Finite Group Databasebeta

← C2×C3×#5190 #11874380

Subgroups — C2×C3×#5190 #11874380

Order 1152.

1257 conjugacy classes of subgroups.

The subgroup lattice has 500 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×#944C211yesmaximal, characteristic, complemented
4C22×C3×#216C211yesmaximal, characteristic, complemented
5C22×C3×#215C211yesmaximal, characteristic
6C22×C3×#204C211yesmaximal, characteristic
7C3×#5190C218yesmaximal, direct product factor
8C2×C3×#815C211yesmaximal, characteristic, complemented
9C2×C3×#497—31nomaximal
10C2×#5190C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×Q16—24no—
16C2×C32×Q16C2214yescomplemented
17C22×C32×C8C2211yescharacteristic
18C3×#947—22no—
19C3×#947C2214yes—
20C3×#944—22no—
21C3×#944C2214yescomplemented
22C2×C3×#216—24no—
23C2×C3×#216C2214yescomplemented
24C2×C3×#215—24no—
25C2×C3×#215C2214yes—
26C2×C3×(C2×D3)⋊C4—22no—
27C2×C3×Dic3⋊C4—22no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×#204—24no—
31C2×C3×#204C2214yes—
32C3×#815C2214yescomplemented
33C22×C3×Q16—21no—
34C22×C3×Q16—31no—
35C22×C3×Q16D311yescharacteristic, complemented
36C22×C3×Q16C2×C311yescharacteristic, complemented
37C22×C3×SD16—31no—
38C22×C3×SD16—31no—
39C2×C3×C4⋊Q8—31no—
40C2×C3×#124—31no—
41C2×#947C2×C311yescharacteristic
42C2×#944C2×C311yescharacteristic, complemented
43C22×#216C2×C311yescharacteristic, complemented
44C22×#215C2×C311yescharacteristic
45C22×#204C2×C311yescharacteristic
46C3×#497—38no—
47C2×C3×#97—31no—
48#5190C2×C318yesdirect product factor
49C2×#815C2×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×Q8—22no—
57C2×C32×Q8C2312yes—
58C2×C32×Q8C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3D412yes—
64C2×C3×D12—22no—
65C2×C3×D12—22no—
66C2×C3×D12—21no—
67C2×C3×D12C2312yes—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2312yes—
72C32×Q16—28no—
73C2×C32×C8D414yes—
74C2×C32×C8C2312yes—
75C3×#216—28no—
76C3×#215—28no—
77C3×(C2×D3)⋊C4—28no—
78C3×Dic3⋊C4—28no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8D414yes—
82C2×C3×C3⋊C8C2312yes—
83C3×#204—28no—
84C2×#497—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8—21no—
87C22×C3×Q8—31no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C2×D311yescharacteristic
90C22×C3×Q8C22×C311yescharacteristic
91C22×C3×Q8C22×C311yescharacteristic
92C22×C3×D4—31no—
93C22×D12C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C2×C3×Q16—28no—
96C2×C3×Q16—24no—
97C2×C3×Q16—24no—
98C2×C3×Q16—24no—
99C2×C3×Q16—34no—
100C2×C3×Q16—64no—
101C2×C3×Q16C2×D314yescomplemented
102C2×C3×Q16C22×C314yescomplemented
103C2×C3×SD16—34no—
104C2×C3×SD16—34no—
105C2×C3×SD16—64no—
106C2×C3×SD16—64no—
107C22×C3×C8—21no—
108C22×C3×C8—31no—
109C22×C3×C8C2×D311yescharacteristic
110C22×C3×C8C22×C311yescharacteristic
111C3×C4⋊Q8—34no—
112C3×C4⋊Q8—62no—
113C3×#124—34no—
114C3×#124—62no—
115C2×C3×C4⋊C4—62no—
116C2×C3×C22⋊C4—62no—
117C2×C3×C42—31no—
118#947—22no—
119#947C22×C314yes—
120#944—22no—
121#944C22×C314yescomplemented
122C2×#216—24no—
123C2×#216C22×C314yescomplemented
124C2×#215—24no—
125C2×#215C22×C314yes—
126C2×(C2×D3)⋊C4—22no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C2×#204—24no—
131C2×#204C22×C314yes—
132C3×#97—34no—
133#815C22×C314yescomplemented
134C23×C32C2×D411yescharacteristic
135C22×C3×D3—22no—
136C22×C3×D3—42no—
137C22×C3×D3—44no—
138C32×Q8—22no—
139C32×Q8—22no—
140C32×Q8—22no—
141C32×Q8—22no—
142C32×Q8—42no—
143C32×Q8—42no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—42no—
147C2×C32×C4—42no—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D412yes—
150C2×C32×C4C2×D412yes—
151C2×C32×C4C2411yescharacteristic
152C2×C3×Dic3—24no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3C2×D414yes—
156C3×D12—22no—
157C3×D12—22no—
158C3×D12—42no—
159C3×Dic6—22no—
160C3×Dic6—22no—
161C3×Dic6—42no—
162C32×C8C2×D414yes—
163C3×C3⋊C8C2×D414yes—
164C22×Q16—31no—
165C22×Q16C3×D311yescharacteristic, complemented
166C22×SD16—31no—
167C22×SD16—31no—
168C2×C4⋊Q8—31no—
169C2×#124—31no—
170#497—38no—
171C2×#97—31no—
172C24×C3—61no—
173C23×D3—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—24no—
179C2×C3×Q8—24no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—32no—
191C2×C3×Q8—41no—
192C2×C3×Q8—41no—
193C2×C3×Q8—42no—
194C2×C3×Q8—42no—
195C2×C3×Q8—61no—
196C2×C3×Q8—62no—
197C2×C3×Q8—62no—
198C2×C3×Q8C22×D312yes—
199C2×C3×Q8C22×D312yes—
200C2×C3×Q8C23×C312yes—
201C2×C3×Q8C23×C312yes—
202C2×C3×D4—32no—
203C2×C3×D4—62no—
204C2×C3×D4—61no—
205C2×C3×D4—62no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—32no—
212C22×C3×C4—41no—
213C22×C3×C4—41no—
214C22×C3×C4—61no—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C22×Dic3—21no—
218C22×Dic3C3×D412yes—
219C2×D12—22no—
220C2×D12—22no—
221C2×D12—21no—
222C2×D12C23×C312yes—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—21no—
226C2×Dic6C23×C312yes—
227C3×Q16—216no—
228C3×Q16—28no—
229C3×Q16—28no—
230C3×Q16—68no—
231C3×SD16—68no—
232C3×SD16—68no—
233C2×C3×C8—24no—
234C2×C3×C8—22no—
235C2×C3×C8—34no—
236C2×C3×C8—32no—
237C2×C3×C8#6714yes—
238C2×C3×C8C3×D414yes—
239C2×C3×C8C22×D312yes—
240C2×C3×C8C23×C312yes—
241C3×C4⋊C4—68no—
242C3×C22⋊C4—68no—
243C3×C42—32no—
244C3×C42—32no—
245#216—28no—
246#215—28no—
247(C2×D3)⋊C4—28no—
248Dic3⋊C4—28no—
249C4×Dic3C3×D412yes—
250C4×Dic3C23×C312yes—
251C2×C3⋊C8C3×D414yes—
252C2×C3⋊C8C23×C312yes—
253#204—28no—
254C22×C32#12712yes—
255C22×C32#13714yes—
256C22×C32C22×D411yescharacteristic
257C2×C3×D3—42no—
258C2×C3×D3—44no—
259C2×C3×D3—84no—
260C32×C4—42no—
261C32×C4—42no—
262C32×C4#12712yes—
263C32×C4C22×D411yescharacteristic
264C32×C4C22×D411yescharacteristic
265C3×Dic3—24no—
266C3×Dic3—42no—
267C22×Q8—31no—
268C22×Q8C2×C3×D311yescharacteristic
269C22×Q8C2×C3×D311yescharacteristic
270C22×D4—31no—
271C2×Q16—24no—
272C2×Q16—34no—
273C2×Q16—64no—
274C2×Q16C2×C3×D314yescomplemented
275C2×SD16—34no—
276C2×SD16—34no—
277C2×SD16—64no—
278C2×SD16—64no—
279C22×C8—31no—
280C22×C8C2×C3×D311yescharacteristic
281C4⋊Q8—34no—
282C4⋊Q8—62no—
283#124—34no—
284#124—62no—
285C2×C4⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C42—31no—
288#97—34no—
289C23×C3—21no—
290C23×C3—62no—
291C23×C3—124no—
292C23×C3—122no—
293C23×C3D3×D411yescharacteristic
294C23×C3C2×C3×D411yescharacteristic
295C22×D3—22no—
296C22×D3—42no—
297C22×D3—44no—
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—22no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—44no—
311C3×Q8—44no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—62no—
317C3×Q8—62no—
318C3×Q8—122no—
319C3×D4—62no—
320C3×D4—62no—
321C3×D4—122no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—34no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—64no—
340C2×C3×C4—62no—
341C2×C3×C4—122no—
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D412yes—
344C2×C3×C4C2×#6712yes—
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C2×C3×D412yes—
347C2×C3×C4C2×C3×D412yes—
348C2×C3×C4C23×D311yescharacteristic
349C2×C3×C4C24×C311yescharacteristic
350C2×Dic3—24no—
351C2×Dic3—22no—
352C2×Dic3—42no—
353C2×Dic3C2×C3×D414yes—
354D12—22no—
355D12—22no—
356D12—42no—
357Dic6—22no—
358Dic6—22no—
359Dic6—42no—
360C3×C8—24no—
361C3×C8—34no—
362C3×C8C2×#6714yes—
363C3×C8C2×C3×D414yes—
364C3⋊C8C2×C3×D414yes—
365C2×C32#49714yescomplemented
366C2×C32C2×#12711yescharacteristic
367C2×C32C2×#13712yes—
368C3×D3—82no—
369C24—61no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8C22×C3×D312yes—
381C2×Q8C22×C3×D312yes—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—61no—
385C2×D4—62no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—61no—
390C22×C4C22×C3×D311yescharacteristic
391Q16—28no—
392Q16—68no—
393SD16—68no—
394SD16—68no—
395C2×C8—32no—
396C2×C8—34no—
397C2×C8C3×#6714yes—
398C2×C8C22×C3×D312yes—
399C4⋊C4—68no—
400C22⋊C4—68no—
401C42—32no—
402C42—32no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—124no—
407C22×C3—122no—
408C22×C3—244no—
409C22×C3#91814yes—
410C22×C3#94012yes—
411C22×C3C3×#12712yes—
412C22×C3C3×#13714yes—
413C22×C3C2×D3×D411yescharacteristic
414C22×C3C22×C3×D411yescharacteristic
415C2×D3—44no—
416C2×D3—42no—
417C2×D3—84no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—44no—
422C3×C4—44no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—64no—
428C3×C4—122no—
429C3×C4#94012yes—
430C3×C4C3×#12712yes—
431C3×C4C2×D3×D411yescharacteristic
432C3×C4C22×#6711yescharacteristic
433C3×C4C22×C3×D411yescharacteristic
434C3×C4C22×C3×D411yescharacteristic
435Dic3—24no—
436Dic3—42no—
437C32C2×#49711yescharacteristic, complemented, Sylow
438C23—62no—
439C23—124no—
440C23—122no—
441C23C3×D3×D411yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—22no—
445Q8—22no—
446Q8—42no—
447Q8—42no—
448Q8—62no—
449Q8—62no—
450Q8—122no—
451D4—62no—
452D4—62no—
453D4—122no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—34no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—122no—
462C2×C4C3×D3×D411yescharacteristic
463C2×C4C3×D3×D412yes—
464C2×C4C2×C3×#6712yes—
465C2×C4C23×C3×D311yescharacteristic
466C8—34no—
467C8C2×C3×#6714yes—
468C2×C3—22no—
469C2×C3—24no—
470C2×C3—21no—
471C2×C3—242no—
472C2×C3#519014yesdirect product factor
473C2×C3C3×#49714yescomplemented
474C2×C3C2×#91812yes—
475C2×C3C2×#94011yescharacteristic
476C2×C3C2×C3×#12711yescharacteristic
477C2×C3C2×C3×#13712yes—
478D3—82no—
479C22—124no—
480C22—122no—
481C22—244no—
482C22C3×#91814yes—
483C22C3×#94012yes—
484C22C2×C3×D3×D411yescharacteristic
485C4—42no—
486C4—42no—
487C4—64no—
488C4—122no—
489C4C3×#94012yes—
490C4C2×C3×D3×D411yescharacteristic
491C4C22×C3×#6711yescharacteristic
492C3—21no—
493C3C2×#519011yescharacteristic, direct product factor
494C3C2×C3×#49711yescharacteristic, complemented
495C2—242no—
496C2C3×#519014yesdirect product factor
497C2C2×C3×#91812yes—
498C2C2×C3×#94011yescharacteristic
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×#497
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×#944C2semidirect
4. C22×C3×#216C2semidirect
7. C3×#5190C2direct (N × H)
8. C2×C3×#815C2semidirect
10. C2×#5190C3direct (N × H)
16. C2×C32×Q16C22semidirect
21. C3×#944C22semidirect
23. C2×C3×#216C22semidirect
32. C3×#815C22semidirect
35. C22×C3×Q16D3semidirect
36. C22×C3×Q16C2×C3semidirect
42. C2×#944C2×C3semidirect
43. C22×#216C2×C3semidirect
48. #5190C2×C3direct (N × H)
49. C2×#815C2×C3semidirect
101. C2×C3×Q16C2×D3semidirect
102. C2×C3×Q16C22×C3semidirect
121. #944C22×C3semidirect
123. C2×#216C22×C3semidirect
133. #815C22×C3semidirect
165. C22×Q16C3×D3semidirect
274. C2×Q16C2×C3×D3semidirect
365. C2×C32#497semidirect
437. C32C2×#497semidirect
472. C2×C3#5190direct (N × H)
473. C2×C3C3×#497semidirect
493. C3C2×#5190direct (N × H)
494. C3C2×C3×#497semidirect
496. C2C3×#5190direct (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×#9442
C22×C3×#2162
C22×C3×#2152
C22×C3×#2042
C3×#51902
C2×C3×#8152
C2×C3×#4973
C2×#51903