Finite Group Databasebeta

← C2×C3×#5242 #11864811

Subgroups — C2×C3×#5242 #11864811

Order 1152.

1969 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#951C212yesmaximal, complemented
3C2×C3×#936C211yesmaximal, characteristic, complemented
4C3×#5242C218yesmaximal, direct product factor
5C2×C3×#837C212yesmaximal, complemented
6C2×C3×#833C211yesmaximal, characteristic, complemented
7C2×C3×#455—31nomaximal
8C2×#5242C311yesmaximal, characteristic, direct product factor
9C32×H(5,𝔽2)—24no—
10C32×H(5,𝔽2)—22no—
11C32×H(5,𝔽2)C2214yes—
12C2×C32×#41—22no—
13C2×C32×#41C2212yes—
14C22×C32×D4—22no—
15C22×C32×D4—22no—
16C22×C32×D4C2211yescharacteristic
17C22×C3×Dic6C2211yescharacteristic, complemented
18C2×C3×#217—22no—
19C3×#951—24no—
20C3×#951C2218yescomplemented
21C2×C3×#216—22no—
22C3×#936—22no—
23C3×#936C2214yescomplemented
24C2×C3×#214—22no—
25C2×C3×C4×Dic3C2211yescharacteristic, complemented
26C2×C3×#208C2212yescomplemented
27C3×#837C2218yescomplemented
28C3×#833C2214yescomplemented
29C2×C3×H(5,𝔽2)—21no—
30C2×C3×H(5,𝔽2)D311yescharacteristic
31C2×C3×H(5,𝔽2)C2×C311yescharacteristic
32C2×C3×#137—32no—
33C2×C3×#124—31no—
34C2×#951C2×C312yescomplemented
35C2×#936C2×C311yescharacteristic, complemented
36C3×#455—38no—
37C2×C3×#104—32no—
38C2×C3×#100—31no—
39#5242C2×C318yesdirect product factor
40C2×#837C2×C312yescomplemented
41C2×#833C2×C311yescharacteristic, complemented
42C24×C32—22no—
43C24×C32D412yes—
44C32×#41—28no—
45C32×#41—28no—
46C32×#41—22no—
47C32×#41—22no—
48C32×#41D414yes—
49C32×#41C2314yes—
50C2×C32×Q8D412yes—
51C2×C32×D4—24no—
52C2×C32×D4—28no—
53C2×C32×D4—28no—
54C2×C32×D4—22no—
55C2×C32×D4—28no—
56C2×C32×D4—28no—
57C2×C32×D4—22no—
58C2×C32×D4—22no—
59C2×C32×D4D412yes—
60C2×C32×D4D412yes—
61C2×C32×D4C2312yes—
62C22×C32×C4—22no—
63C22×C32×C4—22no—
64C22×C32×C4C2311yescharacteristic
65C22×C3×Dic3—21no—
66C22×C3×Dic3—21no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yescomplemented
71C3×#217—28no—
72C3×#216—28no—
73C3×#214—28no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3C2312yescomplemented
76C3×#208—22no—
77C3×#208C2314yescomplemented
78C2×C3×C3⋊C8—22no—
79C2×#455—31noSylow
80C3×H(5,𝔽2)—28no—
81C3×H(5,𝔽2)—24no—
82C3×H(5,𝔽2)—24no—
83C3×H(5,𝔽2)—24no—
84C3×H(5,𝔽2)—22no—
85C3×H(5,𝔽2)—22no—
86C3×H(5,𝔽2)—24no—
87C3×H(5,𝔽2)C2×D314yes—
88C3×H(5,𝔽2)C22×C314yes—
89C2×C3×#41—24no—
90C2×C3×#41—22no—
91C2×C3×#41—22no—
92C2×C3×#41—22no—
93C2×C3×#41C2×D312yes—
94C2×C3×#41C22×C312yes—
95C22×C3×Q8—31no—
96C22×C3×D4—24no—
97C22×C3×D4—24no—
98C22×C3×D4—22no—
99C22×C3×D4—22no—
100C22×C3×D4—22no—
101C22×C3×D4—22no—
102C22×C3×D4—21no—
103C22×C3×D4C2×D311yescharacteristic
104C22×C3×D4C22×C311yescharacteristic
105C22×Dic6C22×C311yescharacteristic, complemented
106C3×#137—38no—
107C3×#137—64no—
108C2×C3×Q16—62no—
109C2×C3×SD16—62no—
110C2×C3×M16—32no—
111C3×#124—34no—
112C3×#124—62no—
113C2×C3×C22⋊C4—62no—
114C2×C3×C42—31no—
115C2×#217—22no—
116#951—24no—
117#951C22×C318yescomplemented
118C2×#216—22no—
119#936—22no—
120#936C22×C314yescomplemented
121C2×#214—22no—
122C2×C4×Dic3C22×C311yescharacteristic, complemented
123C2×#208C22×C312yescomplemented
124C3×#104—38no—
125C3×#100—34no—
126#837C22×C318yescomplemented
127#833C22×C314yescomplemented
128C23×C32—28no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32—44no—
133C23×C32—84no—
134C23×C32C2×D411yescharacteristic
135C23×C32C2×D414yes—
136C32×Q8—22no—
137C32×Q8C2×D414yes—
138C32×D4—216no—
139C32×D4—24no—
140C32×D4—24no—
141C32×D4—24no—
142C32×D4—22no—
143C32×D4—24no—
144C32×D4C2×D414yes—
145C2×C32×C4—28no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D412yes—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2411yescharacteristic
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—41no—
157C2×C3×Dic3—41no—
158C3×Dic6—24no—
159C3×Dic6—42no—
160C3×C3⋊C8—24no—
161C2×H(5,𝔽2)C3×D311yescharacteristic
162C2×#137—32no—
163C2×#124—31no—
164#455—38no—
165C2×#104—32no—
166C2×#100—31no—
167C24×C3—24no—
168C24×C3—22no—
169C24×C3—22no—
170C24×C3—22no—
171C24×C3#6712yes—
172C24×C3C3×D412yes—
173C3×#41—216no—
174C3×#41—28no—
175C3×#41—28no—
176C3×#41—216no—
177C3×#41—24no—
178C3×#41—24no—
179C3×#41—28no—
180C3×#41—28no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—24no—
186C3×#41—24no—
187C3×#41#6714yes—
188C3×#41C3×D414yes—
189C3×#41C22×D314yes—
190C3×#41C23×C314yes—
191C2×C3×Q8—22no—
192C2×C3×Q8—32no—
193C2×C3×Q8—61no—
194C2×C3×Q8—62no—
195C2×C3×Q8—62no—
196C2×C3×Q8#6712yes—
197C2×C3×Q8C3×D412yes—
198C2×C3×D4—28no—
199C2×C3×D4—24no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—216no—
204C2×C3×D4—216no—
205C2×C3×D4—28no—
206C2×C3×D4—28no—
207C2×C3×D4—28no—
208C2×C3×D4—28no—
209C2×C3×D4—216no—
210C2×C3×D4—216no—
211C2×C3×D4—24no—
212C2×C3×D4—22no—
213C2×C3×D4—28no—
214C2×C3×D4—22no—
215C2×C3×D4—28no—
216C2×C3×D4—28no—
217C2×C3×D4—28no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—22no—
224C2×C3×D4—22no—
225C2×C3×D4#6712yes—
226C2×C3×D4#6712yes—
227C2×C3×D4C3×D412yes—
228C2×C3×D4C3×D412yes—
229C2×C3×D4C22×D312yes—
230C2×C3×D4C23×C312yes—
231C22×C3×C4—24no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—24no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—21no—
238C22×C3×C4—61no—
239C22×C3×C4—61no—
240C22×C3×C4C22×D311yescharacteristic
241C22×C3×C4C23×C311yescharacteristic
242C22×Dic3—21no—
243C22×Dic3—21no—
244C2×Dic6—22no—
245C2×Dic6—22no—
246C2×Dic6—21no—
247C2×Dic6C23×C312yescomplemented
248C3×Q16—68no—
249C3×SD16—68no—
250C3×M16—34no—
251C3×M16—62no—
252C2×C3×C8—62no—
253C3×C22⋊C4—68no—
254C3×C42—32no—
255C3×C42—61no—
256#217—28no—
257#216—28no—
258#214—28no—
259C4×Dic3—21no—
260C4×Dic3C23×C312yescomplemented
261#208—22no—
262#208C23×C314yescomplemented
263C2×C3⋊C8—22no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—22no—
267C22×C32—44no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—82no—
271C22×C32—88no—
272C22×C32#12011yescharacteristic
273C22×C32#12011yescharacteristic
274C22×C32C22×D411yescharacteristic
275C32×C4—24no—
276C32×C4#12012yes—
277C32×C4C22×D412yes—
278C3×Dic3—42no—
279C3×Dic3—42no—
280H(5,𝔽2)—24no—
281H(5,𝔽2)—22no—
282H(5,𝔽2)C2×C3×D314yes—
283C2×#41—22no—
284C2×#41C2×C3×D312yes—
285C22×Q8—31no—
286C22×D4—22no—
287C22×D4—22no—
288C22×D4C2×C3×D311yescharacteristic
289#137—38no—
290#137—64no—
291C2×Q16—62no—
292C2×SD16—62no—
293C2×M16—32no—
294#124—34no—
295#124—62no—
296C2×C22⋊C4—62no—
297C2×C42—31no—
298#104—38no—
299#100—34no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—216no—
303C23×C3—28no—
304C23×C3—28no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—21no—
313C23×C3—24no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—84no—
317C23×C3—88no—
318C23×C3—84no—
319C23×C3—84no—
320C23×C3C2×#6711yescharacteristic
321C23×C3C2×#6714yes—
322C23×C3C2×C3×D411yescharacteristic
323C23×C3C2×C3×D414yes—
324C3×Q8—24no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—24no—
328C3×Q8—64no—
329C3×Q8—122no—
330C3×Q8C2×#6714yes—
331C3×Q8C2×C3×D414yes—
332C3×D4—232no—
333C3×D4—216no—
334C3×D4—216no—
335C3×D4—28no—
336C3×D4—28no—
337C3×D4—28no—
338C3×D4—24no—
339C3×D4—28no—
340C3×D4—24no—
341C3×D4—24no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—22no—
346C3×D4—24no—
347C3×D4—22no—
348C3×D4—24no—
349C3×D4—24no—
350C3×D4—24no—
351C3×D4C2×#6714yes—
352C3×D4C2×C3×D414yes—
353C2×C3×C4—24no—
354C2×C3×C4—216no—
355C2×C3×C4—28no—
356C2×C3×C4—28no—
357C2×C3×C4—28no—
358C2×C3×C4—28no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—122no—
372C2×C3×C4—121no—
373C2×C3×C4—121no—
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6712yes—
376C2×C3×C4C2×#6712yes—
377C2×C3×C4C2×C3×D411yescharacteristic
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×D412yes—
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C24×C311yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387Dic6—24no—
388Dic6—42no—
389C3×C8—64no—
390C3⋊C8—24no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—44no—
394C2×C32—44no—
395C2×C32#45512yescomplemented
396C2×C32C2×#12011yescharacteristic
397C24—22no—
398C24C3×#6712yes—
399#41—28no—
400#41—28no—
401#41—22no—
402#41—22no—
403#41C3×#6714yes—
404#41C22×C3×D314yes—
405C2×Q8—32no—
406C2×Q8—61no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8C3×#6712yes—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4C3×#6712yes—
419C2×D4C3×#6712yes—
420C2×D4C22×C3×D312yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4C22×C3×D311yescharacteristic
426Q16—68no—
427SD16—68no—
428M16—34no—
429M16—62no—
430C2×C8—62no—
431C22⋊C4—68no—
432C42—32no—
433C42—61no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—28no—
437C22×C3—28no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—42no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—48no—
449C22×C3—48no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—84no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—84no—
458C22×C3—816no—
459C22×C3—88no—
460C22×C3—88no—
461C22×C3#95311yescharacteristic
462C22×C3#95311yescharacteristic
463C22×C3C3×#12011yescharacteristic
464C22×C3C3×#12011yescharacteristic
465C22×C3C22×#6711yescharacteristic
466C22×C3C22×C3×D411yescharacteristic
467C3×C4—28no—
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—122no—
473C3×C4—122no—
474C3×C4#95312yes—
475C3×C4C3×#12012yes—
476C3×C4C22×#6712yes—
477C3×C4C22×C3×D412yes—
478Dic3—42no—
479Dic3—42no—
480C32C2×#45511yescharacteristic, complemented, Sylow
481C23—28no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—44no—
486C23—84no—
487C23C2×C3×#6711yescharacteristic
488C23C2×C3×#6714yes—
489Q8—22no—
490Q8—64no—
491Q8—122no—
492Q8C2×C3×#6714yes—
493D4—216no—
494D4—24no—
495D4—24no—
496D4—24no—
497D4—22no—
498D4—24no—
499D4C2×C3×#6714yes—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4C2×C3×#6711yescharacteristic
510C2×C4C2×C3×#6712yes—
511C2×C4C2×C3×#6712yes—
512C2×C4C23×C3×D311yescharacteristic
513C8—64no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—48no—
523C2×C3—48no—
524C2×C3—44no—
525C2×C3—44no—
526C2×C3—44no—
527C2×C3—44no—
528C2×C3#524212yesdirect product factor
529C2×C3C3×#45512yescomplemented
530C2×C3C2×#95311yescharacteristic
531C2×C3C2×C3×#12011yescharacteristic
532C22—22no—
533C22—24no—
534C22—24no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—88no—
540C22C3×#95311yescharacteristic
541C22C3×#95311yescharacteristic
542C22C22×C3×#6711yescharacteristic
543C4—24no—
544C4—122no—
545C4—122no—
546C4C3×#95312yes—
547C4C22×C3×#6712yes—
548C3—21no—
549C3C2×#524211yescharacteristic, direct product factor
550C3C2×C3×#45511yescharacteristic, complemented
551C2—21no—
552C2—21no—
553C2—44no—
554C2—44no—
555C2C3×#524212yesdirect product factor
556C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×H(5,𝔽2)C2
socleC22×C32#120
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C2×C3×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×C3×#951C2semidirect
3. C2×C3×#936C2semidirect
4. C3×#5242C2direct (N × H)
5. C2×C3×#837C2semidirect
6. C2×C3×#833C2semidirect
8. C2×#5242C3direct (N × H)
17. C22×C3×Dic6C22semidirect
20. C3×#951C22semidirect
23. C3×#936C22semidirect
25. C2×C3×C4×Dic3C22semidirect
26. C2×C3×#208C22semidirect
27. C3×#837C22semidirect
28. C3×#833C22semidirect
34. C2×#951C2×C3semidirect
35. C2×#936C2×C3semidirect
39. #5242C2×C3direct (N × H)
40. C2×#837C2×C3semidirect
41. C2×#833C2×C3semidirect
70. C2×C3×Dic6C23semidirect
75. C3×C4×Dic3C23semidirect
77. C3×#208C23semidirect
105. C22×Dic6C22×C3semidirect
117. #951C22×C3semidirect
120. #936C22×C3semidirect
122. C2×C4×Dic3C22×C3semidirect
123. C2×#208C22×C3semidirect
126. #837C22×C3semidirect
127. #833C22×C3semidirect
247. C2×Dic6C23×C3semidirect
260. C4×Dic3C23×C3semidirect
262. #208C23×C3semidirect
395. C2×C32#455semidirect
480. C32C2×#455semidirect
528. C2×C3#5242direct (N × H)
529. C2×C3C3×#455semidirect
549. C3C2×#5242direct (N × H)
550. C3C2×C3×#455semidirect
555. C2C3×#5242direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C2×C3×#9512
C2×C3×#9362
C3×#52422
C2×C3×#8372
C2×C3×#8332
C2×C3×#4553
C2×#52423