Finite Group Databasebeta

← C2×C3×#5728 #11900818

Subgroups — C2×C3×#5728 #11900818

Order 1152.

2481 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#946C212yesmaximal
3C2×C3×Q8×Dic3C212yesmaximal, complemented
4C3×#5728C2116yesmaximal, direct product factor
5C2×C3×#897C212yesmaximal
6C2×C3×#892C212yesmaximal
7C2×C3×#890C212yesmaximal, complemented
8C2×C3×#887C212yesmaximal, complemented
9C2×C3×C42×D3C211yesmaximal, characteristic
10C2×C3×#869C211yesmaximal, characteristic, complemented
11C2×C3×#533—31nomaximal
12C2×#5728C311yesmaximal, characteristic, direct product factor
13C22×C32×Q8C2212yes—
14C22×C3×C4×D3C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C22×C3×C4×D3C2211yescharacteristic
17C22×C3×Dic6C2212yes—
18C32×C4⋊Q8C2218yescomplemented
19C2×C32×C4⋊C4C2212yes—
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C42C2211yescharacteristic
22C3×#946C22116yes—
23C3×Q8×Dic3C22116yescomplemented
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4⋊Dic3C2214yes—
27C2×C3×C4⋊Dic3C2212yes—
28C2×C3×Dic3⋊C4C2214yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×C4×Dic3C2214yes—
31C2×C3×C4×Dic3C2211yescharacteristic
32C2×C3×C4×Dic3C2211yescharacteristic
33C3×#897C22116yes—
34C3×#892C22116yes—
35C3×#890C22116yescomplemented
36C3×#887C22116yescomplemented
37C3×C42×D3C2218yes—
38C3×#869C2218yescomplemented
39C3×#533—316no—
40C2×C3×C4⋊Q8—21no—
41C2×C3×C4⋊Q8—31no—
42C2×C3×C4⋊Q8D311yescharacteristic, complemented
43C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
44C2×C3×#125—32no—
45C2×C3×C22⋊Q8—32no—
46C2×C3×C22⋊Q8—32no—
47C2×C3×C4×Q8—32no—
48C2×C3×C4×Q8—32no—
49C2×C3×#117—32no—
50C22×C3×C42—31no—
51C2×#946C2×C312yes—
52C2×Q8×Dic3C2×C312yescomplemented
53#5728C2×C3116yesdirect product factor
54C2×#897C2×C312yes—
55C2×#892C2×C312yes—
56C2×#890C2×C312yescomplemented
57C2×#887C2×C312yescomplemented
58C2×C42×D3C2×C311yescharacteristic
59C2×#869C2×C311yescharacteristic, complemented
60C23×C3×D3C2311yescharacteristic
61C2×C32×Q8—28no—
62C2×C32×Q8C2318yes—
63C22×C32×C4C2314yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C3×Dic3C2311yescharacteristic
68C22×C3×Dic3C2311yescharacteristic
69C22×C3×Dic3C2314yes—
70C22×C3×Dic3C2311yescharacteristic
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3—24no—
73C2×C3×C4×D3Q818yescomplemented
74C2×C3×C4×D3C2314yes—
75C2×C3×C4×D3C2314yes—
76C2×C3×C4×D3C2314yes—
77C2×C3×Dic6—28no—
78C2×C3×Dic6C2318yes—
79C32×C4⋊C4C2318yes—
80C32×C4⋊C4C2318yes—
81C32×C42C2314yes—
82C3×(C2×D3)⋊C4C23116yes—
83C3×C4⋊Dic3C2318yes—
84C3×C4⋊Dic3C23116yes—
85C3×C4⋊Dic3C2318yes—
86C3×Dic3⋊C4C23116yes—
87C3×C4×Dic3C2314yes—
88C3×C4×Dic3C23116yes—
89C3×C4×Dic3C2314yes—
90C3×C4×Dic3C2314yes—
91C2×#533—31noSylow
92C22×C3×Q8—22no—
93C22×C3×Q8—32no—
94C22×C3×Q8C2×D312yes—
95C22×C3×Q8C22×C312yes—
96C23×C3×C4—31no—
97C23×C3×C4—31no—
98C23×C3×C4—31no—
99C22×C4×D3C22×C311yescharacteristic
100C22×C4×D3C22×C311yescharacteristic
101C22×C4×D3C22×C311yescharacteristic
102C22×Dic6C22×C312yes—
103C3×C4⋊Q8—28no—
104C3×C4⋊Q8—38no—
105C3×C4⋊Q8C2×D318yescomplemented
106C3×C4⋊Q8C22×C318yescomplemented
107C3×#125—316no—
108C3×C22⋊Q8—316no—
109C3×C22⋊Q8—316no—
110C3×C4×Q8—316no—
111C3×C4×Q8—316no—
112C3×#117—316no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4—34no—
116C2×C3×C4⋊C4—32no—
117C2×C3×C4⋊C4—34no—
118C2×C3×C4⋊C4—32no—
119C2×C3×C4⋊C4C2×D312yes—
120C2×C3×C4⋊C4C2×D312yes—
121C2×C3×C4⋊C4C22×C312yes—
122C2×C3×C4⋊C4C22×C312yes—
123C2×C3×C22⋊C4—34no—
124C2×C3×C42—21no—
125C2×C3×C42—31no—
126C2×C3×C42—31no—
127C2×C3×C42—31no—
128C2×C3×C42—34no—
129C2×C3×C42—38no—
130C2×C3×C42C2×D311yescharacteristic
131C2×C3×C42C22×C311yescharacteristic
132#946C22×C3116yes—
133Q8×Dic3C22×C3116yescomplemented
134C2×(C2×D3)⋊C4C22×C314yes—
135C2×C4⋊Dic3C22×C312yes—
136C2×C4⋊Dic3C22×C312yes—
137C2×C4⋊Dic3C22×C314yes—
138C2×Dic3⋊C4C22×C314yes—
139C2×C4×Dic3C22×C311yescharacteristic
140C2×C4×Dic3C22×C311yescharacteristic
141C2×C4×Dic3C22×C311yescharacteristic
142C2×C4×Dic3C22×C314yes—
143#897C22×C3116yes—
144#892C22×C3116yes—
145#890C22×C3116yescomplemented
146#887C22×C3116yescomplemented
147C42×D3C22×C318yes—
148#869C22×C318yescomplemented
149C23×C32C2411yescharacteristic
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3C2×Q814yes—
153C22×C3×D3C2412yes—
154C32×Q8—216no—
155C2×C32×C4—28no—
156C2×C32×C4C2×Q814yes—
157C2×C32×C4#4114yes—
158C2×C32×C4#4114yes—
159C2×C32×C4C2418yes—
160C2×C32×C4C2412yes—
161C2×C32×C4C2412yes—
162C2×C32×C4C2412yes—
163C2×C3×Dic3—28no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3C2×Q814yes—
167C2×C3×Dic3C2412yes—
168C2×C3×Dic3C2412yes—
169C2×C3×Dic3C2418yes—
170C2×C3×Dic3C2412yes—
171C3×C4×D3—28no—
172C3×C4×D3—28no—
173C3×C4×D3C2×Q8116yescomplemented
174C3×Dic6—216no—
175#533—316no—
176C2×C4⋊Q8—31no—
177C2×C4⋊Q8C3×D311yescharacteristic, complemented
178C2×#125—32no—
179C2×C22⋊Q8—32no—
180C2×C22⋊Q8—32no—
181C2×C4×Q8—32no—
182C2×C4×Q8—32no—
183C2×#117—32no—
184C22×C42—31no—
185C24×C3—31no—
186C23×D3C23×C311yescharacteristic
187C2×C3×Q8—28no—
188C2×C3×Q8—28no—
189C2×C3×Q8—28no—
190C2×C3×Q8—38no—
191C2×C3×Q8—48no—
192C2×C3×Q8—68no—
193C2×C3×Q8C22×D318yes—
194C2×C3×Q8C23×C318yes—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—24no—
199C22×C3×C4—31no—
200C22×C3×C4—38no—
201C22×C3×C4—34no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—34no—
207C22×C3×C4—64no—
208C22×C3×C4—64no—
209C22×C3×C4C22×D314yes—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C23×C314yes—
214C22×C3×C4C23×C311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C22×Dic3C23×C311yescharacteristic
218C22×Dic3C23×C311yescharacteristic
219C22×Dic3C23×C314yes—
220C22×Dic3C23×C311yescharacteristic
221C2×C4×D3—24no—
222C2×C4×D3—24no—
223C2×C4×D3C3×Q818yescomplemented
224C2×C4×D3C23×C314yes—
225C2×C4×D3C23×C314yes—
226C2×C4×D3C23×C314yes—
227C2×Dic6—28no—
228C2×Dic6C23×C318yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—316no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—316no—
234C3×C4⋊C4—38no—
235C3×C4⋊C4C22×D318yes—
236C3×C4⋊C4C22×D318yes—
237C3×C4⋊C4C23×C318yes—
238C3×C4⋊C4C23×C318yes—
239C3×C22⋊C4—316no—
240C3×C42—24no—
241C3×C42—34no—
242C3×C42—34no—
243C3×C42—34no—
244C3×C42—316no—
245C3×C42C22×D314yes—
246C3×C42C23×C314yes—
247(C2×D3)⋊C4C23×C3116yes—
248C4⋊Dic3C23×C318yes—
249C4⋊Dic3C23×C318yes—
250C4⋊Dic3C23×C3116yes—
251Dic3⋊C4C23×C3116yes—
252C4×Dic3C23×C314yes—
253C4×Dic3C23×C314yes—
254C4×Dic3C23×C314yes—
255C4×Dic3C23×C3116yes—
256C22×C32C22×Q812yes—
257C22×C32C2×#4112yes—
258C22×C32C2×#4112yes—
259C22×C32C2511yescharacteristic
260C2×C3×D3—28no—
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3C22×Q814yes—
264C32×C4—28no—
265C32×C4C22×Q814yes—
266C32×C4C2×#4114yes—
267C32×C4C2×#4114yes—
268C3×Dic3—28no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3C22×Q814yes—
272C22×Q8—32no—
273C22×Q8C2×C3×D312yes—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C4⋊Q8—38no—
278C4⋊Q8C2×C3×D318yescomplemented
279#125—316no—
280C22⋊Q8—316no—
281C22⋊Q8—316no—
282C4×Q8—316no—
283C4×Q8—316no—
284#117—316no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—34no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4C2×C3×D312yes—
290C2×C4⋊C4C2×C3×D312yes—
291C2×C22⋊C4—34no—
292C2×C42—34no—
293C2×C42—31no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—38no—
297C2×C42C2×C3×D311yescharacteristic
298C23×C3—21no—
299C23×C3—34no—
300C23×C3—32no—
301C23×C3—62no—
302C23×C3—62no—
303C23×C3C23×D311yescharacteristic
304C23×C3C24×C311yescharacteristic
305C22×D3—22no—
306C22×D3—22no—
307C22×D3C2×C3×Q814yes—
308C22×D3C24×C312yes—
309C3×Q8—216no—
310C3×Q8—216no—
311C3×Q8—416no—
312C3×Q8—616no—
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—28no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—34no—
323C2×C3×C4—32no—
324C2×C3×C4—316no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—38no—
328C2×C3×C4—48no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—68no—
332C2×C3×C4—68no—
333C2×C3×C4—68no—
334C2×C3×C4#23714yes—
335C2×C3×C4D3×Q814yes—
336C2×C3×C4#23914yes—
337C2×C3×C4C2×C3×Q814yes—
338C2×C3×C4C3×#4114yes—
339C2×C3×C4C3×#4114yes—
340C2×C3×C4C23×D318yes—
341C2×C3×C4C23×D312yes—
342C2×C3×C4C23×D312yes—
343C2×C3×C4C23×D312yes—
344C2×C3×C4C24×C318yes—
345C2×C3×C4C24×C312yes—
346C2×C3×C4C24×C312yes—
347C2×C3×C4C24×C312yes—
348C2×Dic3—28no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3C2×C3×Q814yes—
352C2×Dic3C24×C312yes—
353C2×Dic3C24×C312yes—
354C2×Dic3C24×C318yes—
355C2×Dic3C24×C312yes—
356C4×D3—28no—
357C4×D3—28no—
358C4×D3C2×C3×Q8116yescomplemented
359Dic6—216no—
360C2×C32#53314yescomplemented
361C2×C32C23×Q811yescharacteristic
362C2×C32C22×#4111yescharacteristic
363C2×C32C22×#4111yescharacteristic
364C3×D3—24no—
365C24—31no—
366C2×Q8—28no—
367C2×Q8—38no—
368C2×Q8—68no—
369C2×Q8C22×C3×D318yes—
370C22×C4—31no—
371C22×C4—34no—
372C22×C4—38no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—34no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4C22×C3×D314yes—
381C22×C4C22×C3×D311yescharacteristic
382C22×C4C22×C3×D311yescharacteristic
383C22×C4C22×C3×D311yescharacteristic
384C4⋊C4—38no—
385C4⋊C4—316no—
386C4⋊C4—316no—
387C4⋊C4—38no—
388C4⋊C4C22×C3×D318yes—
389C4⋊C4C22×C3×D318yes—
390C22⋊C4—316no—
391C42—34no—
392C42—34no—
393C42—34no—
394C42—316no—
395C42C22×C3×D314yes—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—34no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—68no—
404C22×C3C2×#23712yes—
405C22×C3C2×D3×Q812yes—
406C22×C3C2×#23912yes—
407C22×C3C22×C3×Q812yes—
408C22×C3C2×C3×#4112yes—
409C22×C3C2×C3×#4112yes—
410C22×C3C24×D311yescharacteristic
411C22×C3C25×C311yescharacteristic
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—28no—
415C2×D3C22×C3×Q814yes—
416C3×C4—28no—
417C3×C4—28no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—24no—
421C3×C4—34no—
422C3×C4—48no—
423C3×C4—62no—
424C3×C4—62no—
425C3×C4—68no—
426C3×C4C2×#23714yes—
427C3×C4C2×D3×Q814yes—
428C3×C4C2×#23914yes—
429C3×C4C22×C3×Q814yes—
430C3×C4C2×C3×#4114yes—
431C3×C4C2×C3×#4114yes—
432Dic3—28no—
433Dic3—22no—
434Dic3—22no—
435Dic3C22×C3×Q814yes—
436C32C2×#53311yescharacteristic, complemented, Sylow
437C23—34no—
438C23—32no—
439C23—62no—
440C23—62no—
441C23C23×C3×D311yescharacteristic
442Q8—216no—
443Q8—616no—
444C2×C4—28no—
445C2×C4—34no—
446C2×C4—32no—
447C2×C4—316no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—38no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—68no—
454C2×C4—68no—
455C2×C4—68no—
456C2×C4C3×#23714yes—
457C2×C4C3×D3×Q814yes—
458C2×C4C3×#23914yes—
459C2×C4C23×C3×D318yes—
460C2×C4C23×C3×D312yes—
461C2×C4C23×C3×D312yes—
462C2×C4C23×C3×D312yes—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—24no—
467C2×C3—64no—
468C2×C3#572814yesdirect product factor
469C2×C3C3×#53314yescomplemented
470C2×C3C22×#23711yescharacteristic
471C2×C3C22×D3×Q811yescharacteristic
472C2×C3C22×#23911yescharacteristic
473C2×C3C23×C3×Q811yescharacteristic
474C2×C3C22×C3×#4111yescharacteristic
475C2×C3C22×C3×#4111yescharacteristic
476D3—24no—
477C22—34no—
478C22—62no—
479C22—62no—
480C22—68no—
481C22C2×C3×#23712yes—
482C22C2×C3×D3×Q812yes—
483C22C2×C3×#23912yes—
484C22C24×C3×D311yescharacteristic
485C4—28no—
486C4—34no—
487C4—62no—
488C4—62no—
489C4—68no—
490C4C2×C3×#23714yes—
491C4C2×C3×D3×Q814yes—
492C4C2×C3×#23914yes—
493C3—21no—
494C3C2×#572811yescharacteristic, direct product factor
495C3C2×C3×#53311yescharacteristic, complemented
496C2—64no—
497C2C3×#572814yesdirect product factor
498C2C22×C3×#23711yescharacteristic
499C2C22×C3×D3×Q811yescharacteristic
500C2C22×C3×#23911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C2×C3×#533
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4⋊Q8C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. C3×#5728C2direct (N × H)
7. C2×C3×#890C2semidirect
8. C2×C3×#887C2semidirect
10. C2×C3×#869C2semidirect
12. C2×#5728C3direct (N × H)
18. C32×C4⋊Q8C22semidirect
23. C3×Q8×Dic3C22semidirect
35. C3×#890C22semidirect
36. C3×#887C22semidirect
38. C3×#869C22semidirect
42. C2×C3×C4⋊Q8D3semidirect
43. C2×C3×C4⋊Q8C2×C3semidirect
52. C2×Q8×Dic3C2×C3semidirect
53. #5728C2×C3direct (N × H)
56. C2×#890C2×C3semidirect
57. C2×#887C2×C3semidirect
59. C2×#869C2×C3semidirect
73. C2×C3×C4×D3Q8semidirect
105. C3×C4⋊Q8C2×D3semidirect
106. C3×C4⋊Q8C22×C3semidirect
133. Q8×Dic3C22×C3semidirect
145. #890C22×C3semidirect
146. #887C22×C3semidirect
148. #869C22×C3semidirect
173. C3×C4×D3C2×Q8semidirect
177. C2×C4⋊Q8C3×D3semidirect
223. C2×C4×D3C3×Q8semidirect
278. C4⋊Q8C2×C3×D3semidirect
358. C4×D3C2×C3×Q8semidirect
360. C2×C32#533semidirect
436. C32C2×#533semidirect
468. C2×C3#5728direct (N × H)
469. C2×C3C3×#533semidirect
494. C3C2×#5728direct (N × H)
495. C3C2×C3×#533semidirect
497. C2C3×#5728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊Q82
C2×C3×#9462
C2×C3×Q8×Dic32
C3×#57282
C2×C3×#8972
C2×C3×#8922
C2×C3×#8902
C2×C3×#8872
C2×C3×C42×D32
C2×C3×#8692
C2×C3×#5333
C2×#57283