Finite Group Databasebeta

← C2×C3×#5046 #11863553

Subgroups — C2×C3×#5046 #11863553

Order 1152.

2153 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D12C211yesmaximal, characteristic
2C2×C32×C22⋊Q8C211yesmaximal, characteristic, complemented
3C22×C3×#215C212yesmaximal, complemented
4C2×C3×#830C211yesmaximal, characteristic, complemented
5C3×#5046C218yesmaximal, direct product factor
6C2×C3×#809C212yesmaximal, complemented
7C2×C3×#450—31nomaximal
8C2×#5046C311yesmaximal, characteristic, direct product factor
9C24×C3×D3—21no—
10C22×C32×Q8C2211yescharacteristic, complemented
11C23×C32×C4C2211yescharacteristic
12C22×C3×D12—21no—
13C22×C3×D12—22no—
14C22×C3×D12—28no—
15C22×C3×D12C2214yes—
16C22×C3×D12C2212yes—
17C32×C22⋊Q8—22no—
18C32×C22⋊Q8C2214yescomplemented
19C2×C32×C4⋊C4—21no—
20C2×C32×C4⋊C4C2211yescharacteristic, complemented
21C2×C32×C22⋊C4—21no—
22C2×C3×#215—28no—
23C2×C3×#215C2218yescomplemented
24C22×C3×C3⋊C8C2212yescomplemented
25C3×#830C2214yescomplemented
26C3×#809C2218yescomplemented
27C23×C3×D4—31no—
28C23×D12C2×C311yescharacteristic
29C22×C3×SD16—32no—
30C2×C3×C22⋊Q8—21no—
31C2×C3×C22⋊Q8D311yescharacteristic, complemented
32C2×C3×C22⋊Q8C2×C311yescharacteristic, complemented
33C22×#215C2×C312yescomplemented
34C3×#450—38no—
35C2×C3×#102—32no—
36C2×C3×C22⋊C8—31no—
37C2×#830C2×C311yescharacteristic, complemented
38#5046C2×C318yesdirect product factor
39C2×#809C2×C312yescomplemented
40C24×C32D411yescharacteristic
41C23×C3×D3—28no—
42C23×C3×D3—24no—
43C23×C3×D3—22no—
44C23×C3×D3—44no—
45C23×C3×D3—44no—
46C2×C32×Q8—21no—
47C2×C32×Q8—24no—
48C2×C32×Q8C2312yescomplemented
49C22×C32×C4—24no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4D411yescharacteristic
53C22×C32×C4D412yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C2×C3×D12—28no—
57C2×C3×D12—216no—
58C2×C3×D12—24no—
59C2×C3×D12—22no—
60C2×C3×D12—24no—
61C2×C3×D12—216no—
62C2×C3×D12D418yes—
63C2×C3×D12C2314yes—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2312yescomplemented
67C32×C22⋊C4—24no—
68C3×#215—216no—
69C2×C3×C3⋊C8—24no—
70C2×C3×C3⋊C8C2314yescomplemented
71C2×#450—31noSylow
72C25×C3—61no—
73C24×D3—21no—
74C22×C3×Q8—21no—
75C22×C3×Q8C2×D311yescharacteristic, complemented
76C22×C3×Q8C22×C311yescharacteristic, complemented
77C22×C3×D4—32no—
78C22×C3×D4—34no—
79C22×C3×D4—68no—
80C22×C3×D4—62no—
81C22×C3×D4—61no—
82C23×C3×C4—21no—
83C23×C3×C4C2×D311yescharacteristic
84C23×C3×C4C22×C311yescharacteristic
85C22×D12—21no—
86C22×D12—22no—
87C22×D12—28no—
88C22×D12C22×C314yes—
89C22×D12C22×C312yes—
90C2×C3×SD16—38no—
91C2×C3×SD16—68no—
92C22×C3×C8—32no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—24no—
97C3×C22⋊Q8C2×D314yescomplemented
98C3×C22⋊Q8C22×C314yescomplemented
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2×D311yescharacteristic, complemented
104C2×C3×C4⋊C4C22×C311yescharacteristic, complemented
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×#215—28no—
109C2×#215C22×C318yescomplemented
110C22×C3⋊C8C22×C312yescomplemented
111C3×#102—38no—
112C3×C22⋊C8—34no—
113#830C22×C314yescomplemented
114#809C22×C318yescomplemented
115C23×C32—22no—
116C23×C32—22no—
117C23×C32SD1614yescomplemented
118C23×C32C2×D412yes—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—216no—
121C22×C3×D3—28no—
122C22×C3×D3—24no—
123C22×C3×D3—48no—
124C22×C3×D3—432no—
125C22×C3×D3—44no—
126C22×C3×D3—48no—
127C22×C3×D3—44no—
128C32×Q8—24no—
129C32×Q8—42no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—42no—
136C2×C32×C4—42no—
137C2×C32×C4C2×D414yes—
138C2×C32×C4C2×D412yes—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2411yescharacteristic
141C3×D12—216no—
142C3×D12—28no—
143C3×D12—24no—
144C3×D12C2×D418yes—
145C3×C3⋊C8—24no—
146C23×D4—31no—
147C22×SD16—32no—
148C2×C22⋊Q8C3×D311yescharacteristic, complemented
149#450—38no—
150C2×#102—32no—
151C2×C22⋊C8—31no—
152C24×C3—21no—
153C24×C3—68no—
154C24×C3—64no—
155C24×C3—62no—
156C24×C3—124no—
157C24×C3—124no—
158C24×C3#6711yescharacteristic
159C24×C3C3×D411yescharacteristic
160C23×D3—28no—
161C23×D3—24no—
162C23×D3—22no—
163C23×D3—44no—
164C23×D3—44no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—24no—
169C2×C3×Q8—24no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8C22×D312yescomplemented
173C2×C3×Q8C23×C312yescomplemented
174C2×C3×D4—38no—
175C2×C3×D4—34no—
176C2×C3×D4—616no—
177C2×C3×D4—68no—
178C2×C3×D4—616no—
179C2×C3×D4—64no—
180C2×C3×D4—64no—
181C2×C3×D4—62no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—44no—
195C22×C3×C4#6712yes—
196C22×C3×C4#6711yescharacteristic
197C22×C3×C4C3×D411yescharacteristic
198C22×C3×C4C3×D412yes—
199C22×C3×C4C22×D312yes—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C312yes—
202C22×C3×C4C23×C311yescharacteristic
203C2×D12—28no—
204C2×D12—216no—
205C2×D12—24no—
206C2×D12—24no—
207C2×D12—22no—
208C2×D12—216no—
209C2×D12C3×D418yes—
210C2×D12C23×C314yes—
211C3×SD16—616no—
212C2×C3×C8—34no—
213C2×C3×C8—64no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4C22×D312yescomplemented
222C3×C4⋊C4C23×C312yescomplemented
223C3×C22⋊C4—28no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226#215—216no—
227C2×C3⋊C8—24no—
228C2×C3⋊C8C23×C314yescomplemented
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—28no—
232C22×C32#12012yes—
233C22×C32C2×SD1612yes—
234C22×C32C2×SD1614yescomplemented
235C22×C32AGL(1,ℤ/8)12yes—
236C22×C32C22×D411yescharacteristic
237C2×C3×D3—28no—
238C2×C3×D3—432no—
239C2×C3×D3—44no—
240C2×C3×D3—44no—
241C2×C3×D3—416no—
242C32×C4—22no—
243C32×C4—42no—
244C32×C4—42no—
245C32×C4#12012yes—
246C32×C4C22×D412yes—
247C25—61no—
248C22×Q8C2×C3×D311yescharacteristic, complemented
249C22×D4—32no—
250C22×D4—34no—
251C22×D4—68no—
252C22×D4—61no—
253C22×D4—62no—
254C23×C4C2×C3×D311yescharacteristic
255C2×SD16—38no—
256C2×SD16—68no—
257C22×C8—32no—
258C22⋊Q8—22no—
259C22⋊Q8C2×C3×D314yescomplemented
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×C3×D311yescharacteristic, complemented
262C2×C22⋊C4—21no—
263#102—38no—
264C22⋊C8—34no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—24no—
270C23×C3—21no—
271C23×C3—22no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—616no—
275C23×C3—68no—
276C23×C3—64no—
277C23×C3—1232no—
278C23×C3—128no—
279C23×C3—124no—
280C23×C3—128no—
281C23×C3—124no—
282C23×C3#21514yescomplemented
283C23×C3C3×SD1614yescomplemented
284C23×C3C2×#6712yes—
285C23×C3C2×#6711yescharacteristic
286C23×C3C2×C3×D412yes—
287C23×C3C2×C3×D411yescharacteristic
288C22×D3—216no—
289C22×D3—28no—
290C22×D3—24no—
291C22×D3—48no—
292C22×D3—432no—
293C22×D3—44no—
294C22×D3—48no—
295C22×D3—44no—
296C3×Q8—24no—
297C3×Q8—24no—
298C3×Q8—44no—
299C3×Q8—44no—
300C3×Q8—42no—
301C3×Q8—42no—
302C3×D4—38no—
303C3×D4—616no—
304C3×D4—68no—
305C3×D4—64no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4—42no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4D3×D414yes—
332C2×C3×C4C2×#6712yes—
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C2×C3×D414yes—
335C2×C3×C4C2×C3×D412yes—
336C2×C3×C4C2×C3×D411yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338C2×C3×C4C24×C311yescharacteristic
339D12—216no—
340D12—28no—
341D12—24no—
342D12C2×C3×D418yes—
343C3×C8—64no—
344C3⋊C8—24no—
345C2×C32—24no—
346C2×C32#45014yescomplemented
347C2×C32C2×#12011yescharacteristic
348C2×C32C22×SD1611yescharacteristic
349C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
350C3×D3—48no—
351C24—68no—
352C24—64no—
353C24—62no—
354C24—124no—
355C24—124no—
356C24C3×#6711yescharacteristic
357C2×Q8—21no—
358C2×Q8—24no—
359C2×Q8C22×C3×D312yescomplemented
360C2×D4—34no—
361C2×D4—38no—
362C2×D4—616no—
363C2×D4—68no—
364C2×D4—616no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—62no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4C3×#6711yescharacteristic
372C22×C4C3×#6712yes—
373C22×C4C22×C3×D312yes—
374C22×C4C22×C3×D311yescharacteristic
375SD16—616no—
376C2×C8—34no—
377C2×C8—64no—
378C4⋊C4—24no—
379C4⋊C4—21no—
380C4⋊C4C22×C3×D312yescomplemented
381C22⋊C4—24no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—28no—
385C22×C3—22no—
386C22×C3—28no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—24no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—48no—
396C22×C3—68no—
397C22×C3—1232no—
398C22×C3—124no—
399C22×C3—1216no—
400C22×C3—124no—
401C22×C3#93712yes—
402C22×C3C2×#21512yes—
403C22×C3C2×#21514yescomplemented
404C22×C3#94912yes—
405C22×C3C3×#12012yes—
406C22×C3C2×C3×SD1612yes—
407C22×C3C2×C3×SD1614yescomplemented
408C22×C3C3×AGL(1,ℤ/8)12yes—
409C22×C3C22×#6711yescharacteristic
410C22×C3C22×C3×D411yescharacteristic
411C2×D3—28no—
412C2×D3—432no—
413C2×D3—44no—
414C2×D3—44no—
415C2×D3—416no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—44no—
422C3×C4—44no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4#93712yes—
428C3×C4C3×#12012yes—
429C3×C4C2×D3×D412yes—
430C3×C4C22×C3×D412yes—
431C32C2×#45011yescharacteristic, complemented, Sylow
432C23—22no—
433C23—22no—
434C23—616no—
435C23—68no—
436C23—64no—
437C23—128no—
438C23—1232no—
439C23—124no—
440C23—128no—
441C23—124no—
442C23C3×#21514yescomplemented
443C23C2×C3×#6712yes—
444C23C2×C3×#6711yescharacteristic
445Q8—24no—
446Q8—42no—
447D4—38no—
448D4—616no—
449D4—68no—
450D4—64no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4C3×D3×D414yes—
459C2×C4C2×C3×#6712yes—
460C2×C4C2×C3×#6711yescharacteristic
461C2×C4C23×C3×D311yescharacteristic
462C8—64no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—24no—
469C2×C3—44no—
470C2×C3—128no—
471C2×C3#504614yesdirect product factor
472C2×C3C3×#45014yescomplemented
473C2×C3C2×#93711yescharacteristic
474C2×C3C22×#21511yescharacteristic
475C2×C3C2×#94911yescharacteristic
476C2×C3C2×C3×#12011yescharacteristic
477C2×C3C22×C3×SD1611yescharacteristic
478C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
479D3—48no—
480C22—22no—
481C22—22no—
482C22—28no—
483C22—68no—
484C22—1232no—
485C22—124no—
486C22—124no—
487C22—1216no—
488C22C3×#93712yes—
489C22C2×C3×#21512yes—
490C22C2×C3×#21514yescomplemented
491C22C3×#94912yes—
492C22C22×C3×#6711yescharacteristic
493C4—22no—
494C4—42no—
495C4—42no—
496C4C3×#93712yes—
497C4C2×C3×D3×D412yes—
498C3—21no—
499C3C2×#504611yescharacteristic, direct product factor
500C3C2×C3×#45011yescharacteristic, complemented
501C2—24no—
502C2—128no—
503C2C3×#504614yesdirect product factor
504C2C2×C3×#93711yescharacteristic
505C2C22×C3×#21511yescharacteristic
506C2C2×C3×#94911yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C2×D4
prime radicalC23×C3×D12C2
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C2×C3×#450
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C22⋊Q8C2semidirect
3. C22×C3×#215C2semidirect
4. C2×C3×#830C2semidirect
5. C3×#5046C2direct (N × H)
6. C2×C3×#809C2semidirect
8. C2×#5046C3direct (N × H)
10. C22×C32×Q8C22semidirect
18. C32×C22⋊Q8C22semidirect
20. C2×C32×C4⋊C4C22semidirect
23. C2×C3×#215C22semidirect
24. C22×C3×C3⋊C8C22semidirect
25. C3×#830C22semidirect
26. C3×#809C22semidirect
31. C2×C3×C22⋊Q8D3semidirect
32. C2×C3×C22⋊Q8C2×C3semidirect
33. C22×#215C2×C3semidirect
37. C2×#830C2×C3semidirect
38. #5046C2×C3direct (N × H)
39. C2×#809C2×C3semidirect
48. C2×C32×Q8C23semidirect
66. C32×C4⋊C4C23semidirect
70. C2×C3×C3⋊C8C23semidirect
75. C22×C3×Q8C2×D3semidirect
76. C22×C3×Q8C22×C3semidirect
97. C3×C22⋊Q8C2×D3semidirect
98. C3×C22⋊Q8C22×C3semidirect
103. C2×C3×C4⋊C4C2×D3semidirect
104. C2×C3×C4⋊C4C22×C3semidirect
109. C2×#215C22×C3semidirect
110. C22×C3⋊C8C22×C3semidirect
113. #830C22×C3semidirect
114. #809C22×C3semidirect
117. C23×C32SD16semidirect
148. C2×C22⋊Q8C3×D3semidirect
172. C2×C3×Q8C22×D3semidirect
173. C2×C3×Q8C23×C3semidirect
221. C3×C4⋊C4C22×D3semidirect
222. C3×C4⋊C4C23×C3semidirect
228. C2×C3⋊C8C23×C3semidirect
234. C22×C32C2×SD16semidirect
248. C22×Q8C2×C3×D3semidirect
259. C22⋊Q8C2×C3×D3semidirect
261. C2×C4⋊C4C2×C3×D3semidirect
282. C23×C3#215semidirect
283. C23×C3C3×SD16semidirect
346. C2×C32#450semidirect
359. C2×Q8C22×C3×D3semidirect
380. C4⋊C4C22×C3×D3semidirect
403. C22×C3C2×#215semidirect
407. C22×C3C2×C3×SD16semidirect
431. C32C2×#450semidirect
442. C23C3×#215semidirect
471. C2×C3#5046direct (N × H)
472. C2×C3C3×#450semidirect
490. C22C2×C3×#215semidirect
499. C3C2×#5046direct (N × H)
500. C3C2×C3×#450semidirect
503. C2C3×#5046direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D122
C2×C32×C22⋊Q82
C22×C3×#2152
C2×C3×#8302
C3×#50462
C2×C3×#8092
C2×C3×#4503
C2×#50463