Finite Group Databasebeta

← C2×C3×#5838 #11918913

Subgroups — C2×C3×#5838 #11918913

Order 1152.

3213 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#1010C211yesmaximal, characteristic, complemented
3C3×#5838C2116yesmaximal, direct product factor
4C2×C3×#951C213yesmaximal, complemented
5C2×C3×#950C213yesmaximal, complemented
6C2×C3×#948C211yesmaximal, characteristic, complemented
7C22×C3×#216C213yesmaximal, complemented
8C2×C3×#942C213yesmaximal, complemented
9C2×C3×#578—31nomaximal
10C2×#5838C311yesmaximal, characteristic, direct product factor
11C32×2−1+4—24no—
12C32×2−1+4C2218yescomplemented
13C2×C32×#41—23no—
14C2×C32×#41C2213yes—
15C2×C32×#41C2211yescharacteristic
16C22×C32×Q8—21no—
17C22×C32×Q8C2213yes—
18C3×#1010—24no—
19C3×#1010C2218yescomplemented
20C2×C3×D3×Q8—21no—
21C2×C3×#237—23no—
22C2×C3×#235C2213yes—
23C22×C3×Dic6C2213yes—
24C3×#951C22124yescomplemented
25C3×#950C22124yescomplemented
26C3×#948C2218yescomplemented
27C2×C3×#216C2216yes—
28C2×C3×#216C2213yes—
29C2×C3×#216C22124yescomplemented
30C3×#942C22124yescomplemented
31C2×C3×#215C2213yes—
32C2×C3×#214C2213yes—
33C2×C3×#213C2211yescharacteristic
34C2×C3×#208C2213yes—
35C22×C3×C3⋊C8C2213yes—
36C2×C3×2−1+4—21no—
37C2×C3×2−1+4—31no—
38C2×C3×2−1+4D311yescharacteristic, complemented
39C2×C3×2−1+4C2×C311yescharacteristic, complemented
40C2×#1010C2×C311yescharacteristic, complemented
41C3×#578—316no—
42C2×C3×#137—33no—
43C2×C3×#137—33no—
44C2×C3×#135—33no—
45C22×C3×Q16—33no—
46C2×C3×#131—31no—
47#5838C2×C3116yesdirect product factor
48C2×#951C2×C313yescomplemented
49C2×#950C2×C313yescomplemented
50C2×#948C2×C311yescharacteristic, complemented
51C22×#216C2×C313yescomplemented
52C2×#942C2×C313yescomplemented
53C32×#41—224no—
54C32×#41—26no—
55C32×#41D414yes—
56C32×#41C23112yes—
57C32×#41C2314yes—
58C2×C32×Q8—28no—
59C2×C32×Q8—23no—
60C2×C32×Q8—26no—
61C2×C32×Q8D411yescharacteristic
62C2×C32×Q8C2313yes—
63C2×C32×Q8C23112yes—
64C2×C32×D4—23no—
65C2×C32×D4D413yes—
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—23no—
68C22×C32×C4—23no—
69C22×C32×C4C2313yes—
70C2×C3×#67—23no—
71C22×C3×Dic3—23no—
72C3×D3×Q8—28no—
73C3×#237—224no—
74C3×#235—26no—
75C3×#235C23112yes—
76C2×C3×D12C2311yescharacteristic
77C2×C3×C4×D3—23no—
78C2×C3×Dic6—23no—
79C2×C3×Dic6—26no—
80C2×C3×Dic6C2313yes—
81C2×C3×Dic6C23112yes—
82C3×#216C23124yes—
83C3×#216C23112yes—
84C3×#215C23112yes—
85C3×#214C23112yes—
86C3×#213C2314yes—
87C3×#208C23112yes—
88C2×C3×C3⋊C8C2311yescharacteristic
89C2×C3×C3⋊C8C2313yes—
90C2×C3×C3⋊C8C23112yes—
91C2×#578—31noSylow
92C3×2−1+4—28no—
93C3×2−1+4—24no—
94C3×2−1+4—24no—
95C3×2−1+4—28no—
96C3×2−1+4—38no—
97C3×2−1+4—64no—
98C3×2−1+4C2×D318yescomplemented
99C3×2−1+4C22×C318yescomplemented
100C2×C3×#41—26no—
101C2×C3×#41—23no—
102C2×C3×#41—23no—
103C2×C3×#41—23no—
104C2×C3×#41—21no—
105C2×C3×#41—33no—
106C2×C3×#41—63no—
107C2×C3×#41C2×D313yes—
108C2×C3×#41C2×D311yescharacteristic
109C2×C3×#41C22×C313yes—
110C2×C3×#41C22×C311yescharacteristic
111C22×C3×Q8—22no—
112C22×C3×Q8—21no—
113C22×C3×Q8—21no—
114C22×C3×Q8—23no—
115C22×C3×Q8—33no—
116C22×C3×Q8—61no—
117C22×C3×Q8C2×D313yes—
118C22×C3×Q8C22×C313yes—
119#1010—24no—
120#1010C22×C318yescomplemented
121C2×D3×Q8—21no—
122C2×#237—23no—
123C2×#235C22×C313yes—
124C22×Dic6C22×C313yes—
125C3×#137—324no—
126C3×#137—324no—
127C3×#135—324no—
128C2×C3×Q16—33no—
129C2×C3×Q16—36no—
130C2×C3×Q16—324no—
131C2×C3×SD16—33no—
132C2×C3×SD16—33no—
133C2×C3×D8—31no—
134C3×#131—38no—
135C2×C3×M16—33no—
136C22×C3×C8—33no—
137#951C22×C3124yescomplemented
138#950C22×C3124yescomplemented
139#948C22×C318yescomplemented
140C2×#216C22×C316yes—
141C2×#216C22×C313yes—
142C2×#216C22×C3124yescomplemented
143#942C22×C3124yescomplemented
144C2×#215C22×C313yes—
145C2×#214C22×C313yes—
146C2×#213C22×C311yescharacteristic
147C2×#208C22×C313yes—
148C22×C3⋊C8C22×C313yes—
149C23×C32—21no—
150C23×C32C2×D413yes—
151C22×C3×D3—21no—
152C32×Q8—23no—
153C32×Q8—212no—
154C32×Q8C2×D414yes—
155C32×Q8C2416yes—
156C32×D4—212no—
157C32×D4—21no—
158C32×D4C2×D4112yes—
159C32×D4C2412yes—
160C2×C32×C4—23no—
161C2×C32×C4—212no—
162C2×C32×C4—212no—
163C2×C32×C4C2×D413yes—
164C2×C32×C4C2×D416yes—
165C2×C32×C4C2411yescharacteristic
166C2×C32×C4C2416yes—
167C3×#67—212no—
168C2×C3×Dic3—23no—
169C2×C3×Dic3—212no—
170C3×D12—21no—
171C3×D12C2412yes—
172C3×C4×D3—212no—
173C3×Dic6—23no—
174C3×Dic6—212no—
175C3×Dic6C2416yes—
176C3×C3⋊C8C2412yes—
177C3×C3⋊C8C2416yes—
178C2×2−1+4—31no—
179C2×2−1+4C3×D311yescharacteristic, complemented
180#578—316no—
181C2×#137—33no—
182C2×#137—33no—
183C2×#135—33no—
184C22×Q16—33no—
185C2×#131—31no—
186C3×#41—248no—
187C3×#41—212no—
188C3×#41—224no—
189C3×#41—224no—
190C3×#41—26no—
191C3×#41—26no—
192C3×#41—24no—
193C3×#41—212no—
194C3×#41—24no—
195C3×#41—312no—
196C3×#41—624no—
197C3×#41—66no—
198C3×#41#6714yes—
199C3×#41C3×D414yes—
200C3×#41C22×D3112yes—
201C3×#41C22×D314yes—
202C3×#41C23×C3112yes—
203C3×#41C23×C314yes—
204C2×C3×Q8—26no—
205C2×C3×Q8—216no—
206C2×C3×Q8—212no—
207C2×C3×Q8—28no—
208C2×C3×Q8—28no—
209C2×C3×Q8—23no—
210C2×C3×Q8—26no—
211C2×C3×Q8—23no—
212C2×C3×Q8—26no—
213C2×C3×Q8—21no—
214C2×C3×Q8—23no—
215C2×C3×Q8—212no—
216C2×C3×Q8—33no—
217C2×C3×Q8—312no—
218C2×C3×Q8—68no—
219C2×C3×Q8—63no—
220C2×C3×Q8—66no—
221C2×C3×Q8#6711yescharacteristic
222C2×C3×Q8C3×D411yescharacteristic
223C2×C3×Q8C22×D313yes—
224C2×C3×Q8C22×D3112yes—
225C2×C3×Q8C23×C313yes—
226C2×C3×Q8C23×C3112yes—
227C2×C3×D4—26no—
228C2×C3×D4—23no—
229C2×C3×D4—23no—
230C2×C3×D4—23no—
231C2×C3×D4—21no—
232C2×C3×D4—31no—
233C2×C3×D4—63no—
234C2×C3×D4#6713yes—
235C2×C3×D4C3×D413yes—
236C2×C3×D4C22×D311yescharacteristic
237C2×C3×D4C23×C311yescharacteristic
238C22×C3×C4—26no—
239C22×C3×C4—26no—
240C22×C3×C4—23no—
241C22×C3×C4—23no—
242C22×C3×C4—23no—
243C22×C3×C4—23no—
244C22×C3×C4—23no—
245C22×C3×C4—63no—
246C22×C3×C4—63no—
247C22×C3×C4C22×D313yes—
248C22×C3×C4C23×C313yes—
249C2×#67—23no—
250C22×Dic3—23no—
251D3×Q8—28no—
252#237—224no—
253#235—26no—
254#235C23×C3112yes—
255C2×D12C23×C311yescharacteristic
256C2×C4×D3—23no—
257C2×Dic6—23no—
258C2×Dic6—26no—
259C2×Dic6C23×C313yes—
260C2×Dic6C23×C3112yes—
261C3×Q16—312no—
262C3×Q16—324no—
263C3×SD16—312no—
264C3×SD16—312no—
265C3×D8—34no—
266C3×M16—312no—
267C2×C3×C8—31no—
268C2×C3×C8—33no—
269C2×C3×C8—312no—
270#216C23×C3124yes—
271#216C23×C3112yes—
272#215C23×C3112yes—
273#214C23×C3112yes—
274#213C23×C314yes—
275#208C23×C3112yes—
276C2×C3⋊C8C23×C311yescharacteristic
277C2×C3⋊C8C23×C313yes—
278C2×C3⋊C8C23×C3112yes—
279C22×C32—22no—
280C22×C32—26no—
281C22×C32—42no—
282C22×C32C22×D416yes—
283C22×C32C22×D411yescharacteristic
284C2×C3×D3—22no—
285C2×C3×D3—42no—
286C32×C4—26no—
287C32×C4C22×D411yescharacteristic
288C32×C4C22×D416yes—
289C32×C4C2511yescharacteristic
290C3×Dic3—26no—
2912−1+4—24no—
2922−1+4—38no—
2932−1+4—64no—
2942−1+4C2×C3×D318yescomplemented
295C2×#41—23no—
296C2×#41—33no—
297C2×#41—63no—
298C2×#41C2×C3×D313yes—
299C2×#41C2×C3×D311yescharacteristic
300C22×Q8—21no—
301C22×Q8—33no—
302C22×Q8—61no—
303C22×Q8C2×C3×D313yes—
304#137—324no—
305#137—324no—
306#135—324no—
307C2×Q16—33no—
308C2×Q16—36no—
309C2×Q16—324no—
310C2×SD16—33no—
311C2×SD16—33no—
312C2×D8—31no—
313#131—38no—
314C2×M16—33no—
315C22×C8—33no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—23no—
320C23×C3—61no—
321C23×C3C2×#6713yes—
322C23×C3C2×C3×D413yes—
323C22×D3—21no—
324C3×Q8—224no—
325C3×Q8—26no—
326C3×Q8—212no—
327C3×Q8—23no—
328C3×Q8—23no—
329C3×Q8—212no—
330C3×Q8—26no—
331C3×Q8—24no—
332C3×Q8—36no—
333C3×Q8—612no—
334C3×Q8—63no—
335C3×Q8C2×#6714yes—
336C3×Q8C2×C3×D414yes—
337C3×Q8C23×D316yes—
338C3×Q8C24×C316yes—
339C3×D4—224no—
340C3×D4—22no—
341C3×D4—212no—
342C3×D4—212no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—22no—
346C3×D4—212no—
347C3×D4—32no—
348C3×D4—612no—
349C3×D4—61no—
350C3×D4C2×#67112yes—
351C3×D4C2×C3×D4112yes—
352C3×D4C23×D312yes—
353C3×D4C24×C312yes—
354C2×C3×C4—26no—
355C2×C3×C4—224no—
356C2×C3×C4—224no—
357C2×C3×C4—212no—
358C2×C3×C4—23no—
359C2×C3×C4—23no—
360C2×C3×C4—212no—
361C2×C3×C4—212no—
362C2×C3×C4—212no—
363C2×C3×C4—21no—
364C2×C3×C4—26no—
365C2×C3×C4—23no—
366C2×C3×C4—26no—
367C2×C3×C4—612no—
368C2×C3×C4—63no—
369C2×C3×C4—612no—
370C2×C3×C4C2×#6716yes—
371C2×C3×C4C2×#6713yes—
372C2×C3×C4C2×C3×D413yes—
373C2×C3×C4C2×C3×D416yes—
374C2×C3×C4C23×D316yes—
375C2×C3×C4C23×D311yescharacteristic
376C2×C3×C4C24×C311yescharacteristic
377C2×C3×C4C24×C316yes—
378#67—212no—
379C2×Dic3—23no—
380C2×Dic3—212no—
381D12—21no—
382D12C24×C312yes—
383C4×D3—212no—
384Dic6—23no—
385Dic6—212no—
386Dic6C24×C316yes—
387C3×C8—32no—
388C3×C8—36no—
389C3⋊C8C24×C312yes—
390C3⋊C8C24×C316yes—
391C2×C32—26no—
392C2×C32—42no—
393C2×C32#57812yescomplemented
394C2×C32C23×D411yescharacteristic
395C3×D3—42no—
396#41—224no—
397#41—26no—
398#41—312no—
399#41—624no—
400#41—66no—
401#41C3×#6714yes—
402#41C22×C3×D3112yes—
403#41C22×C3×D314yes—
404C2×Q8—28no—
405C2×Q8—23no—
406C2×Q8—26no—
407C2×Q8—33no—
408C2×Q8—312no—
409C2×Q8—68no—
410C2×Q8—63no—
411C2×Q8—66no—
412C2×Q8C3×#6711yescharacteristic
413C2×Q8C22×C3×D313yes—
414C2×Q8C22×C3×D3112yes—
415C2×D4—23no—
416C2×D4—31no—
417C2×D4—63no—
418C2×D4C3×#6713yes—
419C2×D4C22×C3×D311yescharacteristic
420C22×C4—23no—
421C22×C4—23no—
422C22×C4—63no—
423C22×C4—63no—
424C22×C4C22×C3×D313yes—
425Q16—312no—
426Q16—324no—
427SD16—312no—
428SD16—312no—
429D8—34no—
430M16—312no—
431C2×C8—31no—
432C2×C8—33no—
433C2×C8—312no—
434C22×C3—26no—
435C22×C3—26no—
436C22×C3—24no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—26no—
441C22×C3—46no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—62no—
446C22×C3—122no—
447C22×C3C22×#6716yes—
448C22×C3C22×#6711yescharacteristic
449C22×C3C22×C3×D416yes—
450C22×C3C22×C3×D411yescharacteristic
451C2×D3—22no—
452C2×D3—42no—
453C3×C4—212no—
454C3×C4—26no—
455C3×C4—26no—
456C3×C4—26no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—66no—
460C3×C4C22×#6716yes—
461C3×C4C22×#6711yescharacteristic
462C3×C4C22×C3×D411yescharacteristic
463C3×C4C22×C3×D416yes—
464C3×C4C24×D311yescharacteristic
465C3×C4C25×C311yescharacteristic
466Dic3—26no—
467C32C2×#57811yescharacteristic, complemented, Sylow
468C23—21no—
469C23—61no—
470C23C2×C3×#6713yes—
471Q8—212no—
472Q8—23no—
473Q8—36no—
474Q8—612no—
475Q8—63no—
476Q8C2×C3×#6714yes—
477Q8C23×C3×D316yes—
478D4—212no—
479D4—21no—
480D4—32no—
481D4—612no—
482D4—61no—
483D4C2×C3×#67112yes—
484D4C23×C3×D312yes—
485C2×C4—23no—
486C2×C4—212no—
487C2×C4—212no—
488C2×C4—63no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4C2×C3×#6713yes—
492C2×C4C2×C3×#6716yes—
493C2×C4C23×C3×D311yescharacteristic
494C2×C4C23×C3×D316yes—
495C8—32no—
496C8—36no—
497C2×C3—26no—
498C2×C3—26no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—46no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—122no—
506C2×C3#583812yesdirect product factor
507C2×C3C3×#57812yescomplemented
508C2×C3C23×#6711yescharacteristic
509C2×C3C23×C3×D411yescharacteristic
510D3—42no—
511C22—26no—
512C22—22no—
513C22—42no—
514C22—62no—
515C22—122no—
516C22C22×C3×#6716yes—
517C22C22×C3×#6711yescharacteristic
518C4—26no—
519C4—66no—
520C4C22×C3×#6716yes—
521C4C22×C3×#6711yescharacteristic
522C4C24×C3×D311yescharacteristic
523C3—21no—
524C3C2×#583811yescharacteristic, direct product factor
525C3C2×C3×#57811yescharacteristic, complemented
526C2—26no—
527C2—42no—
528C2—122no—
529C2C3×#583812yesdirect product factor
530C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC2×C32×2−1+4C2
socleC22×C32C22×D4
hypercenterC2×C3×2−1+4D3
hypocenterC3C2×C3×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C2×C3×#1010C2semidirect
3. C3×#5838C2direct (N × H)
4. C2×C3×#951C2semidirect
5. C2×C3×#950C2semidirect
6. C2×C3×#948C2semidirect
7. C22×C3×#216C2semidirect
8. C2×C3×#942C2semidirect
10. C2×#5838C3direct (N × H)
12. C32×2−1+4C22semidirect
19. C3×#1010C22semidirect
24. C3×#951C22semidirect
25. C3×#950C22semidirect
26. C3×#948C22semidirect
29. C2×C3×#216C22semidirect
30. C3×#942C22semidirect
38. C2×C3×2−1+4D3semidirect
39. C2×C3×2−1+4C2×C3semidirect
40. C2×#1010C2×C3semidirect
47. #5838C2×C3direct (N × H)
48. C2×#951C2×C3semidirect
49. C2×#950C2×C3semidirect
50. C2×#948C2×C3semidirect
51. C22×#216C2×C3semidirect
52. C2×#942C2×C3semidirect
98. C3×2−1+4C2×D3semidirect
99. C3×2−1+4C22×C3semidirect
120. #1010C22×C3semidirect
137. #951C22×C3semidirect
138. #950C22×C3semidirect
139. #948C22×C3semidirect
142. C2×#216C22×C3semidirect
143. #942C22×C3semidirect
179. C2×2−1+4C3×D3semidirect
294. 2−1+4C2×C3×D3semidirect
393. C2×C32#578semidirect
467. C32C2×#578semidirect
506. C2×C3#5838direct (N × H)
507. C2×C3C3×#578semidirect
524. C3C2×#5838direct (N × H)
525. C3C2×C3×#578semidirect
529. C2C3×#5838direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C2×C3×#10102
C3×#58382
C2×C3×#9512
C2×C3×#9502
C2×C3×#9482
C22×C3×#2162
C2×C3×#9422
C2×C3×#5783
C2×#58383