Finite Group Databasebeta

← C2×C3×#5555 #11902506

Subgroups — C2×C3×#5555 #11902506

Order 1152.

3477 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#237C212yesmaximal, complemented
2C2×C32×C4×D4C211yesmaximal, characteristic, complemented
3C2×C3×#930C212yesmaximal, complemented
4C22×C3×C4⋊Dic3C212yesmaximal, complemented
5C3×#5555C2116yesmaximal, direct product factor
6C2×C3×#897C212yesmaximal, complemented
7C2×C3×#886C214yesmaximal, complemented
8C2×C3×C4×D12C211yesmaximal, characteristic, complemented
9C2×C3×#869C211yesmaximal, characteristic, complemented
10C2×C3×#547—31nomaximal
11C2×#5555C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2212yes—
14C22×C3×#67C2214yes—
15C23×C3×Dic3C2214yes—
16C2×C3×#237—216no—
17C2×C3×#237C22116yescomplemented
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3C2212yes—
20C22×C3×Dic6C2212yes—
21C32×C4×D4C2218yescomplemented
22C2×C32×C4⋊C4C2211yescharacteristic
23C2×C32×C22⋊C4C2212yes—
24C2×C32×C42C2211yescharacteristic
25C3×#930C22116yescomplemented
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×(C2×D3)⋊C4C2214yes—
28C2×C3×C4⋊Dic3C2212yes—
29C2×C3×C4⋊Dic3C2214yes—
30C2×C3×C4⋊Dic3C2211yescharacteristic
31C2×C3×C4⋊Dic3C2212yes—
32C2×C3×C4⋊Dic3C22116yescomplemented
33C3×#897C22116yescomplemented
34C3×#886C22132yescomplemented
35C3×C4×D12C2218yescomplemented
36C3×#869C2218yescomplemented
37C22×C3×#41—32no—
38C22×#237C2×C312yescomplemented
39C3×#547—316no—
40C2×C3×C4⋊Q8—31no—
41C2×C3×#123—34no—
42C2×C3×C22⋊Q8—32no—
43C2×C3×#121—32no—
44C2×C3×C4×D4—21no—
45C2×C3×C4×D4—31no—
46C2×C3×C4×D4D311yescharacteristic, complemented
47C2×C3×C4×D4C2×C311yescharacteristic, complemented
48C22×C3×C4⋊C4—32no—
49C2×#930C2×C312yescomplemented
50C22×C4⋊Dic3C2×C312yescomplemented
51#5555C2×C3116yesdirect product factor
52C2×#897C2×C312yescomplemented
53C2×#886C2×C314yescomplemented
54C2×C4×D12C2×C311yescharacteristic, complemented
55C2×#869C2×C311yescharacteristic, complemented
56C24×C32C2312yes—
57C23×C3×D3C2312yes—
58C2×C32×D4D418yescomplemented
59C2×C32×D4C2314yes—
60C22×C32×C4—28no—
61C22×C32×C4C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2318yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—216no—
67C2×C3×#67C23116yes—
68C22×C3×Dic3—216no—
69C22×C3×Dic3C23116yes—
70C22×C3×Dic3C2314yes—
71C22×C3×Dic3C2312yes—
72C3×#237—264no—
73C2×C3×D12—24no—
74C2×C3×D12C2314yes—
75C2×C3×C4×D3—28no—
76C2×C3×C4×D3C2318yes—
77C2×C3×Dic6—28no—
78C2×C3×Dic6C2318yes—
79C32×C4⋊C4C2314yes—
80C32×C22⋊C4C2318yes—
81C32×C42C2314yes—
82C3×(C2×D3)⋊C4C2318yes—
83C3×(C2×D3)⋊C4C23116yes—
84C3×C4⋊Dic3C2318yes—
85C3×C4⋊Dic3C23116yes—
86C3×C4⋊Dic3C2314yes—
87C3×C4⋊Dic3C2318yes—
88C2×#547—31noSylow
89C2×C3×#41—316no—
90C2×C3×#41—616no—
91C22×C3×Q8—32no—
92C22×C3×D4—21no—
93C22×C3×D4—34no—
94C22×C3×D4—31no—
95C22×C3×D4C2×D311yescharacteristic
96C22×C3×D4C22×C311yescharacteristic
97C23×C3×C4—22no—
98C23×C3×C4—34no—
99C23×C3×C4—32no—
100C23×C3×C4C2×D312yes—
101C23×C3×C4C22×C312yes—
102C22×#67C22×C314yes—
103C23×Dic3C22×C314yes—
104C2×#237—216no—
105C2×#237C22×C3116yescomplemented
106C22×D12C22×C311yescharacteristic
107C22×C4×D3C22×C312yes—
108C22×Dic6C22×C312yes—
109C3×C4⋊Q8—38no—
110C3×#123—332no—
111C3×C22⋊Q8—316no—
112C3×#121—316no—
113C3×C4×D4—28no—
114C3×C4×D4—38no—
115C3×C4×D4C2×D318yescomplemented
116C3×C4×D4C22×C318yescomplemented
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—31no—
119C2×C3×C4⋊C4—32no—
120C2×C3×C4⋊C4—34no—
121C2×C3×C4⋊C4—32no—
122C2×C3×C4⋊C4—316no—
123C2×C3×C4⋊C4C2×D311yescharacteristic
124C2×C3×C4⋊C4C22×C311yescharacteristic
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4—32no—
127C2×C3×C22⋊C4—34no—
128C2×C3×C22⋊C4C2×D312yes—
129C2×C3×C22⋊C4C22×C312yes—
130C2×C3×C42—21no—
131C2×C3×C42C2×D311yescharacteristic
132C2×C3×C42C22×C311yescharacteristic
133#930C22×C3116yescomplemented
134C2×(C2×D3)⋊C4C22×C312yes—
135C2×(C2×D3)⋊C4C22×C314yes—
136C2×C4⋊Dic3C22×C312yes—
137C2×C4⋊Dic3C22×C312yes—
138C2×C4⋊Dic3C22×C311yescharacteristic
139C2×C4⋊Dic3C22×C314yes—
140C2×C4⋊Dic3C22×C3116yescomplemented
141#897C22×C3116yescomplemented
142#886C22×C3132yescomplemented
143C4×D12C22×C318yescomplemented
144#869C22×C318yescomplemented
145C23×C32—24no—
146C23×C32—24no—
147C23×C32C2×D418yes—
148C23×C32C2414yes—
149C23×C32C2411yescharacteristic
150C22×C3×D3—24no—
151C22×C3×D3—24no—
152C22×C3×D3—24no—
153C22×C3×D3C2414yes—
154C32×D4C2×D4116yescomplemented
155C2×C32×C4—24no—
156C2×C32×C4—22no—
157C2×C32×C4—216no—
158C2×C32×C4C2×D414yes—
159C2×C32×C4#4114yescomplemented
160C2×C32×C4C2412yes—
161C2×C32×C4C2414yes—
162C2×C32×C4C2412yes—
163C2×C32×C4C2412yes—
164C3×#67—232no—
165C2×C3×Dic3—28no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—232no—
168C2×C3×Dic3C2418yes—
169C2×C3×Dic3C2414yes—
170C3×D12—28no—
171C3×C4×D3—216no—
172C3×Dic6—216no—
173C22×#41—32no—
174#547—316no—
175C2×C4⋊Q8—31no—
176C2×#123—34no—
177C2×C22⋊Q8—32no—
178C2×#121—32no—
179C2×C4×D4—31no—
180C2×C4×D4C3×D311yescharacteristic, complemented
181C22×C4⋊C4—32no—
182C24×C3—22no—
183C24×C3—32no—
184C24×C3C22×D312yes—
185C24×C3C23×C312yes—
186C23×D3C23×C312yes—
187C3×#41—664no—
188C2×C3×Q8—38no—
189C2×C3×Q8—68no—
190C2×C3×D4—28no—
191C2×C3×D4—24no—
192C2×C3×D4—316no—
193C2×C3×D4—34no—
194C2×C3×D4—616no—
195C2×C3×D4—64no—
196C2×C3×D4D1218yescomplemented
197C2×C3×D4C3×D418yescomplemented
198C2×C3×D4C22×D314yes—
199C2×C3×D4C23×C314yes—
200C22×C3×C4—28no—
201C22×C3×C4—28no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—28no—
207C22×C3×C4—316no—
208C22×C3×C4—32no—
209C22×C3×C4—34no—
210C22×C3×C4—38no—
211C22×C3×C4—48no—
212C22×C3×C4—616no—
213C22×C3×C4—68no—
214C22×C3×C4C22×D318yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D312yes—
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C22×C3×C4C23×C312yes—
221C22×C3×C4C23×C318yes—
222C22×C3×C4C23×C311yescharacteristic
223C22×C3×C4C23×C311yescharacteristic
224C2×#67—216no—
225C2×#67C23×C3116yes—
226C22×Dic3—216no—
227C22×Dic3C23×C3116yes—
228C22×Dic3C23×C314yes—
229C22×Dic3C23×C312yes—
230#237—264no—
231C2×D12—24no—
232C2×D12C23×C314yes—
233C2×C4×D3—28no—
234C2×C4×D3C23×C318yes—
235C2×Dic6—28no—
236C2×Dic6C23×C318yes—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—34no—
239C3×C4⋊C4—38no—
240C3×C4⋊C4—316no—
241C3×C4⋊C4—38no—
242C3×C4⋊C4C22×D314yes—
243C3×C4⋊C4C23×C314yes—
244C3×C22⋊C4—28no—
245C3×C22⋊C4—38no—
246C3×C22⋊C4—316no—
247C3×C22⋊C4C22×D318yes—
248C3×C22⋊C4C23×C318yes—
249C3×C42—24no—
250C3×C42C22×D314yes—
251C3×C42C23×C314yes—
252(C2×D3)⋊C4C23×C318yes—
253(C2×D3)⋊C4C23×C3116yes—
254C4⋊Dic3C23×C318yes—
255C4⋊Dic3C23×C318yes—
256C4⋊Dic3C23×C314yes—
257C4⋊Dic3C23×C3116yes—
258C22×C32—216no—
259C22×C32—24no—
260C22×C32—24no—
261C22×C32C22×D418yes—
262C22×C32C22×D412yes—
263C22×C32C2×#4112yes—
264C22×C322−1+412yes—
265C22×C32C2511yescharacteristic
266C2×C3×D3—24no—
267C2×C3×D3—24no—
268C2×C3×D3—24no—
269C2×C3×D3—48no—
270C32×C4—24no—
271C32×C4—22no—
272C32×C4C22×D414yes—
273C32×C4C2×#4114yescomplemented
274C3×Dic3—28no—
275C3×Dic3—24no—
276C2×#41—316no—
277C2×#41—616no—
278C22×Q8—32no—
279C22×D4—34no—
280C22×D4—31no—
281C22×D4C2×C3×D311yescharacteristic
282C23×C4—34no—
283C23×C4—32no—
284C23×C4C2×C3×D312yes—
285C4⋊Q8—38no—
286#123—332no—
287C22⋊Q8—316no—
288#121—316no—
289C4×D4—38no—
290C4×D4C2×C3×D318yescomplemented
291C2×C4⋊C4—31no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—34no—
294C2×C4⋊C4—316no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4C2×C3×D311yescharacteristic
297C2×C22⋊C4—32no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4C2×C3×D312yes—
300C2×C42C2×C3×D311yescharacteristic
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—24no—
307C23×C3—28no—
308C23×C3—34no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—64no—
312C23×C3—64no—
313C23×C3—64no—
314C23×C3C2×D1218yes—
315C23×C3C2×C3×D418yes—
316C23×C3C23×D314yes—
317C23×C3C23×D311yescharacteristic
318C23×C3C24×C314yes—
319C23×C3C24×C311yescharacteristic
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3C24×C314yes—
324C3×Q8—616no—
325C3×D4—216no—
326C3×D4—632no—
327C3×D4—68no—
328C3×D4C2×D12116yescomplemented
329C3×D4C2×C3×D4116yescomplemented
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—216no—
335C2×C3×C4—216no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—34no—
343C2×C3×C4—38no—
344C2×C3×C4—416no—
345C2×C3×C4—42no—
346C2×C3×C4—44no—
347C2×C3×C4—64no—
348C2×C3×C4—68no—
349C2×C3×C4—632no—
350C2×C3×C4—616no—
351C2×C3×C4C2×D1214yes—
352C2×C3×C4#23714yescomplemented
353C2×C3×C4C2×C3×D414yes—
354C2×C3×C4C3×#4114yescomplemented
355C2×C3×C4C23×D312yes—
356C2×C3×C4C23×D314yes—
357C2×C3×C4C23×D312yes—
358C2×C3×C4C23×D312yes—
359C2×C3×C4C24×C312yes—
360C2×C3×C4C24×C314yes—
361C2×C3×C4C24×C312yes—
362C2×C3×C4C24×C312yes—
363#67—232no—
364C2×Dic3—28no—
365C2×Dic3—24no—
366C2×Dic3—232no—
367C2×Dic3C24×C318yes—
368C2×Dic3C24×C314yes—
369D12—28no—
370C4×D3—216no—
371Dic6—216no—
372C2×C32—28no—
373C2×C32#54714yescomplemented
374C2×C32C23×D411yescharacteristic
375C2×C32C22×#4111yescharacteristic
376C2×C32C2×2−1+411yescharacteristic
377C3×D3—44no—
378C24—32no—
379C24C22×C3×D312yes—
380#41—664no—
381C2×Q8—38no—
382C2×Q8—68no—
383C2×D4—316no—
384C2×D4—34no—
385C2×D4—616no—
386C2×D4—64no—
387C2×D4C3×D1218yescomplemented
388C2×D4C22×C3×D314yes—
389C22×C4—28no—
390C22×C4—32no—
391C22×C4—34no—
392C22×C4—316no—
393C22×C4—38no—
394C22×C4—616no—
395C22×C4—68no—
396C22×C4C22×C3×D311yescharacteristic
397C22×C4C22×C3×D312yes—
398C22×C4C22×C3×D318yes—
399C22×C4C22×C3×D311yescharacteristic
400C22×C4C22×C3×D311yescharacteristic
401C4⋊C4—34no—
402C4⋊C4—38no—
403C4⋊C4—316no—
404C4⋊C4—38no—
405C4⋊C4C22×C3×D314yes—
406C22⋊C4—38no—
407C22⋊C4—316no—
408C22⋊C4C22×C3×D318yes—
409C42C22×C3×D314yes—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—216no—
415C22×C3—216no—
416C22×C3—22no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—44no—
422C22×C3—416no—
423C22×C3—44no—
424C22×C3—64no—
425C22×C3—64no—
426C22×C3—64no—
427C22×C3—128no—
428C22×C3C22×D1218yes—
429C22×C3C22×D1212yes—
430C22×C3C2×#23712yes—
431C22×C3#101012yes—
432C22×C3C22×C3×D418yes—
433C22×C3C22×C3×D412yes—
434C22×C3C2×C3×#4112yes—
435C22×C3C3×2−1+412yes—
436C22×C3C24×D311yescharacteristic
437C22×C3C25×C311yescharacteristic
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—24no—
441C2×D3—48no—
442C3×C4—24no—
443C3×C4—22no—
444C3×C4—24no—
445C3×C4—22no—
446C3×C4—24no—
447C3×C4—24no—
448C3×C4—44no—
449C3×C4—42no—
450C3×C4—64no—
451C3×C4—68no—
452C3×C4C22×D1214yes—
453C3×C4C2×#23714yescomplemented
454C3×C4C22×C3×D414yes—
455C3×C4C2×C3×#4114yescomplemented
456Dic3—28no—
457Dic3—24no—
458C32C2×#54711yescharacteristic, complemented, Sylow
459C23—24no—
460C23—24no—
461C23—34no—
462C23—64no—
463C23—64no—
464C23—64no—
465C23C2×C3×D1218yes—
466C23C23×C3×D314yes—
467C23C23×C3×D311yescharacteristic
468Q8—616no—
469D4—632no—
470D4—68no—
471D4C2×C3×D12116yescomplemented
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—216no—
475C2×C4—34no—
476C2×C4—38no—
477C2×C4—64no—
478C2×C4—68no—
479C2×C4—632no—
480C2×C4—616no—
481C2×C4C2×C3×D1214yes—
482C2×C4C3×#23714yescomplemented
483C2×C4C23×C3×D312yes—
484C2×C4C23×C3×D314yes—
485C2×C4C23×C3×D312yes—
486C2×C4C23×C3×D312yes—
487C2×C3—28no—
488C2×C3—28no—
489C2×C3—24no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—48no—
494C2×C3—124no—
495C2×C3#555514yesdirect product factor
496C2×C3C3×#54714yescomplemented
497C2×C3C23×D1211yescharacteristic
498C2×C3C22×#23711yescharacteristic
499C2×C3C2×#101011yescharacteristic
500C2×C3C23×C3×D411yescharacteristic
501C2×C3C22×C3×#4111yescharacteristic
502C2×C3C2×C3×2−1+411yescharacteristic
503D3—44no—
504C22—216no—
505C22—24no—
506C22—24no—
507C22—64no—
508C22—64no—
509C22—64no—
510C22—128no—
511C22C22×C3×D1218yes—
512C22C22×C3×D1212yes—
513C22C2×C3×#23712yes—
514C22C3×#101012yes—
515C22C24×C3×D311yescharacteristic
516C4—24no—
517C4—22no—
518C4—64no—
519C4—68no—
520C4C22×C3×D1214yes—
521C4C2×C3×#23714yescomplemented
522C3—21no—
523C3C2×#555511yescharacteristic, direct product factor
524C3C2×C3×#54711yescharacteristic, complemented
525C2—28no—
526C2—124no—
527C2C3×#555514yesdirect product factor
528C2C23×C3×D1211yescharacteristic
529C2C22×C3×#23711yescharacteristic
530C2C2×C3×#101011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C24
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#547
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#237C2semidirect
2. C2×C32×C4×D4C2semidirect
3. C2×C3×#930C2semidirect
4. C22×C3×C4⋊Dic3C2semidirect
5. C3×#5555C2direct (N × H)
6. C2×C3×#897C2semidirect
7. C2×C3×#886C2semidirect
8. C2×C3×C4×D12C2semidirect
9. C2×C3×#869C2semidirect
11. C2×#5555C3direct (N × H)
17. C2×C3×#237C22semidirect
21. C32×C4×D4C22semidirect
25. C3×#930C22semidirect
32. C2×C3×C4⋊Dic3C22semidirect
33. C3×#897C22semidirect
34. C3×#886C22semidirect
35. C3×C4×D12C22semidirect
36. C3×#869C22semidirect
38. C22×#237C2×C3semidirect
46. C2×C3×C4×D4D3semidirect
47. C2×C3×C4×D4C2×C3semidirect
49. C2×#930C2×C3semidirect
50. C22×C4⋊Dic3C2×C3semidirect
51. #5555C2×C3direct (N × H)
52. C2×#897C2×C3semidirect
53. C2×#886C2×C3semidirect
54. C2×C4×D12C2×C3semidirect
55. C2×#869C2×C3semidirect
58. C2×C32×D4D4semidirect
105. C2×#237C22×C3semidirect
115. C3×C4×D4C2×D3semidirect
116. C3×C4×D4C22×C3semidirect
133. #930C22×C3semidirect
140. C2×C4⋊Dic3C22×C3semidirect
141. #897C22×C3semidirect
142. #886C22×C3semidirect
143. C4×D12C22×C3semidirect
144. #869C22×C3semidirect
154. C32×D4C2×D4semidirect
159. C2×C32×C4#41semidirect
180. C2×C4×D4C3×D3semidirect
196. C2×C3×D4D12semidirect
197. C2×C3×D4C3×D4semidirect
273. C32×C4C2×#41semidirect
290. C4×D4C2×C3×D3semidirect
328. C3×D4C2×D12semidirect
329. C3×D4C2×C3×D4semidirect
352. C2×C3×C4#237semidirect
354. C2×C3×C4C3×#41semidirect
373. C2×C32#547semidirect
387. C2×D4C3×D12semidirect
453. C3×C4C2×#237semidirect
455. C3×C4C2×C3×#41semidirect
458. C32C2×#547semidirect
471. D4C2×C3×D12semidirect
482. C2×C4C3×#237semidirect
495. C2×C3#5555direct (N × H)
496. C2×C3C3×#547semidirect
521. C4C2×C3×#237semidirect
523. C3C2×#5555direct (N × H)
524. C3C2×C3×#547semidirect
527. C2C3×#5555direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2372
C2×C32×C4×D42
C2×C3×#9302
C22×C3×C4⋊Dic32
C3×#55552
C2×C3×#8972
C2×C3×#8862
C2×C3×C4×D122
C2×C3×#8692
C2×C3×#5473
C2×#55553