Finite Group Databasebeta

← C2×C3×#5019 #11865476

Subgroups — C2×C3×#5019 #11865476

Order 1152.

1301 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C22×C3×#214C212yesmaximal
3C2×C3×#869C211yesmaximal, characteristic, complemented
4C3×#5019C218yesmaximal, direct product factor
5C2×C3×#810C212yesmaximal, complemented
6C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C2×C3×#461—31nomaximal
8C2×#5019C311yesmaximal, characteristic, direct product factor
9C22×C32×D4C2211yescharacteristic
10C23×C32×C4—21no—
11C22×C3×Dic6C2212yes—
12C32×C4×D4—22no—
13C32×C4×D4C2214yes—
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C22⋊C4—21no—
16C2×C32×C42C2211yescharacteristic
17C2×C3×#214—28no—
18C2×C3×#214C2218yes—
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4⋊Dic3—21no—
21C22×C3×C3⋊C8C2212yes—
22C3×#869—22no—
23C3×#869C2214yescomplemented
24C3×#810C2218yescomplemented
25C3×C4⋊(C3⋊C8)C2214yescomplemented
26C22×C3×SD16—32no—
27C2×C3×C4⋊Q8—31no—
28C2×C3×C4×D4—21no—
29C2×C3×C4×D4D311yescharacteristic
30C2×C3×C4×D4C2×C311yescharacteristic
31C22×#214C2×C312yes—
32C2×#869C2×C311yescharacteristic, complemented
33C3×#461—38no—
34C2×C3×C4⋊C8—31no—
35C2×C3×#103—32no—
36#5019C2×C318yesdirect product factor
37C2×#810C2×C312yescomplemented
38C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
39C24×C32—21no—
40C2×C32×D4—21no—
41C2×C32×D4—22no—
42C2×C32×D4D414yes—
43C2×C32×D4C2312yes—
44C22×C32×C4—21no—
45C22×C32×C4—24no—
46C22×C32×C4—44no—
47C22×C32×C4D411yescharacteristic
48C22×C32×C4D411yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C22×C3×Dic3—22no—
51C2×C3×Dic6—24no—
52C2×C3×Dic6—24no—
53C2×C3×Dic6—22no—
54C2×C3×Dic6C2314yes—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2312yes—
57C32×C22⋊C4—24no—
58C32×C42D412yes—
59C32×C42C2312yes—
60C3×#214—216no—
61C3×C4⋊Dic3—24no—
62C3×C4⋊Dic3—24no—
63C2×C3×C3⋊C8—24no—
64C2×C3×C3⋊C8C2314yes—
65C2×#461—31noSylow
66C22×C3×Q8—32no—
67C22×C3×D4—21no—
68C22×C3×D4C2×D311yescharacteristic
69C22×C3×D4C22×C311yescharacteristic
70C23×C3×C4—22no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C22×Dic6C22×C312yes—
74C2×C3×SD16—38no—
75C2×C3×SD16—68no—
76C22×C3×C8—32no—
77C3×C4⋊Q8—34no—
78C3×C4⋊Q8—62no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4C2×D314yes—
84C3×C4×D4C22×C314yes—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—61no—
87C2×C3×C4⋊C4—61no—
88C2×C3×C4⋊C4C2×D311yescharacteristic
89C2×C3×C4⋊C4C22×C311yescharacteristic
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2×D311yescharacteristic
95C2×C3×C42C22×C311yescharacteristic
96C2×#214—28no—
97C2×#214C22×C318yes—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C22×C3⋊C8C22×C312yes—
101#869—22no—
102#869C22×C314yescomplemented
103C3×C4⋊C8—34no—
104C3×#103—38no—
105#810C22×C318yescomplemented
106C4⋊(C3⋊C8)C22×C314yescomplemented
107C23×C32—24no—
108C23×C32—22no—
109C23×C32—42no—
110C23×C32—42no—
111C23×C32C2×D411yescharacteristic
112C32×D4—24no—
113C32×D4—22no—
114C32×D4C2×D414yes—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—42no—
118C2×C32×C4—48no—
119C2×C32×C4SD1614yescomplemented
120C2×C32×C4C2×D412yes—
121C2×C32×C4C2×D412yes—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D412yes—
124C2×C32×C4#4112yes—
125C2×C32×C4C2411yescharacteristic
126C2×C3×Dic3—24no—
127C2×C3×Dic3—44no—
128C3×Dic6—24no—
129C3×Dic6—24no—
130C3×Dic6—44no—
131C3×C3⋊C8—24no—
132C22×SD16—32no—
133C2×C4⋊Q8—31no—
134C2×C4×D4C3×D311yescharacteristic
135#461—38no—
136C2×C4⋊C8—31no—
137C2×#103—32no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—21no—
141C2×C3×Q8—34no—
142C2×C3×Q8—62no—
143C2×C3×Q8—64no—
144C2×C3×Q8—64no—
145C2×C3×D4—22no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4D1214yes—
154C2×C3×D4C3×D414yes—
155C2×C3×D4C22×D312yes—
156C2×C3×D4C23×C312yes—
157C22×C3×C4—22no—
158C22×C3×C4—28no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—48no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4—62no—
170C22×C3×C4#6711yescharacteristic
171C22×C3×C4#6711yescharacteristic
172C22×C3×C4C3×D411yescharacteristic
173C22×C3×C4C3×D411yescharacteristic
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C23×C311yescharacteristic
176C22×Dic3—22no—
177C2×Dic6—24no—
178C2×Dic6—24no—
179C2×Dic6—22no—
180C2×Dic6C23×C314yes—
181C3×SD16—616no—
182C2×C3×C8—34no—
183C2×C3×C8—64no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—64no—
189C3×C4⋊C4—64no—
190C3×C4⋊C4C22×D312yes—
191C3×C4⋊C4C23×C312yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42#6712yes—
198C3×C42C3×D412yes—
199C3×C42C22×D312yes—
200C3×C42C23×C312yes—
201#214—216no—
202C4⋊Dic3—24no—
203C4⋊Dic3—24no—
204C2×C3⋊C8—24no—
205C2×C3⋊C8C23×C314yes—
206C22×C32—24no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—48no—
210C22×C32#12112yes—
211C22×C32C2×SD1612yes—
212C22×C32#13712yes—
213C22×C32C22×D411yescharacteristic
214C32×C4—22no—
215C32×C4—42no—
216C32×C4#12112yes—
217C32×C4C2×SD1614yescomplemented
218C32×C4C22×D411yescharacteristic
219C32×C4C2×#4111yescharacteristic
220C3×Dic3—44no—
221C22×Q8—32no—
222C22×D4C2×C3×D311yescharacteristic
223C23×C4—21no—
224C2×SD16—38no—
225C2×SD16—68no—
226C22×C8—32no—
227C4⋊Q8—34no—
228C4⋊Q8—62no—
229C4×D4—22no—
230C4×D4C2×C3×D314yes—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4C2×C3×D311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C42C2×C3×D311yescharacteristic
236C4⋊C8—34no—
237#103—38no—
238C23×C3—24no—
239C23×C3—28no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—21no—
245C23×C3—44no—
246C23×C3—44no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3C2×#6711yescharacteristic
252C23×C3C2×C3×D411yescharacteristic
253C3×Q8—64no—
254C3×Q8—64no—
255C3×Q8—124no—
256C3×D4—28no—
257C3×D4—24no—
258C3×D4—24no—
259C3×D4—24no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—24no—
263C3×D4C2×D1214yes—
264C3×D4C2×C3×D414yes—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—42no—
278C2×C3×C4—416no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—48no—
283C2×C3×C4—48no—
284C2×C3×C4—64no—
285C2×C3×C4—124no—
286C2×C3×C4#21414yescomplemented
287C2×C3×C4C3×SD1614yescomplemented
288C2×C3×C4C2×D1212yes—
289C2×C3×C4#23512yes—
290C2×C3×C4C2×#6712yes—
291C2×C3×C4C2×#6711yescharacteristic
292C2×C3×C4C2×#6712yes—
293C2×C3×C4C2×C3×D412yes—
294C2×C3×C4C2×C3×D412yes—
295C2×C3×C4C2×C3×D411yescharacteristic
296C2×C3×C4C2×C3×D412yes—
297C2×C3×C4C3×#4112yes—
298C2×C3×C4C23×D311yescharacteristic
299C2×C3×C4C24×C311yescharacteristic
300C2×Dic3—24no—
301C2×Dic3—44no—
302Dic6—24no—
303Dic6—24no—
304Dic6—44no—
305C3×C8—64no—
306C3⋊C8—24no—
307C2×C32—44no—
308C2×C32#46114yescomplemented
309C2×C32C2×#12111yescharacteristic
310C2×C32C22×SD1611yescharacteristic
311C2×C32C2×#13711yescharacteristic
312C24—21no—
313C2×Q8—34no—
314C2×Q8—62no—
315C2×Q8—64no—
316C2×Q8—64no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4C3×D1214yes—
320C2×D4C22×C3×D312yes—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—44no—
324C22×C4—62no—
325C22×C4C3×#6711yescharacteristic
326C22×C4C3×#6711yescharacteristic
327C22×C4C22×C3×D311yescharacteristic
328SD16—616no—
329C2×C8—34no—
330C2×C8—64no—
331C4⋊C4—21no—
332C4⋊C4—64no—
333C4⋊C4—64no—
334C4⋊C4C22×C3×D312yes—
335C22⋊C4—24no—
336C42C3×#6712yes—
337C42C22×C3×D312yes—
338C22×C3—28no—
339C22×C3—24no—
340C22×C3—24no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—44no—
346C22×C3—44no—
347C22×C3—416no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—48no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—48no—
354C22×C3#93012yes—
355C22×C3C2×#21412yes—
356C22×C3#95112yes—
357C22×C3C3×#12112yes—
358C22×C3C2×C3×SD1612yes—
359C22×C3C3×#13712yes—
360C22×C3C22×#6711yescharacteristic
361C22×C3C22×C3×D411yescharacteristic
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—24no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—42no—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—124no—
373C3×C4#93012yes—
374C3×C4C2×#21414yescomplemented
375C3×C4C3×#12112yes—
376C3×C4C2×C3×SD1614yescomplemented
377C3×C4C22×D1211yescharacteristic
378C3×C4C2×#23511yescharacteristic
379C3×C4C22×C3×D411yescharacteristic
380C3×C4C2×C3×#4111yescharacteristic
381Dic3—44no—
382C32C2×#46111yescharacteristic, complemented, Sylow
383C23—24no—
384C23—22no—
385C23—42no—
386C23—42no—
387C23C2×C3×#6711yescharacteristic
388Q8—64no—
389Q8—64no—
390Q8—124no—
391D4—24no—
392D4—22no—
393D4C2×C3×D1214yes—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—64no—
399C2×C4—124no—
400C2×C4C3×#21414yescomplemented
401C2×C4C2×C3×D1212yes—
402C2×C4C3×#23512yes—
403C2×C4C2×C3×#6712yes—
404C2×C4C2×C3×#6711yescharacteristic
405C2×C4C2×C3×#6712yes—
406C2×C4C23×C3×D311yescharacteristic
407C8—64no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—24no—
412C2×C3—48no—
413C2×C3—44no—
414C2×C3—44no—
415C2×C3#501914yesdirect product factor
416C2×C3C3×#46114yescomplemented
417C2×C3C2×#93011yescharacteristic
418C2×C3C22×#21411yescharacteristic
419C2×C3C2×#95111yescharacteristic
420C2×C3C2×C3×#12111yescharacteristic
421C2×C3C22×C3×SD1611yescharacteristic
422C2×C3C2×C3×#13711yescharacteristic
423C22—24no—
424C22—42no—
425C22—42no—
426C22—48no—
427C22C3×#93012yes—
428C22C2×C3×#21412yes—
429C22C3×#95112yes—
430C22C22×C3×#6711yescharacteristic
431C4—22no—
432C4—42no—
433C4—124no—
434C4C3×#93012yes—
435C4C2×C3×#21414yescomplemented
436C4C22×C3×D1211yescharacteristic
437C4C2×C3×#23511yescharacteristic
438C3—21no—
439C3C2×#501911yescharacteristic, direct product factor
440C3C2×C3×#46111yescharacteristic, complemented
441C2—44no—
442C2C3×#501914yesdirect product factor
443C2C2×C3×#93011yescharacteristic
444C2C22×C3×#21411yescharacteristic
445C2C2×C3×#95111yescharacteristic
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×#461
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
3. C2×C3×#869C2semidirect
4. C3×#5019C2direct (N × H)
5. C2×C3×#810C2semidirect
6. C2×C3×C4⋊(C3⋊C8)C2semidirect
8. C2×#5019C3direct (N × H)
23. C3×#869C22semidirect
24. C3×#810C22semidirect
25. C3×C4⋊(C3⋊C8)C22semidirect
32. C2×#869C2×C3semidirect
36. #5019C2×C3direct (N × H)
37. C2×#810C2×C3semidirect
38. C2×C4⋊(C3⋊C8)C2×C3semidirect
102. #869C22×C3semidirect
105. #810C22×C3semidirect
106. C4⋊(C3⋊C8)C22×C3semidirect
119. C2×C32×C4SD16semidirect
217. C32×C4C2×SD16semidirect
286. C2×C3×C4#214semidirect
287. C2×C3×C4C3×SD16semidirect
308. C2×C32#461semidirect
374. C3×C4C2×#214semidirect
376. C3×C4C2×C3×SD16semidirect
382. C32C2×#461semidirect
400. C2×C4C3×#214semidirect
415. C2×C3#5019direct (N × H)
416. C2×C3C3×#461semidirect
435. C4C2×C3×#214semidirect
439. C3C2×#5019direct (N × H)
440. C3C2×C3×#461semidirect
442. C2C3×#5019direct (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
C22×C3×#2142
C2×C3×#8692
C3×#50192
C2×C3×#8102
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4613
C2×#50193