Finite Group Databasebeta

← C2×C3×#5579 #11903234

Subgroups — C2×C3×#5579 #11903234

Order 1152.

2549 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#946C212yesmaximal, complemented
3C3×#5579C2116yesmaximal, direct product factor
4C2×C3×#896C214yesmaximal, complemented
5C2×C3×D3×C4⋊C4C212yesmaximal, complemented
6C2×C3×C4×D12C212yesmaximal
7C2×C3×C4×D12C211yesmaximal, characteristic
8C2×C3×#870C212yesmaximal, complemented
9C2×C3×#869C211yesmaximal, characteristic, complemented
10C2×C3×#554—31nomaximal
11C2×#5579C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×C4×D3C2214yes—
16C22×C3×Dic6C2212yes—
17C32×C4×Q8C2218yescomplemented
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C4⋊C4C2212yes—
20C2×C32×C42C2211yescharacteristic
21C2×C32×C42C2212yes—
22C3×#946C22116yescomplemented
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2214yes—
29C2×C3×Dic3⋊C4C2214yes—
30C3×#896C22132yescomplemented
31C3×D3×C4⋊C4C22116yescomplemented
32C3×C4×D12C22116yes—
33C3×C4×D12C2218yes—
34C3×#870C22116yescomplemented
35C3×#869C2218yescomplemented
36C3×#554—316no—
37C2×C3×C4⋊Q8—31no—
38C2×C3×#125—32no—
39C2×C3×C22⋊Q8—32no—
40C2×C3×C22⋊Q8—34no—
41C2×C3×C4×Q8—21no—
42C2×C3×C4×Q8D311yescharacteristic, complemented
43C2×C3×C4×Q8C2×C311yescharacteristic, complemented
44C2×C3×C4×D4—32no—
45C2×C3×C4×D4—31no—
46C22×C3×C4⋊C4—32no—
47C2×#946C2×C312yescomplemented
48#5579C2×C3116yesdirect product factor
49C2×#896C2×C314yescomplemented
50C2×D3×C4⋊C4C2×C312yescomplemented
51C2×C4×D12C2×C312yes—
52C2×C4×D12C2×C311yescharacteristic
53C2×#870C2×C312yescomplemented
54C2×#869C2×C311yescharacteristic, complemented
55C23×C3×D3C2312yes—
56C2×C32×Q8—24no—
57C2×C32×Q8C2314yes—
58C22×C32×C4C2312yes—
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2314yes—
65C2×C3×D12Q818yescomplemented
66C2×C3×D12C2314yes—
67C2×C3×C4×D3—28no—
68C2×C3×C4×D3—48no—
69C2×C3×C4×D3C2318yes—
70C2×C3×C4×D3C23116yes—
71C2×C3×Dic6—44no—
72C2×C3×Dic6C2318yes—
73C32×C4⋊C4C2314yes—
74C32×C4⋊C4C2318yes—
75C32×C42C2314yes—
76C32×C42C2318yes—
77C3×(C2×D3)⋊C4C2318yes—
78C3×(C2×D3)⋊C4C23116yes—
79C3×C4⋊Dic3C2318yes—
80C3×C4⋊Dic3C2318yes—
81C3×C4⋊Dic3C2314yes—
82C3×Dic3⋊C4C23116yes—
83C3×Dic3⋊C4C23116yes—
84C2×#554—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8—32no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—31no—
90C23×C3×C4—32no—
91C23×C3×C4—34no—
92C22×D12C22×C311yescharacteristic
93C22×C4×D3C22×C312yes—
94C22×C4×D3C22×C314yes—
95C22×Dic6C22×C312yes—
96C3×C4⋊Q8—38no—
97C3×#125—316no—
98C3×C22⋊Q8—316no—
99C3×C22⋊Q8—332no—
100C3×C4×Q8—28no—
101C3×C4×Q8C2×D318yescomplemented
102C3×C4×Q8C22×C318yescomplemented
103C3×C4×D4—316no—
104C3×C4×D4—38no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—34no—
108C2×C3×C4⋊C4—34no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—32no—
112C2×C3×C4⋊C4—316no—
113C2×C3×C4⋊C4C2×D311yescharacteristic
114C2×C3×C4⋊C4C2×D312yes—
115C2×C3×C4⋊C4C22×C311yescharacteristic
116C2×C3×C4⋊C4C22×C312yes—
117C2×C3×C22⋊C4—32no—
118C2×C3×C22⋊C4—34no—
119C2×C3×C42—22no—
120C2×C3×C42—21no—
121C2×C3×C42C2×D312yes—
122C2×C3×C42C2×D311yescharacteristic
123C2×C3×C42C22×C311yescharacteristic
124C2×C3×C42C22×C312yes—
125#946C22×C3116yescomplemented
126C2×(C2×D3)⋊C4C22×C312yes—
127C2×(C2×D3)⋊C4C22×C314yes—
128C2×C4⋊Dic3C22×C312yes—
129C2×C4⋊Dic3C22×C312yes—
130C2×C4⋊Dic3C22×C311yescharacteristic
131C2×Dic3⋊C4C22×C314yes—
132C2×Dic3⋊C4C22×C314yes—
133#896C22×C3132yescomplemented
134D3×C4⋊C4C22×C3116yescomplemented
135C4×D12C22×C3116yes—
136C4×D12C22×C318yes—
137#870C22×C3116yescomplemented
138#869C22×C318yescomplemented
139C23×C32C2411yescharacteristic
140C22×C3×D3—24no—
141C22×C3×D3—24no—
142C22×C3×D3C2×Q818yes—
143C22×C3×D3C2414yes—
144C32×Q8—28no—
145C2×C32×C4—24no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4C2×Q814yes—
149C2×C32×C4#4114yes—
150C2×C32×C4C2414yes—
151C2×C32×C4C2414yes—
152C2×C32×C4C2412yes—
153C2×C32×C4C2412yes—
154C2×C32×C4C2412yes—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—28no—
157C2×C3×Dic3C2414yes—
158C2×C3×Dic3C2418yes—
159C3×D12C2×Q8116yescomplemented
160C3×C4×D3—216no—
161C3×C4×D3—416no—
162C3×Dic6—48no—
163#554—316no—
164C2×C4⋊Q8—31no—
165C2×#125—32no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—34no—
168C2×C4×Q8C3×D311yescharacteristic, complemented
169C2×C4×D4—32no—
170C2×C4×D4—31no—
171C22×C4⋊C4—32no—
172C24×C3—32no—
173C23×D3C23×C312yes—
174C2×C3×Q8—28no—
175C2×C3×Q8—24no—
176C2×C3×Q8—24no—
177C2×C3×Q8—24no—
178C2×C3×Q8—38no—
179C2×C3×Q8—124no—
180C2×C3×Q8C22×D314yes—
181C2×C3×Q8C23×C314yes—
182C2×C3×D4—38no—
183C2×C3×D4—34no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—32no—
190C22×C3×C4—34no—
191C22×C3×C4—38no—
192C22×C3×C4—316no—
193C22×C3×C4—68no—
194C22×C3×C4—128no—
195C22×C3×C4C22×D312yes—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C22×D312yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C22×C3×C4C23×C312yes—
201C22×C3×C4C23×C312yes—
202C22×C3×C4C23×C311yescharacteristic
203C22×C3×C4C23×C311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C22×Dic3C23×C312yes—
206C22×Dic3C23×C314yes—
207C2×D12C3×Q818yescomplemented
208C2×D12C23×C314yes—
209C2×C4×D3—28no—
210C2×C4×D3—48no—
211C2×C4×D3C23×C318yes—
212C2×C4×D3C23×C3116yes—
213C2×Dic6—44no—
214C2×Dic6C23×C318yes—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—316no—
218C3×C4⋊C4—316no—
219C3×C4⋊C4—38no—
220C3×C4⋊C4—34no—
221C3×C4⋊C4—38no—
222C3×C4⋊C4C22×D314yes—
223C3×C4⋊C4C22×D318yes—
224C3×C4⋊C4C23×C314yes—
225C3×C4⋊C4C23×C318yes—
226C3×C22⋊C4—38no—
227C3×C22⋊C4—316no—
228C3×C42—28no—
229C3×C42—24no—
230C3×C42C22×D318yes—
231C3×C42C22×D314yes—
232C3×C42C23×C314yes—
233C3×C42C23×C318yes—
234(C2×D3)⋊C4C23×C318yes—
235(C2×D3)⋊C4C23×C3116yes—
236C4⋊Dic3C23×C318yes—
237C4⋊Dic3C23×C318yes—
238C4⋊Dic3C23×C314yes—
239Dic3⋊C4C23×C3116yes—
240Dic3⋊C4C23×C3116yes—
241C22×C32C22×Q812yes—
242C22×C32C2×#4112yes—
243C22×C32H(5,𝔽2)12yes—
244C22×C32C2511yescharacteristic
245C2×C3×D3—216no—
246C2×C3×D3—24no—
247C2×C3×D3—24no—
248C2×C3×D3C22×Q818yes—
249C32×C4—24no—
250C32×C4—22no—
251C32×C4—24no—
252C32×C4C22×Q814yes—
253C32×C4C2×#4114yes—
254C3×Dic3—24no—
255C3×Dic3—28no—
256C22×Q8—32no—
257C22×Q8C2×C3×D311yescharacteristic
258C22×D4—31no—
259C23×C4—32no—
260C23×C4—34no—
261C4⋊Q8—38no—
262#125—316no—
263C22⋊Q8—316no—
264C22⋊Q8—332no—
265C4×Q8C2×C3×D318yescomplemented
266C4×D4—316no—
267C4×D4—38no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—34no—
273C2×C4⋊C4—316no—
274C2×C4⋊C4C2×C3×D311yescharacteristic
275C2×C4⋊C4C2×C3×D312yes—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—34no—
278C2×C42C2×C3×D312yes—
279C2×C42C2×C3×D311yescharacteristic
280C23×C3—21no—
281C23×C3—38no—
282C23×C3—34no—
283C23×C3—64no—
284C23×C3—64no—
285C23×C3C23×D311yescharacteristic
286C23×C3C24×C311yescharacteristic
287C22×D3—24no—
288C22×D3—24no—
289C22×D3C2×C3×Q818yes—
290C22×D3C24×C314yes—
291C3×Q8—216no—
292C3×Q8—28no—
293C3×Q8—28no—
294C3×Q8—128no—
295C3×D4—316no—
296C2×C3×C4—28no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—34no—
311C2×C3×C4—38no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—64no—
315C2×C3×C4—68no—
316C2×C3×C4—616no—
317C2×C3×C4—1216no—
318C2×C3×C4#23514yes—
319C2×C3×C4D3×Q814yes—
320C2×C3×C4C2×C3×Q814yes—
321C2×C3×C4C3×#4114yes—
322C2×C3×C4C23×D314yes—
323C2×C3×C4C23×D312yes—
324C2×C3×C4C23×D314yes—
325C2×C3×C4C23×D312yes—
326C2×C3×C4C23×D312yes—
327C2×C3×C4C24×C314yes—
328C2×C3×C4C24×C314yes—
329C2×C3×C4C24×C312yes—
330C2×C3×C4C24×C312yes—
331C2×C3×C4C24×C312yes—
332C2×Dic3—24no—
333C2×Dic3—28no—
334C2×Dic3C24×C314yes—
335C2×Dic3C24×C318yes—
336D12C2×C3×Q8116yescomplemented
337C4×D3—216no—
338C4×D3—416no—
339Dic6—48no—
340C2×C32#55414yescomplemented
341C2×C32C23×Q811yescharacteristic
342C2×C32C22×#4111yescharacteristic
343C2×C32C2×H(5,𝔽2)11yescharacteristic
344C3×D3—28no—
345C24—32no—
346C2×Q8—24no—
347C2×Q8—38no—
348C2×Q8—124no—
349C2×Q8C22×C3×D314yes—
350C2×D4—34no—
351C2×D4—38no—
352C22×C4—32no—
353C22×C4—34no—
354C22×C4—38no—
355C22×C4—316no—
356C22×C4—68no—
357C22×C4—128no—
358C22×C4C22×C3×D312yes—
359C22×C4C22×C3×D311yescharacteristic
360C22×C4C22×C3×D312yes—
361C22×C4C22×C3×D311yescharacteristic
362C22×C4C22×C3×D311yescharacteristic
363C4⋊C4—316no—
364C4⋊C4—38no—
365C4⋊C4—34no—
366C4⋊C4—38no—
367C4⋊C4—316no—
368C4⋊C4C22×C3×D314yes—
369C4⋊C4C22×C3×D318yes—
370C22⋊C4—38no—
371C22⋊C4—316no—
372C42C22×C3×D318yes—
373C42C22×C3×D314yes—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—38no—
379C22×C3—64no—
380C22×C3—64no—
381C22×C3—616no—
382C22×C3C2×#23512yes—
383C22×C3C2×D3×Q812yes—
384C22×C3#100912yes—
385C22×C3C22×C3×Q812yes—
386C22×C3C2×C3×#4112yes—
387C22×C3C3×H(5,𝔽2)12yes—
388C22×C3C24×D311yescharacteristic
389C22×C3C25×C311yescharacteristic
390C2×D3—24no—
391C2×D3—24no—
392C2×D3—216no—
393C2×D3C22×C3×Q818yes—
394C3×C4—28no—
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—22no—
398C3×C4—24no—
399C3×C4—22no—
400C3×C4—24no—
401C3×C4—24no—
402C3×C4—24no—
403C3×C4—44no—
404C3×C4—42no—
405C3×C4—64no—
406C3×C4—68no—
407C3×C4C2×#23514yes—
408C3×C4C2×D3×Q814yes—
409C3×C4C22×C3×Q814yes—
410C3×C4C2×C3×#4114yes—
411Dic3—24no—
412Dic3—28no—
413C32C2×#55411yescharacteristic, complemented, Sylow
414C23—38no—
415C23—34no—
416C23—64no—
417C23—64no—
418C23C23×C3×D311yescharacteristic
419Q8—28no—
420Q8—128no—
421D4—316no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—34no—
426C2×C4—38no—
427C2×C4—64no—
428C2×C4—68no—
429C2×C4—616no—
430C2×C4—1216no—
431C2×C4C3×#23514yes—
432C2×C4C3×D3×Q814yes—
433C2×C4C23×C3×D314yes—
434C2×C4C23×C3×D312yes—
435C2×C4C23×C3×D314yes—
436C2×C4C23×C3×D312yes—
437C2×C4C23×C3×D312yes—
438C2×C3—24no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—68no—
443C2×C3#557914yesdirect product factor
444C2×C3C3×#55414yescomplemented
445C2×C3C22×#23511yescharacteristic
446C2×C3C22×D3×Q811yescharacteristic
447C2×C3C2×#100911yescharacteristic
448C2×C3C23×C3×Q811yescharacteristic
449C2×C3C22×C3×#4111yescharacteristic
450C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
451D3—28no—
452C22—38no—
453C22—616no—
454C22—64no—
455C22—64no—
456C22C2×C3×#23512yes—
457C22C2×C3×D3×Q812yes—
458C22C3×#100912yes—
459C22C24×C3×D311yescharacteristic
460C4—24no—
461C4—24no—
462C4—22no—
463C4—64no—
464C4—68no—
465C4C2×C3×#23514yes—
466C4C2×C3×D3×Q814yes—
467C3—21no—
468C3C2×#557911yescharacteristic, direct product factor
469C3C2×C3×#55411yescharacteristic, complemented
470C2—68no—
471C2C3×#557914yesdirect product factor
472C2C22×C3×#23511yescharacteristic
473C2C22×C3×D3×Q811yescharacteristic
474C2C2×C3×#100911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C24
prime radicalC22×C3×D12C22
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
2. C2×C3×#946C2semidirect
3. C3×#5579C2direct (N × H)
4. C2×C3×#896C2semidirect
5. C2×C3×D3×C4⋊C4C2semidirect
8. C2×C3×#870C2semidirect
9. C2×C3×#869C2semidirect
11. C2×#5579C3direct (N × H)
17. C32×C4×Q8C22semidirect
22. C3×#946C22semidirect
30. C3×#896C22semidirect
31. C3×D3×C4⋊C4C22semidirect
34. C3×#870C22semidirect
35. C3×#869C22semidirect
42. C2×C3×C4×Q8D3semidirect
43. C2×C3×C4×Q8C2×C3semidirect
47. C2×#946C2×C3semidirect
48. #5579C2×C3direct (N × H)
49. C2×#896C2×C3semidirect
50. C2×D3×C4⋊C4C2×C3semidirect
53. C2×#870C2×C3semidirect
54. C2×#869C2×C3semidirect
65. C2×C3×D12Q8semidirect
101. C3×C4×Q8C2×D3semidirect
102. C3×C4×Q8C22×C3semidirect
125. #946C22×C3semidirect
133. #896C22×C3semidirect
134. D3×C4⋊C4C22×C3semidirect
137. #870C22×C3semidirect
138. #869C22×C3semidirect
159. C3×D12C2×Q8semidirect
168. C2×C4×Q8C3×D3semidirect
207. C2×D12C3×Q8semidirect
265. C4×Q8C2×C3×D3semidirect
336. D12C2×C3×Q8semidirect
340. C2×C32#554semidirect
413. C32C2×#554semidirect
443. C2×C3#5579direct (N × H)
444. C2×C3C3×#554semidirect
468. C3C2×#5579direct (N × H)
469. C3C2×C3×#554semidirect
471. C2C3×#5579direct (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×#9462
C3×#55792
C2×C3×#8962
C2×C3×D3×C4⋊C42
C2×C3×C4×D122
C2×C3×#8702
C2×C3×#8692
C2×C3×#5543
C2×#55793