Finite Group Databasebeta

← C3×#69966 #11788193

Subgroups — C3×#69966 #11788193

Order 1152.

811 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5791C211yesmaximal, characteristic
2C2×C3×#926C211yesmaximal, characteristic, complemented
3C32×#407C211yesmaximal, characteristic, complemented
4C2×C3×#825C212yesmaximal, complemented
5C3×#4555C212yesmaximal, complemented
6C3×#1897—31nomaximal
7#69966C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10C2×C32×M16C2212yes—
11C3×#953—21no—
12C2×C3×#217—21no—
13C3×#930C412yescomplemented
14C3×#929—21no—
15C3×C4×#67—22no—
16C3×#926—24no—
17C3×#926C2211yescharacteristic
18C2×C3×C4⋊Dic3—21no—
19C3×#924C412yescomplemented
20C2×C3×C4×Dic3C2212yes—
21C32×C22⋊C8—22no—
22C3×#825—24no—
23C3×#525—31no—
24C2×C3×#117—31no—
25#5791C2×C311yescharacteristic
26C2×#926C2×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—22no—
44C3×#235—22no—
45C3×#235—22no—
46C2×C3×D12C2×C411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×Dic6C2×C411yescharacteristic
49C32×M16—22no—
50C32×M16—22no—
51C2×C32×C8—22no—
52C3×#217—24no—
53C3×#217—22no—
54C3×#217—21no—
55C3×(C2×D3)⋊C4—22no—
56C3×C4⋊Dic3—24no—
57C3×C4⋊Dic3C2×C412yes—
58C3×Dic3⋊C4—22no—
59C3×C4×Dic3—22no—
60C3×C4×Dic3—21no—
61C3×C4×Dic3D414yescomplemented
62#1897—31noSylow
63C2×C3×#41—31no—
64C23×C3×C4—21no—
65C23×C3×C4C2×D311yescharacteristic
66C23×C3×C4C22×C311yescharacteristic
67C2×#235C22×C311yescharacteristic
68C2×C3×M16—22no—
69C2×C3×M16C2×D312yes—
70C2×C3×M16C22×C312yes—
71C3×#123—61no—
72C3×C22⋊Q8—32no—
73C3×#121—32no—
74C3×#120—61no—
75C3×C4×D4—62no—
76C3×#117—31no—
77C3×#117—64no—
78C2×C3×C4⋊C4—61no—
79C2×C3×C22⋊C4—61no—
80C2×C3×C42—32no—
81#953—21no—
82C2×#217—21no—
83#930C3×C412yescomplemented
84#929—21no—
85C4×#67—22no—
86#926—24no—
87#926C22×C311yescharacteristic
88C2×C4⋊Dic3—21no—
89#924C3×C412yescomplemented
90C2×C4×Dic3C22×C312yes—
91C3×#104—64no—
92C3×C22⋊C8—24no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95#825—24no—
96C23×C32—21no—
97C23×C32—21no—
98C23×C32—42no—
99C23×C32C22⋊C412yes—
100C23×C32C2×D411yescharacteristic
101C22×C3×D3—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4C22⋊C412yes—
107C2×C32×C4C22×C411yescharacteristic
108C2×C32×C4C2×D412yes—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4#4112yes—
111C3×#67—44no—
112C3×#67—41no—
113C3×#67—41no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—44no—
118C3×D12—22no—
119C3×C4×D3—42no—
120C3×Dic6—22no—
121C32×C8—42no—
122#525—31no—
123C2×#117—31no—
124C2×#104—32no—
125#407C3×D311yescharacteristic, complemented
126#339—32no—
127C24×C3—21no—
128C24×C3D1211yescharacteristic
129C24×C3C3×D411yescharacteristic
130C3×#41—62no—
131C3×#41—62no—
132C2×C3×Q8—31no—
133C2×C3×D4—31no—
134C2×C3×D4—62no—
135C2×C3×D4—61no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—62no—
147C22×C3×C4—61no—
148C22×C3×C4C4×D312yes—
149C22×C3×C4D1211yescharacteristic
150C22×C3×C4#6712yes—
151C22×C3×C4C2×C3×C412yes—
152C22×C3×C4C3×D411yescharacteristic
153C22×C3×C4C3×D412yes—
154C22×C3×C4C22×D311yescharacteristic
155C22×C3×C4C23×C311yescharacteristic
156C2×#67—22no—
157C2×#67—21no—
158C22×Dic3—22no—
159#235—22no—
160#235—22no—
161C2×D12C2×C3×C411yescharacteristic
162C2×C4×D3—21no—
163C2×Dic6C2×C3×C411yescharacteristic
164C3×M16—22no—
165C3×M16—22no—
166C3×M16—22no—
167C3×M16—22no—
168C3×M16—42no—
169C3×M16—42no—
170C2×C3×C8—24no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—32no—
174C3×C4⋊C4—64no—
175C3×C4⋊C4—62no—
176C3×C22⋊C4—64no—
177C3×C22⋊C4—62no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—62no—
180C3×C42—34no—
181C3×C42—62no—
182C3×C42—61no—
183#217—24no—
184#217—22no—
185#217—21no—
186(C2×D3)⋊C4—22no—
187C4⋊Dic3—24no—
188C4⋊Dic3C2×C3×C412yes—
189Dic3⋊C4—22no—
190C4×Dic3—22no—
191C4×Dic3—21no—
192C4×Dic3C3×D414yescomplemented
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—42no—
198C22×C32—42no—
199C22×C32C2×C22⋊C411yescharacteristic
200C22×C32#12011yescharacteristic
201C22×C32#12311yescharacteristic
202C2×C3×D3—41no—
203C2×C3×D3—42no—
204C32×C4—22no—
205C32×C4C4×D412yes—
206C32×C4#12112yes—
207C3×Dic3—44no—
208C3×Dic3—41no—
209C3×Dic3—42no—
210C2×#41—31no—
211C23×C4C2×C3×D311yescharacteristic
212C2×M16C2×C3×D312yes—
213#123—61no—
214C22⋊Q8—32no—
215#121—32no—
216#120—61no—
217C4×D4—62no—
218#117—31no—
219#117—64no—
220C2×C4⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C42—32no—
223#104—64no—
224C22⋊C8—22no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—61no—
237C23×C3(C2×D3)⋊C412yes—
238C23×C3C3×C22⋊C412yes—
239C23×C3C2×D1211yescharacteristic
240C23×C3C2×C3×D411yescharacteristic
241C22×D3—21no—
242C3×Q8—62no—
243C3×D4—62no—
244C3×D4—124no—
245C3×D4—121no—
246C3×D4—121no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—64no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—124no—
268C2×C3×C4—122no—
269C2×C3×C4(C2×D3)⋊C412yes—
270C2×C3×C4C3×C22⋊C412yes—
271C2×C3×C4C2×C4×D311yescharacteristic
272C2×C3×C4D3×D412yes—
273C2×C3×C4#23712yes—
274C2×C3×C4C2×#6711yescharacteristic
275C2×C3×C4C22×C3×C411yescharacteristic
276C2×C3×C4C2×C3×D412yes—
277C2×C3×C4C2×C3×D411yescharacteristic
278C2×C3×C4C3×#4112yes—
279#67—44no—
280#67—41no—
281#67—41no—
282C2×Dic3—22no—
283C2×Dic3—21no—
284C2×Dic3—24no—
285C2×Dic3—44no—
286D12—22no—
287C4×D3—42no—
288Dic6—22no—
289C3×C8—44no—
290C3×C8—42no—
291C3×C8—42no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—42no—
295C2×C32#38611yescharacteristic
296C2×C32#42112yes—
297C3×D3—81no—
298C24C3×D1211yescharacteristic
299#41—62no—
300#41—62no—
301C2×Q8—31no—
302C2×D4—31no—
303C2×D4—62no—
304C2×D4—61no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4C3×C4×D312yes—
310C22×C4C3×D1211yescharacteristic
311C22×C4C3×#6712yes—
312C22×C4C22×C3×D311yescharacteristic
313M16—22no—
314M16—22no—
315C2×C8—22no—
316C4⋊C4—32no—
317C4⋊C4—64no—
318C4⋊C4—62no—
319C22⋊C4—64no—
320C22⋊C4—62no—
321C22⋊C4—61no—
322C22⋊C4—62no—
323C42—34no—
324C42—62no—
325C42—61no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—44no—
346C22×C3—44no—
347C22×C3—122no—
348C22×C3—121no—
349C22×C3#88211yescharacteristic
350C22×C3#88611yescharacteristic
351C22×C3C2×(C2×D3)⋊C411yescharacteristic
352C22×C3C2×C3×C22⋊C411yescharacteristic
353C22×C3C3×#12011yescharacteristic
354C22×C3C3×#12311yescharacteristic
355C2×D3—41no—
356C2×D3—42no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—42no—
362C3×C4—124no—
363C3×C4—122no—
364C3×C4—121no—
365C3×C4#67⋊C412yes—
366C3×C4#93912yes—
367C3×C4C3×C4×D412yes—
368C3×C4C3×#12112yes—
369Dic3—44no—
370Dic3—41no—
371Dic3—42no—
372C32#189711yescharacteristic, complemented, Sylow
373C23—21no—
374C23—21no—
375C23—42no—
376C23—61no—
377C23C3×(C2×D3)⋊C412yes—
378C23C2×C3×D1211yescharacteristic
379Q8—62no—
380D4—62no—
381D4—124no—
382D4—121no—
383D4—121no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—64no—
389C2×C4—62no—
390C2×C4—61no—
391C2×C4—124no—
392C2×C4—122no—
393C2×C4C3×(C2×D3)⋊C412yes—
394C2×C4C2×C3×C4×D311yescharacteristic
395C2×C4C3×D3×D412yes—
396C2×C4C3×#23712yes—
397C2×C4C2×C3×#6711yescharacteristic
398C8—42no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—44no—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—241no—
411C2×C3#495811yescharacteristic
412C2×C3#513912yes—
413C2×C3C3×#38611yescharacteristic
414C2×C3C3×#42112yes—
415D3—81no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—121no—
423C22—122no—
424C22C3×#88211yescharacteristic
425C22C3×#88611yescharacteristic
426C22C2×C3×(C2×D3)⋊C411yescharacteristic
427C4—22no—
428C4—124no—
429C4—121no—
430C4—122no—
431C4C3×#67⋊C412yes—
432C4C3×#93912yes—
433C3—21no—
434C3#6996611yescharacteristic, direct product factor
435C3C3×#189711yescharacteristic, complemented
436C2—21no—
437C2—21no—
438C2—42no—
439C2—241no—
440C2C3×#495811yescharacteristic
441C2C3×#513912yes—
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×#1897
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. C2×C3×#926C2semidirect
3. C32×#407C2semidirect
4. C2×C3×#825C2semidirect
5. C3×#4555C2semidirect
7. #69966C3direct (N × H)
13. C3×#930C4semidirect
19. C3×#924C4semidirect
26. C2×#926C2×C3semidirect
29. C3×#407D3semidirect
30. C3×#407C2×C3semidirect
31. C2×#825C2×C3semidirect
33. #4555C2×C3semidirect
61. C3×C4×Dic3D4semidirect
83. #930C3×C4semidirect
89. #924C3×C4semidirect
125. #407C3×D3semidirect
192. C4×Dic3C3×D4semidirect
372. C32#1897semidirect
434. C3#69966direct (N × H)
435. C3C3×#1897semidirect

Maximal subgroups

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

subgroupindex
C3×#57912
C2×C3×#9262
C32×#4072
C2×C3×#8252
C3×#45552
C3×#18973
#699663