Finite Group Databasebeta

← C2×C3×#4839 #11858266

Subgroups — C2×C3×#4839 #11858266

Order 1152.

1093 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C22×C3×D3×C8C211yesmaximal, characteristic
3C2×C3×#872C211yesmaximal, characteristic
4C2×C32×C4⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#4839C218yesmaximal, direct product factor
9C2×C3×#424—31nomaximal
10C2×#4839C311yesmaximal, characteristic, direct product factor
11C22×C3×C4×D3C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4C411yescharacteristic
16C2×C3×(C2×D3)⋊C4C411yescharacteristic
17C2×C3×Dic3⋊C4C411yescharacteristic
18C2×C3×Dic3⋊C4C411yescharacteristic
19C2×C3×C4×Dic3C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×C3×ΓL(1,𝔽25)—24no—
23C2×C3×ΓL(1,𝔽25)C2214yes—
24C2×C3×D3×C8—24no—
25C2×C3×D3×C8C2214yes—
26C3×#872C414yes—
27C3×#872C2214yes—
28C32×C4⋊C8C2214yescomplemented
29C3×Dic3⋊C8C2214yescomplemented
30C3×Dic3⋊C8C2214yescomplemented
31C3×C4⋊(C3⋊C8)C2214yescomplemented
32C22×C3×M16—31no—
33C23×C3×C8—31no—
34C2×C3×#117—31no—
35C22×ΓL(1,𝔽25)C2×C311yescharacteristic
36C22×D3×C8C2×C311yescharacteristic
37C2×#872C2×C311yescharacteristic
38C3×#424—38no—
39C2×C3×C4⋊C8—21no—
40C2×C3×C4⋊C8—31no—
41C2×C3×C4⋊C8—31no—
42C2×C3×C4⋊C8—31no—
43C2×C3×C4⋊C8D311yescharacteristic, complemented
44C2×C3×C4⋊C8C2×C311yescharacteristic, complemented
45C2×Dic3⋊C8C2×C311yescharacteristic, complemented
46C2×Dic3⋊C8C2×C311yescharacteristic, complemented
47C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48#4839C2×C318yesdirect product factor
49C23×C3×D3C2×C411yescharacteristic
50C22×C32×C4C2×C411yescharacteristic
51C22×C32×C4C2×C411yescharacteristic
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3C2×C411yescharacteristic
54C22×C3×Dic3C2×C411yescharacteristic
55C22×C3×Dic3C2×C411yescharacteristic
56C2×C3×C4×D3C2×C412yes—
57C2×C3×C4×D3D414yes—
58C2×C3×C4×D3Q814yes—
59C2×C3×C4×D3C2312yes—
60C2×C32×C8—22no—
61C2×C32×C8—22no—
62C2×C32×C8C2312yes—
63C2×C32×C8C2312yes—
64C32×C42C2×C412yes—
65C32×C42C2312yes—
66C3×(C2×D3)⋊C4C2×C414yes—
67C3×(C2×D3)⋊C4C2×C414yes—
68C3×Dic3⋊C4C2×C414yes—
69C3×Dic3⋊C4C2×C414yes—
70C3×C4×Dic3C2×C412yes—
71C3×C4×Dic3C2312yes—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2312yes—
75C2×C3×C3⋊C8C2312yes—
76C3×ΓL(1,𝔽25)—28no—
77C3×D3×C8—28no—
78C2×#424—31noSylow
79C23×C3×C4—31no—
80C22×C4×D3C22×C311yescharacteristic
81C2×C3×M16—34no—
82C2×C3×M16—64no—
83C22×C3×C8—21no—
84C22×C3×C8—21no—
85C22×C3×C8—31no—
86C22×C3×C8—31no—
87C22×C3×C8—34no—
88C22×C3×C8—64no—
89C22×C3×C8C2×D311yescharacteristic
90C22×C3×C8C2×D311yescharacteristic
91C22×C3×C8C22×C311yescharacteristic
92C22×C3×C8C22×C311yescharacteristic
93C3×#117—34no—
94C3×#117—34no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C22⋊C4—31no—
98C2×C3×C22⋊C4—31no—
99C2×C3×C42—21no—
100C2×C3×C42—31no—
101C2×C3×C42C2×D311yescharacteristic
102C2×C3×C42C22×C311yescharacteristic
103C2×(C2×D3)⋊C4C3×C411yescharacteristic
104C2×(C2×D3)⋊C4C3×C411yescharacteristic
105C2×Dic3⋊C4C3×C411yescharacteristic
106C2×Dic3⋊C4C3×C411yescharacteristic
107C2×C4×Dic3C22×C311yescharacteristic
108C22×C3⋊C8C22×C311yescharacteristic
109C22×C3⋊C8C22×C311yescharacteristic
110C2×ΓL(1,𝔽25)—24no—
111C2×ΓL(1,𝔽25)C22×C314yes—
112C2×D3×C8—24no—
113C2×D3×C8C22×C314yes—
114#872C3×C414yes—
115#872C22×C314yes—
116C3×C4⋊C8—24no—
117C3×C4⋊C8—34no—
118C3×C4⋊C8—34no—
119C3×C4⋊C8—34no—
120C3×C4⋊C8C2×D314yescomplemented
121C3×C4⋊C8C22×C314yescomplemented
122Dic3⋊C8C22×C314yescomplemented
123Dic3⋊C8C22×C314yescomplemented
124C4⋊(C3⋊C8)C22×C314yescomplemented
125C23×C32C22×C411yescharacteristic
126C22×C3×D3—22no—
127C22×C3×D3—22no—
128C22×C3×D3C4⋊C414yes—
129C22×C3×D3C22×C412yes—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4C22×C412yes—
133C2×C32×C4C22×C412yes—
134C2×C32×C4C22×C411yescharacteristic
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×Q812yes—
137C2×C32×C4C2411yescharacteristic
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3C4⋊C414yes—
141C2×C3×Dic3C22×C412yes—
142C2×C3×Dic3C22×C412yes—
143C2×C3×Dic3C22×C412yes—
144C3×C4×D3C4⋊C418yes—
145C3×C4×D3C2×D414yes—
146C3×C4×D3C2×Q814yes—
147C32×C8—22no—
148C32×C8—22no—
149C3×C3⋊C8—22no—
150C3×C3⋊C8—22no—
151C22×M16—31no—
152C23×C8—31no—
153C2×#117—31no—
154#424—38no—
155C2×C4⋊C8—31no—
156C2×C4⋊C8—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8C3×D311yescharacteristic, complemented
159C24×C3—31no—
160C23×D3C2×C3×C411yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—31no—
165C22×C3×C4—32no—
166C22×C3×C4—34no—
167C22×C3×C4—34no—
168C22×C3×C4—32no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4C4×D311yescharacteristic
172C22×C3×C4C4×D311yescharacteristic
173C22×C3×C4C2×C3×C411yescharacteristic
174C22×C3×C4C2×C3×C411yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C22×C3×C4C23×C311yescharacteristic
177C22×Dic3C2×C3×C411yescharacteristic
178C22×Dic3C2×C3×C411yescharacteristic
179C22×Dic3C2×C3×C411yescharacteristic
180C2×C4×D3C2×C3×C412yes—
181C2×C4×D3C3×D414yes—
182C2×C4×D3C3×Q814yes—
183C2×C4×D3C23×C312yes—
184C3×M16—68no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—32no—
192C2×C3×C8—32no—
193C2×C3×C8—42no—
194C2×C3×C8—42no—
195C2×C3×C8—68no—
196C2×C3×C8—62no—
197C2×C3×C8—62no—
198C2×C3×C8C22×D312yes—
199C2×C3×C8C22×D312yes—
200C2×C3×C8C23×C312yes—
201C2×C3×C8C23×C312yes—
202C3×C4⋊C4—34no—
203C3×C4⋊C4—34no—
204C3×C22⋊C4—34no—
205C3×C22⋊C4—34no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—32no—
209C3×C42—32no—
210C3×C42C4×D312yes—
211C3×C42C2×C3×C412yes—
212C3×C42C22×D312yes—
213C3×C42C23×C312yes—
214(C2×D3)⋊C4C2×C3×C414yes—
215(C2×D3)⋊C4C2×C3×C414yes—
216Dic3⋊C4C2×C3×C414yes—
217Dic3⋊C4C2×C3×C414yes—
218C4×Dic3C2×C3×C412yes—
219C4×Dic3C23×C312yes—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8C23×C312yes—
223C2×C3⋊C8C23×C312yes—
224ΓL(1,𝔽25)—28no—
225D3×C8—28no—
226C22×C32C2×C4⋊C412yes—
227C22×C32#13112yes—
228C22×C32#13112yes—
229C22×C32C23×C411yescharacteristic
230C2×C3×D3—22no—
231C2×C3×D3—22no—
232C2×C3×D3—28no—
233C2×C3×D3C2×C4⋊C414yes—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4C2×C4⋊C412yes—
237C32×C4C22×D411yescharacteristic
238C32×C4C22×Q811yescharacteristic
239C3×Dic3—22no—
240C3×Dic3—22no—
241C3×Dic3C2×C4⋊C414yes—
242C23×C4—31no—
243C2×M16—34no—
244C2×M16—64no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—34no—
248C22×C8—64no—
249C22×C8C2×C3×D311yescharacteristic
250C22×C8C2×C3×D311yescharacteristic
251#117—34no—
252#117—34no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C42—31no—
258C2×C42C2×C3×D311yescharacteristic
259C4⋊C8—34no—
260C4⋊C8—34no—
261C4⋊C8—34no—
262C4⋊C8C2×C3×D314yescomplemented
263C23×C3—21no—
264C23×C3—34no—
265C23×C3—32no—
266C23×C3—62no—
267C23×C3—62no—
268C23×C3C2×C4×D311yescharacteristic
269C23×C3C22×C3×C411yescharacteristic
270C22×D3—22no—
271C22×D3—22no—
272C22×D3C3×C4⋊C414yes—
273C22×D3C22×C3×C412yes—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—34no—
286C2×C3×C4—32no—
287C2×C3×C4—38no—
288C2×C3×C4—34no—
289C2×C3×C4—34no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—42no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4C2×C4×D312yes—
296C2×C3×C4C2×C4×D312yes—
297C2×C3×C4C2×C4×D311yescharacteristic
298C2×C3×C4D3×D412yes—
299C2×C3×C4D3×Q812yes—
300C2×C3×C4C22×C3×C412yes—
301C2×C3×C4C22×C3×C412yes—
302C2×C3×C4C22×C3×C411yescharacteristic
303C2×C3×C4C2×C3×D412yes—
304C2×C3×C4C2×C3×Q812yes—
305C2×C3×C4C23×D311yescharacteristic
306C2×C3×C4C24×C311yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3C3×C4⋊C414yes—
310C2×Dic3C22×C3×C412yes—
311C2×Dic3C22×C3×C412yes—
312C2×Dic3C22×C3×C412yes—
313C4×D3C3×C4⋊C418yes—
314C4×D3C2×C3×D414yes—
315C4×D3C2×C3×Q814yes—
316C3×C8—22no—
317C3×C8—22no—
318C3×C8—22no—
319C3×C8—22no—
320C3×C8—42no—
321C3×C8—42no—
322C3×C8—62no—
323C3×C8—62no—
324C3⋊C8—22no—
325C3⋊C8—22no—
326C2×C32#42414yescomplemented
327C2×C32C22×C4⋊C411yescharacteristic
328C2×C32C2×#13111yescharacteristic
329C2×C32C2×#13111yescharacteristic
330C3×D3—24no—
331C24—31no—
332C22×C4—31no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—34no—
336C22×C4—34no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4C3×C4×D311yescharacteristic
340C22×C4C3×C4×D311yescharacteristic
341C22×C4C22×C3×D311yescharacteristic
342M16—68no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—32no—
346C2×C8—32no—
347C2×C8—68no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8C22×C3×D312yes—
351C2×C8C22×C3×D312yes—
352C4⋊C4—34no—
353C4⋊C4—34no—
354C22⋊C4—34no—
355C22⋊C4—34no—
356C42—32no—
357C42—32no—
358C42C3×C4×D312yes—
359C42C22×C3×D312yes—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—34no—
365C22×C3—62no—
366C22×C3—62no—
367C22×C3—68no—
368C22×C3D3×C4⋊C412yes—
369C22×C3#90112yes—
370C22×C3#90712yes—
371C22×C3C2×C3×C4⋊C412yes—
372C22×C3C3×#13112yes—
373C22×C3C3×#13112yes—
374C22×C3C22×C4×D311yescharacteristic
375C22×C3C23×C3×C411yescharacteristic
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—28no—
379C2×D3C2×C3×C4⋊C414yes—
380C3×C4—24no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—34no—
389C3×C4—42no—
390C3×C4—62no—
391C3×C4—62no—
392C3×C4D3×C4⋊C412yes—
393C3×C4C2×C3×C4⋊C412yes—
394C3×C4C2×D3×D411yescharacteristic
395C3×C4C2×D3×Q811yescharacteristic
396C3×C4C22×C3×D411yescharacteristic
397C3×C4C22×C3×Q811yescharacteristic
398Dic3—22no—
399Dic3—22no—
400Dic3C2×C3×C4⋊C414yes—
401C32C2×#42411yescharacteristic, complemented, Sylow
402C23—34no—
403C23—32no—
404C23—62no—
405C23—62no—
406C23C2×C3×C4×D311yescharacteristic
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—34no—
410C2×C4—32no—
411C2×C4—38no—
412C2×C4—34no—
413C2×C4—34no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4C2×C3×C4×D312yes—
419C2×C4C2×C3×C4×D312yes—
420C2×C4C2×C3×C4×D311yescharacteristic
421C2×C4C3×D3×D412yes—
422C2×C4C3×D3×Q812yes—
423C2×C4C23×C3×D311yescharacteristic
424C8—22no—
425C8—22no—
426C8—62no—
427C8—62no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—24no—
431C2×C3—21no—
432C2×C3—64no—
433C2×C3#483914yesdirect product factor
434C2×C3C3×#42414yescomplemented
435C2×C3C2×D3×C4⋊C411yescharacteristic
436C2×C3C2×#90111yescharacteristic
437C2×C3C2×#90711yescharacteristic
438C2×C3C22×C3×C4⋊C411yescharacteristic
439C2×C3C2×C3×#13111yescharacteristic
440C2×C3C2×C3×#13111yescharacteristic
441D3—24no—
442C22—34no—
443C22—62no—
444C22—68no—
445C22—62no—
446C22C3×D3×C4⋊C412yes—
447C22C3×#90112yes—
448C22C3×#90712yes—
449C22C22×C3×C4×D311yescharacteristic
450C4—22no—
451C4—22no—
452C4—34no—
453C4—62no—
454C4—62no—
455C4C3×D3×C4⋊C412yes—
456C4C2×C3×D3×D411yescharacteristic
457C4C2×C3×D3×Q811yescharacteristic
458C3—21no—
459C3C2×#483911yescharacteristic, direct product factor
460C3C2×C3×#42411yescharacteristic, complemented
461C2—64no—
462C2C3×#483914yesdirect product factor
463C2C2×C3×D3×C4⋊C411yescharacteristic
464C2C2×C3×#90111yescharacteristic
465C2C2×C3×#90711yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4⋊C8C2
socleC23×C32C22×C4
prime radicalC23×C3×D3C2×C4
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×C4⋊C8D3
hypocenterC3C2×C3×#424
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×C32×C4⋊C8C2semidirect
5. C2×C3×Dic3⋊C8C2semidirect
6. C2×C3×Dic3⋊C8C2semidirect
7. C2×C3×C4⋊(C3⋊C8)C2semidirect
8. C3×#4839C2direct (N × H)
10. C2×#4839C3direct (N × H)
28. C32×C4⋊C8C22semidirect
29. C3×Dic3⋊C8C22semidirect
30. C3×Dic3⋊C8C22semidirect
31. C3×C4⋊(C3⋊C8)C22semidirect
43. C2×C3×C4⋊C8D3semidirect
44. C2×C3×C4⋊C8C2×C3semidirect
45. C2×Dic3⋊C8C2×C3semidirect
46. C2×Dic3⋊C8C2×C3semidirect
47. C2×C4⋊(C3⋊C8)C2×C3semidirect
48. #4839C2×C3direct (N × H)
120. C3×C4⋊C8C2×D3semidirect
121. C3×C4⋊C8C22×C3semidirect
122. Dic3⋊C8C22×C3semidirect
123. Dic3⋊C8C22×C3semidirect
124. C4⋊(C3⋊C8)C22×C3semidirect
158. C2×C4⋊C8C3×D3semidirect
262. C4⋊C8C2×C3×D3semidirect
326. C2×C32#424semidirect
401. C32C2×#424semidirect
433. C2×C3#4839direct (N × H)
434. C2×C3C3×#424semidirect
459. C3C2×#4839direct (N × H)
460. C3C2×C3×#424semidirect
462. C2C3×#4839direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽25)2
C22×C3×D3×C82
C2×C3×#8722
C2×C32×C4⋊C82
C2×C3×Dic3⋊C82
C2×C3×C4⋊(C3⋊C8)2
C3×#48392
C2×C3×#4243
C2×#48393