Finite Group Databasebeta

← C3×#69965 #11788159

Subgroups — C3×#69965 #11788159

Order 1152.

979 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5791C211yesmaximal, characteristic
2C22×C3×C4×Dic3C211yesmaximal, characteristic, complemented
3C32×#407C211yesmaximal, characteristic, complemented
4C2×C3×#825C212yesmaximal, complemented
5C3×#4555C212yesmaximal, complemented
6C3×#1896—31nomaximal
7#69965C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C23×C3×Dic3—21no—
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2212yes—
12C3×#953—21no—
13C3×#930C412yescomplemented
14C3×#929—21no—
15C3×C4×#67—22no—
16C3×#926C2211yescharacteristic
17C3×#924C412yescomplemented
18C2×C3×C4×Dic3—21no—
19C2×C3×C4×Dic3—24no—
20C2×C3×C4×Dic3C2212yes—
21C32×C22⋊C8—22no—
22C3×#825—24no—
23C3×#525—31no—
24C22×C3×C42—31no—
25#5791C2×C311yescharacteristic
26C22×C4×Dic3C2×C311yescharacteristic, complemented
27C2×C3×#104—32no—
28C3×#407—21no—
29C3×#407D311yescharacteristic, complemented
30C3×#407C2×C311yescharacteristic, complemented
31C2×#825C2×C312yescomplemented
32C3×#339—32no—
33#4555C2×C312yescomplemented
34C24×C32D411yescharacteristic
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—22no—
42C2×C3×#67—21no—
43C22×C3×Dic3—24no—
44C22×C3×Dic3—28no—
45C22×C3×Dic3—22no—
46C3×#235—22no—
47C3×#235—22no—
48C2×C3×D12C2×C411yescharacteristic, complemented
49C2×C3×C4×D3—21no—
50C2×C3×Dic6C2×C411yescharacteristic, complemented
51C32×M16—22no—
52C32×M16—22no—
53C2×C32×C8—22no—
54C3×#217—22no—
55C3×#217—21no—
56C3×(C2×D3)⋊C4—22no—
57C3×C4⋊Dic3C2×C412yescomplemented
58C3×Dic3⋊C4—22no—
59C3×C4×Dic3—24no—
60C3×C4×Dic3—22no—
61C3×C4×Dic3—21no—
62C3×C4×Dic3D414yescomplemented
63#1896—31noSylow
64C2×C3×#41—31no—
65C23×C3×C4—21no—
66C23×C3×C4—61no—
67C23×C3×C4C2×D311yescharacteristic
68C23×C3×C4C22×C311yescharacteristic
69C23×Dic3—21no—
70C2×#235C22×C311yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16C2×D312yes—
73C2×C3×M16C22×C312yes—
74C3×#123—61no—
75C3×C22⋊Q8—32no—
76C3×#121—32no—
77C3×#120—61no—
78C3×C4×D4—62no—
79C3×#117—31no—
80C2×C3×C42—32no—
81C2×C3×C42—64no—
82C2×C3×C42—61no—
83#953—21no—
84#930C3×C412yescomplemented
85#929—21no—
86C4×#67—22no—
87#926C22×C311yescharacteristic
88#924C3×C412yescomplemented
89C2×C4×Dic3—21no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3C22×C312yes—
92C3×#104—64no—
93C3×C22⋊C8—24no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96#825—24no—
97C23×C32—24no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32C22⋊C412yes—
101C23×C32C2×D411yescharacteristic
102C22×C3×D3—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4C22⋊C412yes—
108C2×C32×C4C22×C411yescharacteristic
109C2×C32×C4C2×D412yes—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4#4112yes—
112C3×#67—44no—
113C3×#67—41no—
114C3×#67—41no—
115C2×C3×Dic3—216no—
116C2×C3×Dic3—28no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—21no—
120C3×D12—22no—
121C3×C4×D3—42no—
122C3×Dic6—22no—
123C32×C8—42no—
124#525—31no—
125C22×C42—31no—
126C2×#104—32no—
127#407C3×D311yescharacteristic, complemented
128#339—32no—
129C24×C3—21no—
130C24×C3D1211yescharacteristic
131C24×C3C3×D411yescharacteristic
132C3×#41—62no—
133C3×#41—62no—
134C2×C3×Q8—31no—
135C2×C3×D4—31no—
136C2×C3×D4—62no—
137C2×C3×D4—61no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—64no—
149C22×C3×C4—68no—
150C22×C3×C4—62no—
151C22×C3×C4—61no—
152C22×C3×C4C4×D312yes—
153C22×C3×C4D1211yescharacteristic
154C22×C3×C4#6712yes—
155C22×C3×C4C2×C3×C412yes—
156C22×C3×C4C3×D411yescharacteristic
157C22×C3×C4C3×D412yes—
158C22×C3×C4C22×D311yescharacteristic
159C22×C3×C4C23×C311yescharacteristic
160C2×#67—22no—
161C2×#67—21no—
162C22×Dic3—24no—
163C22×Dic3—28no—
164C22×Dic3—22no—
165#235—22no—
166#235—22no—
167C2×D12C2×C3×C411yescharacteristic, complemented
168C2×C4×D3—21no—
169C2×Dic6C2×C3×C411yescharacteristic, complemented
170C3×M16—22no—
171C3×M16—22no—
172C3×M16—22no—
173C3×M16—22no—
174C3×M16—42no—
175C3×M16—42no—
176C2×C3×C8—24no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—32no—
180C3×C4⋊C4—62no—
181C3×C22⋊C4—62no—
182C3×C22⋊C4—61no—
183C3×C22⋊C4—62no—
184C3×C42—34no—
185C3×C42—64no—
186C3×C42—62no—
187C3×C42—61no—
188#217—22no—
189#217—21no—
190(C2×D3)⋊C4—22no—
191C4⋊Dic3C2×C3×C412yescomplemented
192Dic3⋊C4—22no—
193C4×Dic3—24no—
194C4×Dic3—22no—
195C4×Dic3—21no—
196C4×Dic3C3×D414yescomplemented
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32#10414yescomplemented
203C22×C32C2×C22⋊C411yescharacteristic
204C22×C32#12011yescharacteristic
205C22×C32#12311yescharacteristic
206C2×C3×D3—41no—
207C2×C3×D3—42no—
208C32×C4—22no—
209C32×C4C4×D412yes—
210C32×C4#12112yes—
211C3×Dic3—28no—
212C3×Dic3—41no—
213C3×Dic3—42no—
214C2×#41—31no—
215C23×C4—61no—
216C23×C4C2×C3×D311yescharacteristic
217C2×M16C2×C3×D312yes—
218#123—61no—
219C22⋊Q8—32no—
220#121—32no—
221#120—61no—
222C4×D4—62no—
223#117—31no—
224C2×C42—32no—
225C2×C42—64no—
226C2×C42—61no—
227#104—64no—
228C22⋊C8—22no—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—22no—
236C23×C3—21no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—44no—
240C23×C3—61no—
241C23×C3(C2×D3)⋊C412yes—
242C23×C3C3×C22⋊C412yes—
243C23×C3C2×D1211yescharacteristic
244C23×C3C2×C3×D411yescharacteristic
245C22×D3—21no—
246C3×Q8—62no—
247C3×D4—62no—
248C3×D4—124no—
249C3×D4—121no—
250C3×D4—121no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—616no—
269C2×C3×C4—68no—
270C2×C3×C4—64no—
271C2×C3×C4—62no—
272C2×C3×C4—61no—
273C2×C3×C4—122no—
274C2×C3×C4(C2×D3)⋊C412yes—
275C2×C3×C4C3×C22⋊C412yes—
276C2×C3×C4C2×C4×D311yescharacteristic
277C2×C3×C4D3×D412yes—
278C2×C3×C4#23712yes—
279C2×C3×C4C2×#6711yescharacteristic
280C2×C3×C4C22×C3×C411yescharacteristic
281C2×C3×C4C2×C3×D412yes—
282C2×C3×C4C2×C3×D411yescharacteristic
283C2×C3×C4C3×#4112yes—
284#67—44no—
285#67—41no—
286#67—41no—
287C2×Dic3—216no—
288C2×Dic3—28no—
289C2×Dic3—24no—
290C2×Dic3—22no—
291C2×Dic3—21no—
292D12—22no—
293C4×D3—42no—
294Dic6—22no—
295C3×C8—44no—
296C3×C8—42no—
297C3×C8—42no—
298C2×C32—21no—
299C2×C32—24no—
300C2×C32—21no—
301C2×C32#38611yescharacteristic
302C2×C32C2×#10412yes—
303C3×D3—81no—
304C24C3×D1211yescharacteristic
305#41—62no—
306#41—62no—
307C2×Q8—31no—
308C2×D4—31no—
309C2×D4—62no—
310C2×D4—61no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—64no—
314C22×C4—68no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4C3×C4×D312yes—
318C22×C4C3×D1211yescharacteristic
319C22×C4C3×#6712yes—
320C22×C4C22×C3×D311yescharacteristic
321M16—22no—
322M16—22no—
323C2×C8—22no—
324C4⋊C4—32no—
325C4⋊C4—62no—
326C22⋊C4—62no—
327C22⋊C4—61no—
328C22⋊C4—62no—
329C42—34no—
330C42—64no—
331C42—62no—
332C42—61no—
333C22×C3—24no—
334C22×C3—24no—
335C22×C3—24no—
336C22×C3—24no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—24no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—122no—
353C22×C3—121no—
354C22×C3#82514yescomplemented
355C22×C3C3×#10414yescomplemented
356C22×C3#88211yescharacteristic
357C22×C3#88611yescharacteristic
358C22×C3C2×(C2×D3)⋊C411yescharacteristic
359C22×C3C2×C3×C22⋊C411yescharacteristic
360C22×C3C3×#12011yescharacteristic
361C22×C3C3×#12311yescharacteristic
362C2×D3—41no—
363C2×D3—42no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—68no—
370C3×C4—122no—
371C3×C4—121no—
372C3×C4#67⋊C412yes—
373C3×C4#93912yes—
374C3×C4C3×C4×D412yes—
375C3×C4C3×#12112yes—
376Dic3—28no—
377Dic3—41no—
378Dic3—42no—
379C32#189611yescharacteristic, complemented, Sylow
380C23—21no—
381C23—21no—
382C23—24no—
383C23—61no—
384C23C3×(C2×D3)⋊C412yes—
385C23C2×C3×D1211yescharacteristic
386Q8—62no—
387D4—62no—
388D4—121no—
389D4—121no—
390D4—124no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—616no—
396C2×C4—68no—
397C2×C4—64no—
398C2×C4—62no—
399C2×C4—61no—
400C2×C4—122no—
401C2×C4C3×(C2×D3)⋊C412yes—
402C2×C4C2×C3×C4×D311yescharacteristic
403C2×C4C3×D3×D412yes—
404C2×C4C3×#23712yes—
405C2×C4C2×C3×#6711yescharacteristic
406C8—42no—
407C2×C3—21no—
408C2×C3—24no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—24no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—44no—
418C2×C3—241no—
419C2×C3#495811yescharacteristic
420C2×C3C2×#82512yes—
421C2×C3C3×#38611yescharacteristic
422C2×C3C2×C3×#10412yes—
423D3—81no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—121no—
430C22—122no—
431C22C3×#82514yescomplemented
432C22C3×#88211yescharacteristic
433C22C3×#88611yescharacteristic
434C22C2×C3×(C2×D3)⋊C411yescharacteristic
435C4—22no—
436C4—68no—
437C4—121no—
438C4—122no—
439C4C3×#67⋊C412yes—
440C4C3×#93912yes—
441C3—21no—
442C3#6996511yescharacteristic, direct product factor
443C3C3×#189611yescharacteristic, complemented
444C2—21no—
445C2—21no—
446C2—24no—
447C2—241no—
448C2C3×#495811yescharacteristic
449C2C2×C3×#82512yes—
subgroupG/H
centerC2×C3×C4C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#407C2
socleC22×C32C2×C22⋊C4
prime radicalC3×#5791C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#407D3
hypocenterC3C3×#1896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C3×#407

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×Dic3C2semidirect
3. C32×#407C2semidirect
4. C2×C3×#825C2semidirect
5. C3×#4555C2semidirect
7. #69965C3direct (N × H)
13. C3×#930C4semidirect
17. C3×#924C4semidirect
26. C22×C4×Dic3C2×C3semidirect
29. C3×#407D3semidirect
30. C3×#407C2×C3semidirect
31. C2×#825C2×C3semidirect
33. #4555C2×C3semidirect
48. C2×C3×D12C2×C4semidirect
50. C2×C3×Dic6C2×C4semidirect
57. C3×C4⋊Dic3C2×C4semidirect
62. C3×C4×Dic3D4semidirect
84. #930C3×C4semidirect
88. #924C3×C4semidirect
127. #407C3×D3semidirect
167. C2×D12C2×C3×C4semidirect
169. C2×Dic6C2×C3×C4semidirect
191. C4⋊Dic3C2×C3×C4semidirect
196. C4×Dic3C3×D4semidirect
202. C22×C32#104semidirect
354. C22×C3#825semidirect
355. C22×C3C3×#104semidirect
379. C32#1896semidirect
431. C22C3×#825semidirect
442. C3#69965direct (N × H)
443. C3C3×#1896semidirect

Maximal subgroups

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

subgroupindex
C3×#57912
C22×C3×C4×Dic32
C32×#4072
C2×C3×#8252
C3×#45552
C3×#18963
#699653