Finite Group Databasebeta

← C2×C3×#5522 #11899950

Subgroups — C2×C3×#5522 #11899950

Order 1152.

3621 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C2×C32×#117C211yesmaximal, characteristic, complemented
3C22×C3×C4×Dic3C211yesmaximal, characteristic, complemented
4C3×#5522C2116yesmaximal, direct product factor
5C2×C3×#893C212yesmaximal, complemented
6C2×C3×#887C212yesmaximal, complemented
7C2×C3×#67⋊C4C214yesmaximal, complemented
8C2×C3×#872C212yesmaximal, complemented
9C2×C3×C42×D3C212yesmaximal, complemented
10C2×C3×C4×#41—31nomaximal
11C2×#5522C311yesmaximal, characteristic, direct product factor
12C23×C32×C4C2211yescharacteristic
13C22×C3×#67C2212yes—
14C23×C3×Dic3C2212yes—
15C2×C3×#235C4116yescomplemented
16C2×C3×#235C2218yes—
17C22×C3×D12C2211yescharacteristic
18C22×C3×C4×D3C2214yes—
19C22×C3×C4×D3C2212yes—
20C22×C3×Dic6C2211yescharacteristic
21C32×#117C2218yescomplemented
22C2×C32×C4⋊C4C2212yes—
23C2×C32×C22⋊C4C2212yes—
24C2×C32×C42C2212yes—
25C2×C3×(C2×D3)⋊C4C2214yes—
26C2×C3×Dic3⋊C4C2214yes—
27C2×C3×C4×Dic3C2212yes—
28C2×C3×C4×Dic3C2214yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×C4×Dic3C2211yescharacteristic
31C2×C3×C4×Dic3C2218yescomplemented
32C3×#893C22116yescomplemented
33C3×#887C22116yescomplemented
34C3×#67⋊C4C22132yescomplemented
35C3×#872C22116yescomplemented
36C3×C42×D3C22116yescomplemented
37C22×C3×#41—31no—
38C22×#235C2×C311yescharacteristic
39C3×C4×#41—316no—
40C2×C3×C4×Q8—32no—
41C2×C3×C4×D4—32no—
42C2×C3×C4×D4—34no—
43C2×C3×#117—21no—
44C2×C3×#117—32no—
45C2×C3×#117D311yescharacteristic, complemented
46C2×C3×#117C2×C311yescharacteristic, complemented
47C22×C3×C42—31no—
48C22×C3×C42—32no—
49C22×C4×Dic3C2×C311yescharacteristic, complemented
50#5522C2×C3116yesdirect product factor
51C2×#893C2×C312yescomplemented
52C2×#887C2×C312yescomplemented
53C2×#67⋊C4C2×C314yescomplemented
54C2×#872C2×C312yescomplemented
55C2×C42×D3C2×C312yescomplemented
56C24×C32C2311yescharacteristic
57C23×C3×D3C2312yes—
58C22×C32×C4C2×C418yescomplemented
59C22×C32×C4C2314yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2314yes—
63C2×C3×#67C2×C4116yescomplemented
64C2×C3×#67C2318yes—
65C22×C3×Dic3—28no—
66C22×C3×Dic3C2318yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C3×#235C2×C4164yescomplemented
71C2×C3×D12C2×C418yescomplemented
72C2×C3×D12C2314yes—
73C2×C3×C4×D3—216no—
74C2×C3×C4×D3C2×C4116yescomplemented
75C2×C3×C4×D3C23116yes—
76C2×C3×C4×D3C2318yes—
77C2×C3×Dic6C2×C418yescomplemented
78C2×C3×Dic6C2314yes—
79C32×C4⋊C4C2318yes—
80C32×C22⋊C4C2318yes—
81C32×C42C2318yes—
82C3×(C2×D3)⋊C4C23116yes—
83C3×Dic3⋊C4C23116yes—
84C3×C4×Dic3C23116yes—
85C3×C4×Dic3C2318yes—
86C3×C4×Dic3C2314yes—
87C3×C4×Dic3C2314yes—
88C2×C4×#41—31noSylow
89C2×C3×#41—316no—
90C2×C3×#41—38no—
91C22×C3×Q8—31no—
92C22×C3×D4—32no—
93C22×C3×D4—31no—
94C23×C3×C4—21no—
95C23×C3×C4—32no—
96C23×C3×C4—32no—
97C23×C3×C4—34no—
98C23×C3×C4C2×D311yescharacteristic
99C23×C3×C4C22×C311yescharacteristic
100C22×#67C22×C312yes—
101C23×Dic3C22×C312yes—
102C2×#235C3×C4116yescomplemented
103C2×#235C22×C318yes—
104C22×D12C22×C311yescharacteristic
105C22×C4×D3C22×C314yes—
106C22×C4×D3C22×C312yes—
107C22×Dic6C22×C311yescharacteristic
108C3×C4×Q8—316no—
109C3×C4×D4—316no—
110C3×C4×D4—332no—
111C3×#117—28no—
112C3×#117—316no—
113C3×#117C2×D318yescomplemented
114C3×#117C22×C318yescomplemented
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—34no—
117C2×C3×C4⋊C4C2×D312yes—
118C2×C3×C4⋊C4C22×C312yes—
119C2×C3×C22⋊C4—22no—
120C2×C3×C22⋊C4—34no—
121C2×C3×C22⋊C4C2×D312yes—
122C2×C3×C22⋊C4C22×C312yes—
123C2×C3×C42—22no—
124C2×C3×C42—34no—
125C2×C3×C42—32no—
126C2×C3×C42—38no—
127C2×C3×C42—31no—
128C2×C3×C42—31no—
129C2×C3×C42—316no—
130C2×C3×C42C2×D312yes—
131C2×C3×C42C22×C312yes—
132C2×(C2×D3)⋊C4C22×C314yes—
133C2×Dic3⋊C4C22×C314yes—
134C2×C4×Dic3C22×C311yescharacteristic
135C2×C4×Dic3C22×C314yes—
136C2×C4×Dic3C22×C311yescharacteristic
137C2×C4×Dic3C22×C312yes—
138C2×C4×Dic3C22×C318yescomplemented
139#893C22×C3116yescomplemented
140#887C22×C3116yescomplemented
141#67⋊C4C22×C3132yescomplemented
142#872C22×C3116yescomplemented
143C42×D3C22×C3116yescomplemented
144C23×C32—24no—
145C23×C32C22×C414yes—
146C23×C32C2412yes—
147C23×C32C2411yescharacteristic
148C22×C3×D3—28no—
149C22×C3×D3C22×C418yes—
150C22×C3×D3C2414yes—
151C2×C32×C4—28no—
152C2×C32×C4C22×C414yes—
153C2×C32×C4C22×C414yes—
154C2×C32×C4C22×C4116yescomplemented
155C2×C32×C4C2418yes—
156C2×C32×C4C2412yes—
157C2×C32×C4C2412yes—
158C3×#67C22×C4132yescomplemented
159C2×C3×Dic3—24no—
160C2×C3×Dic3—216no—
161C2×C3×Dic3C22×C418yes—
162C2×C3×Dic3#4118yescomplemented
163C2×C3×Dic3C2414yes—
164C2×C3×Dic3C2414yes—
165C2×C3×Dic3C2414yes—
166C3×D12C22×C4116yescomplemented
167C3×C4×D3—232no—
168C3×C4×D3C22×C4132yescomplemented
169C3×Dic6C22×C4116yescomplemented
170C22×#41—31no—
171C4×#41—316no—
172C2×C4×Q8—32no—
173C2×C4×D4—32no—
174C2×C4×D4—34no—
175C2×#117—32no—
176C2×#117C3×D311yescharacteristic, complemented
177C22×C42—31no—
178C22×C42—32no—
179C24×C3—21no—
180C24×C3—32no—
181C24×C3C22×D311yescharacteristic
182C24×C3C23×C311yescharacteristic
183C23×D3C23×C312yes—
184C3×#41—364no—
185C2×C3×Q8—38no—
186C2×C3×Q8—34no—
187C2×C3×D4—316no—
188C2×C3×D4—38no—
189C2×C3×D4—38no—
190C2×C3×D4—34no—
191C22×C3×C4—28no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—32no—
197C22×C3×C4—32no—
198C22×C3×C4—316no—
199C22×C3×C4—38no—
200C22×C3×C4—32no—
201C22×C3×C4—38no—
202C22×C3×C4—316no—
203C22×C3×C4—616no—
204C22×C3×C4—68no—
205C22×C3×C4C4×D318yescomplemented
206C22×C3×C4C2×C3×C418yescomplemented
207C22×C3×C4C22×D314yes—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C22×D314yes—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C23×C314yes—
212C22×C3×C4C23×C311yescharacteristic
213C22×C3×C4C23×C311yescharacteristic
214C22×C3×C4C23×C314yes—
215C2×#67C2×C3×C4116yescomplemented
216C2×#67C23×C318yes—
217C22×Dic3—28no—
218C22×Dic3C23×C318yes—
219C22×Dic3C23×C312yes—
220C22×Dic3C23×C312yes—
221C22×Dic3C23×C312yes—
222#235C2×C3×C4164yescomplemented
223C2×D12C2×C3×C418yescomplemented
224C2×D12C23×C314yes—
225C2×C4×D3—216no—
226C2×C4×D3C2×C3×C4116yescomplemented
227C2×C4×D3C23×C3116yes—
228C2×C4×D3C23×C318yes—
229C2×Dic6C2×C3×C418yescomplemented
230C2×Dic6C23×C314yes—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—316no—
233C3×C4⋊C4C22×D318yes—
234C3×C4⋊C4C23×C318yes—
235C3×C22⋊C4—28no—
236C3×C22⋊C4—316no—
237C3×C22⋊C4C22×D318yes—
238C3×C22⋊C4C23×C318yes—
239C3×C42—28no—
240C3×C42—34no—
241C3×C42—34no—
242C3×C42—316no—
243C3×C42—38no—
244C3×C42C22×D318yes—
245C3×C42C23×C318yes—
246(C2×D3)⋊C4C23×C3116yes—
247Dic3⋊C4C23×C3116yes—
248C4×Dic3C23×C314yes—
249C4×Dic3C23×C3116yes—
250C4×Dic3C23×C314yes—
251C4×Dic3C23×C318yes—
252C22×C32—28no—
253C22×C32—24no—
254C22×C32C23×C414yes—
255C22×C32C23×C412yes—
256C22×C32C2×#4114yes—
257C22×C32C2511yescharacteristic
258C2×C3×D3—216no—
259C2×C3×D3—28no—
260C2×C3×D3C23×C418yes—
261C32×C4—28no—
262C32×C4C23×C414yes—
263C32×C4C23×C414yes—
264C3×Dic3—24no—
265C3×Dic3C23×C418yes—
266C3×Dic3C2×#4118yescomplemented
267C2×#41—316no—
268C2×#41—38no—
269C22×Q8—31no—
270C22×D4—32no—
271C22×D4—31no—
272C23×C4—32no—
273C23×C4—32no—
274C23×C4—34no—
275C23×C4C2×C3×D311yescharacteristic
276C4×Q8—316no—
277C4×D4—316no—
278C4×D4—332no—
279#117—316no—
280#117C2×C3×D318yescomplemented
281C2×C4⋊C4—34no—
282C2×C4⋊C4C2×C3×D312yes—
283C2×C22⋊C4—34no—
284C2×C22⋊C4C2×C3×D312yes—
285C2×C42—31no—
286C2×C42—34no—
287C2×C42—32no—
288C2×C42—38no—
289C2×C42—31no—
290C2×C42—316no—
291C2×C42C2×C3×D312yes—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—38no—
298C23×C3—34no—
299C23×C3—44no—
300C23×C3—68no—
301C23×C3C2×C4×D314yes—
302C23×C3C22×C3×C414yes—
303C23×C3C23×D312yes—
304C23×C3C23×D311yescharacteristic
305C23×C3C24×C312yes—
306C23×C3C24×C311yescharacteristic
307C22×D3—28no—
308C22×D3C22×C3×C418yes—
309C22×D3C24×C314yes—
310C3×Q8—316no—
311C3×D4—332no—
312C3×D4—316no—
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—28no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—216no—
321C2×C3×C4—38no—
322C2×C3×C4—38no—
323C2×C3×C4—34no—
324C2×C3×C4—34no—
325C2×C3×C4—332no—
326C2×C3×C4—34no—
327C2×C3×C4—48no—
328C2×C3×C4—632no—
329C2×C3×C4—616no—
330C2×C3×C4—64no—
331C2×C3×C4C2×C4×D314yes—
332C2×C3×C4C2×C4×D314yes—
333C2×C3×C4C2×C4×D3116yescomplemented
334C2×C3×C4C22×C3×C414yes—
335C2×C3×C4C22×C3×C414yes—
336C2×C3×C4C22×C3×C4116yescomplemented
337C2×C3×C4C23×D318yes—
338C2×C3×C4C23×D312yes—
339C2×C3×C4C23×D312yes—
340C2×C3×C4C24×C318yes—
341C2×C3×C4C24×C312yes—
342C2×C3×C4C24×C312yes—
343#67C22×C3×C4132yescomplemented
344C2×Dic3—24no—
345C2×Dic3—216no—
346C2×Dic3C22×C3×C418yes—
347C2×Dic3C3×#4118yescomplemented
348C2×Dic3C24×C314yes—
349C2×Dic3C24×C314yes—
350C2×Dic3C24×C314yes—
351D12C22×C3×C4116yescomplemented
352C4×D3—232no—
353C4×D3C22×C3×C4132yescomplemented
354Dic6C22×C3×C4116yescomplemented
355C2×C32—24no—
356C2×C32C4×#4114yescomplemented
357C2×C32C24×C411yescharacteristic
358C2×C32C22×#4112yes—
359C3×D3—28no—
360C24—32no—
361C24C22×C3×D311yescharacteristic
362#41—364no—
363C2×Q8—34no—
364C2×Q8—38no—
365C2×D4—38no—
366C2×D4—316no—
367C2×D4—34no—
368C2×D4—38no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—38no—
372C22×C4—316no—
373C22×C4—32no—
374C22×C4—38no—
375C22×C4—316no—
376C22×C4—616no—
377C22×C4—68no—
378C22×C4C3×C4×D318yescomplemented
379C22×C4C22×C3×D314yes—
380C22×C4C22×C3×D311yescharacteristic
381C22×C4C22×C3×D311yescharacteristic
382C22×C4C22×C3×D314yes—
383C4⋊C4—316no—
384C4⋊C4C22×C3×D318yes—
385C22⋊C4—316no—
386C22⋊C4C22×C3×D318yes—
387C42—34no—
388C42—34no—
389C42—316no—
390C42—38no—
391C42C22×C3×D318yes—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—28no—
395C22×C3—28no—
396C22×C3—22no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—21no—
400C22×C3—38no—
401C22×C3—48no—
402C22×C3—44no—
403C22×C3—68no—
404C22×C3—616no—
405C22×C3C22×C4×D314yes—
406C22×C3C22×C4×D312yes—
407C22×C3D3×#4114yes—
408C22×C3C23×C3×C414yes—
409C22×C3C23×C3×C412yes—
410C22×C3C2×C3×#4114yes—
411C22×C3C24×D311yescharacteristic
412C22×C3C25×C311yescharacteristic
413C2×D3—28no—
414C2×D3—216no—
415C2×D3C23×C3×C418yes—
416C3×C4—28no—
417C3×C4—28no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—38no—
421C3×C4—38no—
422C3×C4—48no—
423C3×C4—64no—
424C3×C4C22×C4×D314yes—
425C3×C4C22×C4×D314yes—
426C3×C4C23×C3×C414yes—
427C3×C4C23×C3×C414yes—
428Dic3—24no—
429Dic3C23×C3×C418yes—
430Dic3C2×C3×#4118yescomplemented
431C32C2×C4×#4111yescharacteristic, complemented, Sylow
432C23—24no—
433C23—38no—
434C23—34no—
435C23—68no—
436C23C2×C3×C4×D314yes—
437C23C23×C3×D312yes—
438C23C23×C3×D311yescharacteristic
439Q8—316no—
440D4—332no—
441D4—316no—
442C2×C4—28no—
443C2×C4—38no—
444C2×C4—38no—
445C2×C4—34no—
446C2×C4—34no—
447C2×C4—332no—
448C2×C4—34no—
449C2×C4—632no—
450C2×C4—616no—
451C2×C4—64no—
452C2×C4C2×C3×C4×D314yes—
453C2×C4C2×C3×C4×D314yes—
454C2×C4C2×C3×C4×D3116yescomplemented
455C2×C4C23×C3×D318yes—
456C2×C4C23×C3×D312yes—
457C2×C4C23×C3×D312yes—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—44no—
464C2×C3—68no—
465C2×C3#552214yesdirect product factor
466C2×C3C3×C4×#4114yescomplemented
467C2×C3C23×C4×D311yescharacteristic
468C2×C3C2×D3×#4112yes—
469C2×C3C24×C3×C411yescharacteristic
470C2×C3C22×C3×#4112yes—
471D3—28no—
472C22—28no—
473C22—24no—
474C22—38no—
475C22—616no—
476C22—68no—
477C22C22×C3×C4×D314yes—
478C22C22×C3×C4×D312yes—
479C22C3×D3×#4114yes—
480C22C24×C3×D311yescharacteristic
481C4—28no—
482C4—38no—
483C4—38no—
484C4—64no—
485C4C22×C3×C4×D314yes—
486C4C22×C3×C4×D314yes—
487C3—21no—
488C3C2×#552211yescharacteristic, direct product factor
489C3C2×C3×C4×#4111yescharacteristic, complemented
490C2—24no—
491C2—68no—
492C2C3×#552214yesdirect product factor
493C2C23×C3×C4×D311yescharacteristic
494C2C2×C3×D3×#4112yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC2×C3C24×C3×C4
FrattiniC22C24×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC22×C3×#235C2
cosocleC22×C3C25×C3
hypercenterC2×C3×#117D3
hypocenterC3C2×C3×C4×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
2. C2×C32×#117C2semidirect
3. C22×C3×C4×Dic3C2semidirect
4. C3×#5522C2direct (N × H)
5. C2×C3×#893C2semidirect
6. C2×C3×#887C2semidirect
7. C2×C3×#67⋊C4C2semidirect
8. C2×C3×#872C2semidirect
9. C2×C3×C42×D3C2semidirect
11. C2×#5522C3direct (N × H)
15. C2×C3×#235C4semidirect
21. C32×#117C22semidirect
31. C2×C3×C4×Dic3C22semidirect
32. C3×#893C22semidirect
33. C3×#887C22semidirect
34. C3×#67⋊C4C22semidirect
35. C3×#872C22semidirect
36. C3×C42×D3C22semidirect
45. C2×C3×#117D3semidirect
46. C2×C3×#117C2×C3semidirect
49. C22×C4×Dic3C2×C3semidirect
50. #5522C2×C3direct (N × H)
51. C2×#893C2×C3semidirect
52. C2×#887C2×C3semidirect
53. C2×#67⋊C4C2×C3semidirect
54. C2×#872C2×C3semidirect
55. C2×C42×D3C2×C3semidirect
58. C22×C32×C4C2×C4semidirect
63. C2×C3×#67C2×C4semidirect
70. C3×#235C2×C4semidirect
71. C2×C3×D12C2×C4semidirect
74. C2×C3×C4×D3C2×C4semidirect
77. C2×C3×Dic6C2×C4semidirect
102. C2×#235C3×C4semidirect
113. C3×#117C2×D3semidirect
114. C3×#117C22×C3semidirect
138. C2×C4×Dic3C22×C3semidirect
139. #893C22×C3semidirect
140. #887C22×C3semidirect
141. #67⋊C4C22×C3semidirect
142. #872C22×C3semidirect
143. C42×D3C22×C3semidirect
154. C2×C32×C4C22×C4semidirect
158. C3×#67C22×C4semidirect
162. C2×C3×Dic3#41semidirect
166. C3×D12C22×C4semidirect
168. C3×C4×D3C22×C4semidirect
169. C3×Dic6C22×C4semidirect
176. C2×#117C3×D3semidirect
205. C22×C3×C4C4×D3semidirect
206. C22×C3×C4C2×C3×C4semidirect
215. C2×#67C2×C3×C4semidirect
222. #235C2×C3×C4semidirect
223. C2×D12C2×C3×C4semidirect
226. C2×C4×D3C2×C3×C4semidirect
229. C2×Dic6C2×C3×C4semidirect
266. C3×Dic3C2×#41semidirect
280. #117C2×C3×D3semidirect
333. C2×C3×C4C2×C4×D3semidirect
336. C2×C3×C4C22×C3×C4semidirect
343. #67C22×C3×C4semidirect
347. C2×Dic3C3×#41semidirect
351. D12C22×C3×C4semidirect
353. C4×D3C22×C3×C4semidirect
354. Dic6C22×C3×C4semidirect
356. C2×C32C4×#41semidirect
378. C22×C4C3×C4×D3semidirect
430. Dic3C2×C3×#41semidirect
431. C32C2×C4×#41semidirect
454. C2×C4C2×C3×C4×D3semidirect
465. C2×C3#5522direct (N × H)
466. C2×C3C3×C4×#41semidirect
488. C3C2×#5522direct (N × H)
489. C3C2×C3×C4×#41semidirect
492. C2C3×#5522direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C2×C32×#1172
C22×C3×C4×Dic32
C3×#55222
C2×C3×#8932
C2×C3×#8872
C2×C3×#67⋊C42
C2×C3×#8722
C2×C3×C42×D32
C2×C3×C4×#413
C2×#55223