Finite Group Databasebeta

← C2×C3×#5009 #11868329

Subgroups — C2×C3×#5009 #11868329

Order 1152.

1109 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×#870C211yesmaximal, characteristic, complemented
3C2×C3×#832C211yesmaximal, characteristic
4C2×C3×#832C211yesmaximal, characteristic
5C3×#5009C218yesmaximal, direct product factor
6C2×C3×#808C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×C3×#478—31nomaximal
10C2×#5009C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4—21no—
13C32×C4×D4—22no—
14C32×C4×D4C2214yes—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C22⋊C4—21no—
17C2×C32×C42C2211yescharacteristic
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4—21no—
21C2×C3×Dic3⋊C4—21no—
22C22×C3×C3⋊C8C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#870—22no—
25C3×#870C2214yescomplemented
26C3×#832C2214yes—
27C3×#832C2214yes—
28C3×#808C2214yescomplemented
29C3×#807C2214yescomplemented
30C3×C4⋊(C3⋊C8)C2214yescomplemented
31C2×C3×#125—31no—
32C2×C3×C4×D4—21no—
33C2×C3×C4×D4D311yescharacteristic
34C2×C3×C4×D4C2×C311yescharacteristic
35C2×#870C2×C311yescharacteristic, complemented
36C3×#478—38no—
37C2×C3×#107—31no—
38C2×C3×#106—31no—
39C2×C3×C4⋊C8—31no—
40C2×C3×#102—31no—
41C2×C3×#102—31no—
42C2×#832C2×C311yescharacteristic
43C2×#832C2×C311yescharacteristic
44#5009C2×C318yesdirect product factor
45C2×#808C2×C311yescharacteristic, complemented
46C2×#807C2×C311yescharacteristic, complemented
47C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C24×C32—21no—
49C2×C32×D4—21no—
50C2×C32×D4—22no—
51C2×C32×D4Q814yes—
52C2×C32×D4C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4—24no—
55C22×C32×C4—44no—
56C22×C32×C4D411yescharacteristic
57C22×C32×C4D411yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C22×C3×Dic3—21no—
60C22×C3×Dic3—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2312yes—
63C32×C22⋊C4—24no—
64C32×C42D412yes—
65C32×C42C2312yes—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3C2312yes—
69C3×C4⋊Dic3C2312yes—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—24no—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2312yes—
75C2×C3×C3⋊C8C2312yes—
76C2×#478—31noSylow
77C22×C3×D4—21no—
78C22×C3×D4C2×D311yescharacteristic
79C22×C3×D4C22×C311yescharacteristic
80C23×C3×C4—22no—
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C22×C3×C8—31no—
84C22×C3×C8—31no—
85C3×#125—34no—
86C3×#125—62no—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—24no—
91C3×C4×D4C2×D314yes—
92C3×C4×D4C22×C314yes—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—61no—
97C2×C3×C4⋊C4—61no—
98C2×C3×C4⋊C4C2×D311yescharacteristic
99C2×C3×C4⋊C4C22×C311yescharacteristic
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2×D311yescharacteristic
105C2×C3×C42C22×C311yescharacteristic
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×C4⋊Dic3C22×C311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C22×C3⋊C8C22×C311yescharacteristic
111C22×C3⋊C8C22×C311yescharacteristic
112#870—22no—
113#870C22×C314yescomplemented
114C3×#107—34no—
115C3×#106—34no—
116C3×C4⋊C8—34no—
117C3×#102—34no—
118C3×#102—34no—
119#832C22×C314yes—
120#832C22×C314yes—
121#808C22×C314yescomplemented
122#807C22×C314yescomplemented
123C4⋊(C3⋊C8)C22×C314yescomplemented
124C23×C32—24no—
125C23×C32—22no—
126C23×C32—42no—
127C23×C32—42no—
128C23×C32C2×D411yescharacteristic
129C32×D4—24no—
130C32×D4—22no—
131C32×D4C2×Q814yes—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—48no—
136C2×C32×C4—42no—
137C2×C32×C4C2×D412yes—
138C2×C32×C4C2×D412yes—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×Q812yes—
141C2×C32×C4#4112yes—
142C2×C32×C4C2411yescharacteristic
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—42no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149C2×#125—31no—
150C2×C4×D4C3×D311yescharacteristic
151#478—38no—
152C2×#107—31no—
153C2×#106—31no—
154C2×C4⋊C8—31no—
155C2×#102—31no—
156C2×#102—31no—
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—21no—
160C2×C3×D4—22no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4Dic614yes—
169C2×C3×D4C3×Q814yes—
170C2×C3×D4C22×D312yes—
171C2×C3×D4C23×C312yes—
172C22×C3×C4—22no—
173C22×C3×C4—28no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—48no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—61no—
185C22×C3×C4—61no—
186C22×C3×C4#6711yescharacteristic
187C22×C3×C4#6711yescharacteristic
188C22×C3×C4C3×D411yescharacteristic
189C22×C3×C4C3×D411yescharacteristic
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C23×C311yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×C3×C8—32no—
195C2×C3×C8—32no—
196C2×C3×C8—62no—
197C2×C3×C8—62no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—32no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—64no—
205C3×C4⋊C4—64no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4C22×D312yes—
209C3×C4⋊C4C23×C312yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42#6712yes—
216C3×C42C3×D412yes—
217C3×C42C22×D312yes—
218C3×C42C23×C312yes—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3C23×C312yes—
222C4⋊Dic3C23×C312yes—
223Dic3⋊C4—24no—
224Dic3⋊C4—24no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8C23×C312yes—
228C2×C3⋊C8C23×C312yes—
229C22×C32—24no—
230C22×C32—42no—
231C22×C32—42no—
232C22×C32—48no—
233C22×C32C22⋊Q812yes—
234C22×C32#13512yes—
235C22×C32AGL(1,ℤ/8)12yes—
236C22×C32C22×D411yescharacteristic
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—42no—
240C32×C4C22⋊Q812yes—
241C32×C4C22×Q811yescharacteristic
242C32×C4C2×#4111yescharacteristic
243C3×Dic3—42no—
244C3×Dic3—42no—
245C22×D4C2×C3×D311yescharacteristic
246C23×C4—21no—
247C22×C8—31no—
248C22×C8—31no—
249#125—34no—
250#125—62no—
251C4×D4—22no—
252C4×D4C2×C3×D314yes—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C2×C3×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C42C2×C3×D311yescharacteristic
260#107—34no—
261#106—34no—
262C4⋊C8—34no—
263#102—34no—
264#102—34no—
265C23×C3—24no—
266C23×C3—28no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3C2×#6711yescharacteristic
279C23×C3C2×C3×D411yescharacteristic
280C3×D4—28no—
281C3×D4—24no—
282C3×D4—24no—
283C3×D4—24no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—24no—
287C3×D4C2×Dic614yes—
288C3×D4C2×C3×Q814yes—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—416no—
305C2×C3×C4—48no—
306C2×C3×C4—48no—
307C2×C3×C4—44no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—122no—
313C2×C3×C4—122no—
314C2×C3×C4C2×Dic612yes—
315C2×C3×C4#23512yes—
316C2×C3×C4C2×#6712yes—
317C2×C3×C4C2×#6712yes—
318C2×C3×C4C2×#6711yescharacteristic
319C2×C3×C4C2×C3×D412yes—
320C2×C3×C4C2×C3×D412yes—
321C2×C3×C4C2×C3×D411yescharacteristic
322C2×C3×C4C2×C3×Q812yes—
323C2×C3×C4C3×#4112yes—
324C2×C3×C4C23×D311yescharacteristic
325C2×C3×C4C24×C311yescharacteristic
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C3×C8—62no—
331C3×C8—62no—
332C3⋊C8—22no—
333C3⋊C8—22no—
334C2×C32—44no—
335C2×C32#47814yescomplemented
336C2×C32C2×C22⋊Q811yescharacteristic
337C2×C32C2×#13511yescharacteristic
338C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
339C24—21no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4C3×Dic614yes—
343C2×D4C22×C3×D312yes—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—44no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4C3×#6711yescharacteristic
350C22×C4C3×#6711yescharacteristic
351C22×C4C22×C3×D311yescharacteristic
352C2×C8—32no—
353C2×C8—32no—
354C2×C8—62no—
355C2×C8—62no—
356C4⋊C4—21no—
357C4⋊C4—32no—
358C4⋊C4—32no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—64no—
362C4⋊C4—64no—
363C4⋊C4C22×C3×D312yes—
364C22⋊C4—24no—
365C42C3×#6712yes—
366C42C22×C3×D312yes—
367C22×C3—28no—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—416no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—48no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—48no—
383C22×C3#92412yes—
384C22×C3#93212yes—
385C22×C3#95012yes—
386C22×C3C3×C22⋊Q812yes—
387C22×C3C3×#13512yes—
388C22×C3C3×AGL(1,ℤ/8)12yes—
389C22×C3C22×#6711yescharacteristic
390C22×C3C22×C3×D411yescharacteristic
391C3×C4—24no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—42no—
400C3×C4—44no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—122no—
404C3×C4—122no—
405C3×C4#92412yes—
406C3×C4C3×C22⋊Q812yes—
407C3×C4C22×Dic611yescharacteristic
408C3×C4C2×#23511yescharacteristic
409C3×C4C22×C3×Q811yescharacteristic
410C3×C4C2×C3×#4111yescharacteristic
411Dic3—42no—
412Dic3—42no—
413C32C2×#47811yescharacteristic, complemented, Sylow
414C23—24no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23C2×C3×#6711yescharacteristic
419D4—24no—
420D4—22no—
421D4C2×C3×Dic614yes—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4C2×C3×Dic612yes—
432C2×C4C3×#23512yes—
433C2×C4C2×C3×#6712yes—
434C2×C4C2×C3×#6712yes—
435C2×C4C2×C3×#6711yescharacteristic
436C2×C4C23×C3×D311yescharacteristic
437C8—62no—
438C8—62no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—24no—
443C2×C3—48no—
444C2×C3—44no—
445C2×C3—44no—
446C2×C3#500914yesdirect product factor
447C2×C3C3×#47814yescomplemented
448C2×C3C2×#92411yescharacteristic
449C2×C3C2×#93211yescharacteristic
450C2×C3C2×#95011yescharacteristic
451C2×C3C2×C3×C22⋊Q811yescharacteristic
452C2×C3C2×C3×#13511yescharacteristic
453C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
454C22—24no—
455C22—42no—
456C22—42no—
457C22—48no—
458C22C3×#92412yes—
459C22C3×#93212yes—
460C22C3×#95012yes—
461C22C22×C3×#6711yescharacteristic
462C4—22no—
463C4—22no—
464C4—42no—
465C4—122no—
466C4—122no—
467C4C3×#92412yes—
468C4C22×C3×Dic611yescharacteristic
469C4C2×C3×#23511yescharacteristic
470C3—21no—
471C3C2×#500911yescharacteristic, direct product factor
472C3C2×C3×#47811yescharacteristic, complemented
473C2—44no—
474C2C3×#500914yesdirect product factor
475C2C2×C3×#92411yescharacteristic
476C2C2×C3×#93211yescharacteristic
477C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#478
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#870C2semidirect
5. C3×#5009C2direct (N × H)
6. C2×C3×#808C2semidirect
7. C2×C3×#807C2semidirect
8. C2×C3×C4⋊(C3⋊C8)C2semidirect
10. C2×#5009C3direct (N × H)
25. C3×#870C22semidirect
28. C3×#808C22semidirect
29. C3×#807C22semidirect
30. C3×C4⋊(C3⋊C8)C22semidirect
35. C2×#870C2×C3semidirect
44. #5009C2×C3direct (N × H)
45. C2×#808C2×C3semidirect
46. C2×#807C2×C3semidirect
47. C2×C4⋊(C3⋊C8)C2×C3semidirect
113. #870C22×C3semidirect
121. #808C22×C3semidirect
122. #807C22×C3semidirect
123. C4⋊(C3⋊C8)C22×C3semidirect
335. C2×C32#478semidirect
413. C32C2×#478semidirect
446. C2×C3#5009direct (N × H)
447. C2×C3C3×#478semidirect
471. C3C2×#5009direct (N × H)
472. C3C2×C3×#478semidirect
474. C2C3×#5009direct (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×D42
C2×C3×#8702
C2×C3×#8322
C3×#50092
C2×C3×#8082
C2×C3×#8072
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4783
C2×#50093