Finite Group Databasebeta

← C2×C3×#5113 #11857783

Subgroups — C2×C3×#5113 #11857783

Order 1152.

1753 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C23×C32×C8C211yesmaximal, characteristic, complemented
3C2×C3×#926C211yesmaximal, characteristic, complemented
4C3×#5113C218yesmaximal, direct product factor
5C2×C3×#821C212yesmaximal, complemented
6C2×C3×#816C212yesmaximal, complemented
7C2×C3×#416—31nomaximal
8C2×#5113C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2211yescharacteristic
10C22×C3×#67—21no—
11C2×C3×#235—24no—
12C2×C3×#235—22no—
13C2×C3×#235C418yescomplemented
14C2×C3×#235C2214yes—
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3—21no—
17C22×C3×Dic6C2211yescharacteristic
18C22×C32×C8—24no—
19C22×C32×C8C2212yes—
20C22×C32×C8C2214yescomplemented
21C2×C3×#217—21no—
22C3×#926—22no—
23C3×#926C2214yescomplemented
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4×Dic3—21no—
26C3×#821C2218yescomplemented
27C3×#816C2218yescomplemented
28C22×C3×#41—31no—
29C22×#235C2×C311yescharacteristic
30C23×C3×C8—21no—
31C23×C3×C8D311yescharacteristic, complemented
32C23×C3×C8C2×C311yescharacteristic, complemented
33C2×C3×#117—31no—
34C2×#926C2×C311yescharacteristic, complemented
35C3×#416—38no—
36C2×C3×#103—32no—
37C2×C3×#102—32no—
38#5113C2×C318yesdirect product factor
39C2×#821C2×C312yescomplemented
40C2×#816C2×C312yescomplemented
41C24×C32D411yescharacteristic
42C23×C3×D3—21no—
43C22×C32×C4C2×C414yes—
44C22×C32×C4D412yes—
45C22×C32×C4D411yescharacteristic
46C22×C32×C4D414yes—
47C22×C32×C4C2311yescharacteristic
48C22×C32×C4C2312yes—
49C2×C3×#67—28no—
50C2×C3×#67—24no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—22no—
53C3×#235—216no—
54C3×#235—28no—
55C3×#235C2×C4116yescomplemented
56C2×C3×D12—21no—
57C2×C3×D12—22no—
58C2×C3×D12C2×C414yes—
59C2×C3×D12C2312yes—
60C2×C3×C4×D3—28no—
61C2×C3×C4×D3—24no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—22no—
64C2×C3×Dic6C2×C414yes—
65C2×C3×Dic6C2312yes—
66C2×C32×C8—24no—
67C2×C32×C8—28no—
68C2×C32×C8C2314yes—
69C3×#217—24no—
70C3×C4⋊Dic3—22no—
71C3×C4⋊Dic3C2314yes—
72C3×C4×Dic3—24no—
73C2×#416—31noSylow
74C2×C3×#41—38no—
75C2×C3×#41—34no—
76C2×C3×#41—62no—
77C2×C3×#41—64no—
78C22×C3×Q8—31no—
79C22×C3×D4—31no—
80C22×C3×D4—61no—
81C23×C3×C4—21no—
82C23×C3×C4—61no—
83C23×C3×C4C2×D311yescharacteristic
84C23×C3×C4C22×C311yescharacteristic
85C22×#67—21no—
86C2×#235—22no—
87C2×#235—24no—
88C2×#235C3×C418yescomplemented
89C2×#235C22×C314yes—
90C22×D12C22×C311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6C22×C311yescharacteristic
93C22×C3×C8—28no—
94C22×C3×C8—24no—
95C22×C3×C8—24no—
96C22×C3×C8—22no—
97C22×C3×C8—24no—
98C22×C3×C8C2×D312yes—
99C22×C3×C8C2×D314yescomplemented
100C22×C3×C8C22×C312yes—
101C22×C3×C8C22×C314yescomplemented
102C3×#117—34no—
103C3×#117—62no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C22⋊C4—61no—
106C2×C3×C42—61no—
107C2×#217—21no—
108#926—22no—
109#926C22×C314yescomplemented
110C2×C4⋊Dic3C22×C312yes—
111C2×C4×Dic3—21no—
112C3×#103—38no—
113C3×#102—38no—
114#821C22×C318yescomplemented
115#816C22×C318yescomplemented
116C23×C32—24no—
117C23×C32C22⋊C414yes—
118C23×C32C2×D412yes—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—24no—
121C22×C3×D3—22no—
122C22×C3×D3—44no—
123C2×C32×C4C22⋊C414yes—
124C2×C32×C4C22⋊C418yes—
125C2×C32×C4C22×C412yes—
126C2×C32×C4C22×C414yes—
127C2×C32×C4C2×D412yes—
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D412yes—
130C2×C32×C4C2×D414yes—
131C2×C32×C4C2411yescharacteristic
132C3×#67—216no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—44no—
137C3×D12—24no—
138C3×D12—22no—
139C3×D12C22×C414yes—
140C3×C4×D3—216no—
141C3×Dic6—24no—
142C3×Dic6—22no—
143C3×Dic6C22×C414yes—
144C32×C8—24no—
145C22×#41—31no—
146C23×C8C3×D311yescharacteristic, complemented
147C2×#117—31no—
148#416—38no—
149C2×#103—32no—
150C2×#102—32no—
151C24×C3—21no—
152C24×C3—61no—
153C24×C3D1211yescharacteristic
154C24×C3C3×D411yescharacteristic
155C23×D3—21no—
156C3×#41—316no—
157C3×#41—616no—
158C3×#41—68no—
159C2×C3×Q8—34no—
160C2×C3×Q8—32no—
161C2×C3×Q8—61no—
162C2×C3×Q8—62no—
163C2×C3×D4—34no—
164C2×C3×D4—32no—
165C2×C3×D4—68no—
166C2×C3×D4—64no—
167C2×C3×D4—61no—
168C2×C3×D4—62no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—61no—
176C22×C3×C4—68no—
177C22×C3×C4—64no—
178C22×C3×C4—62no—
179C22×C3×C4C4×D314yes—
180C22×C3×C4D1212yes—
181C22×C3×C4D1211yescharacteristic
182C22×C3×C4#6714yes—
183C22×C3×C4C2×C3×C414yes—
184C22×C3×C4C3×D411yescharacteristic
185C22×C3×C4C3×D412yes—
186C22×C3×C4C3×D414yes—
187C22×C3×C4C22×D311yescharacteristic
188C22×C3×C4C22×D312yes—
189C22×C3×C4C23×C311yescharacteristic
190C22×C3×C4C23×C312yes—
191C2×#67—28no—
192C2×#67—24no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195#235—216no—
196#235—28no—
197#235C2×C3×C4116yescomplemented
198C2×D12—21no—
199C2×D12—22no—
200C2×D12C2×C3×C414yes—
201C2×D12C23×C312yes—
202C2×C4×D3—28no—
203C2×C4×D3—24no—
204C2×Dic6—21no—
205C2×Dic6—22no—
206C2×Dic6C2×C3×C414yes—
207C2×Dic6C23×C312yes—
208C2×C3×C8—28no—
209C2×C3×C8—216no—
210C2×C3×C8—24no—
211C2×C3×C8—28no—
212C2×C3×C8—24no—
213C2×C3×C8—28no—
214C2×C3×C8—24no—
215C2×C3×C8C22×D314yes—
216C2×C3×C8C23×C314yes—
217C3×C4⋊C4—34no—
218C3×C4⋊C4—62no—
219C3×C22⋊C4—64no—
220C3×C42—64no—
221#217—24no—
222C4⋊Dic3—22no—
223C4⋊Dic3C23×C314yes—
224C4×Dic3—24no—
225C22×C32—24no—
226C22×C32—28no—
227C22×C32C2×C22⋊C414yes—
228C22×C32C2×C22⋊C412yes—
229C22×C32#13514yes—
230C22×C32C22×D411yescharacteristic
231C2×C3×D3—24no—
232C2×C3×D3—44no—
233C2×C3×D3—48no—
234C32×C4C2×C22⋊C412yes—
235C32×C4C2×C22⋊C414yes—
236C32×C4C23×C411yescharacteristic
237C32×C4C22×D411yescharacteristic
238C3×Dic3—24no—
239C3×Dic3—44no—
240C2×#41—38no—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—64no—
244C22×Q8—31no—
245C22×D4—31no—
246C22×D4—61no—
247C23×C4—61no—
248C23×C4C2×C3×D311yescharacteristic
249C22×C8—24no—
250C22×C8C2×C3×D312yes—
251C22×C8C2×C3×D314yescomplemented
252#117—34no—
253#117—62no—
254C2×C4⋊C4—32no—
255C2×C22⋊C4—61no—
256C2×C42—61no—
257#103—38no—
258#102—38no—
259C23×C3—28no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—21no—
265C23×C3—64no—
266C23×C3—62no—
267C23×C3—124no—
268C23×C3(C2×D3)⋊C414yes—
269C23×C3C3×C22⋊C414yes—
270C23×C3C2×D1212yes—
271C23×C3C2×D1211yescharacteristic
272C23×C3C2×C3×D412yes—
273C23×C3C2×C3×D411yescharacteristic
274C22×D3—24no—
275C22×D3—22no—
276C22×D3—44no—
277C3×Q8—34no—
278C3×Q8—64no—
279C3×Q8—62no—
280C3×D4—34no—
281C3×D4—616no—
282C3×D4—64no—
283C3×D4—62no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—64no—
294C2×C3×C4—62no—
295C2×C3×C4—616no—
296C2×C3×C4—64no—
297C2×C3×C4—124no—
298C2×C3×C4(C2×D3)⋊C414yes—
299C2×C3×C4(C2×D3)⋊C418yes—
300C2×C3×C4C3×C22⋊C414yes—
301C2×C3×C4C3×C22⋊C418yes—
302C2×C3×C4C2×C4×D312yes—
303C2×C3×C4C2×C4×D314yes—
304C2×C3×C4C2×D1212yes—
305C2×C3×C4C2×D1211yescharacteristic
306C2×C3×C4C2×#6712yes—
307C2×C3×C4C2×#6714yes—
308C2×C3×C4C22×C3×C412yes—
309C2×C3×C4C22×C3×C414yes—
310C2×C3×C4C2×C3×D412yes—
311C2×C3×C4C2×C3×D411yescharacteristic
312C2×C3×C4C2×C3×D412yes—
313C2×C3×C4C2×C3×D414yes—
314C2×C3×C4C23×D311yescharacteristic
315C2×C3×C4C24×C311yescharacteristic
316#67—216no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—44no—
321D12—24no—
322D12—22no—
323D12C22×C3×C414yes—
324C4×D3—216no—
325Dic6—24no—
326Dic6—22no—
327Dic6C22×C3×C414yes—
328C3×C8—28no—
329C3×C8—24no—
330C3×C8—24no—
331C2×C32—24no—
332C2×C32#41614yescomplemented
333C2×C32C22×C22⋊C411yescharacteristic
334C2×C32C2×#13512yes—
335C3×D3—44no—
336C24—61no—
337C24C3×D1211yescharacteristic
338#41—316no—
339#41—616no—
340#41—68no—
341C2×Q8—32no—
342C2×Q8—34no—
343C2×Q8—61no—
344C2×Q8—62no—
345C2×D4—32no—
346C2×D4—34no—
347C2×D4—68no—
348C2×D4—64no—
349C2×D4—61no—
350C2×D4—62no—
351C22×C4—61no—
352C22×C4—68no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4C3×C4×D314yes—
356C22×C4C3×D1211yescharacteristic
357C22×C4C3×D1212yes—
358C22×C4C3×#6714yes—
359C22×C4C22×C3×D311yescharacteristic
360C22×C4C22×C3×D312yes—
361C2×C8—24no—
362C2×C8—28no—
363C2×C8C22×C3×D314yes—
364C4⋊C4—34no—
365C4⋊C4—62no—
366C22⋊C4—64no—
367C42—64no—
368C22×C3—216no—
369C22×C3—28no—
370C22×C3—24no—
371C22×C3—24no—
372C22×C3—28no—
373C22×C3—28no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—64no—
379C22×C3—124no—
380C22×C3—128no—
381C22×C3#90414yes—
382C22×C3C2×(C2×D3)⋊C414yes—
383C22×C3C2×(C2×D3)⋊C412yes—
384C22×C3C2×C3×C22⋊C414yes—
385C22×C3C2×C3×C22⋊C412yes—
386C22×C3C3×#13514yes—
387C22×C3C22×D1211yescharacteristic
388C22×C3C22×C3×D411yescharacteristic
389C2×D3—24no—
390C2×D3—44no—
391C2×D3—48no—
392C3×C4—24no—
393C3×C4—22no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—64no—
397C3×C4—124no—
398C3×C4C2×(C2×D3)⋊C414yes—
399C3×C4C2×(C2×D3)⋊C412yes—
400C3×C4C2×C3×C22⋊C412yes—
401C3×C4C2×C3×C22⋊C414yes—
402C3×C4C22×C4×D311yescharacteristic
403C3×C4C22×#6711yescharacteristic
404C3×C4C23×C3×C411yescharacteristic
405C3×C4C22×C3×D411yescharacteristic
406Dic3—24no—
407Dic3—44no—
408C32C2×#41611yescharacteristic, complemented, Sylow
409C23—24no—
410C23—64no—
411C23—62no—
412C23—124no—
413C23C3×(C2×D3)⋊C414yes—
414C23C2×C3×D1212yes—
415C23C2×C3×D1211yescharacteristic
416Q8—34no—
417Q8—64no—
418Q8—62no—
419D4—34no—
420D4—616no—
421D4—64no—
422D4—62no—
423C2×C4—64no—
424C2×C4—62no—
425C2×C4—616no—
426C2×C4—64no—
427C2×C4—124no—
428C2×C4C3×(C2×D3)⋊C414yes—
429C2×C4C3×(C2×D3)⋊C418yes—
430C2×C4C2×C3×C4×D312yes—
431C2×C4C2×C3×C4×D314yes—
432C2×C4C2×C3×D1212yes—
433C2×C4C2×C3×D1211yescharacteristic
434C2×C4C2×C3×#6712yes—
435C2×C4C2×C3×#6714yes—
436C2×C4C23×C3×D311yescharacteristic
437C8—24no—
438C2×C3—28no—
439C2×C3—24no—
440C2×C3—24no—
441C2×C3—21no—
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—124no—
445C2×C3#511314yesdirect product factor
446C2×C3C3×#41614yescomplemented
447C2×C3C2×#90412yes—
448C2×C3C22×(C2×D3)⋊C411yescharacteristic
449C2×C3C22×C3×C22⋊C411yescharacteristic
450C2×C3C2×C3×#13512yes—
451D3—44no—
452C22—28no—
453C22—24no—
454C22—64no—
455C22—128no—
456C22—124no—
457C22C3×#90414yes—
458C22C2×C3×(C2×D3)⋊C414yes—
459C22C2×C3×(C2×D3)⋊C412yes—
460C22C22×C3×D1211yescharacteristic
461C4—64no—
462C4—124no—
463C4C2×C3×(C2×D3)⋊C414yes—
464C4C2×C3×(C2×D3)⋊C412yes—
465C4C22×C3×C4×D311yescharacteristic
466C4C22×C3×#6711yescharacteristic
467C3—21no—
468C3C2×#511311yescharacteristic, direct product factor
469C3C2×C3×#41611yescharacteristic, complemented
470C2—24no—
471C2—124no—
472C2C3×#511314yesdirect product factor
473C2C2×C3×#90412yes—
474C2C22×C3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22×C3×C4D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC23×C32×C8C2
socleC23×C32C2×D4
prime radicalC22×C3×#235C2
cosocleC2×C3×C4C24×C3
hypercenterC23×C3×C8D3
hypocenterC3C2×C3×#416
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8

Complements

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

NH ≅ G/Ntype
2. C23×C32×C8C2semidirect
3. C2×C3×#926C2semidirect
4. C3×#5113C2direct (N × H)
5. C2×C3×#821C2semidirect
6. C2×C3×#816C2semidirect
8. C2×#5113C3direct (N × H)
13. C2×C3×#235C4semidirect
20. C22×C32×C8C22semidirect
23. C3×#926C22semidirect
26. C3×#821C22semidirect
27. C3×#816C22semidirect
31. C23×C3×C8D3semidirect
32. C23×C3×C8C2×C3semidirect
34. C2×#926C2×C3semidirect
38. #5113C2×C3direct (N × H)
39. C2×#821C2×C3semidirect
40. C2×#816C2×C3semidirect
55. C3×#235C2×C4semidirect
88. C2×#235C3×C4semidirect
99. C22×C3×C8C2×D3semidirect
101. C22×C3×C8C22×C3semidirect
109. #926C22×C3semidirect
114. #821C22×C3semidirect
115. #816C22×C3semidirect
146. C23×C8C3×D3semidirect
197. #235C2×C3×C4semidirect
251. C22×C8C2×C3×D3semidirect
332. C2×C32#416semidirect
408. C32C2×#416semidirect
445. C2×C3#5113direct (N × H)
446. C2×C3C3×#416semidirect
468. C3C2×#5113direct (N × H)
469. C3C2×C3×#416semidirect
472. C2C3×#5113direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C23×C32×C82
C2×C3×#9262
C3×#51132
C2×C3×#8212
C2×C3×#8162
C2×C3×#4163
C2×#51133