Finite Group Databasebeta

← C2×C3×#5771 #11918910

Subgroups — C2×C3×#5771 #11918910

Order 1152.

2949 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C212yesmaximal, complemented
2C2×C32×#135C211yesmaximal, characteristic, complemented
3C2×C3×#951C212yesmaximal, complemented
4C3×#5771C2116yesmaximal, direct product factor
5C2×C3×D3×Q16C212yesmaximal, complemented
6C2×C3×#915C214yesmaximal, complemented
7C2×C3×#912C212yesmaximal, complemented
8C22×C3×Dic12C211yesmaximal, characteristic, complemented
9C2×C3×#901C211yesmaximal, characteristic, complemented
10C2×C3×#578—31nomaximal
11C2×#5771C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2212yes—
13C3×#1010—28no—
14C3×#1010C22116yescomplemented
15C2×C3×D3×Q8C2214yes—
16C2×C3×#237—24no—
17C2×C3×#237C2214yes—
18C2×C3×#235—22no—
19C2×C3×#235C2211yescharacteristic
20C22×C3×Dic6—22no—
21C22×C3×Dic6C2212yes—
22C32×#135C2218yescomplemented
23C2×C32×Q16C2211yescharacteristic
24C2×C32×SD16C2212yes—
25C2×C32×D8C2211yescharacteristic
26C22×C32×C8C2211yescharacteristic
27C3×#951C22116yescomplemented
28C2×C3×#216C2214yes—
29C2×C3×#214C2214yes—
30C2×C3×#208C2211yescharacteristic
31C3×D3×Q16C22116yescomplemented
32C3×#915C22132yescomplemented
33C3×#912C22116yescomplemented
34C2×C3×Dic12C2212yes—
35C2×C3×Dic12C2212yes—
36C2×C3×Dic12C2218yescomplemented
37C3×#901C2218yescomplemented
38C2×C3×ΓL(1,𝔽25)C2212yes—
39C2×C3×D3×C8C2212yes—
40C2×C3×2−1+4—32no—
41C2×#1010C2×C312yescomplemented
42C3×#578—316no—
43C2×C3×#137—32no—
44C2×C3×#137—34no—
45C2×C3×#135—21no—
46C2×C3×#135—32no—
47C2×C3×#135D311yescharacteristic, complemented
48C2×C3×#135C2×C311yescharacteristic, complemented
49C22×C3×Q16—32no—
50C22×C3×Q16—31no—
51C2×C3×#131—31no—
52C2×#951C2×C312yescomplemented
53#5771C2×C3116yesdirect product factor
54C2×D3×Q16C2×C312yescomplemented
55C2×#915C2×C314yescomplemented
56C2×#912C2×C312yescomplemented
57C22×Dic12C2×C311yescharacteristic, complemented
58C2×#901C2×C311yescharacteristic, complemented
59C32×#41—24no—
60C32×#41C2318yes—
61C2×C32×Q8C2312yes—
62C2×C32×D4—22no—
63C2×C32×D4C2312yes—
64C22×C32×C4—22no—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—24no—
67C2×C3×#67D412yes—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—24no—
70C3×D3×Q8—28no—
71C3×D3×Q8C23116yes—
72C3×#237—232no—
73C3×#237—28no—
74C3×#237C23116yes—
75C3×#235—216no—
76C3×#235D414yes—
77C3×#235C2314yes—
78C2×C3×D12D411yescharacteristic
79C2×C3×C4×D3—24no—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6—22no—
82C2×C3×Dic6—24no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6—216no—
85C2×C3×Dic6D411yescharacteristic
86C2×C3×Dic6C2314yes—
87C2×C3×Dic6C2318yes—
88C32×Q16C2314yes—
89C32×SD16C2318yes—
90C32×D8C2314yes—
91C2×C32×C8C2311yescharacteristic
92C2×C32×C8C2311yescharacteristic
93C2×C32×C8C2314yes—
94C3×#216C23116yes—
95C3×#214C23116yes—
96C3×#208C2314yes—
97C2×C3×C3⋊C8C2312yes—
98C3×Dic12C2318yes—
99C3×Dic12C2318yes—
100C3×ΓL(1,𝔽25)C2318yes—
101C3×D3×C8C2318yes—
102C2×#578—31noSylow
103C3×2−1+4—316no—
104C3×2−1+4—68no—
105C2×C3×#41—22no—
106C2×C3×#41—31no—
107C2×C3×#41—34no—
108C2×C3×#41—62no—
109C2×C3×#41—64no—
110C2×C3×#41C2×D312yes—
111C2×C3×#41C22×C312yes—
112C22×C3×Q8—34no—
113C22×C3×Q8—32no—
114C22×C3×Q8—62no—
115#1010—28no—
116#1010C22×C3116yescomplemented
117C2×D3×Q8C22×C314yes—
118C2×#237—24no—
119C2×#237C22×C314yes—
120C2×#235—22no—
121C2×#235C22×C311yescharacteristic
122C22×Dic6—22no—
123C22×Dic6C22×C312yes—
124C3×#137—316no—
125C3×#137—332no—
126C3×#135—28no—
127C3×#135—316no—
128C3×#135C2×D318yescomplemented
129C3×#135C22×C318yescomplemented
130C2×C3×Q16—21no—
131C2×C3×Q16—34no—
132C2×C3×Q16—316no—
133C2×C3×Q16—32no—
134C2×C3×Q16—32no—
135C2×C3×Q16—38no—
136C2×C3×Q16C2×D311yescharacteristic
137C2×C3×Q16C22×C311yescharacteristic
138C2×C3×SD16—22no—
139C2×C3×SD16—34no—
140C2×C3×SD16C2×D312yes—
141C2×C3×SD16C22×C312yes—
142C2×C3×D8—21no—
143C2×C3×D8C2×D311yescharacteristic
144C2×C3×D8C22×C311yescharacteristic
145C3×#131—38no—
146C2×C3×M16—31no—
147C2×C3×M16—32no—
148C22×C3×C8—21no—
149C22×C3×C8—32no—
150C22×C3×C8C2×D311yescharacteristic
151C22×C3×C8C22×C311yescharacteristic
152#951C22×C3116yescomplemented
153C2×#216C22×C314yes—
154C2×#214C22×C314yes—
155C2×#208C22×C311yescharacteristic
156D3×Q16C22×C3116yescomplemented
157#915C22×C3132yescomplemented
158#912C22×C3116yescomplemented
159C2×Dic12C22×C312yes—
160C2×Dic12C22×C312yes—
161C2×Dic12C22×C318yescomplemented
162#901C22×C318yescomplemented
163C2×ΓL(1,𝔽25)C22×C312yes—
164C2×D3×C8C22×C312yes—
165C23×C32—22no—
166C23×C32C2×D411yescharacteristic
167C22×C3×D3C2×D412yes—
168C32×Q8—22no—
169C32×Q8C2414yes—
170C32×D4—28no—
171C32×D4—22no—
172C32×D4C2414yes—
173C2×C32×C4—22no—
174C2×C32×C4—28no—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2411yescharacteristic
178C2×C32×C4C2412yes—
179C3×#67—216no—
180C3×#67C2×D418yes—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—28no—
183C2×C3×Dic3—216no—
184C2×C3×Dic3C2×D412yes—
185C3×D12C2×D414yes—
186C3×C4×D3—216no—
187C3×C4×D3C2×D414yes—
188C3×C4×D3C2414yes—
189C3×Dic6—28no—
190C3×Dic6—24no—
191C3×Dic6—216no—
192C3×Dic6C2×D414yes—
193C3×Dic6C2418yes—
194C32×C8C2412yes—
195C32×C8C2412yes—
196C3×C3⋊C8C2414yes—
197C2×2−1+4—32no—
198#578—316no—
199C2×#137—32no—
200C2×#137—34no—
201C2×#135—32no—
202C2×#135C3×D311yescharacteristic, complemented
203C22×Q16—32no—
204C22×Q16—31no—
205C2×#131—31no—
206C3×#41—24no—
207C3×#41—24no—
208C3×#41—28no—
209C3×#41—34no—
210C3×#41—34no—
211C3×#41—316no—
212C3×#41—44no—
213C3×#41—616no—
214C3×#41—68no—
215C3×#41—632no—
216C3×#41C22×D318yes—
217C3×#41C23×C318yes—
218C2×C3×Q8—22no—
219C2×C3×Q8—31no—
220C2×C3×Q8—316no—
221C2×C3×Q8—34no—
222C2×C3×Q8—38no—
223C2×C3×Q8—64no—
224C2×C3×Q8—616no—
225C2×C3×Q8—68no—
226C2×C3×Q8—62no—
227C2×C3×Q8—64no—
228C2×C3×Q8C22×D312yes—
229C2×C3×Q8C23×C312yes—
230C2×C3×D4—22no—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4—32no—
234C2×C3×D4—31no—
235C2×C3×D4—42no—
236C2×C3×D4—64no—
237C2×C3×D4C22×D312yes—
238C2×C3×D4C23×C312yes—
239C22×C3×C4—22no—
240C22×C3×C4—22no—
241C22×C3×C4—21no—
242C22×C3×C4—32no—
243C22×C3×C4—42no—
244C22×C3×C4—64no—
245C22×C3×C4—64no—
246C22×C3×C4—62no—
247C22×C3×C4C22×D311yescharacteristic
248C22×C3×C4C23×C311yescharacteristic
249C2×#67—24no—
250C2×#67C3×D412yes—
251C22×Dic3—22no—
252C22×Dic3—24no—
253D3×Q8—28no—
254D3×Q8C23×C3116yes—
255#237—232no—
256#237—28no—
257#237C23×C3116yes—
258#235—216no—
259#235C3×D414yes—
260#235C23×C314yes—
261C2×D12C3×D411yescharacteristic
262C2×C4×D3—24no—
263C2×C4×D3C23×C312yes—
264C2×Dic6—22no—
265C2×Dic6—24no—
266C2×Dic6—24no—
267C2×Dic6—216no—
268C2×Dic6C3×D411yescharacteristic
269C2×Dic6C23×C314yes—
270C2×Dic6C23×C318yes—
271C3×Q16—24no—
272C3×Q16—316no—
273C3×Q16—38no—
274C3×Q16—38no—
275C3×Q16C22×D314yes—
276C3×Q16C23×C314yes—
277C3×SD16—28no—
278C3×SD16—316no—
279C3×SD16C22×D318yes—
280C3×SD16C23×C318yes—
281C3×D8—24no—
282C3×D8C22×D314yes—
283C3×D8C23×C314yes—
284C3×M16—34no—
285C3×M16—38no—
286C2×C3×C8—21no—
287C2×C3×C8—21no—
288C2×C3×C8—24no—
289C2×C3×C8—32no—
290C2×C3×C8—38no—
291C2×C3×C8C22×D311yescharacteristic
292C2×C3×C8C22×D311yescharacteristic
293C2×C3×C8C22×D314yes—
294C2×C3×C8C23×C311yescharacteristic
295C2×C3×C8C23×C311yescharacteristic
296C2×C3×C8C23×C314yes—
297#216C23×C3116yes—
298#214C23×C3116yes—
299#208C23×C314yes—
300C2×C3⋊C8C23×C312yes—
301Dic12C23×C318yes—
302Dic12C23×C318yes—
303ΓL(1,𝔽25)C23×C318yes—
304D3×C8C23×C318yes—
305C22×C32—22no—
306C22×C32—24no—
307C22×C32—44no—
308C22×C32C22×D412yes—
309C22×C32C22×D411yescharacteristic
310C2×C3×D3—24no—
311C2×C3×D3C22×D414yes—
312C32×C4—24no—
313C32×C4C22×D412yes—
314C32×C4C22×D411yescharacteristic
315C32×C4C2511yescharacteristic
316C3×Dic3—28no—
317C3×Dic3C22×D414yes—
3182−1+4—316no—
3192−1+4—68no—
320C2×#41—31no—
321C2×#41—34no—
322C2×#41—62no—
323C2×#41—64no—
324C2×#41C2×C3×D312yes—
325C22×Q8—34no—
326C22×Q8—32no—
327C22×Q8—62no—
328#137—316no—
329#137—332no—
330#135—316no—
331#135C2×C3×D318yescomplemented
332C2×Q16—34no—
333C2×Q16—316no—
334C2×Q16—32no—
335C2×Q16—32no—
336C2×Q16—38no—
337C2×Q16C2×C3×D311yescharacteristic
338C2×SD16—34no—
339C2×SD16C2×C3×D312yes—
340C2×D8C2×C3×D311yescharacteristic
341#131—38no—
342C2×M16—31no—
343C2×M16—32no—
344C22×C8—32no—
345C22×C8C2×C3×D311yescharacteristic
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—21no—
349C23×C3—32no—
350C23×C3—42no—
351C23×C3D3×D411yescharacteristic
352C23×C3C2×C3×D411yescharacteristic
353C22×D3C2×C3×D412yes—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—24no—
357C3×Q8—34no—
358C3×Q8—38no—
359C3×Q8—42no—
360C3×Q8—616no—
361C3×Q8—68no—
362C3×Q8—64no—
363C3×Q8C23×D314yes—
364C3×Q8C24×C314yes—
365C3×D4—28no—
366C3×D4—28no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—24no—
370C3×D4—38no—
371C3×D4—34no—
372C3×D4—42no—
373C3×D4—48no—
374C3×D4—616no—
375C3×D4C23×D314yes—
376C3×D4C24×C314yes—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—28no—
380C2×C3×C4—28no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—32no—
386C2×C3×C4—34no—
387C2×C3×C4—34no—
388C2×C3×C4—48no—
389C2×C3×C4—42no—
390C2×C3×C4—616no—
391C2×C3×C4—616no—
392C2×C3×C4—68no—
393C2×C3×C4—64no—
394C2×C3×C4D3×D412yes—
395C2×C3×C4D3×D411yescharacteristic
396C2×C3×C4C2×C3×D411yescharacteristic
397C2×C3×C4C2×C3×D412yes—
398C2×C3×C4C23×D312yes—
399C2×C3×C4C23×D311yescharacteristic
400C2×C3×C4C24×C311yescharacteristic
401C2×C3×C4C24×C312yes—
402#67—216no—
403#67C2×C3×D418yes—
404C2×Dic3—24no—
405C2×Dic3—28no—
406C2×Dic3—216no—
407C2×Dic3C2×C3×D412yes—
408D12C2×C3×D414yes—
409C4×D3—216no—
410C4×D3C2×C3×D414yes—
411C4×D3C24×C314yes—
412Dic6—28no—
413Dic6—24no—
414Dic6—216no—
415Dic6C2×C3×D414yes—
416Dic6C24×C318yes—
417C3×C8—22no—
418C3×C8—22no—
419C3×C8—34no—
420C3×C8C23×D312yes—
421C3×C8C23×D312yes—
422C3×C8C24×C312yes—
423C3×C8C24×C312yes—
424C3⋊C8C24×C314yes—
425C2×C32—22no—
426C2×C32—44no—
427C2×C32#57812yescomplemented
428C2×C32C23×D411yescharacteristic
429C3×D3—24no—
430#41—24no—
431#41—34no—
432#41—34no—
433#41—316no—
434#41—616no—
435#41—68no—
436#41—632no—
437#41C22×C3×D318yes—
438C2×Q8—31no—
439C2×Q8—316no—
440C2×Q8—34no—
441C2×Q8—38no—
442C2×Q8—64no—
443C2×Q8—616no—
444C2×Q8—68no—
445C2×Q8—62no—
446C2×Q8—64no—
447C2×Q8C22×C3×D312yes—
448C2×D4—22no—
449C2×D4—32no—
450C2×D4—31no—
451C2×D4—64no—
452C2×D4C22×C3×D312yes—
453C22×C4—22no—
454C22×C4—32no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—62no—
458C22×C4C22×C3×D311yescharacteristic
459Q16—316no—
460Q16—38no—
461Q16—38no—
462Q16C22×C3×D314yes—
463SD16—316no—
464SD16C22×C3×D318yes—
465D8C22×C3×D314yes—
466M16—34no—
467M16—38no—
468C2×C8—32no—
469C2×C8—38no—
470C2×C8C22×C3×D311yescharacteristic
471C2×C8C22×C3×D311yescharacteristic
472C2×C8C22×C3×D314yes—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—24no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—34no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—64no—
485C22×C3—84no—
486C22×C3C2×D3×D412yes—
487C22×C3C2×D3×D411yescharacteristic
488C22×C3C22×C3×D412yes—
489C22×C3C22×C3×D411yescharacteristic
490C2×D3—24no—
491C2×D3C22×C3×D414yes—
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—34no—
498C3×C4—44no—
499C3×C4—68no—
500C3×C4C2×D3×D412yes—
501C3×C4C2×D3×D411yescharacteristic
502C3×C4C22×C3×D412yes—
503C3×C4C22×C3×D411yescharacteristic
504C3×C4C24×D311yescharacteristic
505C3×C4C25×C311yescharacteristic
506Dic3—28no—
507Dic3C22×C3×D414yes—
508C32C2×#57811yescharacteristic, complemented, Sylow
509C23—22no—
510C23—32no—
511C23C3×D3×D411yescharacteristic
512Q8—22no—
513Q8—34no—
514Q8—38no—
515Q8—616no—
516Q8—68no—
517Q8—64no—
518Q8C23×C3×D314yes—
519D4—28no—
520D4—22no—
521D4—38no—
522D4—34no—
523D4—616no—
524D4C23×C3×D314yes—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—32no—
528C2×C4—34no—
529C2×C4—34no—
530C2×C4—616no—
531C2×C4—616no—
532C2×C4—64no—
533C2×C4—68no—
534C2×C4C3×D3×D411yescharacteristic
535C2×C4C3×D3×D412yes—
536C2×C4C23×C3×D311yescharacteristic
537C2×C4C23×C3×D312yes—
538C8—34no—
539C8C23×C3×D312yes—
540C8C23×C3×D312yes—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—42no—
546C2×C3—44no—
547C2×C3—44no—
548C2×C3—64no—
549C2×C3—84no—
550C2×C3#577112yesdirect product factor
551C2×C3C3×#57812yescomplemented
552C2×C3C22×D3×D411yescharacteristic
553C2×C3C23×C3×D411yescharacteristic
554D3—24no—
555C22—22no—
556C22—24no—
557C22—34no—
558C22—44no—
559C22—64no—
560C22C2×C3×D3×D412yes—
561C22C2×C3×D3×D411yescharacteristic
562C4—24no—
563C4—34no—
564C4—68no—
565C4C2×C3×D3×D412yes—
566C4C2×C3×D3×D411yescharacteristic
567C4C24×C3×D311yescharacteristic
568C3—21no—
569C3C2×#577111yescharacteristic, direct product factor
570C3C2×C3×#57811yescharacteristic, complemented
571C2—22no—
572C2—44no—
573C2—64no—
574C2C3×#577112yesdirect product factor
575C2C22×C3×D3×D411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC2×C32×#135C2
socleC22×C32C22×D4
hypercenterC2×C3×#135D3
hypocenterC3C2×C3×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#135

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C2×C32×#135C2semidirect
3. C2×C3×#951C2semidirect
4. C3×#5771C2direct (N × H)
5. C2×C3×D3×Q16C2semidirect
6. C2×C3×#915C2semidirect
7. C2×C3×#912C2semidirect
8. C22×C3×Dic12C2semidirect
9. C2×C3×#901C2semidirect
11. C2×#5771C3direct (N × H)
14. C3×#1010C22semidirect
22. C32×#135C22semidirect
27. C3×#951C22semidirect
31. C3×D3×Q16C22semidirect
32. C3×#915C22semidirect
33. C3×#912C22semidirect
36. C2×C3×Dic12C22semidirect
37. C3×#901C22semidirect
41. C2×#1010C2×C3semidirect
47. C2×C3×#135D3semidirect
48. C2×C3×#135C2×C3semidirect
52. C2×#951C2×C3semidirect
53. #5771C2×C3direct (N × H)
54. C2×D3×Q16C2×C3semidirect
55. C2×#915C2×C3semidirect
56. C2×#912C2×C3semidirect
57. C22×Dic12C2×C3semidirect
58. C2×#901C2×C3semidirect
116. #1010C22×C3semidirect
128. C3×#135C2×D3semidirect
129. C3×#135C22×C3semidirect
152. #951C22×C3semidirect
156. D3×Q16C22×C3semidirect
157. #915C22×C3semidirect
158. #912C22×C3semidirect
161. C2×Dic12C22×C3semidirect
162. #901C22×C3semidirect
202. C2×#135C3×D3semidirect
331. #135C2×C3×D3semidirect
427. C2×C32#578semidirect
508. C32C2×#578semidirect
550. C2×C3#5771direct (N × H)
551. C2×C3C3×#578semidirect
569. C3C2×#5771direct (N × H)
570. C3C2×C3×#578semidirect
574. C2C3×#5771direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C2×C32×#1352
C2×C3×#9512
C3×#57712
C2×C3×D3×Q162
C2×C3×#9152
C2×C3×#9122
C22×C3×Dic122
C2×C3×#9012
C2×C3×#5783
C2×#57713