Finite Group Databasebeta

← C2×C3×#4928 #11801570

Subgroups — C2×C3×#4928 #11801570

Order 1152.

1849 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C42C211yesmaximal, characteristic
2C22×C3×C4⋊Dic3C211yesmaximal, characteristic
3C22×C3×C4⋊Dic3C211yesmaximal, characteristic
4C22×C3×Dic3⋊C4C212yesmaximal
5C2×C3×C3⋊#95C212yesmaximal
6C3×#4928C218yesmaximal, direct product factor
7C2×C3×#389—31nomaximal
8C2×#4928C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C32×C4C2211yescharacteristic
12C23×C3×Dic3C2212yes—
13C23×C3×Dic3C2212yes—
14C2×C32×C42—24no—
15C2×C32×C42C2214yes—
16C2×C3×C4⋊Dic3—24no—
17C2×C3×C4⋊Dic3C418yescomplemented
18C2×C3×C4⋊Dic3C2214yes—
19C2×C3×C4⋊Dic3C2214yes—
20C2×C3×Dic3⋊C4—28no—
21C2×C3×Dic3⋊C4C2218yes—
22C3×C3⋊#95C2218yes—
23C22×C3×C4⋊C4—31no—
24C22×C3×C4⋊C4—32no—
25C22×C3×C4⋊C4—31no—
26C22×C3×C42—21no—
27C22×C3×C42D311yescharacteristic
28C22×C3×C42C2×C311yescharacteristic
29C22×C4⋊Dic3C2×C311yescharacteristic
30C22×C4⋊Dic3C2×C311yescharacteristic
31C22×Dic3⋊C4C2×C312yes—
32C3×#389—38no—
33C2×C3×#95—32no—
34C2×C3⋊#95C2×C312yes—
35#4928C2×C318yesdirect product factor
36C24×C32C2311yescharacteristic
37C22×C32×C4—22no—
38C22×C32×C4—22no—
39C22×C32×C4—22no—
40C22×C32×C4—22no—
41C22×C32×C4C2×C414yes—
42C22×C32×C4D414yes—
43C22×C32×C4D414yes—
44C22×C32×C4Q814yes—
45C22×C32×C4Q814yes—
46C22×C32×C4C2312yes—
47C22×C32×C4C2312yes—
48C22×C32×C4C2312yes—
49C22×C3×Dic3—24no—
50C22×C3×Dic3—24no—
51C22×C3×Dic3—24no—
52C22×C3×Dic3—24no—
53C22×C3×Dic3—24no—
54C22×C3×Dic3C2×C418yes—
55C22×C3×Dic3C2314yes—
56C22×C3×Dic3C2314yes—
57C32×C42—28no—
58C3×C4⋊Dic3—28no—
59C3×C4⋊Dic3C2×C4116yescomplemented
60C3×Dic3⋊C4—216no—
61C2×#389—31noSylow
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4—32no—
66C23×C3×C4—32no—
67C23×C3×C4C2×D311yescharacteristic
68C23×C3×C4C2×D311yescharacteristic
69C23×C3×C4C2×D311yescharacteristic
70C23×C3×C4C22×C311yescharacteristic
71C23×C3×C4C22×C311yescharacteristic
72C23×C3×C4C22×C311yescharacteristic
73C23×Dic3C22×C312yes—
74C23×Dic3C22×C312yes—
75C2×C3×C4⋊C4—38no—
76C2×C3×C4⋊C4—34no—
77C2×C3×C4⋊C4—38no—
78C2×C3×C4⋊C4—34no—
79C2×C3×C4⋊C4—68no—
80C2×C3×C4⋊C4—64no—
81C2×C3×C42—28no—
82C2×C3×C42—24no—
83C2×C3×C42—24no—
84C2×C3×C42—24no—
85C2×C3×C42C2×D314yes—
86C2×C3×C42C22×C314yes—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3C3×C418yescomplemented
89C2×C4⋊Dic3C22×C314yes—
90C2×C4⋊Dic3C22×C314yes—
91C2×Dic3⋊C4—28no—
92C2×Dic3⋊C4C22×C318yes—
93C3×#95—38no—
94C3⋊#95C22×C318yes—
95C23×C32C22×C412yes—
96C23×C32C2×D412yes—
97C23×C32C2×D412yes—
98C23×C32C2×Q812yes—
99C23×C32C2×Q812yes—
100C23×C32#4112yes—
101C23×C32#4112yes—
102C23×C32C2411yescharacteristic
103C2×C32×C4—28no—
104C2×C32×C4—28no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4C4⋊C4116yescomplemented
110C2×C32×C4C22×C414yes—
111C2×C32×C4C2×D414yes—
112C2×C32×C4C2×D414yes—
113C2×C32×C4C2×Q814yes—
114C2×C32×C4C2×Q814yes—
115C2×C3×Dic3—216no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—48no—
122C2×C3×Dic3C22×C418yes—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C42C3×D311yescharacteristic
127#389—38no—
128C2×#95—32no—
129C24×C3—21no—
130C24×C3C22×D311yescharacteristic
131C24×C3C23×C311yescharacteristic
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—38no—
153C22×C3×C4—34no—
154C22×C3×C4—34no—
155C22×C3×C4—64no—
156C22×C3×C4—64no—
157C22×C3×C4—64no—
158C22×C3×C4—64no—
159C22×C3×C4—64no—
160C22×C3×C4Dic614yes—
161C22×C3×C4Dic614yes—
162C22×C3×C4C4×D314yes—
163C22×C3×C4D1214yes—
164C22×C3×C4#6714yes—
165C22×C3×C4C2×C3×C414yes—
166C22×C3×C4C3×D414yes—
167C22×C3×C4C3×D414yes—
168C22×C3×C4C3×Q814yes—
169C22×C3×C4C3×Q814yes—
170C22×C3×C4C22×D312yes—
171C22×C3×C4C22×D312yes—
172C22×C3×C4C22×D312yes—
173C22×C3×C4C23×C312yes—
174C22×C3×C4C23×C312yes—
175C22×C3×C4C23×C312yes—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3C2×C3×C418yes—
182C22×Dic3C23×C314yes—
183C22×Dic3C23×C314yes—
184C3×C4⋊C4—316no—
185C3×C4⋊C4—616no—
186C3×C4⋊C4—68no—
187C3×C42—216no—
188C3×C42—28no—
189C3×C42—28no—
190C4⋊Dic3—28no—
191C4⋊Dic3C2×C3×C4116yescomplemented
192Dic3⋊C4—216no—
193C22×C32C2×C4⋊C414yes—
194C22×C32C4×D414yes—
195C22×C32C4×Q814yes—
196C22×C32#12114yes—
197C22×C32C22⋊Q814yes—
198C22×C32#12514yes—
199C22×C32C4⋊Q814yes—
200C22×C32C23×C411yescharacteristic
201C22×C32C22×D411yescharacteristic
202C22×C32C22×D411yescharacteristic
203C22×C32C22×Q811yescharacteristic
204C22×C32C22×Q811yescharacteristic
205C22×C32C2×#4111yescharacteristic
206C22×C32C2×#4111yescharacteristic
207C32×C4—24no—
208C32×C4—24no—
209C32×C4C2×C4⋊C418yescomplemented
210C3×Dic3—28no—
211C3×Dic3—44no—
212C23×C4—32no—
213C23×C4—32no—
214C23×C4C2×C3×D311yescharacteristic
215C23×C4C2×C3×D311yescharacteristic
216C23×C4C2×C3×D311yescharacteristic
217C2×C4⋊C4—38no—
218C2×C4⋊C4—34no—
219C2×C4⋊C4—38no—
220C2×C4⋊C4—34no—
221C2×C4⋊C4—68no—
222C2×C4⋊C4—64no—
223C2×C42—24no—
224C2×C42C2×C3×D314yes—
225#95—38no—
226C23×C3—21no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3C2×Dic612yes—
235C23×C3C2×Dic612yes—
236C23×C3C2×C4×D312yes—
237C23×C3C2×D1212yes—
238C23×C3#23512yes—
239C23×C3#23512yes—
240C23×C3C2×#6712yes—
241C23×C3C22×C3×C412yes—
242C23×C3C2×C3×D412yes—
243C23×C3C2×C3×D412yes—
244C23×C3C2×C3×Q812yes—
245C23×C3C2×C3×Q812yes—
246C23×C3C3×#4112yes—
247C23×C3C3×#4112yes—
248C23×C3C23×D311yescharacteristic
249C23×C3C24×C311yescharacteristic
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—216no—
253C2×C3×C4—216no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—28no—
257C2×C3×C4—28no—
258C2×C3×C4—28no—
259C2×C3×C4—28no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—216no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—38no—
275C2×C3×C4—616no—
276C2×C3×C4—64no—
277C2×C3×C4—64no—
278C2×C3×C4—64no—
279C2×C3×C4—64no—
280C2×C3×C4—64no—
281C2×C3×C4—128no—
282C2×C3×C4Dic3⋊C4116yescomplemented
283C2×C3×C4C3×C4⋊C4116yescomplemented
284C2×C3×C4C2×Dic614yes—
285C2×C3×C4C2×Dic614yes—
286C2×C3×C4C2×C4×D314yes—
287C2×C3×C4C2×D1214yes—
288C2×C3×C4C2×#6714yes—
289C2×C3×C4C22×C3×C414yes—
290C2×C3×C4C2×C3×D414yes—
291C2×C3×C4C2×C3×D414yes—
292C2×C3×C4C2×C3×Q814yes—
293C2×C3×C4C2×C3×Q814yes—
294C2×Dic3—216no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—48no—
301C2×Dic3C22×C3×C418yes—
302C2×C32#38918yescomplemented
303C2×C32C22×C4⋊C411yescharacteristic
304C2×C32C2×C4×D411yescharacteristic
305C2×C32C2×C4×Q811yescharacteristic
306C2×C32C2×#12111yescharacteristic
307C2×C32C2×C22⋊Q811yescharacteristic
308C2×C32C2×#12511yescharacteristic
309C2×C32C2×C4⋊Q811yescharacteristic
310C24C22×C3×D311yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—34no—
316C22×C4—38no—
317C22×C4—34no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4C3×Dic614yes—
324C22×C4C3×Dic614yes—
325C22×C4C3×C4×D314yes—
326C22×C4C3×D1214yes—
327C22×C4C3×#6714yes—
328C22×C4C22×C3×D312yes—
329C22×C4C22×C3×D312yes—
330C22×C4C22×C3×D312yes—
331C4⋊C4—316no—
332C4⋊C4—616no—
333C4⋊C4—68no—
334C42—28no—
335C22×C3—24no—
336C22×C3—24no—
337C22×C3—24no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3C4×Dic614yes—
350C22×C3#86914yes—
351C22×C3#87014yes—
352C22×C3C4×D1214yes—
353C22×C3C2×Dic3⋊C414yes—
354C22×C3#92414yes—
355C22×C3#93014yes—
356C22×C3C2×C3×C4⋊C414yes—
357C22×C3C3×C4×D414yes—
358C22×C3C3×C4×Q814yes—
359C22×C3C3×#12114yes—
360C22×C3C3×C22⋊Q814yes—
361C22×C3C3×#12514yes—
362C22×C3C3×C4⋊Q814yes—
363C22×C3C22×Dic611yescharacteristic
364C22×C3C22×Dic611yescharacteristic
365C22×C3C22×C4×D311yescharacteristic
366C22×C3C22×D1211yescharacteristic
367C22×C3C2×#23511yescharacteristic
368C22×C3C2×#23511yescharacteristic
369C22×C3C22×#6711yescharacteristic
370C22×C3C23×C3×C411yescharacteristic
371C22×C3C22×C3×D411yescharacteristic
372C22×C3C22×C3×D411yescharacteristic
373C22×C3C22×C3×Q811yescharacteristic
374C22×C3C22×C3×Q811yescharacteristic
375C22×C3C2×C3×#4111yescharacteristic
376C22×C3C2×C3×#4111yescharacteristic
377C3×C4—28no—
378C3×C4—28no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—24no—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4—68no—
385C3×C4—124no—
386C3×C4C2×Dic3⋊C418yescomplemented
387C3×C4C2×C3×C4⋊C418yescomplemented
388Dic3—28no—
389Dic3—44no—
390C32C2×#38911yescharacteristic, complemented, Sylow
391C23C2×C3×Dic612yes—
392C23C2×C3×Dic612yes—
393C23C2×C3×C4×D312yes—
394C23C2×C3×D1212yes—
395C23C3×#23512yes—
396C23C3×#23512yes—
397C23C2×C3×#6712yes—
398C23C23×C3×D311yescharacteristic
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—38no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—616no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—128no—
413C2×C4C3×Dic3⋊C4116yescomplemented
414C2×C4C2×C3×Dic614yes—
415C2×C4C2×C3×Dic614yes—
416C2×C4C2×C3×C4×D314yes—
417C2×C4C2×C3×D1214yes—
418C2×C4C2×C3×#6714yes—
419C2×C3—28no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3#492818yesdirect product factor
428C2×C3C3×#38918yescomplemented
429C2×C3C2×C4×Dic611yescharacteristic
430C2×C3C2×#86911yescharacteristic
431C2×C3C2×#87011yescharacteristic
432C2×C3C2×C4×D1211yescharacteristic
433C2×C3C22×Dic3⋊C411yescharacteristic
434C2×C3C2×#92411yescharacteristic
435C2×C3C2×#93011yescharacteristic
436C2×C3C22×C3×C4⋊C411yescharacteristic
437C2×C3C2×C3×C4×D411yescharacteristic
438C2×C3C2×C3×C4×Q811yescharacteristic
439C2×C3C2×C3×#12111yescharacteristic
440C2×C3C2×C3×C22⋊Q811yescharacteristic
441C2×C3C2×C3×#12511yescharacteristic
442C2×C3C2×C3×C4⋊Q811yescharacteristic
443C22C3×C4×Dic614yes—
444C22C3×#86914yes—
445C22C3×#87014yes—
446C22C3×C4×D1214yes—
447C22C2×C3×Dic3⋊C414yes—
448C22C3×#92414yes—
449C22C3×#93014yes—
450C22C22×C3×Dic611yescharacteristic
451C22C22×C3×Dic611yescharacteristic
452C22C22×C3×C4×D311yescharacteristic
453C22C22×C3×D1211yescharacteristic
454C22C2×C3×#23511yescharacteristic
455C22C2×C3×#23511yescharacteristic
456C22C22×C3×#6711yescharacteristic
457C4—24no—
458C4—24no—
459C4—68no—
460C4—124no—
461C4C2×C3×Dic3⋊C418yescomplemented
462C3—21no—
463C3C2×#492811yescharacteristic, direct product factor
464C3C2×C3×#38911yescharacteristic, complemented
465C2C3×#492818yesdirect product factor
466C2C2×C3×C4×Dic611yescharacteristic
467C2C2×C3×#86911yescharacteristic
468C2C2×C3×#87011yescharacteristic
469C2C2×C3×C4×D1211yescharacteristic
470C2C22×C3×Dic3⋊C411yescharacteristic
471C2C2×C3×#92411yescharacteristic
472C2C2×C3×#93011yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC22×C32×C42C2
socle, prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC22×C3×C42D3
hypocenterC3C2×C3×#389
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C22×C3×C42

Complements

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

NH ≅ G/Ntype
6. C3×#4928C2direct (N × H)
8. C2×#4928C3direct (N × H)
17. C2×C3×C4⋊Dic3C4semidirect
35. #4928C2×C3direct (N × H)
59. C3×C4⋊Dic3C2×C4semidirect
88. C2×C4⋊Dic3C3×C4semidirect
109. C2×C32×C4C4⋊C4semidirect
191. C4⋊Dic3C2×C3×C4semidirect
209. C32×C4C2×C4⋊C4semidirect
282. C2×C3×C4Dic3⋊C4semidirect
283. C2×C3×C4C3×C4⋊C4semidirect
302. C2×C32#389semidirect
386. C3×C4C2×Dic3⋊C4semidirect
387. C3×C4C2×C3×C4⋊C4semidirect
390. C32C2×#389semidirect
413. C2×C4C3×Dic3⋊C4semidirect
427. C2×C3#4928direct (N × H)
428. C2×C3C3×#389semidirect
461. C4C2×C3×Dic3⋊C4semidirect
463. C3C2×#4928direct (N × H)
464. C3C2×C3×#389semidirect
465. C2C3×#4928direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C422
C22×C3×C4⋊Dic32
C22×C3×Dic3⋊C42
C2×C3×C3⋊#952
C3×#49282
C2×C3×#3893
C2×#49283