Finite Group Databasebeta

← C2×C3×#5719 #11901476

Subgroups — C2×C3×#5719 #11901476

Order 1152.

2769 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#127C211yesmaximal, characteristic, complemented
2C2×C3×#939C214yesmaximal, complemented
3C2×C3×#938C212yesmaximal, complemented
4C2×C3×#935C214yesmaximal, complemented
5C2×C3×D4×Dic3C212yesmaximal, complemented
6C3×#5719C2116yesmaximal, direct product factor
7C2×C3×#872C211yesmaximal, characteristic, complemented
8C2×C3×#870C211yesmaximal, characteristic, complemented
9C2×C3×#540—31nomaximal
10C2×#5719C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2212yes—
12C22×C32×D4C2214yes—
13C22×C3×#67C2214yes—
14C23×C3×Dic3C2214yes—
15C22×C3×C4×D3C2211yescharacteristic
16C32×#127C2218yescomplemented
17C2×C32×C42C2211yescharacteristic
18C2×C3×#217C2214yes—
19C2×C3×#217C2214yes—
20C3×#939C22132yescomplemented
21C3×#938C22116yescomplemented
22C3×#935C22132yescomplemented
23C3×D4×Dic3C22116yescomplemented
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×Dic3⋊C4C2214yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C3×#872C2218yescomplemented
30C3×#870C2218yescomplemented
31C3×#540—316no—
32C2×C3×#127—21no—
33C2×C3×#127D311yescharacteristic, complemented
34C2×C3×#127C2×C311yescharacteristic, complemented
35C2×C3×#125—31no—
36C2×C3×#123—34no—
37C2×C3×#121—32no—
38C2×C3×#121—34no—
39C2×C3×C4×D4—32no—
40C2×C3×#117—31no—
41C2×#939C2×C314yescomplemented
42C2×#938C2×C312yescomplemented
43C2×#935C2×C314yescomplemented
44C2×D4×Dic3C2×C312yescomplemented
45#5719C2×C3116yesdirect product factor
46C2×#872C2×C311yescharacteristic, complemented
47C2×#870C2×C311yescharacteristic, complemented
48C24×C32C2314yes—
49C23×C3×D3C2311yescharacteristic
50C2×C32×D4—28no—
51C2×C32×D4—48no—
52C2×C32×D4C2318yes—
53C2×C32×D4C23116yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C2×C3×#67—48no—
57C2×C3×#67C23116yes—
58C22×C3×Dic3—48no—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2314yes—
61C22×C3×Dic3C2311yescharacteristic
62C22×C3×Dic3C23116yes—
63C2×C3×C4×D3—24no—
64C2×C3×C4×D3C2314yes—
65C32×C42C2314yes—
66C3×#217C23116yes—
67C3×#217C23116yes—
68C3×(C2×D3)⋊C4C2318yes—
69C3×C4⋊Dic3C2318yes—
70C3×Dic3⋊C4C2318yes—
71C3×Dic3⋊C4C23116yes—
72C3×C4×Dic3C2314yes—
73C2×#540—31noSylow
74C22×C3×D4—22no—
75C22×C3×D4—24no—
76C22×C3×D4—34no—
77C22×C3×D4C2×D312yes—
78C22×C3×D4C2×D314yes—
79C22×C3×D4C22×C312yes—
80C22×C3×D4C22×C314yes—
81C23×C3×C4—34no—
82C23×C3×C4—31no—
83C22×#67C22×C314yes—
84C23×Dic3C22×C314yes—
85C22×C4×D3C22×C311yescharacteristic
86C3×#127—28no—
87C3×#127C2×D318yescomplemented
88C3×#127C22×C318yescomplemented
89C3×#125—38no—
90C3×#123—332no—
91C3×#121—316no—
92C3×#121—332no—
93C3×C4×D4—316no—
94C3×#117—38no—
95C2×C3×C4⋊C4—32no—
96C2×C3×C4⋊C4—34no—
97C2×C3×C4⋊C4—32no—
98C2×C3×C22⋊C4—34no—
99C2×C3×C22⋊C4—34no—
100C2×C3×C22⋊C4—32no—
101C2×C3×C42—21no—
102C2×C3×C42—31no—
103C2×C3×C42C2×D311yescharacteristic
104C2×C3×C42C22×C311yescharacteristic
105C2×#217C22×C314yes—
106C2×#217C22×C314yes—
107#939C22×C3132yescomplemented
108#938C22×C3116yescomplemented
109#935C22×C3132yescomplemented
110D4×Dic3C22×C3116yescomplemented
111C2×(C2×D3)⋊C4C22×C312yes—
112C2×C4⋊Dic3C22×C312yes—
113C2×Dic3⋊C4C22×C314yes—
114C2×Dic3⋊C4C22×C312yes—
115C2×C4×Dic3C22×C311yescharacteristic
116#872C22×C318yescomplemented
117#870C22×C318yescomplemented
118C23×C32—28no—
119C23×C32—28no—
120C23×C32—28no—
121C23×C32C2418yes—
122C23×C32C2411yescharacteristic
123C22×C3×D3—22no—
124C22×C3×D3—24no—
125C22×C3×D3C2412yes—
126C32×D4—216no—
127C32×D4—416no—
128C2×C32×C4—24no—
129C2×C32×C4#4114yes—
130C2×C32×C4C2414yes—
131C2×C32×C4C2412yes—
132C3×#67—416no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—28no—
136C2×C3×Dic3—416no—
137C2×C3×Dic3C2414yes—
138C2×C3×Dic3C2418yes—
139C2×C3×Dic3C2412yes—
140C3×C4×D3—28no—
141#540—316no—
142C2×#127C3×D311yescharacteristic, complemented
143C2×#125—31no—
144C2×#123—34no—
145C2×#121—32no—
146C2×#121—34no—
147C2×C4×D4—32no—
148C2×#117—31no—
149C24×C3—24no—
150C24×C3—31no—
151C24×C3C22×D314yes—
152C24×C3C23×C314yes—
153C23×D3C23×C311yescharacteristic
154C2×C3×D4—28no—
155C2×C3×D4—28no—
156C2×C3×D4—28no—
157C2×C3×D4—216no—
158C2×C3×D4—316no—
159C2×C3×D4—48no—
160C2×C3×D4—48no—
161C2×C3×D4—48no—
162C2×C3×D4—416no—
163C2×C3×D4—128no—
164C2×C3×D4C22×D318yes—
165C2×C3×D4C22×D3116yes—
166C2×C3×D4C23×C318yes—
167C2×C3×D4C23×C3116yes—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—32no—
171C22×C3×C4—31no—
172C22×C3×C4—34no—
173C22×C3×C4—316no—
174C22×C3×C4—34no—
175C22×C3×C4—64no—
176C22×C3×C4—128no—
177C22×C3×C4C22×D312yes—
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C23×C312yes—
180C22×C3×C4C23×C311yescharacteristic
181C2×#67—48no—
182C2×#67C23×C3116yes—
183C22×Dic3—48no—
184C22×Dic3C23×C312yes—
185C22×Dic3C23×C314yes—
186C22×Dic3C23×C311yescharacteristic
187C22×Dic3C23×C3116yes—
188C2×C4×D3—24no—
189C2×C4×D3C23×C314yes—
190C3×C4⋊C4—38no—
191C3×C4⋊C4—316no—
192C3×C4⋊C4—38no—
193C3×C22⋊C4—316no—
194C3×C22⋊C4—316no—
195C3×C22⋊C4—38no—
196C3×C42—24no—
197C3×C42—34no—
198C3×C42C22×D314yes—
199C3×C42C23×C314yes—
200#217C23×C3116yes—
201#217C23×C3116yes—
202(C2×D3)⋊C4C23×C318yes—
203C4⋊Dic3C23×C318yes—
204Dic3⋊C4C23×C3116yes—
205Dic3⋊C4C23×C318yes—
206C4×Dic3C23×C314yes—
207C22×C32—28no—
208C22×C32—28no—
209C22×C32—28no—
210C22×C32—416no—
211C22×C32C2×#4112yes—
212C22×C32H(5,𝔽2)14yes—
213C22×C32C2511yescharacteristic
214C2×C3×D3—22no—
215C2×C3×D3—24no—
216C2×C3×D3—44no—
217C32×C4—24no—
218C32×C4C2×#4114yes—
219C3×Dic3—22no—
220C3×Dic3—24no—
221C3×Dic3—28no—
222C22×D4—34no—
223C22×D4C2×C3×D312yes—
224C22×D4C2×C3×D314yes—
225C23×C4—34no—
226C23×C4—31no—
227#127C2×C3×D318yescomplemented
228#125—38no—
229#123—332no—
230#121—316no—
231#121—332no—
232C4×D4—316no—
233#117—38no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—34no—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—34no—
239C2×C22⋊C4—32no—
240C2×C42—31no—
241C2×C42C2×C3×D311yescharacteristic
242C23×C3—28no—
243C23×C3—28no—
244C23×C3—28no—
245C23×C3—28no—
246C23×C3—28no—
247C23×C3—28no—
248C23×C3—21no—
249C23×C3—28no—
250C23×C3—32no—
251C23×C3—48no—
252C23×C3—48no—
253C23×C3—48no—
254C23×C3—62no—
255C23×C3—64no—
256C23×C3C23×D318yes—
257C23×C3C23×D311yescharacteristic
258C23×C3C24×C318yes—
259C23×C3C24×C311yescharacteristic
260C22×D3—22no—
261C22×D3—24no—
262C22×D3C24×C312yes—
263C3×D4—216no—
264C3×D4—216no—
265C3×D4—416no—
266C3×D4—416no—
267C3×D4—416no—
268C3×D4—432no—
269C3×D4—1216no—
270C2×C3×C4—28no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—34no—
277C2×C3×C4—32no—
278C2×C3×C4—38no—
279C2×C3×C4—62no—
280C2×C3×C4—64no—
281C2×C3×C4—68no—
282C2×C3×C4—68no—
283C2×C3×C4—1216no—
284C2×C3×C4#23714yes—
285C2×C3×C4C3×#4114yes—
286C2×C3×C4C23×D314yes—
287C2×C3×C4C23×D312yes—
288C2×C3×C4C24×C314yes—
289C2×C3×C4C24×C312yes—
290#67—416no—
291C2×Dic3—22no—
292C2×Dic3—24no—
293C2×Dic3—28no—
294C2×Dic3—416no—
295C2×Dic3C24×C314yes—
296C2×Dic3C24×C318yes—
297C2×Dic3C24×C312yes—
298C4×D3—28no—
299C2×C32—48no—
300C2×C32#54014yescomplemented
301C2×C32C22×#4111yescharacteristic
302C2×C32C2×H(5,𝔽2)12yes—
303C3×D3—42no—
304C24—31no—
305C24C22×C3×D314yes—
306C2×D4—28no—
307C2×D4—316no—
308C2×D4—48no—
309C2×D4—128no—
310C2×D4C22×C3×D318yes—
311C2×D4C22×C3×D3116yes—
312C22×C4—32no—
313C22×C4—31no—
314C22×C4—34no—
315C22×C4—316no—
316C22×C4—34no—
317C22×C4—64no—
318C22×C4—128no—
319C22×C4C22×C3×D312yes—
320C22×C4C22×C3×D311yescharacteristic
321C4⋊C4—38no—
322C4⋊C4—38no—
323C4⋊C4—316no—
324C22⋊C4—316no—
325C22⋊C4—316no—
326C22⋊C4—38no—
327C42—34no—
328C42C22×C3×D314yes—
329C22×C3—28no—
330C22×C3—28no—
331C22×C3—28no—
332C22×C3—28no—
333C22×C3—28no—
334C22×C3—28no—
335C22×C3—22no—
336C22×C3—24no—
337C22×C3—21no—
338C22×C3—48no—
339C22×C3—48no—
340C22×C3—48no—
341C22×C3—416no—
342C22×C3—416no—
343C22×C3—62no—
344C22×C3—64no—
345C22×C3—816no—
346C22×C3—124no—
347C22×C3C2×#23712yes—
348C22×C3#100414yes—
349C22×C3C2×C3×#4112yes—
350C22×C3C3×H(5,𝔽2)14yes—
351C22×C3C24×D311yescharacteristic
352C22×C3C25×C311yescharacteristic
353C2×D3—22no—
354C2×D3—24no—
355C2×D3—44no—
356C3×C4—28no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—24no—
360C3×C4—62no—
361C3×C4—64no—
362C3×C4—68no—
363C3×C4C2×#23714yes—
364C3×C4C2×C3×#4114yes—
365Dic3—22no—
366Dic3—24no—
367Dic3—28no—
368C32C2×#54011yescharacteristic, complemented, Sylow
369C23—28no—
370C23—28no—
371C23—28no—
372C23—32no—
373C23—62no—
374C23—64no—
375C23C23×C3×D318yes—
376C23C23×C3×D311yescharacteristic
377D4—216no—
378D4—416no—
379D4—1216no—
380C2×C4—24no—
381C2×C4—34no—
382C2×C4—32no—
383C2×C4—38no—
384C2×C4—62no—
385C2×C4—64no—
386C2×C4—68no—
387C2×C4—68no—
388C2×C4—1216no—
389C2×C4C3×#23714yes—
390C2×C4C23×C3×D314yes—
391C2×C4C23×C3×D312yes—
392C2×C3—21no—
393C2×C3—22no—
394C2×C3—24no—
395C2×C3—48no—
396C2×C3—48no—
397C2×C3—88no—
398C2×C3—122no—
399C2×C3#571914yesdirect product factor
400C2×C3C3×#54014yescomplemented
401C2×C3C22×#23711yescharacteristic
402C2×C3C2×#100412yes—
403C2×C3C22×C3×#4111yescharacteristic
404C2×C3C2×C3×H(5,𝔽2)12yes—
405D3—42no—
406C22—28no—
407C22—28no—
408C22—28no—
409C22—416no—
410C22—62no—
411C22—64no—
412C22—124no—
413C22C2×C3×#23712yes—
414C22C3×#100414yes—
415C22C24×C3×D311yescharacteristic
416C4—24no—
417C4—62no—
418C4—64no—
419C4—68no—
420C4C2×C3×#23714yes—
421C3—21no—
422C3C2×#571911yescharacteristic, direct product factor
423C3C2×C3×#54011yescharacteristic, complemented
424C2—48no—
425C2—122no—
426C2C3×#571914yesdirect product factor
427C2C22×C3×#23711yescharacteristic
428C2C2×C3×#100412yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#127C2
socleC23×C32C24
hypercenterC2×C3×#127D3
hypocenterC3C2×C3×#540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#127

Complements

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

NH ≅ G/Ntype
1. C2×C32×#127C2semidirect
2. C2×C3×#939C2semidirect
3. C2×C3×#938C2semidirect
4. C2×C3×#935C2semidirect
5. C2×C3×D4×Dic3C2semidirect
6. C3×#5719C2direct (N × H)
7. C2×C3×#872C2semidirect
8. C2×C3×#870C2semidirect
10. C2×#5719C3direct (N × H)
16. C32×#127C22semidirect
20. C3×#939C22semidirect
21. C3×#938C22semidirect
22. C3×#935C22semidirect
23. C3×D4×Dic3C22semidirect
29. C3×#872C22semidirect
30. C3×#870C22semidirect
33. C2×C3×#127D3semidirect
34. C2×C3×#127C2×C3semidirect
41. C2×#939C2×C3semidirect
42. C2×#938C2×C3semidirect
43. C2×#935C2×C3semidirect
44. C2×D4×Dic3C2×C3semidirect
45. #5719C2×C3direct (N × H)
46. C2×#872C2×C3semidirect
47. C2×#870C2×C3semidirect
87. C3×#127C2×D3semidirect
88. C3×#127C22×C3semidirect
107. #939C22×C3semidirect
108. #938C22×C3semidirect
109. #935C22×C3semidirect
110. D4×Dic3C22×C3semidirect
116. #872C22×C3semidirect
117. #870C22×C3semidirect
142. C2×#127C3×D3semidirect
227. #127C2×C3×D3semidirect
300. C2×C32#540semidirect
368. C32C2×#540semidirect
399. C2×C3#5719direct (N × H)
400. C2×C3C3×#540semidirect
422. C3C2×#5719direct (N × H)
423. C3C2×C3×#540semidirect
426. C2C3×#5719direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1272
C2×C3×#9392
C2×C3×#9382
C2×C3×#9352
C2×C3×D4×Dic32
C3×#57192
C2×C3×#8722
C2×C3×#8702
C2×C3×#5403
C2×#57193