Finite Group Databasebeta

← C2×C3×#4525 #11773712

Subgroups — C2×C3×#4525 #11773712

Order 1152.

1249 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C22⋊C4C211yesmaximal, characteristic
2C2×C3×#830C212yesmaximal, complemented
3C3×#4525C214yesmaximal, direct product factor
4C2×C3×#323—31nomaximal
5C2×#4525C311yesmaximal, characteristic, direct product factor
6C25×C32C411yescharacteristic
7C23×C32×C4C411yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C2×C32×C22⋊C4—22no—
10C2×C32×C22⋊C4—22no—
11C2×C32×C22⋊C4C412yes—
12C2×C32×C22⋊C4C2212yes—
13C22×C3×C3⋊C8—22no—
14C3×#830—22no—
15C3×#830C2214yescomplemented
16C22×C3×C22⋊C4—21no—
17C22×C3×C22⋊C4D311yescharacteristic
18C22×C3×C22⋊C4C2×C311yescharacteristic
19C2×C3×C22⋊C8—32no—
20C2×#830C2×C312yescomplemented
21C3×#323—34no—
22#4525C2×C314yesdirect product factor
23C24×C32—21no—
24C24×C32—21no—
25C24×C32—22no—
26C24×C32—42no—
27C24×C32—42no—
28C24×C32C814yescomplemented
29C24×C32C2×C411yescharacteristic
30C24×C32C2×C412yes—
31C22×C32×C4—21no—
32C22×C32×C4—21no—
33C22×C32×C4—42no—
34C22×C32×C4—42no—
35C22×C32×C4C2×C412yes—
36C22×C32×C4C2×C411yescharacteristic
37C22×C32×C4D412yes—
38C22×C32×C4D412yes—
39C22×C32×C4C2311yescharacteristic
40C32×C22⋊C4—24no—
41C32×C22⋊C4—24no—
42C2×C3×C3⋊C8—24no—
43C2×C3×C3⋊C8—44no—
44C2×#323—31noSylow
45C25×C3—21no—
46C25×C3Dic311yescharacteristic
47C25×C3C3×C411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4—21no—
50C23×C3×C4Dic311yescharacteristic
51C23×C3×C4C3×C411yescharacteristic
52C23×C3×C4C2×D311yescharacteristic
53C23×C3×C4C22×C311yescharacteristic
54C22×C3×C8—62no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4Dic312yes—
64C2×C3×C22⋊C4C3×C412yes—
65C2×C3×C22⋊C4C2×D312yes—
66C2×C3×C22⋊C4C22×C312yes—
67C22×C3⋊C8—22no—
68C3×C22⋊C8—34no—
69C3×C22⋊C8—62no—
70#830—22no—
71#830C22×C314yescomplemented
72C23×C32—22no—
73C23×C32—22no—
74C23×C32—24no—
75C23×C32—24no—
76C23×C32—22no—
77C23×C32—22no—
78C23×C32—48no—
79C23×C32—42no—
80C23×C32—42no—
81C23×C32—48no—
82C23×C32—42no—
83C23×C32—42no—
84C23×C32—42no—
85C23×C32—42no—
86C23×C32C22⋊C411yescharacteristic
87C23×C32C22⋊C411yescharacteristic
88C23×C32C2×C812yes—
89C23×C32C2×C814yescomplemented
90C23×C32M1612yes—
91C23×C32C22×C411yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—24no—
94C2×C32×C4—41no—
95C2×C32×C4—42no—
96C2×C32×C4—41no—
97C2×C32×C4—42no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4C22⋊C412yes—
101C2×C32×C4C2×D412yes—
102C3×C3⋊C8—44no—
103C22×C22⋊C4C3×D311yescharacteristic
104C2×C22⋊C8—32no—
105#323—34no—
106C24×C3—24no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—22no—
115C24×C3—24no—
116C24×C3—21no—
117C24×C3—22no—
118C24×C3—42no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3—44no—
123C24×C3—44no—
124C24×C3C3⋊C814yescomplemented
125C24×C3C3×C814yescomplemented
126C24×C3C2×Dic311yescharacteristic
127C24×C3C2×Dic312yes—
128C24×C3C2×C3×C411yescharacteristic
129C24×C3C2×C3×C412yes—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—44no—
144C22×C3×C4—44no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4C2×Dic312yes—
148C22×C3×C4C2×Dic311yescharacteristic
149C22×C3×C4#6712yes—
150C22×C3×C4#6712yes—
151C22×C3×C4C2×C3×C412yes—
152C22×C3×C4C2×C3×C411yescharacteristic
153C22×C3×C4C3×D412yes—
154C22×C3×C4C3×D412yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C23×C311yescharacteristic
157C2×C3×C8—64no—
158C2×C3×C8—124no—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—28no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—24no—
164C3×C22⋊C4—24no—
165C2×C3⋊C8—24no—
166C2×C3⋊C8—44no—
167C22×C32—24no—
168C22×C32—24no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—24no—
174C22×C32—44no—
175C22×C32—48no—
176C22×C32—44no—
177C22×C32—48no—
178C22×C32—42no—
179C22×C32—42no—
180C22×C32C22⋊C812yes—
181C22×C32C22⋊C812yes—
182C22×C32#9912yes—
183C22×C32#10012yes—
184C22×C32C2×C22⋊C411yescharacteristic
185C22×C32C22×C811yescharacteristic
186C22×C32C2×M1611yescharacteristic
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C25C3×Dic311yescharacteristic
191C23×C4C3×Dic311yescharacteristic
192C23×C4C2×C3×D311yescharacteristic
193C22×C8—62no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C3×Dic312yes—
197C2×C22⋊C4C2×C3×D312yes—
198C22⋊C8—34no—
199C22⋊C8—62no—
200C23×C3—24no—
201C23×C3—24no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—24no—
207C23×C3—24no—
208C23×C3—28no—
209C23×C3—28no—
210C23×C3—24no—
211C23×C3—24no—
212C23×C3—24no—
213C23×C3—24no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—24no—
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—48no—
225C23×C3—48no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—416no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—416no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—48no—
241C23×C3—48no—
242C23×C3—44no—
243C23×C3—44no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3C2×C3⋊C812yes—
249C23×C3C2×C3⋊C814yescomplemented
250C23×C3#20812yes—
251C23×C3#21711yescharacteristic
252C23×C3#21711yescharacteristic
253C23×C3C3×C22⋊C411yescharacteristic
254C23×C3C3×C22⋊C411yescharacteristic
255C23×C3C2×C3×C812yes—
256C23×C3C2×C3×C814yescomplemented
257C23×C3C3×M1612yes—
258C23×C3C22×Dic311yescharacteristic
259C23×C3C22×C3×C411yescharacteristic
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—44no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#21712yes—
287C2×C3×C4C3×C22⋊C412yes—
288C2×C3×C4C2×#6712yes—
289C2×C3×C4C2×C3×D412yes—
290C3×C8—124no—
291C3⋊C8—44no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—44no—
296C2×C32#32314yescomplemented
297C2×C32C2×C22⋊C811yescharacteristic
298C2×C32C2×#9911yescharacteristic
299C2×C32C2×#10011yescharacteristic
300C24—22no—
301C24—21no—
302C24—21no—
303C24—42no—
304C24—42no—
305C24C3×C3⋊C814yescomplemented
306C24C2×C3×Dic311yescharacteristic
307C24C2×C3×Dic312yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C2×C3×Dic312yes—
313C22×C4C2×C3×Dic311yescharacteristic
314C22×C4C3×#6712yes—
315C22×C4C3×#6712yes—
316C22×C4C22×C3×D311yescharacteristic
317C2×C8—64no—
318C2×C8—124no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—28no—
326C22×C3—28no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—24no—
336C22×C3—24no—
337C22×C3—24no—
338C22×C3—24no—
339C22×C3—28no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—48no—
352C22×C3—48no—
353C22×C3—416no—
354C22×C3—48no—
355C22×C3—44no—
356C22×C3—48no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—416no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—48no—
366C22×C3—48no—
367C22×C3#83012yes—
368C22×C3#83012yes—
369C22×C3#83312yes—
370C22×C3#83412yes—
371C22×C3C3×C22⋊C812yes—
372C22×C3C3×C22⋊C812yes—
373C22×C3C3×#9912yes—
374C22×C3C3×#10012yes—
375C22×C3C22×C3⋊C811yescharacteristic
376C22×C3C2×#20811yescharacteristic
377C22×C3C2×#21711yescharacteristic
378C22×C3C2×C3×C22⋊C411yescharacteristic
379C22×C3C22×C3×C811yescharacteristic
380C22×C3C2×C3×M1611yescharacteristic
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—44no—
388C3×C4—42no—
389C3×C4—42no—
390C32C2×#32311yescharacteristic, complemented, Sylow
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—48no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—48no—
403C23—42no—
404C23—42no—
405C23C2×C3×C3⋊C812yes—
406C23C2×C3×C3⋊C814yescomplemented
407C23C3×#20812yes—
408C23C3×#21711yescharacteristic
409C23C3×#21711yescharacteristic
410C23C22×C3×Dic311yescharacteristic
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4C3×#21712yes—
420C2×C4C2×C3×#6712yes—
421C8—124no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—24no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—24no—
435C2×C3—48no—
436C2×C3—44no—
437C2×C3—44no—
438C2×C3#452514yesdirect product factor
439C2×C3C3×#32314yescomplemented
440C2×C3C2×#83011yescharacteristic
441C2×C3C2×#83311yescharacteristic
442C2×C3C2×#83411yescharacteristic
443C2×C3C2×C3×C22⋊C811yescharacteristic
444C2×C3C2×C3×#9911yescharacteristic
445C2×C3C2×C3×#10011yescharacteristic
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—44no—
454C22—44no—
455C22—48no—
456C22—42no—
457C22—42no—
458C22—48no—
459C22C3×#83012yes—
460C22C3×#83012yes—
461C22C3×#83312yes—
462C22C3×#83412yes—
463C22C22×C3×C3⋊C811yescharacteristic
464C22C2×C3×#20811yescharacteristic
465C22C2×C3×#21711yescharacteristic
466C4—22no—
467C4—22no—
468C4—42no—
469C3—21no—
470C3C2×#452511yescharacteristic, direct product factor
471C3C2×C3×#32311yescharacteristic, complemented
472C2—21no—
473C2—21no—
474C2—22no—
475C2—44no—
476C2C3×#452514yesdirect product factor
477C2C2×C3×#83011yescharacteristic
478C2C2×C3×#83311yescharacteristic
479C2C2×C3×#83411yescharacteristic
subgroupG/H
centerC23×C3#217
commutatorC22×C3C22×C3×C8
FrattiniC22×C4C22×C3×D3
FittingC22×C32×C22⋊C4C2
socleC23×C32C22⋊C4
prime radicalC25×C32C4
cosocleC22×C3×C4C23×C3
hypercenterC22×C3×C22⋊C4D3
hypocenterC3C2×C3×#323
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#830C2semidirect
3. C3×#4525C2direct (N × H)
5. C2×#4525C3direct (N × H)
15. C3×#830C22semidirect
20. C2×#830C2×C3semidirect
22. #4525C2×C3direct (N × H)
28. C24×C32C8semidirect
71. #830C22×C3semidirect
89. C23×C32C2×C8semidirect
124. C24×C3C3⋊C8semidirect
125. C24×C3C3×C8semidirect
249. C23×C3C2×C3⋊C8semidirect
256. C23×C3C2×C3×C8semidirect
296. C2×C32#323semidirect
305. C24C3×C3⋊C8semidirect
390. C32C2×#323semidirect
406. C23C2×C3×C3⋊C8semidirect
438. C2×C3#4525direct (N × H)
439. C2×C3C3×#323semidirect
470. C3C2×#4525direct (N × H)
471. C3C2×C3×#323semidirect
476. C2C3×#4525direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊C42
C2×C3×#8302
C3×#45252
C2×C3×#3233
C2×#45253