Finite Group Databasebeta

← C2×C3×#5730 #11903237

Subgroups — C2×C3×#5730 #11903237

Order 1152.

2545 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#946C214yesmaximal, complemented
3C3×#5730C2116yesmaximal, direct product factor
4C2×C3×#897C212yesmaximal, complemented
5C2×C3×#893C212yesmaximal
6C2×C3×D3×C4⋊C4C212yesmaximal, complemented
7C2×C3×#890C212yesmaximal, complemented
8C2×C3×C4×D12C211yesmaximal, characteristic
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#554—31nomaximal
11C2×#5730C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2212yes—
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×C4×D3C2214yes—
16C22×C3×Dic6C2211yescharacteristic
17C32×C4⋊Q8C2218yescomplemented
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C4⋊C4C2212yes—
20C2×C32×C42C2211yescharacteristic
21C3×#946C22132yescomplemented
22C2×C3×(C2×D3)⋊C4C2214yes—
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic3C2214yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×Dic3⋊C4C2214yes—
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×#897C22116yescomplemented
31C3×#893C22116yes—
32C3×D3×C4⋊C4C22116yescomplemented
33C3×#890C22116yescomplemented
34C3×C4×D12C2218yes—
35C3×C4×Dic6C2218yescomplemented
36C3×#554—316no—
37C2×C3×C4⋊Q8—21no—
38C2×C3×C4⋊Q8D311yescharacteristic, complemented
39C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
40C2×C3×#125—32no—
41C2×C3×C22⋊Q8—34no—
42C2×C3×C22⋊Q8—32no—
43C2×C3×C4×Q8—31no—
44C2×C3×C4×D4—32no—
45C2×C3×C4×D4—31no—
46C22×C3×C4⋊C4—32no—
47C2×#946C2×C314yescomplemented
48#5730C2×C3116yesdirect product factor
49C2×#897C2×C312yescomplemented
50C2×#893C2×C312yes—
51C2×D3×C4⋊C4C2×C312yescomplemented
52C2×#890C2×C312yescomplemented
53C2×C4×D12C2×C311yescharacteristic
54C2×C4×Dic6C2×C311yescharacteristic, complemented
55C23×C3×D3C2312yes—
56C2×C32×Q8—44no—
57C2×C32×Q8C2318yes—
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2314yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
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—24no—
72C2×C3×Dic6C2314yes—
73C32×C4⋊C4C2318yes—
74C32×C4⋊C4C2318yes—
75C32×C42C2314yes—
76C3×(C2×D3)⋊C4C23116yes—
77C3×(C2×D3)⋊C4C2318yes—
78C3×C4⋊Dic3C2314yes—
79C3×C4⋊Dic3C23116yes—
80C3×C4⋊Dic3C2314yes—
81C3×Dic3⋊C4C23116yes—
82C3×Dic3⋊C4C2318yes—
83C3×C4×Dic3C2318yes—
84C2×#554—31noSylow
85C22×C3×Q8—22no—
86C22×C3×Q8—31no—
87C22×C3×Q8C2×D312yes—
88C22×C3×Q8C22×C312yes—
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×C311yescharacteristic
96C3×C4⋊Q8—28no—
97C3×C4⋊Q8C2×D318yescomplemented
98C3×C4⋊Q8C22×C318yescomplemented
99C3×#125—316no—
100C3×C22⋊Q8—332no—
101C3×C22⋊Q8—316no—
102C3×C4×Q8—38no—
103C3×C4×D4—316no—
104C3×C4×D4—38no—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—34no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4—31no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—34no—
112C2×C3×C4⋊C4—316no—
113C2×C3×C4⋊C4C2×D312yes—
114C2×C3×C4⋊C4C2×D312yes—
115C2×C3×C4⋊C4C22×C312yes—
116C2×C3×C4⋊C4C22×C312yes—
117C2×C3×C22⋊C4—34no—
118C2×C3×C22⋊C4—32no—
119C2×C3×C42—21no—
120C2×C3×C42—32no—
121C2×C3×C42C2×D311yescharacteristic
122C2×C3×C42C22×C311yescharacteristic
123#946C22×C3132yescomplemented
124C2×(C2×D3)⋊C4C22×C312yes—
125C2×(C2×D3)⋊C4C22×C314yes—
126C2×C4⋊Dic3C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C311yescharacteristic
128C2×C4⋊Dic3C22×C314yes—
129C2×Dic3⋊C4C22×C312yes—
130C2×Dic3⋊C4C22×C314yes—
131C2×C4×Dic3C22×C312yes—
132#897C22×C3116yescomplemented
133#893C22×C3116yes—
134D3×C4⋊C4C22×C3116yescomplemented
135#890C22×C3116yescomplemented
136C4×D12C22×C318yes—
137C4×Dic6C22×C318yescomplemented
138C23×C32C2411yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3C2×Q818yes—
142C22×C3×D3C2414yes—
143C32×Q8—48no—
144C2×C32×C4—22no—
145C2×C32×C4—28no—
146C2×C32×C4C2×Q814yes—
147C2×C32×C4#4114yes—
148C2×C32×C4C2412yes—
149C2×C32×C4C2418yes—
150C2×C32×C4C2412yes—
151C2×C32×C4C2412yes—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3C2414yes—
156C2×C3×Dic3C2414yes—
157C2×C3×Dic3C2414yes—
158C3×D12C2×Q8116yescomplemented
159C3×C4×D3—216no—
160C3×C4×D3—416no—
161C3×Dic6—28no—
162#554—316no—
163C2×C4⋊Q8C3×D311yescharacteristic, complemented
164C2×#125—32no—
165C2×C22⋊Q8—34no—
166C2×C22⋊Q8—32no—
167C2×C4×Q8—31no—
168C2×C4×D4—32no—
169C2×C4×D4—31no—
170C22×C4⋊C4—32no—
171C24×C3—32no—
172C23×D3C23×C312yes—
173C2×C3×Q8—28no—
174C2×C3×Q8—34no—
175C2×C3×Q8—44no—
176C2×C3×Q8—44no—
177C2×C3×Q8—48no—
178C2×C3×Q8—64no—
179C2×C3×Q8C22×D318yes—
180C2×C3×Q8C23×C318yes—
181C2×C3×D4—38no—
182C2×C3×D4—34no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—38no—
191C22×C3×C4—316no—
192C22×C3×C4—68no—
193C22×C3×C4—128no—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D314yes—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C23×C311yescharacteristic
199C22×C3×C4C23×C314yes—
200C22×C3×C4C23×C311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C22×Dic3C23×C312yes—
203C22×Dic3C23×C312yes—
204C22×Dic3C23×C312yes—
205C2×D12C3×Q818yescomplemented
206C2×D12C23×C314yes—
207C2×C4×D3—28no—
208C2×C4×D3—48no—
209C2×C4×D3C23×C318yes—
210C2×C4×D3C23×C3116yes—
211C2×Dic6—24no—
212C2×Dic6C23×C314yes—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—316no—
216C3×C4⋊C4—38no—
217C3×C4⋊C4—34no—
218C3×C4⋊C4—34no—
219C3×C4⋊C4—316no—
220C3×C4⋊C4C22×D318yes—
221C3×C4⋊C4C22×D318yes—
222C3×C4⋊C4C23×C318yes—
223C3×C4⋊C4C23×C318yes—
224C3×C22⋊C4—316no—
225C3×C22⋊C4—38no—
226C3×C42—24no—
227C3×C42—38no—
228C3×C42C22×D314yes—
229C3×C42C23×C314yes—
230(C2×D3)⋊C4C23×C318yes—
231(C2×D3)⋊C4C23×C3116yes—
232C4⋊Dic3C23×C314yes—
233C4⋊Dic3C23×C314yes—
234C4⋊Dic3C23×C3116yes—
235Dic3⋊C4C23×C318yes—
236Dic3⋊C4C23×C3116yes—
237C4×Dic3C23×C318yes—
238C22×C32C22×Q812yes—
239C22×C32C2×#4112yes—
240C22×C32H(5,𝔽2)12yes—
241C22×C32C2511yescharacteristic
242C2×C3×D3—216no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3C22×Q818yes—
246C32×C4—22no—
247C32×C4—28no—
248C32×C4C22×Q814yes—
249C32×C4C2×#4114yes—
250C3×Dic3—24no—
251C3×Dic3—24no—
252C3×Dic3—24no—
253C22×Q8—31no—
254C22×Q8C2×C3×D312yes—
255C22×D4—31no—
256C23×C4—32no—
257C23×C4—34no—
258C4⋊Q8C2×C3×D318yescomplemented
259#125—316no—
260C22⋊Q8—332no—
261C22⋊Q8—316no—
262C4×Q8—38no—
263C4×D4—316no—
264C4×D4—38no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—34no—
270C2×C4⋊C4—316no—
271C2×C4⋊C4C2×C3×D312yes—
272C2×C4⋊C4C2×C3×D312yes—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—32no—
275C2×C42—32no—
276C2×C42C2×C3×D311yescharacteristic
277C23×C3—21no—
278C23×C3—38no—
279C23×C3—34no—
280C23×C3—64no—
281C23×C3—64no—
282C23×C3C23×D311yescharacteristic
283C23×C3C24×C311yescharacteristic
284C22×D3—24no—
285C22×D3—24no—
286C22×D3C2×C3×Q818yes—
287C22×D3C24×C314yes—
288C3×Q8—48no—
289C3×Q8—48no—
290C3×Q8—416no—
291C3×Q8—68no—
292C3×D4—316no—
293C2×C3×C4—24no—
294C2×C3×C4—28no—
295C2×C3×C4—22no—
296C2×C3×C4—28no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—28no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—34no—
307C2×C3×C4—48no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—616no—
312C2×C3×C4—1216no—
313C2×C3×C4D3×Q814yes—
314C2×C3×C4#23914yes—
315C2×C3×C4C2×C3×Q814yes—
316C2×C3×C4C3×#4114yes—
317C2×C3×C4C23×D312yes—
318C2×C3×C4C23×D318yes—
319C2×C3×C4C23×D312yes—
320C2×C3×C4C23×D312yes—
321C2×C3×C4C24×C312yes—
322C2×C3×C4C24×C318yes—
323C2×C3×C4C24×C312yes—
324C2×C3×C4C24×C312yes—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3C24×C314yes—
329C2×Dic3C24×C314yes—
330C2×Dic3C24×C314yes—
331D12C2×C3×Q8116yescomplemented
332C4×D3—216no—
333C4×D3—416no—
334Dic6—28no—
335C2×C32#55414yescomplemented
336C2×C32C23×Q811yescharacteristic
337C2×C32C22×#4111yescharacteristic
338C2×C32C2×H(5,𝔽2)11yescharacteristic
339C3×D3—28no—
340C24—32no—
341C2×Q8—34no—
342C2×Q8—44no—
343C2×Q8—64no—
344C2×Q8C22×C3×D318yes—
345C2×D4—34no—
346C2×D4—38no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—38no—
351C22×C4—316no—
352C22×C4—68no—
353C22×C4—128no—
354C22×C4C22×C3×D311yescharacteristic
355C22×C4C22×C3×D314yes—
356C22×C4C22×C3×D311yescharacteristic
357C22×C4C22×C3×D311yescharacteristic
358C4⋊C4—34no—
359C4⋊C4—316no—
360C4⋊C4—34no—
361C4⋊C4—38no—
362C4⋊C4—316no—
363C4⋊C4C22×C3×D318yes—
364C4⋊C4C22×C3×D318yes—
365C22⋊C4—316no—
366C22⋊C4—38no—
367C42—38no—
368C42C22×C3×D314yes—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—38no—
374C22×C3—64no—
375C22×C3—64no—
376C22×C3—616no—
377C22×C3#100412yes—
378C22×C3C2×D3×Q812yes—
379C22×C3C2×#23912yes—
380C22×C3C22×C3×Q812yes—
381C22×C3C2×C3×#4112yes—
382C22×C3C3×H(5,𝔽2)12yes—
383C22×C3C24×D311yescharacteristic
384C22×C3C25×C311yescharacteristic
385C2×D3—24no—
386C2×D3—24no—
387C2×D3—216no—
388C2×D3C22×C3×Q818yes—
389C3×C4—24no—
390C3×C4—28no—
391C3×C4—22no—
392C3×C4—28no—
393C3×C4—22no—
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—48no—
397C3×C4—64no—
398C3×C4—64no—
399C3×C4—64no—
400C3×C4C2×D3×Q814yes—
401C3×C4C2×#23914yes—
402C3×C4C22×C3×Q814yes—
403C3×C4C2×C3×#4114yes—
404Dic3—24no—
405Dic3—24no—
406Dic3—24no—
407C32C2×#55411yescharacteristic, complemented, Sylow
408C23—38no—
409C23—34no—
410C23—64no—
411C23—64no—
412C23C23×C3×D311yescharacteristic
413Q8—48no—
414Q8—68no—
415D4—316no—
416C2×C4—28no—
417C2×C4—22no—
418C2×C4—34no—
419C2×C4—34no—
420C2×C4—34no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—64no—
424C2×C4—616no—
425C2×C4—1216no—
426C2×C4C3×D3×Q814yes—
427C2×C4C3×#23914yes—
428C2×C4C23×C3×D312yes—
429C2×C4C23×C3×D318yes—
430C2×C4C23×C3×D312yes—
431C2×C4C23×C3×D312yes—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—24no—
436C2×C3—68no—
437C2×C3#573014yesdirect product factor
438C2×C3C3×#55414yescomplemented
439C2×C3C2×#100411yescharacteristic
440C2×C3C22×D3×Q811yescharacteristic
441C2×C3C22×#23911yescharacteristic
442C2×C3C23×C3×Q811yescharacteristic
443C2×C3C22×C3×#4111yescharacteristic
444C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
445D3—28no—
446C22—38no—
447C22—64no—
448C22—64no—
449C22—616no—
450C22C3×#100412yes—
451C22C2×C3×D3×Q812yes—
452C22C2×C3×#23912yes—
453C22C24×C3×D311yescharacteristic
454C4—28no—
455C4—22no—
456C4—64no—
457C4—64no—
458C4—64no—
459C4C2×C3×D3×Q814yes—
460C4C2×C3×#23914yes—
461C3—21no—
462C3C2×#573011yescharacteristic, direct product factor
463C3C2×C3×#55411yescharacteristic, complemented
464C2—68no—
465C2C3×#573014yesdirect product factor
466C2C2×C3×#100411yescharacteristic
467C2C22×C3×D3×Q811yescharacteristic
468C2C22×C3×#23911yescharacteristic
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×#5730C2direct (N × H)
4. C2×C3×#897C2semidirect
6. C2×C3×D3×C4⋊C4C2semidirect
7. C2×C3×#890C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5730C3direct (N × H)
17. C32×C4⋊Q8C22semidirect
21. C3×#946C22semidirect
30. C3×#897C22semidirect
32. C3×D3×C4⋊C4C22semidirect
33. C3×#890C22semidirect
35. C3×C4×Dic6C22semidirect
38. C2×C3×C4⋊Q8D3semidirect
39. C2×C3×C4⋊Q8C2×C3semidirect
47. C2×#946C2×C3semidirect
48. #5730C2×C3direct (N × H)
49. C2×#897C2×C3semidirect
51. C2×D3×C4⋊C4C2×C3semidirect
52. C2×#890C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
65. C2×C3×D12Q8semidirect
97. C3×C4⋊Q8C2×D3semidirect
98. C3×C4⋊Q8C22×C3semidirect
123. #946C22×C3semidirect
132. #897C22×C3semidirect
134. D3×C4⋊C4C22×C3semidirect
135. #890C22×C3semidirect
137. C4×Dic6C22×C3semidirect
158. C3×D12C2×Q8semidirect
163. C2×C4⋊Q8C3×D3semidirect
205. C2×D12C3×Q8semidirect
258. C4⋊Q8C2×C3×D3semidirect
331. D12C2×C3×Q8semidirect
335. C2×C32#554semidirect
407. C32C2×#554semidirect
437. C2×C3#5730direct (N × H)
438. C2×C3C3×#554semidirect
462. C3C2×#5730direct (N × H)
463. C3C2×C3×#554semidirect
465. C2C3×#5730direct (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×#57302
C2×C3×#8972
C2×C3×#8932
C2×C3×D3×C4⋊C42
C2×C3×#8902
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×C3×#5543
C2×#57303