Finite Group Databasebeta

← C2×C3×#5185 #11865682

Subgroups — C2×C3×#5185 #11865682

Order 1152.

1129 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic
3C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4C22×C3×#215C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#5185C218yesmaximal, direct product factor
7C2×C3×#821C211yesmaximal, characteristic
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#464—31nomaximal
10C2×#5185C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C2×C32×Q16—24no—
15C2×C32×Q16C2214yescomplemented
16C22×C32×C8C2211yescharacteristic
17C3×#947—22no—
18C3×#947C2214yes—
19C3×Q8×Dic3—22no—
20C3×Q8×Dic3C2214yescomplemented
21C2×C3×#215—24no—
22C2×C3×#215C2214yes—
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×(C2×D3)⋊C4—21no—
25C2×C3×C4⋊Dic3—21no—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4×Dic3—21no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#835C2214yescomplemented
31C3×#821C2214yes—
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×Q16—21no—
34C22×C3×Q16D311yescharacteristic, complemented
35C22×C3×Q16C2×C311yescharacteristic, complemented
36C22×C3×SD16—31no—
37C2×C3×#124—31no—
38C2×C3×C4×Q8—31no—
39C2×#947C2×C311yescharacteristic
40C2×Q8×Dic3C2×C311yescharacteristic, complemented
41C22×#215C2×C311yescharacteristic
42C3×#464—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#835C2×C311yescharacteristic, complemented
47#5185C2×C318yesdirect product factor
48C2×#821C2×C311yescharacteristic
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C23×C3×D3—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8—22no—
56C2×C32×Q8D414yes—
57C2×C32×Q8C2312yes—
58C2×C32×Q8C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3D411yescharacteristic
64C22×C3×Dic3D411yescharacteristic
65C2×C3×D12—22no—
66C2×C3×D12—22no—
67C2×C3×D12—21no—
68C2×C3×D12C2312yes—
69C32×Q16—28no—
70C2×C32×C8—22no—
71C2×C32×C8C2312yes—
72C3×#215—28no—
73C3×(C2×D3)⋊C4—24no—
74C3×(C2×D3)⋊C4—24no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—24no—
77C3×C4⋊Dic3C2312yes—
78C3×C4×Dic3—24no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2312yes—
83C2×#464—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—31no—
91C22×D12C22×C311yescharacteristic
92C2×C3×Q16—28no—
93C2×C3×Q16—24no—
94C2×C3×Q16—24no—
95C2×C3×Q16—24no—
96C2×C3×Q16C2×D314yescomplemented
97C2×C3×Q16C22×C314yescomplemented
98C2×C3×SD16—34no—
99C2×C3×SD16—64no—
100C22×C3×C8—21no—
101C22×C3×C8—31no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C3×#124—34no—
105C3×#124—62no—
106C3×C4×Q8—34no—
107C3×C4×Q8—62no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—61no—
110C2×C3×C22⋊C4—61no—
111C2×C3×C22⋊C4—61no—
112C2×C3×C42—31no—
113C2×C3×C42—61no—
114#947—22no—
115#947C22×C314yes—
116Q8×Dic3—22no—
117Q8×Dic3C22×C314yescomplemented
118C2×#215—24no—
119C2×#215C22×C314yes—
120C2×(C2×D3)⋊C4—21no—
121C2×(C2×D3)⋊C4—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C22×C311yescharacteristic
124C2×C4×Dic3—21no—
125C2×C4×Dic3C22×C311yescharacteristic
126C22×C3⋊C8C22×C311yescharacteristic
127C3×C4⋊C8—34no—
128C3×#103—34no—
129C3×#102—34no—
130#835C22×C314yescomplemented
131#821C22×C314yes—
132Dic3⋊C8C22×C314yescomplemented
133C23×C32C2×D411yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—42no—
136C22×C3×D3—42no—
137C22×C3×D3—42no—
138C32×Q8—22no—
139C32×Q8—24no—
140C32×Q8—22no—
141C32×Q8—22no—
142C32×Q8—42no—
143C32×Q8C2×D414yes—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—42no—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D412yes—
150C2×C32×C4#4112yes—
151C2×C32×C4C2411yescharacteristic
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3C2×D412yes—
157C2×C3×Dic3C2×D412yes—
158C3×D12—22no—
159C3×D12—22no—
160C3×D12—42no—
161C32×C8—22no—
162C3×C3⋊C8—22no—
163C22×Q16C3×D311yescharacteristic, complemented
164C22×SD16—31no—
165C2×#124—31no—
166C2×C4×Q8—31no—
167#464—38no—
168C2×C4⋊C8—31no—
169C2×#103—31no—
170C2×#102—31no—
171C24×C3—61no—
172C23×D3—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—22no—
179C2×C3×Q8—24no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—24no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—41no—
188C2×C3×Q8—41no—
189C2×C3×Q8—42no—
190C2×C3×Q8—42no—
191C2×C3×Q8#6714yes—
192C2×C3×Q8C3×D414yes—
193C2×C3×Q8C22×D312yes—
194C2×C3×Q8C22×D312yes—
195C2×C3×Q8C23×C312yes—
196C2×C3×Q8C23×C312yes—
197C2×C3×D4—32no—
198C2×C3×D4—61no—
199C2×C3×D4—62no—
200C2×C3×D4—62no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—41no—
209C22×C3×C4—41no—
210C22×C3×C4—61no—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C311yescharacteristic
213C22×Dic3—21no—
214C22×Dic3C3×D411yescharacteristic
215C22×Dic3C3×D411yescharacteristic
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12C23×C312yes—
220C3×Q16—216no—
221C3×Q16—28no—
222C3×Q16—28no—
223C3×SD16—68no—
224C2×C3×C8—24no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—32no—
229C2×C3×C8—62no—
230C2×C3×C8C22×D312yes—
231C2×C3×C8C23×C312yes—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—64no—
234C3×C4⋊C4—61no—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—64no—
237C3×C42—32no—
238C3×C42—32no—
239C3×C42—64no—
240#215—28no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—24no—
243C4⋊Dic3—21no—
244C4⋊Dic3—24no—
245C4⋊Dic3C23×C312yes—
246C4×Dic3—24no—
247C4×Dic3C3×D412yes—
248C4×Dic3C23×C312yes—
249C2×C3⋊C8—22no—
250C2×C3⋊C8C23×C312yes—
251C22×C32#12112yes—
252C22×C32#13512yes—
253C22×C32#13712yes—
254C22×C32C22×D411yescharacteristic
255C2×C3×D3—42no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—84no—
259C32×C4—24no—
260C32×C4—42no—
261C32×C4#12112yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C2×#4111yescharacteristic
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—42no—
267C22×Q8C2×C3×D311yescharacteristic
268C22×Q8C2×C3×D311yescharacteristic
269C22×D4—31no—
270C2×Q16—24no—
271C2×Q16C2×C3×D314yescomplemented
272C2×SD16—34no—
273C2×SD16—64no—
274C22×C8—31no—
275C22×C8C2×C3×D311yescharacteristic
276#124—34no—
277#124—62no—
278C4×Q8—34no—
279C4×Q8—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—61no—
286C4⋊C8—34no—
287#103—34no—
288#102—34no—
289C23×C3—21no—
290C23×C3—62no—
291C23×C3—122no—
292C23×C3—122no—
293C23×C3—122no—
294C23×C3D3×D411yescharacteristic
295C23×C3C2×C3×D411yescharacteristic
296C22×D3—22no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C3×Q8—24no—
301C3×Q8—24no—
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—24no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—24no—
310C3×Q8—44no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—44no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8C2×#6714yes—
317C3×Q8C2×C3×D414yes—
318C3×D4—62no—
319C3×D4—62no—
320C3×D4—122no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—44no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—122no—
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4#23712yes—
345C2×C3×C4C2×#6712yes—
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D412yes—
348C2×C3×C4C3×#4112yes—
349C2×C3×C4C23×D311yescharacteristic
350C2×C3×C4C24×C311yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—42no—
355C2×Dic3C2×C3×D412yes—
356C2×Dic3C2×C3×D412yes—
357D12—22no—
358D12—22no—
359D12—42no—
360C3×C8—24no—
361C3×C8—22no—
362C3×C8—22no—
363C3×C8—62no—
364C3⋊C8—22no—
365C2×C32#46414yescomplemented
366C2×C32C2×#12111yescharacteristic
367C2×C32C2×#13511yescharacteristic
368C2×C32C2×#13711yescharacteristic
369C3×D3—82no—
370C24—61no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C3×#6714yes—
377C2×Q8C22×C3×D312yes—
378C2×Q8C22×C3×D312yes—
379C2×D4—32no—
380C2×D4—61no—
381C2×D4—62no—
382C2×D4—62no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—61no—
388C22×C4C22×C3×D311yescharacteristic
389Q16—28no—
390SD16—68no—
391C2×C8—22no—
392C2×C8—32no—
393C2×C8—62no—
394C2×C8C22×C3×D312yes—
395C4⋊C4—32no—
396C4⋊C4—64no—
397C4⋊C4—61no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C42—32no—
401C42—32no—
402C42—64no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3—122no—
410C22×C3—244no—
411C22×C3#91812yes—
412C22×C3#91912yes—
413C22×C3#93912yes—
414C22×C3C3×#12112yes—
415C22×C3C3×#13512yes—
416C22×C3C3×#13712yes—
417C22×C3C2×D3×D411yescharacteristic
418C22×C3C22×C3×D411yescharacteristic
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—84no—
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—122no—
435C3×C4#93912yes—
436C3×C4C3×#12112yes—
437C3×C4C2×#23711yescharacteristic
438C3×C4C22×#6711yescharacteristic
439C3×C4C22×C3×D411yescharacteristic
440C3×C4C2×C3×#4111yescharacteristic
441Dic3—22no—
442Dic3—22no—
443Dic3—42no—
444C32C2×#46411yescharacteristic, complemented, Sylow
445C23—62no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23C3×D3×D411yescharacteristic
450Q8—24no—
451Q8—22no—
452Q8—22no—
453Q8—22no—
454Q8—42no—
455Q8C2×C3×#6714yes—
456D4—62no—
457D4—62no—
458D4—122no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—42no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—122no—
469C2×C4C3×D3×D411yescharacteristic
470C2×C4C3×#23712yes—
471C2×C4C2×C3×#6712yes—
472C2×C4C23×C3×D311yescharacteristic
473C8—22no—
474C8—62no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—24no—
478C2×C3—21no—
479C2×C3—242no—
480C2×C3#518514yesdirect product factor
481C2×C3C3×#46414yescomplemented
482C2×C3C2×#91811yescharacteristic
483C2×C3C2×#91911yescharacteristic
484C2×C3C2×#93911yescharacteristic
485C2×C3C2×C3×#12111yescharacteristic
486C2×C3C2×C3×#13511yescharacteristic
487C2×C3C2×C3×#13711yescharacteristic
488D3—82no—
489C22—122no—
490C22—122no—
491C22—122no—
492C22—244no—
493C22C3×#91812yes—
494C22C3×#91912yes—
495C22C3×#93912yes—
496C22C2×C3×D3×D411yescharacteristic
497C4—24no—
498C4—42no—
499C4—62no—
500C4—62no—
501C4—122no—
502C4C3×#93912yes—
503C4C2×C3×#23711yescharacteristic
504C4C22×C3×#6711yescharacteristic
505C3—21no—
506C3C2×#518511yescharacteristic, direct product factor
507C3C2×C3×#46411yescharacteristic, complemented
508C2—242no—
509C2C3×#518514yesdirect product factor
510C2C2×C3×#91811yescharacteristic
511C2C2×C3×#91911yescharacteristic
512C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×Q16C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
hypercenterC22×C3×Q16D3
hypocenterC3C2×C3×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C22×C32×Q16C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#5185C2direct (N × H)
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5185C3direct (N × H)
15. C2×C32×Q16C22semidirect
20. C3×Q8×Dic3C22semidirect
30. C3×#835C22semidirect
32. C3×Dic3⋊C8C22semidirect
34. C22×C3×Q16D3semidirect
35. C22×C3×Q16C2×C3semidirect
40. C2×Q8×Dic3C2×C3semidirect
46. C2×#835C2×C3semidirect
47. #5185C2×C3direct (N × H)
49. C2×Dic3⋊C8C2×C3semidirect
96. C2×C3×Q16C2×D3semidirect
97. C2×C3×Q16C22×C3semidirect
117. Q8×Dic3C22×C3semidirect
130. #835C22×C3semidirect
132. Dic3⋊C8C22×C3semidirect
163. C22×Q16C3×D3semidirect
271. C2×Q16C2×C3×D3semidirect
365. C2×C32#464semidirect
444. C32C2×#464semidirect
480. C2×C3#5185direct (N × H)
481. C2×C3C3×#464semidirect
506. C3C2×#5185direct (N × H)
507. C3C2×C3×#464semidirect
509. C2C3×#5185direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Q162
C2×C3×#9472
C2×C3×Q8×Dic32
C22×C3×#2152
C2×C3×#8352
C3×#51852
C2×C3×#8212
C2×C3×Dic3⋊C82
C2×C3×#4643
C2×#51853