456⋊C3C218yesmaximal, direct product factor
9C2×C3×#894—31nomaximal
10C2×#63456C311yesmaximal, characteristic, complemented
11C23×AGL(1,ℤ/9)C2211yescharacteristic
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
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×C4×AGL(1,ℤ/9)—24no—
21C2×C4×AGL(1,ℤ/9)C2214yescomplemented
22C4⋊C4×3−1+2C2214yescomplemented
23(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
24(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
25(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
26(3−1+2⋊C4)⋊C4C2214yescomplemented
27C22×C3×D12—31no—
28C22×C3×C4×D3—31no—
29C2×C32×C4⋊C4D311yescharacteristic
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×(C2×D3)⋊C4—31no—
32C2×C3×(C2×D3)⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#894—38no—
35C22×D36C2×C311yescharacteristic
36C22×C4×D9C2×C311yescharacteristic, complemented
37C2×C9×C4⋊C4—21no—
38C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
39C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
40C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
41C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63456C2×C318yescomplemented
44C23×3−1+2C2311yescharacteristic
45C22×AGL(1,ℤ/9)—22no—
46C22×AGL(1,ℤ/9)—22no—
47C22×AGL(1,ℤ/9)—22no—
48C22×AGL(1,ℤ/9)—22no—
49C22×AGL(1,ℤ/9)—22no—
50C22×AGL(1,ℤ/9)D414yes—
51C22×AGL(1,ℤ/9)C2312yes—
52C22×AGL(1,ℤ/9)C2312yes—
53C2×C4×3−1+2—22no—
54C2×C4×3−1+2—22no—
55C2×C4×3−1+2—22no—
56C2×C4×3−1+2C2312yes—
57C2×C4×3−1+2C2312yes—
58C2×C4×3−1+2C2312yes—
59C2×3−1+2⋊C4—22no—
60C2×3−1+2⋊C4—22no—
61C2×3−1+2⋊C4C2312yes—
62C2×3−1+2⋊C4C2312yes—
63C4⋊AGL(1,ℤ/9)—28no—
64C4×AGL(1,ℤ/9)—28no—
65C2×C3×#123—91no—
66C2×#894—31no—
67C23×C3×D3—31no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C2×C3×D12—34no—
75C2×C3×D12—64no—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—64no—
78C23×D9C22×C311yescharacteristic
79C23×D9C22×C311yescharacteristic
80C32×C4⋊C4C2×D314yes—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×Dic3⋊C4—34no—
85C22×C4×C9—21no—
86C22×C4×C9—21no—
87C22×C4×C9—21no—
88C22×C4×C9C22×C311yescharacteristic
89C22×C4×C9C22×C311yescharacteristic
90C22×C4×C9C22×C311yescharacteristic
91C22×Dic9C22×C311yescharacteristic
92C22×Dic9C22×C311yescharacteristic
93C2×D36—24no—
94C2×D36C22×C314yes—
95C2×C4×D9—24no—
96C2×C4×D9C22×C314yescomplemented
97C9×C4⋊C4—24no—
98C9×C4⋊C4C22×C314yescomplemented
99(C2×D9)⋊C4C22×C314yescomplemented
100(C2×D9)⋊C4C22×C314yescomplemented
101(C2×D9)⋊C4C22×C314yescomplemented
102Dic9⋊C4C22×C314yescomplemented
103C22×3−1+2C2×D412yes—
104C22×3−1+2#4112yes—
105C22×3−1+2#4112yes—
106C22×3−1+2C2411yescharacteristic
107C2×AGL(1,ℤ/9)—28no—
108C2×AGL(1,ℤ/9)—22no—
109C2×AGL(1,ℤ/9)—22no—
110C2×AGL(1,ℤ/9)—22no—
111C2×AGL(1,ℤ/9)—22no—
112C2×AGL(1,ℤ/9)—22no—
113C2×AGL(1,ℤ/9)—44no—
114C2×AGL(1,ℤ/9)C2×D414yes—
115C4×3−1+2—22no—
116C4×3−1+2—22no—
117C4×3−1+2—22no—
1183−1+2⋊C4—22no—
1193−1+2⋊C4—22no—
120C22×C3×D4—91no—
121C23×C3×C4—91no—
122C22×D12—31no—
123C22×C4×D3—31no—
124C3×#123—98no—
125C2×C3×C4⋊C4—31no—
126C2×C3×C4⋊C4—91no—
127C2×C3×C4⋊C4C3×D311yescharacteristic
128C2×C3×C22⋊C4—91no—
129C2×C3×C22⋊C4—91no—
130C2×C3×C22⋊C4—91no—
131C2×(C2×D3)⋊C4—31no—
132C2×(C2×D3)⋊C4—31no—
133C2×(C2×D3)⋊C4—31no—
134C2×Dic3⋊C4—31no—
135#894—38no—
136C23×C32C22×D311yescharacteristic
137C22×C3×D3—34no—
138C22×C3×D3—32no—
139C22×C3×D3—32no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D312yes—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C3×D12—68no—
156C3×C4×D3—68no—
157C23×C9—21no—
158C23×C9C23×C311yescharacteristic
159C22×D9—22no—
160C22×D9—22no—
161C22×D9—22no—
162C22×D9—22no—
163C22×D9—22no—
164C22×D9C3×D414yes—
165C22×D9C23×C312yes—
166C22×D9C23×C312yes—
167C2×C4×C9—24no—
168C2×C4×C9—22no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—42no—
175C2×C4×C9—42no—
176C2×C4×C9C23×C312yes—
177C2×C4×C9C23×C312yes—
178C2×C4×C9C23×C312yes—
179C2×Dic9—22no—
180C2×Dic9—22no—
181C2×Dic9C23×C312yes—
182C2×Dic9C23×C312yes—
183D36—28no—
184C4×D9—28no—
185C2×#123—91noSylow
186C2×3−1+2#12314yescomplemented
187C2×3−1+2C22×D411yescharacteristic
188C2×3−1+2C2×#4111yescharacteristic
189C2×3−1+2C2×#4111yescharacteristic
190AGL(1,ℤ/9)—24no—
191AGL(1,ℤ/9)—42no—
192C24×C3—91no—
193C24×C3—91no—
194C23×D3—31no—
195C23×D3—31no—
196C2×C3×D4—94no—
197C2×C3×D4—184no—
198C22×C3×C4—31no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—94no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—184no—
205C22×C3×C4C2×C3×D311yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×Dic3—31no—
209C22×Dic3—31no—
210C2×D12—34no—
211C2×D12—64no—
212C2×C4×D3—34no—
213C2×C4×D3—64no—
214C3×C4⋊C4—34no—
215C3×C4⋊C4—94no—
216C3×C4⋊C4C2×C3×D314yes—
217C3×C22⋊C4—94no—
218C3×C22⋊C4—94no—
219C3×C22⋊C4—94no—
220(C2×D3)⋊C4—34no—
221(C2×D3)⋊C4—34no—
222(C2×D3)⋊C4—34no—
223Dic3⋊C4—34no—
224C22×C32#23512yes—
225C22×C32D3×D412yes—
226C22×C32#23912yes—
227C22×C32C23×D311yescharacteristic
228C2×C3×D3—34no—
229C2×C3×D3—68no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—124no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—22no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C22×C9—21no—
242C22×C9—22no—
243C22×C9—22no—
244C22×C9—22no—
245C22×C9C2×C3×D412yes—
246C22×C9C3×#4112yes—
247C22×C9C3×#4112yes—
248C22×C9C24×C311yescharacteristic
249C2×D9—28no—
250C2×D9—22no—
251C2×D9—22no—
252C2×D9—22no—
253C2×D9—22no—
254C2×D9—22no—
255C2×D9—44no—
256C2×D9C2×C3×D414yes—
257C4×C9—24no—
258C4×C9—22no—
259C4×C9—22no—
260C4×C9—22no—
261C4×C9—42no—
262C4×C9—42no—
263Dic9—22no—
264Dic9—22no—
265C22×D4—91no—
266C23×C4—91no—
267#123—98no—
268C2×C4⋊C4—91no—
269C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
270C2×C22⋊C4—91no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4—91no—
2733−1+2C2×#12311yescharacteristic, complemented, Sylow
274C23×C3—31no—
275C23×C3—92no—
276C23×C3—94no—
277C23×C3—92no—
278C23×C3—182no—
279C23×C3—182no—
280C23×C3—182no—
281C23×C3—182no—
282C23×C3—182no—
283C23×C3C22×C3×D311yescharacteristic
284C22×D3—32no—
285C22×D3—34no—
286C22×D3—32no—
287C22×D3—62no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—62no—
292C3×D4—188no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—188no—
305C2×C3×C4—182no—
306C2×C3×C4—182no—
307C2×C3×C4C22×C3×D312yes—
308C2×C3×C4C22×C3×D312yes—
309C2×C3×C4C22×C3×D312yes—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314D12—68no—
315C4×D3—68no—
316C2×C32#89414yes—
317C2×C32C2×#23511yescharacteristic
318C2×C32C2×D3×D411yescharacteristic
319C2×C32C2×#23911yescharacteristic
320C3×D3—64no—
321C3×D3—122no—
322C2×C9—24no—
323C2×C9—21no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9C3×#12314yescomplemented
327C2×C9C22×C3×D411yescharacteristic
328C2×C9C2×C3×#4111yescharacteristic
329C2×C9C2×C3×#4111yescharacteristic
330D9—24no—
331D9—42no—
332C24—91no—
333C24—91no—
334C2×D4—94no—
335C2×D4—184no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—94no—
339C22×C4—184no—
340C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
341C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
342C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
343C4⋊C4—94no—
344C4⋊C4C2×AGL(1,ℤ/9)14yescomplemented
345C22⋊C4—94no—
346C22⋊C4—94no—
347C22⋊C4—94no—
348C22×C3—31no—
349C22×C3—32no—
350C22×C3—32no—
351C22×C3—32no—
352C22×C3—94no—
353C22×C3—188no—
354C22×C3—182no—
355C22×C3—182no—
356C22×C3—182no—
357C22×C3—182no—
358C22×C3—182no—
359C22×C3—364no—
360C22×C3C3×#23512yes—
361C22×C3C3×D3×D412yes—
362C22×C3C3×#23912yes—
363C22×C3C23×C3×D311yescharacteristic
364C2×D3—34no—
365C2×D3—68no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—124no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—62no—
376C3×C4—62no—
377C3×C4—62no—
378C3×C4—182no—
379C3×C4—182no—
380Dic3—62no—
381Dic3—62no—
382C32C2×#89411yescharacteristic
383C9—21no—
384C9C2×C3×#12311yescharacteristic, complemented
385C23—92no—
386C23—94no—
387C23—92no—
388C23—182no—
389C23—182no—
390C23—182no—
391C23—182no—
392C23—182no—
393C23C22×AGL(1,ℤ/9)11yescharacteristic
394D4—188no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4—188no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4C22×AGL(1,ℤ/9)12yes—
404C2×C4C22×AGL(1,ℤ/9)12yes—
405C2×C4C22×AGL(1,ℤ/9)12yes—
406C2×C3—34no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—184no—
411C2×C3—362no—
412C2×C3C3×#89414yes—
413C2×C3C2×C3×#23511yescharacteristic
414C2×C3C2×C3×D3×D411yescharacteristic
415C2×C3C2×C3×#23911yescharacteristic
416D3—64no—
417D3—122no—
418C22—94no—
419C22—188no—
420C22—182no—
421C22—182no—
422C22—182no—
423C22—182no—
424C22—182no—
425C22—364no—
426C22#3697⋊C312yes—
427C22D4×AGL(1,ℤ/9)12yes—
428C22#8872412yes—
429C22C23×AGL(1,ℤ/9)11yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—182no—
434C4—182no—
435C3—31no—
436C3C2×C3×#89411yescharacteristic
437C2—184no—
438C2—362no—
439C2#63456⋊C314yesdirect product factor
440C2C2×#3697⋊C311yescharacteristic
441C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
442C2C2×#8872411yescharacteristic
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 radicalC22×C4⋊AGL(1,ℤ/9)C2
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#123
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×C4×AGL(1,ℤ/9)C2semidirect
3. C2×C4⋊C4×3−1+2C2semidirect
4. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
5. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
6. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
7. C2×(3−1+2⋊C4)⋊C4C2semidirect
8. #63456⋊C3C2direct (N × H)
10. C2×#63456C3semidirect
21. C2×C4×AGL(1,ℤ/9)C22semidirect
22. C4⋊C4×3−1+2C22semidirect
23. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
24. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
25. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
26. (3−1+2⋊C4)⋊C4C22semidirect
36. C22×C4×D9C2×C3semidirect
38. C2×C9×C4⋊C4C2×C3semidirect
39. C2×(C2×D9)⋊C4C2×C3semidirect
40. C2×(C2×D9)⋊C4C2×C3semidirect
41. C2×(C2×D9)⋊C4C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63456C2×C3semidirect
96. C2×C4×D9C22×C3semidirect
98. C9×C4⋊C4C22×C3semidirect
99. (C2×D9)⋊C4C22×C3semidirect
100. (C2×D9)⋊C4C22×C3semidirect
101. (C2×D9)⋊C4C22×C3semidirect
102. Dic9⋊C4C22×C3semidirect
186. C2×3−1+2#123semidirect
269. C2×C4⋊C4AGL(1,ℤ/9)semidirect
273. 3−1+2C2×#123semidirect
326. C2×C9C3×#123semidirect
344. C4⋊C4C2×AGL(1,ℤ/9)semidirect
384. C9C2×C3×#123semidirect
439. C2#63456⋊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×C4×AGL(1,ℤ/9)2
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
#63456⋊C32
C2×C3×#8943
C2×#634563