Finite Group Databasebeta

← C2×C3×#5726 #11902503

Subgroups — C2×C3×#5726 #11902503

Order 1152.

3409 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C212yesmaximal, complemented
2C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#946C212yesmaximal, complemented
4C3×#5726C2116yesmaximal, direct product factor
5C2×C3×#895C212yesmaximal, complemented
6C2×C3×#894C214yesmaximal, complemented
7C2×C3×D3×C4⋊C4C212yesmaximal, complemented
8C2×C3×C4×D12C211yesmaximal, characteristic, complemented
9C2×C3×C4×D12C211yesmaximal, characteristic, complemented
10C2×C3×#547—31nomaximal
11C2×#5726C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2212yes—
13C2×C3×#239—216no—
14C2×C3×#239C22116yescomplemented
15C22×C3×D12C2214yes—
16C22×C3×D12C2211yescharacteristic
17C22×C3×D12C2211yescharacteristic
18C22×C3×C4×D3C2214yes—
19C22×C3×C4×D3C2212yes—
20C22×C3×C4×D3C2212yes—
21C32×C4⋊Q8C2218yescomplemented
22C2×C32×C4⋊C4C2212yes—
23C2×C32×C4⋊C4C2212yes—
24C2×C32×C42C2211yescharacteristic
25C3×#946C22116yescomplemented
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×(C2×D3)⋊C4C2212yes—
28C2×C3×(C2×D3)⋊C4C2214yes—
29C2×C3×C4⋊Dic3C2211yescharacteristic
30C2×C3×C4⋊Dic3C2211yescharacteristic
31C2×C3×Dic3⋊C4C2214yes—
32C3×#895C22116yescomplemented
33C3×#894C22132yescomplemented
34C3×D3×C4⋊C4C22116yescomplemented
35C3×C4×D12C2218yescomplemented
36C3×C4×D12C2218yescomplemented
37C22×C3×#41—32no—
38C22×#239C2×C312yescomplemented
39C3×#547—316no—
40C2×C3×C4⋊Q8—21no—
41C2×C3×C4⋊Q8D311yescharacteristic, complemented
42C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
43C2×C3×#123—34no—
44C2×C3×C22⋊Q8—32no—
45C2×C3×#121—32no—
46C2×C3×C4×D4—31no—
47C2×C3×C4×D4—31no—
48C22×C3×C4⋊C4—32no—
49C2×#946C2×C312yescomplemented
50#5726C2×C3116yesdirect product factor
51C2×#895C2×C312yescomplemented
52C2×#894C2×C314yescomplemented
53C2×D3×C4⋊C4C2×C312yescomplemented
54C2×C4×D12C2×C311yescharacteristic, complemented
55C2×C4×D12C2×C311yescharacteristic, complemented
56C23×C3×D3C2312yes—
57C23×C3×D3C2312yes—
58C2×C32×Q8—28no—
59C2×C32×Q8C2318yes—
60C22×C32×C4C2314yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C3×#239—264no—
67C2×C3×D12—216no—
68C2×C3×D12—24no—
69C2×C3×D12D418yescomplemented
70C2×C3×D12C23116yes—
71C2×C3×D12C2314yes—
72C2×C3×D12C2314yes—
73C2×C3×C4×D3—216no—
74C2×C3×C4×D3—28no—
75C2×C3×C4×D3—28no—
76C2×C3×C4×D3C23116yes—
77C2×C3×C4×D3C2318yes—
78C2×C3×C4×D3C2318yes—
79C32×C4⋊C4C2318yes—
80C32×C4⋊C4C2318yes—
81C32×C42C2314yes—
82C3×(C2×D3)⋊C4C2318yes—
83C3×(C2×D3)⋊C4C2318yes—
84C3×(C2×D3)⋊C4C23116yes—
85C3×C4⋊Dic3C2314yes—
86C3×C4⋊Dic3C2314yes—
87C3×Dic3⋊C4C23116yes—
88C2×#547—31noSylow
89C2×C3×#41—316no—
90C2×C3×#41—616no—
91C22×C3×Q8—22no—
92C22×C3×Q8C2×D312yes—
93C22×C3×Q8C22×C312yes—
94C22×C3×D4—31no—
95C22×C3×D4—34no—
96C22×C3×D4—31no—
97C23×C3×C4—34no—
98C23×C3×C4—32no—
99C23×C3×C4—32no—
100C2×#239—216no—
101C2×#239C22×C3116yescomplemented
102C22×D12C22×C314yes—
103C22×D12C22×C311yescharacteristic
104C22×D12C22×C311yescharacteristic
105C22×C4×D3C22×C312yes—
106C22×C4×D3C22×C314yes—
107C22×C4×D3C22×C312yes—
108C3×C4⋊Q8—28no—
109C3×C4⋊Q8C2×D318yescomplemented
110C3×C4⋊Q8C22×C318yescomplemented
111C3×#123—332no—
112C3×C22⋊Q8—316no—
113C3×#121—316no—
114C3×C4×D4—38no—
115C3×C4×D4—38no—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4—34no—
119C2×C3×C4⋊C4—31no—
120C2×C3×C4⋊C4—31no—
121C2×C3×C4⋊C4—316no—
122C2×C3×C4⋊C4C2×D312yes—
123C2×C3×C4⋊C4C2×D312yes—
124C2×C3×C4⋊C4C22×C312yes—
125C2×C3×C4⋊C4C22×C312yes—
126C2×C3×C22⋊C4—32no—
127C2×C3×C22⋊C4—32no—
128C2×C3×C22⋊C4—34no—
129C2×C3×C42—21no—
130C2×C3×C42C2×D311yescharacteristic
131C2×C3×C42C22×C311yescharacteristic
132#946C22×C3116yescomplemented
133C2×(C2×D3)⋊C4C22×C312yes—
134C2×(C2×D3)⋊C4C22×C312yes—
135C2×(C2×D3)⋊C4C22×C314yes—
136C2×C4⋊Dic3C22×C311yescharacteristic
137C2×C4⋊Dic3C22×C311yescharacteristic
138C2×Dic3⋊C4C22×C314yes—
139#895C22×C3116yescomplemented
140#894C22×C3132yescomplemented
141D3×C4⋊C4C22×C3116yescomplemented
142C4×D12C22×C318yescomplemented
143C4×D12C22×C318yescomplemented
144C23×C32C2411yescharacteristic
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—24no—
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3C2×D418yes—
151C22×C3×D3C2414yes—
152C22×C3×D3C2414yes—
153C32×Q8—216no—
154C2×C32×C4—22no—
155C2×C32×C4—28no—
156C2×C32×C4C2×D414yes—
157C2×C32×C4#4114yescomplemented
158C2×C32×C4C2418yes—
159C2×C32×C4C2412yes—
160C2×C32×C4C2412yes—
161C2×C32×C4C2412yes—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3C2414yes—
165C2×C3×Dic3C2414yes—
166C3×D12—232no—
167C3×D12—28no—
168C3×D12C2×D4116yescomplemented
169C3×C4×D3—232no—
170C3×C4×D3—216no—
171C3×C4×D3—216no—
172C22×#41—32no—
173#547—316no—
174C2×C4⋊Q8C3×D311yescharacteristic, complemented
175C2×#123—34no—
176C2×C22⋊Q8—32no—
177C2×#121—32no—
178C2×C4×D4—31no—
179C2×C4×D4—31no—
180C22×C4⋊C4—32no—
181C24×C3—32no—
182C24×C3—32no—
183C23×D3C23×C312yes—
184C23×D3C23×C312yes—
185C3×#41—664no—
186C2×C3×Q8—28no—
187C2×C3×Q8—28no—
188C2×C3×Q8—28no—
189C2×C3×Q8—48no—
190C2×C3×Q8C22×D318yes—
191C2×C3×Q8C23×C318yes—
192C2×C3×D4—38no—
193C2×C3×D4—34no—
194C2×C3×D4—34no—
195C2×C3×D4—316no—
196C2×C3×D4—616no—
197C2×C3×D4—64no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—24no—
202C22×C3×C4—32no—
203C22×C3×C4—32no—
204C22×C3×C4—38no—
205C22×C3×C4—316no—
206C22×C3×C4—38no—
207C22×C3×C4—616no—
208C22×C3×C4—68no—
209C22×C3×C4—68no—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D314yes—
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C23×C314yes—
215C22×C3×C4C23×C311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C22×C3×C4C23×C311yescharacteristic
218C22×Dic3C23×C312yes—
219C22×Dic3C23×C312yes—
220#239—264no—
221C2×D12—216no—
222C2×D12—24no—
223C2×D12C3×D418yescomplemented
224C2×D12C23×C314yes—
225C2×D12C23×C3116yes—
226C2×D12C23×C314yes—
227C2×C4×D3—216no—
228C2×C4×D3—28no—
229C2×C4×D3—28no—
230C2×C4×D3C23×C318yes—
231C2×C4×D3C23×C318yes—
232C2×C4×D3C23×C3116yes—
233C3×C4⋊C4—28no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—316no—
236C3×C4⋊C4—34no—
237C3×C4⋊C4—34no—
238C3×C4⋊C4C22×D318yes—
239C3×C4⋊C4C22×D318yes—
240C3×C4⋊C4C23×C318yes—
241C3×C4⋊C4C23×C318yes—
242C3×C22⋊C4—38no—
243C3×C22⋊C4—38no—
244C3×C22⋊C4—316no—
245C3×C42—24no—
246C3×C42C22×D314yes—
247C3×C42C23×C314yes—
248(C2×D3)⋊C4C23×C318yes—
249(C2×D3)⋊C4C23×C318yes—
250(C2×D3)⋊C4C23×C3116yes—
251C4⋊Dic3C23×C314yes—
252C4⋊Dic3C23×C314yes—
253Dic3⋊C4C23×C3116yes—
254C22×C32C22×D412yes—
255C22×C32C2×#4112yes—
256C22×C322−1+412yes—
257C22×C32C2511yescharacteristic
258C2×C3×D3—216no—
259C2×C3×D3—24no—
260C2×C3×D3—24no—
261C2×C3×D3—24no—
262C2×C3×D3—24no—
263C2×C3×D3—24no—
264C2×C3×D3—48no—
265C2×C3×D3C22×D418yes—
266C32×C4—22no—
267C32×C4—28no—
268C32×C4C22×D414yes—
269C32×C4C2×#4114yescomplemented
270C3×Dic3—24no—
271C3×Dic3—24no—
272C2×#41—316no—
273C2×#41—616no—
274C22×Q8C2×C3×D312yes—
275C22×D4—31no—
276C22×D4—31no—
277C22×D4—34no—
278C23×C4—32no—
279C23×C4—34no—
280C23×C4—32no—
281C4⋊Q8C2×C3×D318yescomplemented
282#123—332no—
283C22⋊Q8—316no—
284#121—316no—
285C4×D4—38no—
286C4×D4—38no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—34no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—316no—
291C2×C4⋊C4C2×C3×D312yes—
292C2×C4⋊C4C2×C3×D312yes—
293C2×C22⋊C4—32no—
294C2×C22⋊C4—32no—
295C2×C22⋊C4—34no—
296C2×C42C2×C3×D311yescharacteristic
297C23×C3—21no—
298C23×C3—38no—
299C23×C3—34no—
300C23×C3—34no—
301C23×C3—64no—
302C23×C3—64no—
303C23×C3—64no—
304C23×C3—64no—
305C23×C3—64no—
306C23×C3C23×D311yescharacteristic
307C23×C3C24×C311yescharacteristic
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3C2×C3×D418yes—
314C22×D3C24×C314yes—
315C22×D3C24×C314yes—
316C3×Q8—216no—
317C3×Q8—216no—
318C3×Q8—416no—
319C3×D4—316no—
320C3×D4—632no—
321C3×D4—68no—
322C2×C3×C4—24no—
323C2×C3×C4—28no—
324C2×C3×C4—22no—
325C2×C3×C4—28no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—28no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—48no—
336C2×C3×C4—632no—
337C2×C3×C4—64no—
338C2×C3×C4—64no—
339C2×C3×C4—616no—
340C2×C3×C4—616no—
341C2×C3×C4D3×D414yes—
342C2×C3×C4#23914yescomplemented
343C2×C3×C4C2×C3×D414yes—
344C2×C3×C4C3×#4114yescomplemented
345C2×C3×C4C23×D312yes—
346C2×C3×C4C23×D318yes—
347C2×C3×C4C23×D312yes—
348C2×C3×C4C23×D312yes—
349C2×C3×C4C24×C318yes—
350C2×C3×C4C24×C312yes—
351C2×C3×C4C24×C312yes—
352C2×C3×C4C24×C312yes—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3C24×C314yes—
356C2×Dic3C24×C314yes—
357D12—232no—
358D12—28no—
359D12C2×C3×D4116yescomplemented
360C4×D3—232no—
361C4×D3—216no—
362C4×D3—216no—
363C2×C32#54714yescomplemented
364C2×C32C23×D411yescharacteristic
365C2×C32C22×#4111yescharacteristic
366C2×C32C2×2−1+411yescharacteristic
367C3×D3—28no—
368C3×D3—44no—
369C24—32no—
370C24—32no—
371#41—664no—
372C2×Q8—28no—
373C2×Q8C22×C3×D318yes—
374C2×D4—34no—
375C2×D4—34no—
376C2×D4—316no—
377C2×D4—38no—
378C2×D4—616no—
379C2×D4—64no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—38no—
383C22×C4—316no—
384C22×C4—38no—
385C22×C4—616no—
386C22×C4—68no—
387C22×C4—68no—
388C22×C4C22×C3×D311yescharacteristic
389C22×C4C22×C3×D314yes—
390C22×C4C22×C3×D311yescharacteristic
391C22×C4C22×C3×D311yescharacteristic
392C4⋊C4—34no—
393C4⋊C4—316no—
394C4⋊C4—34no—
395C4⋊C4C22×C3×D318yes—
396C4⋊C4C22×C3×D318yes—
397C22⋊C4—38no—
398C22⋊C4—38no—
399C22⋊C4—316no—
400C42C22×C3×D314yes—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—38no—
406C22×C3—64no—
407C22×C3—64no—
408C22×C3—616no—
409C22×C3—64no—
410C22×C3—64no—
411C22×C3—64no—
412C22×C3—128no—
413C22×C3C2×D3×D412yes—
414C22×C3C2×#23912yes—
415C22×C3#100712yes—
416C22×C3C22×C3×D412yes—
417C22×C3C2×C3×#4112yes—
418C22×C3C3×2−1+412yes—
419C22×C3C24×D311yescharacteristic
420C22×C3C25×C311yescharacteristic
421C2×D3—24no—
422C2×D3—24no—
423C2×D3—216no—
424C2×D3—24no—
425C2×D3—24no—
426C2×D3—24no—
427C2×D3—48no—
428C2×D3C22×C3×D418yes—
429C3×C4—24no—
430C3×C4—28no—
431C3×C4—22no—
432C3×C4—28no—
433C3×C4—22no—
434C3×C4—24no—
435C3×C4—24no—
436C3×C4—48no—
437C3×C4—64no—
438C3×C4—64no—
439C3×C4C2×D3×D414yes—
440C3×C4C2×#23914yescomplemented
441C3×C4C22×C3×D414yes—
442C3×C4C2×C3×#4114yescomplemented
443Dic3—24no—
444Dic3—24no—
445C32C2×#54711yescharacteristic, complemented, Sylow
446C23—38no—
447C23—34no—
448C23—34no—
449C23—64no—
450C23—64no—
451C23—64no—
452C23—64no—
453C23—64no—
454C23C23×C3×D311yescharacteristic
455Q8—216no—
456D4—316no—
457D4—632no—
458D4—68no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—34no—
462C2×C4—34no—
463C2×C4—632no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—616no—
467C2×C4—616no—
468C2×C4C3×D3×D414yes—
469C2×C4C3×#23914yescomplemented
470C2×C4C23×C3×D312yes—
471C2×C4C23×C3×D318yes—
472C2×C4C23×C3×D312yes—
473C2×C4C23×C3×D312yes—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—24no—
478C2×C3—68no—
479C2×C3—124no—
480C2×C3#572614yesdirect product factor
481C2×C3C3×#54714yescomplemented
482C2×C3C22×D3×D411yescharacteristic
483C2×C3C22×#23911yescharacteristic
484C2×C3C2×#100711yescharacteristic
485C2×C3C23×C3×D411yescharacteristic
486C2×C3C22×C3×#4111yescharacteristic
487C2×C3C2×C3×2−1+411yescharacteristic
488D3—28no—
489D3—44no—
490C22—38no—
491C22—64no—
492C22—64no—
493C22—616no—
494C22—64no—
495C22—64no—
496C22—64no—
497C22—128no—
498C22C2×C3×D3×D412yes—
499C22C2×C3×#23912yes—
500C22C3×#100712yes—
501C22C24×C3×D311yescharacteristic
502C4—28no—
503C4—22no—
504C4—64no—
505C4—64no—
506C4C2×C3×D3×D414yes—
507C4C2×C3×#23914yescomplemented
508C3—21no—
509C3C2×#572611yescharacteristic, direct product factor
510C3C2×C3×#54711yescharacteristic, complemented
511C2—68no—
512C2—124no—
513C2C3×#572614yesdirect product factor
514C2C22×C3×D3×D411yescharacteristic
515C2C22×C3×#23911yescharacteristic
516C2C2×C3×#100711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC23×C32C24
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C2×C3×#547
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. C22×C3×#239C2semidirect
2. C2×C32×C4⋊Q8C2semidirect
3. C2×C3×#946C2semidirect
4. C3×#5726C2direct (N × H)
5. C2×C3×#895C2semidirect
6. C2×C3×#894C2semidirect
7. C2×C3×D3×C4⋊C4C2semidirect
8. C2×C3×C4×D12C2semidirect
9. C2×C3×C4×D12C2semidirect
11. C2×#5726C3direct (N × H)
14. C2×C3×#239C22semidirect
21. C32×C4⋊Q8C22semidirect
25. C3×#946C22semidirect
32. C3×#895C22semidirect
33. C3×#894C22semidirect
34. C3×D3×C4⋊C4C22semidirect
35. C3×C4×D12C22semidirect
36. C3×C4×D12C22semidirect
38. C22×#239C2×C3semidirect
41. C2×C3×C4⋊Q8D3semidirect
42. C2×C3×C4⋊Q8C2×C3semidirect
49. C2×#946C2×C3semidirect
50. #5726C2×C3direct (N × H)
51. C2×#895C2×C3semidirect
52. C2×#894C2×C3semidirect
53. C2×D3×C4⋊C4C2×C3semidirect
54. C2×C4×D12C2×C3semidirect
55. C2×C4×D12C2×C3semidirect
69. C2×C3×D12D4semidirect
101. C2×#239C22×C3semidirect
109. C3×C4⋊Q8C2×D3semidirect
110. C3×C4⋊Q8C22×C3semidirect
132. #946C22×C3semidirect
139. #895C22×C3semidirect
140. #894C22×C3semidirect
141. D3×C4⋊C4C22×C3semidirect
142. C4×D12C22×C3semidirect
143. C4×D12C22×C3semidirect
157. C2×C32×C4#41semidirect
168. C3×D12C2×D4semidirect
174. C2×C4⋊Q8C3×D3semidirect
223. C2×D12C3×D4semidirect
269. C32×C4C2×#41semidirect
281. C4⋊Q8C2×C3×D3semidirect
342. C2×C3×C4#239semidirect
344. C2×C3×C4C3×#41semidirect
359. D12C2×C3×D4semidirect
363. C2×C32#547semidirect
440. C3×C4C2×#239semidirect
442. C3×C4C2×C3×#41semidirect
445. C32C2×#547semidirect
469. C2×C4C3×#239semidirect
480. C2×C3#5726direct (N × H)
481. C2×C3C3×#547semidirect
507. C4C2×C3×#239semidirect
509. C3C2×#5726direct (N × H)
510. C3C2×C3×#547semidirect
513. C2C3×#5726direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C32×C4⋊Q82
C2×C3×#9462
C3×#57262
C2×C3×#8952
C2×C3×#8942
C2×C3×D3×C4⋊C42
C2×C3×C4×D122
C2×C3×#5473
C2×#57263