Finite Group Databasebeta

← C3×#68601 #11783595

Subgroups — C3×#68601 #11783595

Order 1152.

731 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic
2C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
3C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
4C2×C3×#815C211yesmaximal, characteristic, complemented
5C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
6C32×#336C211yesmaximal, characteristic, complemented
7C3×#4524C211yesmaximal, characteristic, complemented
8C3×#1750—31nomaximal
9#68601C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×D3C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4C412yes—
15C2×C3×Dic3⋊C4C412yes—
16C2×C3×C4×Dic3C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C22×C3×C3⋊C8C2211yescharacteristic
19C3×#872—22no—
20C3×#872—22no—
21C3×(C2×D3)⋊C8C414yescomplemented
22C3×(C2×D3)⋊C8C414yescomplemented
23C3×#815—22no—
24C3×C8×Dic3—22no—
25C2×C3×#117—31no—
26C2×#872C2×C311yescharacteristic
27C2×C3×C22⋊C8—31no—
28C2×C3×C22⋊C8—31no—
29C2×C3×#97—31no—
30C2×C3×C4×C8—31no—
31C2×(C2×D3)⋊C8C2×C311yescharacteristic
32C2×(C2×D3)⋊C8C2×C311yescharacteristic
33C2×#815C2×C311yescharacteristic, complemented
34C2×C8×Dic3C2×C311yescharacteristic, complemented
35C3×#336—21no—
36C3×#336—31no—
37C3×#336D311yescharacteristic, complemented
38C3×#336C2×C311yescharacteristic, complemented
39#4524C2×C311yescharacteristic, complemented
40C23×C3×D3C2×C411yescharacteristic
41C22×C32×C4C2×C412yes—
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3C2×C411yescharacteristic
44C22×C3×Dic3C2×C412yes—
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—21no—
47C2×C3×C4×D3—21no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3C2×C412yes—
50C2×C3×C4×D3C2×C412yes—
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8C2×C412yes—
56C2×C32×C8C2×C412yes—
57C32×C42—21no—
58C32×C42—21no—
59C3×(C2×D3)⋊C4—24no—
60C3×Dic3⋊C4—24no—
61C3×C4×Dic3D412yes—
62C3×C4×Dic3D412yes—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8C2×C412yes—
68C2×C3×C3⋊C8C2×C412yes—
69#1750—31noSylow
70C23×C3×C4—31no—
71C22×C4×D3C22×C311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8—31no—
75C22×C3×C8—31no—
76C22×C3×C8C2×D311yescharacteristic
77C22×C3×C8C2×D311yescharacteristic
78C22×C3×C8C22×C311yescharacteristic
79C22×C3×C8C22×C311yescharacteristic
80C3×#117—62no—
81C3×#117—62no—
82C2×C3×C4⋊C4—32no—
83C2×C3×C22⋊C4—32no—
84C2×C3×C42—21no—
85C2×C3×C42—31no—
86C2×C3×C42C2×D311yescharacteristic
87C2×C3×C42C22×C311yescharacteristic
88C2×(C2×D3)⋊C4C3×C412yes—
89C2×Dic3⋊C4C3×C412yes—
90C2×C4×Dic3C22×C311yescharacteristic
91C22×C3⋊C8C22×C311yescharacteristic
92C22×C3⋊C8C22×C311yescharacteristic
93#872—22no—
94#872—22no—
95C3×C22⋊C8—34no—
96C3×C22⋊C8—34no—
97C3×#97—62no—
98C3×C4×C8—62no—
99(C2×D3)⋊C8C3×C414yescomplemented
100(C2×D3)⋊C8C3×C414yescomplemented
101#815—22no—
102C8×Dic3—22no—
103C23×C32C22×C411yescharacteristic
104C22×C3×D3—21no—
105C22×C3×D3—21no—
106C22×C3×D3—22no—
107C22×C3×D3—22no—
108C22×C3×D3C4212yes—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4C22×C411yescharacteristic
113C2×C32×C4C22×C411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4#4111yescharacteristic
117C2×C32×C4#4111yescharacteristic
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3C4212yes—
123C2×C3×Dic3C22⋊C414yes—
124C3×C4×D3—22no—
125C3×C4×D3—22no—
126C3×C4×D3—22no—
127C3×C4×D3—22no—
128C32×C8—22no—
129C32×C8—22no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C2×#117—31no—
133C2×C22⋊C8—31no—
134C2×C22⋊C8—31no—
135C2×#97—31no—
136C2×C4×C8—31no—
137#336—31no—
138#336C3×D311yescharacteristic, complemented
139C24×C3—31no—
140C23×D3C2×C3×C411yescharacteristic
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—32no—
144C22×C3×C4—31no—
145C22×C3×C4—32no—
146C22×C3×C4—32no—
147C22×C3×C4—61no—
148C22×C3×C4—61no—
149C22×C3×C4—61no—
150C22×C3×C4—61no—
151C22×C3×C4C4×D312yes—
152C22×C3×C4C2×C3×C412yes—
153C22×C3×C4C22×D311yescharacteristic
154C22×C3×C4C23×C311yescharacteristic
155C22×Dic3C2×C3×C411yescharacteristic
156C22×Dic3C2×C3×C412yes—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3C2×C3×C412yes—
162C2×C4×D3C2×C3×C412yes—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—32no—
174C2×C3×C8—32no—
175C2×C3×C8—41no—
176C2×C3×C8—41no—
177C2×C3×C8—41no—
178C2×C3×C8—41no—
179C2×C3×C8—61no—
180C2×C3×C8—61no—
181C2×C3×C8—61no—
182C2×C3×C8—61no—
183C2×C3×C8C4×D312yes—
184C2×C3×C8C4×D312yes—
185C2×C3×C8C2×C3×C412yes—
186C2×C3×C8C2×C3×C412yes—
187C3×C4⋊C4—64no—
188C3×C22⋊C4—64no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—32no—
194C3×C42—32no—
195C3×C42—41no—
196C3×C42—41no—
197(C2×D3)⋊C4—24no—
198Dic3⋊C4—24no—
199C4×Dic3C3×D412yes—
200C4×Dic3C3×D412yes—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8C2×C3×C412yes—
206C2×C3⋊C8C2×C3×C412yes—
207C22×C32C2×C4211yescharacteristic
208C22×C32C2×C22⋊C411yescharacteristic
209C22×C32#11711yescharacteristic
210C22×C32#13112yes—
211C22×C32#13112yes—
212C2×C3×D3—22no—
213C2×C3×D3—22no—
214C2×C3×D3—22no—
215C2×C3×D3—42no—
216C2×C3×D3—42no—
217C32×C4—42no—
218C32×C4C4×D411yescharacteristic
219C32×C4C4×D411yescharacteristic
220C32×C4C4×D411yescharacteristic
221C32×C4C4×D411yescharacteristic
222C3×Dic3—22no—
223C3×Dic3—24no—
224C23×C4—31no—
225C22×C8—31no—
226C22×C8—31no—
227C22×C8C2×C3×D311yescharacteristic
228C22×C8C2×C3×D311yescharacteristic
229#117—62no—
230#117—62no—
231C2×C4⋊C4—32no—
232C2×C22⋊C4—32no—
233C2×C42—31no—
234C2×C42C2×C3×D311yescharacteristic
235C22⋊C8—34no—
236C22⋊C8—34no—
237#97—62no—
238C4×C8—62no—
239C23×C3—21no—
240C23×C3—32no—
241C23×C3—61no—
242C23×C3—61no—
243C23×C3—62no—
244C23×C3—62no—
245C23×C3C2×C4×D311yescharacteristic
246C23×C3C22×C3×C411yescharacteristic
247C22×D3—21no—
248C22×D3—21no—
249C22×D3—22no—
250C22×D3—22no—
251C22×D3C3×C4212yes—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—32no—
266C2×C3×C4—34no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4C2×C4×D311yescharacteristic
278C2×C3×C4C2×C4×D311yescharacteristic
279C2×C3×C4D3×D411yescharacteristic
280C2×C3×C4D3×D411yescharacteristic
281C2×C3×C4#23711yescharacteristic
282C2×C3×C4#23911yescharacteristic
283C2×C3×C4C22×C3×C411yescharacteristic
284C2×C3×C4C22×C3×C411yescharacteristic
285C2×C3×C4C2×C3×D411yescharacteristic
286C2×C3×C4C2×C3×D411yescharacteristic
287C2×C3×C4C3×#4111yescharacteristic
288C2×C3×C4C3×#4111yescharacteristic
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3C3×C4212yes—
294C2×Dic3C3×C22⋊C414yes—
295C4×D3—22no—
296C4×D3—22no—
297C4×D3—22no—
298C4×D3—22no—
299C3×C8—22no—
300C3×C8—22no—
301C3×C8—22no—
302C3×C8—22no—
303C3×C8—42no—
304C3×C8—42no—
305C3×C8—62no—
306C3×C8—62no—
307C3⋊C8—22no—
308C3⋊C8—22no—
309C2×C32C4×C22⋊C411yescharacteristic
310C2×C32#40511yescharacteristic
311C2×C32#40511yescharacteristic
312C2×C32#40812yes—
313C2×C32#43312yes—
314C3×D3—42no—
315C24—31no—
316C22×C4—32no—
317C22×C4—31no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4C3×C4×D312yes—
325C22×C4C22×C3×D311yescharacteristic
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—32no—
331C2×C8—32no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8C3×C4×D312yes—
337C2×C8C3×C4×D312yes—
338C4⋊C4—64no—
339C22⋊C4—64no—
340C42—21no—
341C42—21no—
342C42—32no—
343C42—32no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—62no—
350C22×C3—62no—
351C22×C3—62no—
352C22×C3—122no—
353C22×C3—122no—
354C22×C3C42×D311yescharacteristic
355C22×C3D3×C22⋊C411yescharacteristic
356C22×C3#89211yescharacteristic
357C22×C3#90112yes—
358C22×C3#90712yes—
359C22×C3C2×C3×C4211yescharacteristic
360C22×C3C2×C3×C22⋊C411yescharacteristic
361C22×C3C3×#11711yescharacteristic
362C22×C3C3×#13112yes—
363C22×C3C3×#13112yes—
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—42no—
368C2×D3—42no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—44no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—62no—
377C3×C4—64no—
378C3×C4#67⋊C411yescharacteristic
379C3×C4#67⋊C411yescharacteristic
380C3×C4#89311yescharacteristic
381C3×C4#89311yescharacteristic
382C3×C4C3×C4×D411yescharacteristic
383C3×C4C3×C4×D411yescharacteristic
384C3×C4C3×C4×D411yescharacteristic
385C3×C4C3×C4×D411yescharacteristic
386Dic3—22no—
387Dic3—24no—
388C32#175011yescharacteristic, complemented, Sylow
389C23—32no—
390C23—61no—
391C23—61no—
392C23—62no—
393C23—62no—
394C23C2×C3×C4×D311yescharacteristic
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—34no—
399C2×C4—32no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4C2×C3×C4×D311yescharacteristic
409C2×C4C2×C3×C4×D311yescharacteristic
410C2×C4C3×D3×D411yescharacteristic
411C2×C4C3×D3×D411yescharacteristic
412C2×C4C3×#23711yescharacteristic
413C2×C4C3×#23911yescharacteristic
414C8—22no—
415C8—22no—
416C8—62no—
417C8—62no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—122no—
424C2×C3#466611yescharacteristic
425C2×C3#468911yescharacteristic
426C2×C3#470811yescharacteristic
427C2×C3#472712yes—
428C2×C3#484012yes—
429C2×C3C3×C4×C22⋊C411yescharacteristic
430C2×C3C3×#40511yescharacteristic
431C2×C3C3×#40511yescharacteristic
432C2×C3C3×#40812yes—
433C2×C3C3×#43312yes—
434D3—42no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—122no—
439C22—122no—
440C22C3×C42×D311yescharacteristic
441C22C3×D3×C22⋊C411yescharacteristic
442C22C3×#89211yescharacteristic
443C22C3×#90112yes—
444C22C3×#90712yes—
445C4—42no—
446C4—62no—
447C4—64no—
448C4C3×#67⋊C411yescharacteristic
449C4C3×#67⋊C411yescharacteristic
450C4C3×#89311yescharacteristic
451C4C3×#89311yescharacteristic
452C3—21no—
453C3#6860111yescharacteristic, direct product factor
454C3C3×#175011yescharacteristic, complemented
455C2—122no—
456C2C3×#466611yescharacteristic
457C2C3×#468911yescharacteristic
458C2C3×#470811yescharacteristic
459C2C3×#472712yes—
460C2C3×#484012yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C2×C3×C42
FrattiniC22×C4C22×C3×D3
FittingC32×#336C2
socleC23×C32C22×C4
prime radicalC23×C3×D3C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC3×#336D3
hypocenterC3C3×#1750
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C3×#336

Complements

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

NH ≅ G/Ntype
4. C2×C3×#815C2semidirect
5. C2×C3×C8×Dic3C2semidirect
6. C32×#336C2semidirect
7. C3×#4524C2semidirect
9. #68601C3direct (N × H)
21. C3×(C2×D3)⋊C8C4semidirect
22. C3×(C2×D3)⋊C8C4semidirect
33. C2×#815C2×C3semidirect
34. C2×C8×Dic3C2×C3semidirect
37. C3×#336D3semidirect
38. C3×#336C2×C3semidirect
39. #4524C2×C3semidirect
99. (C2×D3)⋊C8C3×C4semidirect
100. (C2×D3)⋊C8C3×C4semidirect
138. #336C3×D3semidirect
388. C32#1750semidirect
453. C3#68601direct (N × H)
454. C3C3×#1750semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C2×C3×(C2×D3)⋊C82
C2×C3×#8152
C2×C3×C8×Dic32
C32×#3362
C3×#45242
C3×#17503
#686013