Finite Group Databasebeta

← C2×C3×#5553 #11902702

Subgroups — C2×C3×#5553 #11902702

Order 1152.

3685 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C212yesmaximal, complemented
2C2×C32×C4×D4C211yesmaximal, characteristic, complemented
3C2×C3×#940C212yesmaximal, complemented
4C2×C3×#930C212yesmaximal, complemented
5C3×#5553C2116yesmaximal, direct product factor
6C2×C3×#893C212yesmaximal, complemented
7C2×C3×#884C214yesmaximal, complemented
8C2×C3×#874C211yesmaximal, characteristic, complemented
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#550—31nomaximal
11C2×#5553C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2212yes—
14C22×C3×#67C2214yes—
15C22×C3×#67C2214yes—
16C2×C3×#235—216no—
17C2×C3×#235C22116yescomplemented
18C22×C3×D12C2212yes—
19C22×C3×D12C2212yes—
20C22×C3×D12C2212yes—
21C22×C3×C4×D3C2214yes—
22C22×C3×Dic6C2211yescharacteristic
23C32×C4×D4C2218yescomplemented
24C2×C32×C4⋊C4C2211yescharacteristic
25C2×C32×C22⋊C4C2212yes—
26C2×C32×C42C2211yescharacteristic
27C3×#940C22116yescomplemented
28C3×#930C22116yescomplemented
29C2×C3×(C2×D3)⋊C4C2214yes—
30C2×C3×C4⋊Dic3C2211yescharacteristic
31C2×C3×Dic3⋊C4C2212yes—
32C2×C3×C4×Dic3C2212yes—
33C3×#893C22116yescomplemented
34C3×#884C22132yescomplemented
35C3×#874C2218yescomplemented
36C3×C4×Dic6C2218yescomplemented
37C22×C3×#41—32no—
38C22×#235C2×C312yescomplemented
39C3×#550—316no—
40C2×C3×#127—32no—
41C2×C3×#127—31no—
42C2×C3×#121—32no—
43C2×C3×#121—34no—
44C2×C3×C4×Q8—31no—
45C2×C3×C4×D4—21no—
46C2×C3×C4×D4—32no—
47C2×C3×C4×D4D311yescharacteristic, complemented
48C2×C3×C4×D4C2×C311yescharacteristic, complemented
49C2×#940C2×C312yescomplemented
50C2×#930C2×C312yescomplemented
51#5553C2×C3116yesdirect product factor
52C2×#893C2×C312yescomplemented
53C2×#884C2×C314yescomplemented
54C2×#874C2×C311yescharacteristic, complemented
55C2×C4×Dic6C2×C311yescharacteristic, complemented
56C24×C32C2312yes—
57C23×C3×D3C2314yes—
58C2×C32×D4—24no—
59C2×C32×D4C2314yes—
60C22×C32×C4—28no—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2312yes—
63C22×C32×C4C2318yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—216no—
67C2×C3×#67—48no—
68C2×C3×#67C23116yes—
69C2×C3×#67C23116yes—
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C3×#235—264no—
73C2×C3×D12—28no—
74C2×C3×D12—28no—
75C2×C3×D12—44no—
76C2×C3×D12C2318yes—
77C2×C3×D12C2318yes—
78C2×C3×D12C2318yes—
79C2×C3×C4×D3—216no—
80C2×C3×C4×D3C23116yes—
81C2×C3×Dic6D418yescomplemented
82C2×C3×Dic6C2314yes—
83C32×C4⋊C4C2314yes—
84C32×C22⋊C4C2318yes—
85C32×C42C2314yes—
86C3×(C2×D3)⋊C4C23116yes—
87C3×C4⋊Dic3C2314yes—
88C3×Dic3⋊C4C2318yes—
89C3×C4×Dic3C2318yes—
90C2×#550—31noSylow
91C2×C3×#41—316no—
92C2×C3×#41—616no—
93C22×C3×Q8—31no—
94C22×C3×D4—21no—
95C22×C3×D4—34no—
96C22×C3×D4—34no—
97C22×C3×D4—32no—
98C22×C3×D4—32no—
99C22×C3×D4—32no—
100C22×C3×D4C2×D311yescharacteristic
101C22×C3×D4C22×C311yescharacteristic
102C23×C3×C4—22no—
103C23×C3×C4—34no—
104C23×C3×C4C2×D312yes—
105C23×C3×C4C22×C312yes—
106C22×#67C22×C314yes—
107C22×#67C22×C314yes—
108C2×#235—216no—
109C2×#235C22×C3116yescomplemented
110C22×D12C22×C312yes—
111C22×D12C22×C312yes—
112C22×D12C22×C312yes—
113C22×C4×D3C22×C314yes—
114C22×Dic6C22×C311yescharacteristic
115C3×#127—316no—
116C3×#127—38no—
117C3×#121—316no—
118C3×#121—332no—
119C3×C4×Q8—38no—
120C3×C4×D4—28no—
121C3×C4×D4—316no—
122C3×C4×D4C2×D318yescomplemented
123C3×C4×D4C22×C318yescomplemented
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4—32no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C4⋊C4C2×D311yescharacteristic
128C2×C3×C4⋊C4C22×C311yescharacteristic
129C2×C3×C22⋊C4—22no—
130C2×C3×C22⋊C4—34no—
131C2×C3×C22⋊C4C2×D312yes—
132C2×C3×C22⋊C4C22×C312yes—
133C2×C3×C42—21no—
134C2×C3×C42—32no—
135C2×C3×C42C2×D311yescharacteristic
136C2×C3×C42C22×C311yescharacteristic
137#940C22×C3116yescomplemented
138#930C22×C3116yescomplemented
139C2×(C2×D3)⋊C4C22×C314yes—
140C2×C4⋊Dic3C22×C311yescharacteristic
141C2×Dic3⋊C4C22×C312yes—
142C2×C4×Dic3C22×C312yes—
143#893C22×C3116yescomplemented
144#884C22×C3132yescomplemented
145#874C22×C318yescomplemented
146C4×Dic6C22×C318yescomplemented
147C23×C32—24no—
148C23×C32—24no—
149C23×C32—24no—
150C23×C32C2414yes—
151C23×C32C2411yescharacteristic
152C22×C3×D3—28no—
153C22×C3×D3—28no—
154C22×C3×D3—28no—
155C22×C3×D3C2418yes—
156C32×D4—28no—
157C2×C32×C4—24no—
158C2×C32×C4—22no—
159C2×C32×C4—216no—
160C2×C32×C4C2×D414yes—
161C2×C32×C4#4114yescomplemented
162C2×C32×C4C2412yes—
163C2×C32×C4C2414yes—
164C2×C32×C4C2412yes—
165C2×C32×C4C2412yes—
166C3×#67—232no—
167C3×#67—416no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3C2×D418yes—
170C2×C3×Dic3C2414yes—
171C2×C3×Dic3C2414yes—
172C3×D12—216no—
173C3×D12—216no—
174C3×D12—48no—
175C3×C4×D3—232no—
176C3×Dic6C2×D4116yescomplemented
177C22×#41—32no—
178#550—316no—
179C2×#127—32no—
180C2×#127—31no—
181C2×#121—32no—
182C2×#121—34no—
183C2×C4×Q8—31no—
184C2×C4×D4—32no—
185C2×C4×D4C3×D311yescharacteristic, complemented
186C24×C3—22no—
187C24×C3—34no—
188C24×C3C22×D312yes—
189C24×C3C23×C312yes—
190C23×D3C23×C314yes—
191C3×#41—664no—
192C2×C3×Q8—38no—
193C2×C3×Q8—34no—
194C2×C3×D4—28no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—24no—
198C2×C3×D4—316no—
199C2×C3×D4—316no—
200C2×C3×D4—38no—
201C2×C3×D4—38no—
202C2×C3×D4—38no—
203C2×C3×D4—616no—
204C2×C3×D4—68no—
205C2×C3×D4—68no—
206C2×C3×D4—124no—
207C2×C3×D4—128no—
208C2×C3×D4C22×D314yes—
209C2×C3×D4C23×C314yes—
210C22×C3×C4—28no—
211C22×C3×C4—28no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—28no—
217C22×C3×C4—32no—
218C22×C3×C4—32no—
219C22×C3×C4—316no—
220C22×C3×C4—48no—
221C22×C3×C4—616no—
222C22×C3×C4C22×D318yes—
223C22×C3×C4C22×D311yescharacteristic
224C22×C3×C4C22×D312yes—
225C22×C3×C4C22×D311yescharacteristic
226C22×C3×C4C22×D311yescharacteristic
227C22×C3×C4C23×C311yescharacteristic
228C22×C3×C4C23×C312yes—
229C22×C3×C4C23×C318yes—
230C22×C3×C4C23×C311yescharacteristic
231C22×C3×C4C23×C311yescharacteristic
232C2×#67—216no—
233C2×#67—48no—
234C2×#67C23×C3116yes—
235C2×#67C23×C3116yes—
236C22×Dic3C23×C312yes—
237C22×Dic3C23×C312yes—
238#235—264no—
239C2×D12—28no—
240C2×D12—28no—
241C2×D12—44no—
242C2×D12C23×C318yes—
243C2×D12C23×C318yes—
244C2×D12C23×C318yes—
245C2×C4×D3—216no—
246C2×C4×D3C23×C3116yes—
247C2×Dic6C3×D418yescomplemented
248C2×Dic6C23×C314yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—38no—
251C3×C4⋊C4—34no—
252C3×C4⋊C4C22×D314yes—
253C3×C4⋊C4C23×C314yes—
254C3×C22⋊C4—28no—
255C3×C22⋊C4—316no—
256C3×C22⋊C4C22×D318yes—
257C3×C22⋊C4C23×C318yes—
258C3×C42—24no—
259C3×C42—38no—
260C3×C42C22×D314yes—
261C3×C42C23×C314yes—
262(C2×D3)⋊C4C23×C3116yes—
263C4⋊Dic3C23×C314yes—
264Dic3⋊C4C23×C318yes—
265C4×Dic3C23×C318yes—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—24no—
269C22×C32—48no—
270C22×C32C22×D412yes—
271C22×C32C2×#4112yes—
272C22×C32H(5,𝔽2)12yes—
273C22×C32C2511yescharacteristic
274C2×C3×D3—28no—
275C2×C3×D3—28no—
276C2×C3×D3—28no—
277C2×C3×D3—416no—
278C32×C4—24no—
279C32×C4—22no—
280C32×C4C22×D414yes—
281C32×C4C2×#4114yescomplemented
282C3×Dic3—24no—
283C3×Dic3C22×D418yes—
284C2×#41—316no—
285C2×#41—616no—
286C22×Q8—31no—
287C22×D4—34no—
288C22×D4—34no—
289C22×D4—32no—
290C22×D4—32no—
291C22×D4—32no—
292C22×D4C2×C3×D311yescharacteristic
293C23×C4—34no—
294C23×C4C2×C3×D312yes—
295#127—316no—
296#127—38no—
297#121—316no—
298#121—332no—
299C4×Q8—38no—
300C4×D4—316no—
301C4×D4C2×C3×D318yescomplemented
302C2×C4⋊C4—32no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4C2×C3×D311yescharacteristic
305C2×C22⋊C4—34no—
306C2×C22⋊C4C2×C3×D312yes—
307C2×C42—32no—
308C2×C42C2×C3×D311yescharacteristic
309C23×C3—28no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—21no—
317C23×C3—24no—
318C23×C3—38no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—68no—
322C23×C3—68no—
323C23×C3—68no—
324C23×C3C23×D314yes—
325C23×C3C23×D311yescharacteristic
326C23×C3C24×C314yes—
327C23×C3C24×C311yescharacteristic
328C22×D3—28no—
329C22×D3—28no—
330C22×D3—28no—
331C22×D3C24×C318yes—
332C3×Q8—316no—
333C3×D4—216no—
334C3×D4—28no—
335C3×D4—28no—
336C3×D4—632no—
337C3×D4—616no—
338C3×D4—616no—
339C3×D4—128no—
340C3×D4—1216no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—216no—
346C2×C3×C4—216no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—38no—
354C2×C3×C4—34no—
355C2×C3×C4—34no—
356C2×C3×C4—416no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4—632no—
360C2×C3×C4—64no—
361C2×C3×C4#23514yescomplemented
362C2×C3×C4D3×D414yes—
363C2×C3×C4C2×C3×D414yes—
364C2×C3×C4C3×#4114yescomplemented
365C2×C3×C4C23×D312yes—
366C2×C3×C4C23×D314yes—
367C2×C3×C4C23×D312yes—
368C2×C3×C4C23×D312yes—
369C2×C3×C4C24×C312yes—
370C2×C3×C4C24×C314yes—
371C2×C3×C4C24×C312yes—
372C2×C3×C4C24×C312yes—
373#67—232no—
374#67—416no—
375C2×Dic3—24no—
376C2×Dic3C2×C3×D418yes—
377C2×Dic3C24×C314yes—
378C2×Dic3C24×C314yes—
379D12—216no—
380D12—216no—
381D12—48no—
382C4×D3—232no—
383Dic6C2×C3×D4116yescomplemented
384C2×C32—44no—
385C2×C32#55014yescomplemented
386C2×C32C23×D411yescharacteristic
387C2×C32C22×#4111yescharacteristic
388C2×C32C2×H(5,𝔽2)11yescharacteristic
389C3×D3—48no—
390C24—34no—
391C24C22×C3×D312yes—
392#41—664no—
393C2×Q8—34no—
394C2×Q8—38no—
395C2×D4—24no—
396C2×D4—316no—
397C2×D4—316no—
398C2×D4—38no—
399C2×D4—38no—
400C2×D4—38no—
401C2×D4—616no—
402C2×D4—68no—
403C2×D4—68no—
404C2×D4—124no—
405C2×D4—128no—
406C2×D4C22×C3×D314yes—
407C22×C4—28no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—316no—
411C22×C4—616no—
412C22×C4C22×C3×D311yescharacteristic
413C22×C4C22×C3×D312yes—
414C22×C4C22×C3×D318yes—
415C22×C4C22×C3×D311yescharacteristic
416C22×C4C22×C3×D311yescharacteristic
417C4⋊C4—38no—
418C4⋊C4—34no—
419C4⋊C4C22×C3×D314yes—
420C22⋊C4—316no—
421C22⋊C4C22×C3×D318yes—
422C42—38no—
423C42C22×C3×D314yes—
424C22×C3—28no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—416no—
438C22×C3—48no—
439C22×C3—48no—
440C22×C3—68no—
441C22×C3—68no—
442C22×C3—68no—
443C22×C3—1216no—
444C22×C3C2×#23512yes—
445C22×C3C2×D3×D412yes—
446C22×C3#100912yes—
447C22×C3C22×C3×D412yes—
448C22×C3C2×C3×#4112yes—
449C22×C3C3×H(5,𝔽2)12yes—
450C22×C3C24×D311yescharacteristic
451C22×C3C25×C311yescharacteristic
452C2×D3—28no—
453C2×D3—28no—
454C2×D3—28no—
455C2×D3—416no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—24no—
459C3×C4—22no—
460C3×C4—24no—
461C3×C4—24no—
462C3×C4—38no—
463C3×C4—44no—
464C3×C4—42no—
465C3×C4—64no—
466C3×C4C2×#23514yescomplemented
467C3×C4C2×D3×D414yes—
468C3×C4C22×C3×D414yes—
469C3×C4C2×C3×#4114yescomplemented
470Dic3—24no—
471Dic3C22×C3×D418yes—
472C32C2×#55011yescharacteristic, complemented, Sylow
473C23—24no—
474C23—24no—
475C23—24no—
476C23—38no—
477C23—68no—
478C23—68no—
479C23—68no—
480C23C23×C3×D314yes—
481C23C23×C3×D311yescharacteristic
482Q8—316no—
483D4—28no—
484D4—632no—
485D4—616no—
486D4—616no—
487D4—128no—
488D4—1216no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—216no—
492C2×C4—38no—
493C2×C4—34no—
494C2×C4—34no—
495C2×C4—632no—
496C2×C4—64no—
497C2×C4C3×#23514yescomplemented
498C2×C4C3×D3×D414yes—
499C2×C4C23×C3×D312yes—
500C2×C4C23×C3×D314yes—
501C2×C4C23×C3×D312yes—
502C2×C4C23×C3×D312yes—
503C2×C3—24no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—48no—
508C2×C3—44no—
509C2×C3—44no—
510C2×C3—128no—
511C2×C3#555314yesdirect product factor
512C2×C3C3×#55014yescomplemented
513C2×C3C22×#23511yescharacteristic
514C2×C3C22×D3×D411yescharacteristic
515C2×C3C2×#100911yescharacteristic
516C2×C3C23×C3×D411yescharacteristic
517C2×C3C22×C3×#4111yescharacteristic
518C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
519D3—48no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—48no—
524C22—68no—
525C22—68no—
526C22—68no—
527C22—1216no—
528C22C2×C3×#23512yes—
529C22C2×C3×D3×D412yes—
530C22C3×#100912yes—
531C22C24×C3×D311yescharacteristic
532C4—24no—
533C4—22no—
534C4—38no—
535C4—64no—
536C4C2×C3×#23514yescomplemented
537C4C2×C3×D3×D414yes—
538C3—21no—
539C3C2×#555311yescharacteristic, direct product factor
540C3C2×C3×#55011yescharacteristic, complemented
541C2—44no—
542C2—128no—
543C2C3×#555314yesdirect product factor
544C2C22×C3×#23511yescharacteristic
545C2C22×C3×D3×D411yescharacteristic
546C2C2×C3×#100911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C24
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#550
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×#235C2semidirect
2. C2×C32×C4×D4C2semidirect
3. C2×C3×#940C2semidirect
4. C2×C3×#930C2semidirect
5. C3×#5553C2direct (N × H)
6. C2×C3×#893C2semidirect
7. C2×C3×#884C2semidirect
8. C2×C3×#874C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5553C3direct (N × H)
17. C2×C3×#235C22semidirect
23. C32×C4×D4C22semidirect
27. C3×#940C22semidirect
28. C3×#930C22semidirect
33. C3×#893C22semidirect
34. C3×#884C22semidirect
35. C3×#874C22semidirect
36. C3×C4×Dic6C22semidirect
38. C22×#235C2×C3semidirect
47. C2×C3×C4×D4D3semidirect
48. C2×C3×C4×D4C2×C3semidirect
49. C2×#940C2×C3semidirect
50. C2×#930C2×C3semidirect
51. #5553C2×C3direct (N × H)
52. C2×#893C2×C3semidirect
53. C2×#884C2×C3semidirect
54. C2×#874C2×C3semidirect
55. C2×C4×Dic6C2×C3semidirect
81. C2×C3×Dic6D4semidirect
109. C2×#235C22×C3semidirect
122. C3×C4×D4C2×D3semidirect
123. C3×C4×D4C22×C3semidirect
137. #940C22×C3semidirect
138. #930C22×C3semidirect
143. #893C22×C3semidirect
144. #884C22×C3semidirect
145. #874C22×C3semidirect
146. C4×Dic6C22×C3semidirect
161. C2×C32×C4#41semidirect
176. C3×Dic6C2×D4semidirect
185. C2×C4×D4C3×D3semidirect
247. C2×Dic6C3×D4semidirect
281. C32×C4C2×#41semidirect
301. C4×D4C2×C3×D3semidirect
361. C2×C3×C4#235semidirect
364. C2×C3×C4C3×#41semidirect
383. Dic6C2×C3×D4semidirect
385. C2×C32#550semidirect
466. C3×C4C2×#235semidirect
469. C3×C4C2×C3×#41semidirect
472. C32C2×#550semidirect
497. C2×C4C3×#235semidirect
511. C2×C3#5553direct (N × H)
512. C2×C3C3×#550semidirect
536. C4C2×C3×#235semidirect
539. C3C2×#5553direct (N × H)
540. C3C2×C3×#550semidirect
543. C2C3×#5553direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C2×C32×C4×D42
C2×C3×#9402
C2×C3×#9302
C3×#55532
C2×C3×#8932
C2×C3×#8842
C2×C3×#8742
C2×C3×C4×Dic62
C2×C3×#5503
C2×#55533