Finite Group Databasebeta

← C2×C3×#5732 #11903373

Subgroups — C2×C3×#5732 #11903373

Order 1152.

2369 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#947C212yesmaximal
3C2×C3×Q8×Dic3C212yesmaximal, complemented
4C3×#5732C2116yesmaximal, direct product factor
5C2×C3×#898C214yesmaximal, complemented
6C2×C3×#897C212yesmaximal, complemented
7C2×C3×#892C212yesmaximal, complemented
8C2×C3×C4×D12C211yesmaximal, characteristic
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#556—31nomaximal
11C2×#5732C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2212yes—
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×Dic6C2211yescharacteristic
16C32×C4⋊Q8C2218yescomplemented
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C42C2211yescharacteristic
20C3×#947C22116yes—
21C3×Q8×Dic3C22116yescomplemented
22C2×C3×(C2×D3)⋊C4C2214yes—
23C2×C3×(C2×D3)⋊C4C2214yes—
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2214yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2214yes—
30C2×C3×C4×Dic3C2212yes—
31C3×#898C22132yescomplemented
32C3×#897C22116yescomplemented
33C3×#892C22116yescomplemented
34C3×C4×D12C2218yes—
35C3×C4×Dic6C2218yescomplemented
36C3×#556—316no—
37C2×C3×C4⋊Q8—21no—
38C2×C3×C4⋊Q8D311yescharacteristic, complemented
39C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
40C2×C3×#126—34no—
41C2×C3×#124—32no—
42C2×C3×C22⋊Q8—32no—
43C2×C3×C4×Q8—32no—
44C2×C3×C4×Q8—31no—
45C2×C3×C4×D4—31no—
46C2×C3×#117—32no—
47C2×#947C2×C312yes—
48C2×Q8×Dic3C2×C312yescomplemented
49#5732C2×C3116yesdirect product factor
50C2×#898C2×C314yescomplemented
51C2×#897C2×C312yescomplemented
52C2×#892C2×C312yescomplemented
53C2×C4×D12C2×C311yescharacteristic
54C2×C4×Dic6C2×C311yescharacteristic, complemented
55C23×C3×D3C2312yes—
56C2×C32×Q8—28no—
57C2×C32×Q8C2318yes—
58C22×C32×C4C2314yes—
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
65C2×C3×D12—24no—
66C2×C3×D12C2314yes—
67C2×C3×C4×D3—28no—
68C2×C3×C4×D3C2318yes—
69C2×C3×Dic6—24no—
70C2×C3×Dic6C2314yes—
71C32×C4⋊C4C2318yes—
72C32×C4⋊C4C2318yes—
73C32×C42C2314yes—
74C3×(C2×D3)⋊C4C23116yes—
75C3×(C2×D3)⋊C4C23116yes—
76C3×(C2×D3)⋊C4C2318yes—
77C3×C4⋊Dic3C2314yes—
78C3×C4⋊Dic3C23116yes—
79C3×C4⋊Dic3C2314yes—
80C3×Dic3⋊C4C2318yes—
81C3×C4×Dic3C23116yes—
82C3×C4×Dic3C2318yes—
83C2×#556—31noSylow
84C22×C3×Q8—22no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D312yes—
87C22×C3×Q8C22×C312yes—
88C22×C3×D4—31no—
89C23×C3×C4—32no—
90C22×D12C22×C311yescharacteristic
91C22×C4×D3C22×C312yes—
92C22×Dic6C22×C311yescharacteristic
93C3×C4⋊Q8—28no—
94C3×C4⋊Q8C2×D318yescomplemented
95C3×C4⋊Q8C22×C318yescomplemented
96C3×#126—332no—
97C3×#124—316no—
98C3×C22⋊Q8—316no—
99C3×C4×Q8—316no—
100C3×C4×Q8—38no—
101C3×C4×D4—38no—
102C3×#117—316no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—34no—
109C2×C3×C4⋊C4C2×D312yes—
110C2×C3×C4⋊C4C2×D312yes—
111C2×C3×C4⋊C4C22×C312yes—
112C2×C3×C4⋊C4C22×C312yes—
113C2×C3×C22⋊C4—34no—
114C2×C3×C22⋊C4—32no—
115C2×C3×C22⋊C4—34no—
116C2×C3×C42—21no—
117C2×C3×C42—34no—
118C2×C3×C42—32no—
119C2×C3×C42C2×D311yescharacteristic
120C2×C3×C42C22×C311yescharacteristic
121#947C22×C3116yes—
122Q8×Dic3C22×C3116yescomplemented
123C2×(C2×D3)⋊C4C22×C312yes—
124C2×(C2×D3)⋊C4C22×C314yes—
125C2×(C2×D3)⋊C4C22×C314yes—
126C2×C4⋊Dic3C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C311yescharacteristic
128C2×C4⋊Dic3C22×C314yes—
129C2×Dic3⋊C4C22×C312yes—
130C2×C4×Dic3C22×C312yes—
131C2×C4×Dic3C22×C314yes—
132#898C22×C3132yescomplemented
133#897C22×C3116yescomplemented
134#892C22×C3116yescomplemented
135C4×D12C22×C318yes—
136C4×Dic6C22×C318yescomplemented
137C23×C32C2411yescharacteristic
138C22×C3×D3—24no—
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3C2414yes—
142C32×Q8—216no—
143C2×C32×C4—22no—
144C2×C32×C4—28no—
145C2×C32×C4#4114yes—
146C2×C32×C4#4114yes—
147C2×C32×C4C2418yes—
148C2×C32×C4C2412yes—
149C2×C32×C4C2412yes—
150C2×C32×C4C2412yes—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3C2414yes—
155C2×C3×Dic3C2414yes—
156C2×C3×Dic3C2414yes—
157C3×D12—28no—
158C3×C4×D3—216no—
159C3×Dic6—28no—
160#556—316no—
161C2×C4⋊Q8C3×D311yescharacteristic, complemented
162C2×#126—34no—
163C2×#124—32no—
164C2×C22⋊Q8—32no—
165C2×C4×Q8—32no—
166C2×C4×Q8—31no—
167C2×C4×D4—31no—
168C2×#117—32no—
169C24×C3—32no—
170C23×D3C23×C312yes—
171C2×C3×Q8—28no—
172C2×C3×Q8—28no—
173C2×C3×Q8—28no—
174C2×C3×Q8—34no—
175C2×C3×Q8—48no—
176C2×C3×Q8—64no—
177C2×C3×Q8C22×D318yes—
178C2×C3×Q8C23×C318yes—
179C2×C3×D4—34no—
180C2×C3×D4—64no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—38no—
189C22×C3×C4—68no—
190C22×C3×C4C22×D314yes—
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C22×D311yescharacteristic
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C23×C314yes—
195C22×C3×C4C23×C311yescharacteristic
196C22×C3×C4C23×C311yescharacteristic
197C22×C3×C4C23×C311yescharacteristic
198C22×Dic3C23×C312yes—
199C22×Dic3C23×C312yes—
200C22×Dic3C23×C312yes—
201C2×D12—24no—
202C2×D12C23×C314yes—
203C2×C4×D3—28no—
204C2×C4×D3C23×C318yes—
205C2×Dic6—24no—
206C2×Dic6C23×C314yes—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—38no—
210C3×C4⋊C4—34no—
211C3×C4⋊C4—34no—
212C3×C4⋊C4—316no—
213C3×C4⋊C4C22×D318yes—
214C3×C4⋊C4C22×D318yes—
215C3×C4⋊C4C23×C318yes—
216C3×C4⋊C4C23×C318yes—
217C3×C22⋊C4—316no—
218C3×C22⋊C4—38no—
219C3×C22⋊C4—316no—
220C3×C42—24no—
221C3×C42—316no—
222C3×C42—38no—
223C3×C42C22×D314yes—
224C3×C42C23×C314yes—
225(C2×D3)⋊C4C23×C318yes—
226(C2×D3)⋊C4C23×C3116yes—
227(C2×D3)⋊C4C23×C3116yes—
228C4⋊Dic3C23×C314yes—
229C4⋊Dic3C23×C314yes—
230C4⋊Dic3C23×C3116yes—
231Dic3⋊C4C23×C318yes—
232C4×Dic3C23×C318yes—
233C4×Dic3C23×C3116yes—
234C22×C32C2×#4112yes—
235C22×C32C2×#4112yes—
236C22×C322−1+412yes—
237C22×C32C2511yescharacteristic
238C2×C3×D3—24no—
239C2×C3×D3—24no—
240C2×C3×D3—24no—
241C2×C3×D3—48no—
242C32×C4—22no—
243C32×C4—28no—
244C32×C4C2×#4114yes—
245C32×C4C2×#4114yes—
246C3×Dic3—24no—
247C3×Dic3—24no—
248C3×Dic3—24no—
249C22×Q8—31no—
250C22×Q8C2×C3×D312yes—
251C22×D4—31no—
252C23×C4—32no—
253C4⋊Q8C2×C3×D318yescomplemented
254#126—332no—
255#124—316no—
256C22⋊Q8—316no—
257C4×Q8—316no—
258C4×Q8—38no—
259C4×D4—38no—
260#117—316no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4C2×C3×D312yes—
266C2×C4⋊C4C2×C3×D312yes—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—34no—
270C2×C42—34no—
271C2×C42—32no—
272C2×C42C2×C3×D311yescharacteristic
273C23×C3—21no—
274C23×C3—34no—
275C23×C3—64no—
276C23×C3—64no—
277C23×C3—64no—
278C23×C3C23×D311yescharacteristic
279C23×C3C24×C311yescharacteristic
280C22×D3—24no—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3C24×C314yes—
284C3×Q8—216no—
285C3×Q8—216no—
286C3×Q8—416no—
287C3×Q8—68no—
288C3×D4—68no—
289C2×C3×C4—24no—
290C2×C3×C4—28no—
291C2×C3×C4—22no—
292C2×C3×C4—28no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—34no—
301C2×C3×C4—34no—
302C2×C3×C4—34no—
303C2×C3×C4—48no—
304C2×C3×C4—64no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—616no—
308C2×C3×C4#23714yes—
309C2×C3×C4#23914yes—
310C2×C3×C4C3×#4114yes—
311C2×C3×C4C3×#4114yes—
312C2×C3×C4C23×D318yes—
313C2×C3×C4C23×D312yes—
314C2×C3×C4C23×D312yes—
315C2×C3×C4C23×D312yes—
316C2×C3×C4C24×C318yes—
317C2×C3×C4C24×C312yes—
318C2×C3×C4C24×C312yes—
319C2×C3×C4C24×C312yes—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3C24×C314yes—
324C2×Dic3C24×C314yes—
325C2×Dic3C24×C314yes—
326D12—28no—
327C4×D3—216no—
328Dic6—28no—
329C2×C32#55614yescomplemented
330C2×C32C22×#4111yescharacteristic
331C2×C32C22×#4111yescharacteristic
332C2×C32C2×2−1+411yescharacteristic
333C3×D3—44no—
334C24—32no—
335C2×Q8—28no—
336C2×Q8—34no—
337C2×Q8—64no—
338C2×Q8C22×C3×D318yes—
339C2×D4—34no—
340C2×D4—64no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—38no—
345C22×C4—68no—
346C22×C4C22×C3×D314yes—
347C22×C4C22×C3×D311yescharacteristic
348C22×C4C22×C3×D311yescharacteristic
349C22×C4C22×C3×D311yescharacteristic
350C4⋊C4—34no—
351C4⋊C4—34no—
352C4⋊C4—316no—
353C4⋊C4—38no—
354C4⋊C4C22×C3×D318yes—
355C4⋊C4C22×C3×D318yes—
356C22⋊C4—316no—
357C22⋊C4—38no—
358C22⋊C4—316no—
359C42—316no—
360C42—38no—
361C42C22×C3×D314yes—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—21no—
366C22×C3—64no—
367C22×C3—64no—
368C22×C3—64no—
369C22×C3—128no—
370C22×C3C2×#23712yes—
371C22×C3C2×#23912yes—
372C22×C3#100712yes—
373C22×C3C2×C3×#4112yes—
374C22×C3C2×C3×#4112yes—
375C22×C3C3×2−1+412yes—
376C22×C3C24×D311yescharacteristic
377C22×C3C25×C311yescharacteristic
378C2×D3—24no—
379C2×D3—24no—
380C2×D3—24no—
381C2×D3—48no—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4—22no—
385C3×C4—28no—
386C3×C4—22no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—48no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—64no—
393C3×C4C2×#23714yes—
394C3×C4C2×#23914yes—
395C3×C4C2×C3×#4114yes—
396C3×C4C2×C3×#4114yes—
397Dic3—24no—
398Dic3—24no—
399Dic3—24no—
400C32C2×#55611yescharacteristic, complemented, Sylow
401C23—34no—
402C23—64no—
403C23—64no—
404C23—64no—
405C23C23×C3×D311yescharacteristic
406Q8—216no—
407Q8—68no—
408D4—68no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—34no—
412C2×C4—34no—
413C2×C4—34no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—64no—
417C2×C4—616no—
418C2×C4C3×#23714yes—
419C2×C4C3×#23914yes—
420C2×C4C23×C3×D318yes—
421C2×C4C23×C3×D312yes—
422C2×C4C23×C3×D312yes—
423C2×C4C23×C3×D312yes—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—24no—
428C2×C3—124no—
429C2×C3#573214yesdirect product factor
430C2×C3C3×#55614yescomplemented
431C2×C3C22×#23711yescharacteristic
432C2×C3C22×#23911yescharacteristic
433C2×C3C2×#100711yescharacteristic
434C2×C3C22×C3×#4111yescharacteristic
435C2×C3C22×C3×#4111yescharacteristic
436C2×C3C2×C3×2−1+411yescharacteristic
437D3—44no—
438C22—64no—
439C22—64no—
440C22—64no—
441C22—128no—
442C22C2×C3×#23712yes—
443C22C2×C3×#23912yes—
444C22C3×#100712yes—
445C22C24×C3×D311yescharacteristic
446C4—28no—
447C4—22no—
448C4—64no—
449C4—64no—
450C4—64no—
451C4C2×C3×#23714yes—
452C4C2×C3×#23914yes—
453C3—21no—
454C3C2×#573211yescharacteristic, direct product factor
455C3C2×C3×#55611yescharacteristic, complemented
456C2—124no—
457C2C3×#573214yesdirect product factor
458C2C22×C3×#23711yescharacteristic
459C2C22×C3×#23911yescharacteristic
460C2C2×C3×#100711yescharacteristic
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×#556
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
3. C2×C3×Q8×Dic3C2semidirect
4. C3×#5732C2direct (N × H)
5. C2×C3×#898C2semidirect
6. C2×C3×#897C2semidirect
7. C2×C3×#892C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5732C3direct (N × H)
16. C32×C4⋊Q8C22semidirect
21. C3×Q8×Dic3C22semidirect
31. C3×#898C22semidirect
32. C3×#897C22semidirect
33. C3×#892C22semidirect
35. C3×C4×Dic6C22semidirect
38. C2×C3×C4⋊Q8D3semidirect
39. C2×C3×C4⋊Q8C2×C3semidirect
48. C2×Q8×Dic3C2×C3semidirect
49. #5732C2×C3direct (N × H)
50. C2×#898C2×C3semidirect
51. C2×#897C2×C3semidirect
52. C2×#892C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
94. C3×C4⋊Q8C2×D3semidirect
95. C3×C4⋊Q8C22×C3semidirect
122. Q8×Dic3C22×C3semidirect
132. #898C22×C3semidirect
133. #897C22×C3semidirect
134. #892C22×C3semidirect
136. C4×Dic6C22×C3semidirect
161. C2×C4⋊Q8C3×D3semidirect
253. C4⋊Q8C2×C3×D3semidirect
329. C2×C32#556semidirect
400. C32C2×#556semidirect
429. C2×C3#5732direct (N × H)
430. C2×C3C3×#556semidirect
454. C3C2×#5732direct (N × H)
455. C3C2×C3×#556semidirect
457. C2C3×#5732direct (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×#9472
C2×C3×Q8×Dic32
C3×#57322
C2×C3×#8982
C2×C3×#8972
C2×C3×#8922
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×C3×#5563
C2×#57323