Finite Group Databasebeta

← C2×C3×#5022 #11868483

Subgroups — C2×C3×#5022 #11868483

Order 1152.

917 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic
2C2×C3×#870C211yesmaximal, characteristic
3C2×C3×#835C211yesmaximal, characteristic
4C2×C3×#835C211yesmaximal, characteristic
5C3×#5022C218yesmaximal, direct product factor
6C2×C3×#808C211yesmaximal, characteristic
7C2×C3×#807C211yesmaximal, characteristic
8C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic
9C2×C3×#479—31nomaximal
10C2×#5022C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C32×C4×Q8—22no—
13C32×C4×Q8C2214yes—
14C2×C32×C4⋊C4—21no—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C42—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×#870C2214yes—
26C3×#835C2214yes—
27C3×#835C2214yes—
28C3×#808C2214yes—
29C3×#807C2214yes—
30C3×C4⋊(C3⋊C8)C2214yes—
31C2×C3×#125—31no—
32C2×C3×C4×Q8—21no—
33C2×C3×C4×Q8D311yescharacteristic
34C2×C3×C4×Q8C2×C311yescharacteristic
35C2×#870C2×C311yescharacteristic
36C3×#479—38no—
37C2×C3×#107—31no—
38C2×C3×#106—31no—
39C2×C3×C4⋊C8—31no—
40C2×C3×#103—31no—
41C2×C3×#103—31no—
42C2×#835C2×C311yescharacteristic
43C2×#835C2×C311yescharacteristic
44#5022C2×C318yesdirect product factor
45C2×#808C2×C311yescharacteristic
46C2×#807C2×C311yescharacteristic
47C2×C4⋊(C3⋊C8)C2×C311yescharacteristic
48C2×C32×Q8—21no—
49C2×C32×Q8—22no—
50C2×C32×Q8Q814yes—
51C2×C32×Q8C2312yes—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D411yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C22×C3×Dic3—21no—
58C22×C3×Dic3—21no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2312yes—
62C32×C42—24no—
63C32×C42D412yes—
64C32×C42C2312yes—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3C2312yes—
68C3×C4⋊Dic3C2312yes—
69C3×Dic3⋊C4—24no—
70C3×Dic3⋊C4—24no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8C2312yes—
74C2×C3×C3⋊C8C2312yes—
75C2×#479—31noSylow
76C22×C3×Q8—21no—
77C22×C3×Q8C2×D311yescharacteristic
78C22×C3×Q8C22×C311yescharacteristic
79C22×C3×C8—31no—
80C22×C3×C8—31no—
81C3×#125—34no—
82C3×#125—62no—
83C3×C4×Q8—24no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—24no—
87C3×C4×Q8C2×D314yes—
88C3×C4×Q8C22×C314yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—61no—
96C2×C3×C4⋊C4—61no—
97C2×C3×C4⋊C4C2×D311yescharacteristic
98C2×C3×C4⋊C4C22×C311yescharacteristic
99C2×C3×C42—22no—
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2×D311yescharacteristic
104C2×C3×C42C22×C311yescharacteristic
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C22×C3⋊C8C22×C311yescharacteristic
110C22×C3⋊C8C22×C311yescharacteristic
111#870—22no—
112#870C22×C314yes—
113C3×#107—34no—
114C3×#106—34no—
115C3×C4⋊C8—34no—
116C3×#103—34no—
117C3×#103—34no—
118#835C22×C314yes—
119#835C22×C314yes—
120#808C22×C314yes—
121#807C22×C314yes—
122C4⋊(C3⋊C8)C22×C314yes—
123C23×C32C2×D411yescharacteristic
124C32×Q8—24no—
125C32×Q8—22no—
126C32×Q8C2×Q814yes—
127C2×C32×C4—24no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—42no—
133C2×C32×C4C2×D412yes—
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×Q812yes—
137C2×C32×C4#4112yes—
138C2×C32×C4C2411yescharacteristic
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—42no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145C2×#125—31no—
146C2×C4×Q8C3×D311yescharacteristic
147#479—38no—
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8Dic614yes—
162C2×C3×Q8C3×Q814yes—
163C2×C3×Q8C22×D312yes—
164C2×C3×Q8C23×C312yes—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—61no—
175C22×C3×C4—61no—
176C22×C3×C4#6711yescharacteristic
177C22×C3×C4#6711yescharacteristic
178C22×C3×C4C3×D411yescharacteristic
179C22×C3×C4C3×D411yescharacteristic
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C23×C311yescharacteristic
182C22×Dic3—21no—
183C22×Dic3—21no—
184C2×C3×C8—32no—
185C2×C3×C8—32no—
186C2×C3×C8—62no—
187C2×C3×C8—62no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—32no—
197C3×C4⋊C4—64no—
198C3×C4⋊C4—64no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4C22×D312yes—
202C3×C4⋊C4C23×C312yes—
203C3×C42—28no—
204C3×C42—24no—
205C3×C42—24no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42#6712yes—
209C3×C42C3×D412yes—
210C3×C42C22×D312yes—
211C3×C42C23×C312yes—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C23×C312yes—
215C4⋊Dic3C23×C312yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C23×C312yes—
221C2×C3⋊C8C23×C312yes—
222C22×C32C22⋊Q812yes—
223C22×C32#13512yes—
224C22×C32#13712yes—
225C22×C32C22×D411yescharacteristic
226C32×C4—24no—
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—42no—
230C32×C4C22⋊Q812yes—
231C32×C4C22×Q811yescharacteristic
232C32×C4C2×#4111yescharacteristic
233C3×Dic3—42no—
234C3×Dic3—42no—
235C22×Q8C2×C3×D311yescharacteristic
236C22×C8—31no—
237C22×C8—31no—
238#125—34no—
239#125—62no—
240C4×Q8—22no—
241C4×Q8C2×C3×D314yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4C2×C3×D311yescharacteristic
248C2×C42—21no—
249C2×C42C2×C3×D311yescharacteristic
250#107—34no—
251#106—34no—
252C4⋊C8—34no—
253#103—34no—
254#103—34no—
255C23×C3—21no—
256C23×C3C2×#6711yescharacteristic
257C23×C3C2×C3×D411yescharacteristic
258C3×Q8—28no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—22no—
263C3×Q8—22no—
264C3×Q8—24no—
265C3×Q8C2×Dic614yes—
266C3×Q8C2×C3×Q814yes—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—122no—
294C2×C3×C4—122no—
295C2×C3×C4C2×Dic612yes—
296C2×C3×C4#23512yes—
297C2×C3×C4C2×#6712yes—
298C2×C3×C4C2×#6712yes—
299C2×C3×C4C2×#6711yescharacteristic
300C2×C3×C4C2×C3×D412yes—
301C2×C3×C4C2×C3×D412yes—
302C2×C3×C4C2×C3×D411yescharacteristic
303C2×C3×C4C2×C3×Q812yes—
304C2×C3×C4C3×#4112yes—
305C2×C3×C4C23×D311yescharacteristic
306C2×C3×C4C24×C311yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C3×C8—62no—
312C3×C8—62no—
313C3⋊C8—22no—
314C3⋊C8—22no—
315C2×C32#47914yescomplemented
316C2×C32C2×C22⋊Q811yescharacteristic
317C2×C32C2×#13511yescharacteristic
318C2×C32C2×#13711yescharacteristic
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8C3×Dic614yes—
322C2×Q8C22×C3×D312yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4C3×#6711yescharacteristic
328C22×C4C3×#6711yescharacteristic
329C22×C4C22×C3×D311yescharacteristic
330C2×C8—32no—
331C2×C8—32no—
332C2×C8—62no—
333C2×C8—62no—
334C4⋊C4—24no—
335C4⋊C4—21no—
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C4⋊C4—64no—
341C4⋊C4—64no—
342C4⋊C4C22×C3×D312yes—
343C42—24no—
344C42C3×#6712yes—
345C42C22×C3×D312yes—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3#92412yes—
351C22×C3#94212yes—
352C22×C3#95012yes—
353C22×C3C3×C22⋊Q812yes—
354C22×C3C3×#13512yes—
355C22×C3C3×#13712yes—
356C22×C3C22×#6711yescharacteristic
357C22×C3C22×C3×D411yescharacteristic
358C3×C4—28no—
359C3×C4—24no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—24no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—24no—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—42no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—122no—
374C3×C4—122no—
375C3×C4#92412yes—
376C3×C4C3×C22⋊Q812yes—
377C3×C4C22×Dic611yescharacteristic
378C3×C4C2×#23511yescharacteristic
379C3×C4C22×C3×Q811yescharacteristic
380C3×C4C2×C3×#4111yescharacteristic
381Dic3—42no—
382Dic3—42no—
383C32C2×#47911yescharacteristic, complemented, Sylow
384C23C2×C3×#6711yescharacteristic
385Q8—24no—
386Q8—22no—
387Q8C2×C3×Dic614yes—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4C2×C3×Dic612yes—
399C2×C4C3×#23512yes—
400C2×C4C2×C3×#6712yes—
401C2×C4C2×C3×#6712yes—
402C2×C4C2×C3×#6711yescharacteristic
403C2×C4C23×C3×D311yescharacteristic
404C8—62no—
405C8—62no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—24no—
410C2×C3#502214yesdirect product factor
411C2×C3C3×#47914yescomplemented
412C2×C3C2×#92411yescharacteristic
413C2×C3C2×#94211yescharacteristic
414C2×C3C2×#95011yescharacteristic
415C2×C3C2×C3×C22⋊Q811yescharacteristic
416C2×C3C2×C3×#13511yescharacteristic
417C2×C3C2×C3×#13711yescharacteristic
418C22C3×#92412yes—
419C22C3×#94212yes—
420C22C3×#95012yes—
421C22C22×C3×#6711yescharacteristic
422C4—24no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—122no—
427C4—122no—
428C4C3×#92412yes—
429C4C22×C3×Dic611yescharacteristic
430C4C2×C3×#23511yescharacteristic
431C3—21no—
432C3C2×#502211yescharacteristic, direct product factor
433C3C2×C3×#47911yescharacteristic, complemented
434C2C3×#502214yesdirect product factor
435C2C2×C3×#92411yescharacteristic
436C2C2×C3×#94211yescharacteristic
437C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×Q8C2
socle, prime radicalC23×C32C2×D4
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#479
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
5. C3×#5022C2direct (N × H)
10. C2×#5022C3direct (N × H)
44. #5022C2×C3direct (N × H)
315. C2×C32#479semidirect
383. C32C2×#479semidirect
410. C2×C3#5022direct (N × H)
411. C2×C3C3×#479semidirect
432. C3C2×#5022direct (N × H)
433. C3C2×C3×#479semidirect
434. C2C3×#5022direct (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×Q82
C2×C3×#8702
C2×C3×#8352
C3×#50222
C2×C3×#8082
C2×C3×#8072
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4793
C2×#50223