Finite Group Databasebeta

← C2×C3×#5735 #11910283

Subgroups — C2×C3×#5735 #11910283

Order 1152.

2193 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic
3C2×C3×#946C212yesmaximal, complemented
4C2×C3×#944C211yesmaximal, characteristic, complemented
5C3×#5735C2116yesmaximal, direct product factor
6C2×C3×#898C212yesmaximal, complemented
7C2×C3×#896C212yesmaximal, complemented
8C2×C3×#894C212yesmaximal
9C2×C3×#889C212yesmaximal, complemented
10C2×C3×#876C212yesmaximal, complemented
11C2×C3×#563—31nomaximal
12C2×#5735C311yesmaximal, characteristic, direct product factor
13C22×C32×Q8C2211yescharacteristic
14C22×C32×Q8C2211yescharacteristic
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3C2212yes—
17C22×C3×Dic6C2211yescharacteristic
18C32×C4⋊Q8C2218yescomplemented
19C2×C32×C4⋊C4C2212yes—
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C42C2211yescharacteristic
22C3×#947C2218yes—
23C3×#946C22116yescomplemented
24C3×#944C2218yescomplemented
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×(C2×D3)⋊C4C2212yes—
28C2×C3×(C2×D3)⋊C4C2212yes—
29C2×C3×(C2×D3)⋊C4C2212yes—
30C2×C3×C4⋊Dic3C2212yes—
31C2×C3×Dic3⋊C4C2212yes—
32C2×C3×Dic3⋊C4C2212yes—
33C2×C3×Dic3⋊C4C2212yes—
34C2×C3×Dic3⋊C4C2212yes—
35C2×C3×Dic3⋊C4C2212yes—
36C2×C3×C4×Dic3C2211yescharacteristic
37C2×C3×C4×Dic3C2211yescharacteristic
38C3×#898C22116yescomplemented
39C3×#896C22116yescomplemented
40C3×#894C22116yes—
41C3×#889C22116yescomplemented
42C3×#876C22116yescomplemented
43C3×#563—316no—
44C2×C3×C4⋊Q8—21no—
45C2×C3×C4⋊Q8—31no—
46C2×C3×C4⋊Q8D311yescharacteristic, complemented
47C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
48C2×C3×#126—32no—
49C2×C3×#126—32no—
50C2×C3×#125—32no—
51C2×C3×#124—31no—
52C2×C3×#123—32no—
53C2×C3×C22⋊Q8—32no—
54C2×C3×C22⋊Q8—32no—
55C2×#947C2×C311yescharacteristic
56C2×#946C2×C312yescomplemented
57C2×#944C2×C311yescharacteristic, complemented
58#5735C2×C3116yesdirect product factor
59C2×#898C2×C312yescomplemented
60C2×#896C2×C312yescomplemented
61C2×#894C2×C312yes—
62C2×#889C2×C312yescomplemented
63C2×#876C2×C312yescomplemented
64C23×C3×D3C2312yes—
65C2×C32×Q8—42no—
66C2×C32×Q8—42no—
67C2×C32×Q8C2314yes—
68C2×C32×Q8C2314yes—
69C22×C32×C4C2311yescharacteristic
70C22×C32×C4C2311yescharacteristic
71C22×C32×C4C2311yescharacteristic
72C22×C32×C4C2312yes—
73C22×C32×C4C2312yes—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2312yes—
77C2×C3×D12—42no—
78C2×C3×D12C2314yes—
79C2×C3×C4×D3—44no—
80C2×C3×C4×D3C2318yes—
81C2×C3×Dic6—42no—
82C2×C3×Dic6C2314yes—
83C32×C4⋊C4C2318yes—
84C32×C4⋊C4C2318yes—
85C32×C42C2314yes—
86C3×(C2×D3)⋊C4C2318yes—
87C3×(C2×D3)⋊C4C2318yes—
88C3×(C2×D3)⋊C4C2318yes—
89C3×(C2×D3)⋊C4C2318yes—
90C3×(C2×D3)⋊C4C2318yes—
91C3×C4⋊Dic3C2318yes—
92C3×Dic3⋊C4C2318yes—
93C3×Dic3⋊C4C2318yes—
94C3×Dic3⋊C4C2318yes—
95C3×Dic3⋊C4C2318yes—
96C3×Dic3⋊C4C2318yes—
97C3×C4×Dic3C2314yes—
98C3×C4×Dic3C2314yes—
99C2×#563—31noSylow
100C22×C3×Q8—21no—
101C22×C3×Q8—21no—
102C22×C3×Q8—31no—
103C22×C3×Q8C2×D311yescharacteristic
104C22×C3×Q8C2×D311yescharacteristic
105C22×C3×Q8C22×C311yescharacteristic
106C22×C3×Q8C22×C311yescharacteristic
107C22×C3×D4—31no—
108C23×C3×C4—32no—
109C22×D12C22×C311yescharacteristic
110C22×C4×D3C22×C312yes—
111C22×Dic6C22×C311yescharacteristic
112C3×C4⋊Q8—28no—
113C3×C4⋊Q8—38no—
114C3×C4⋊Q8C2×D318yescomplemented
115C3×C4⋊Q8C22×C318yescomplemented
116C3×#126—316no—
117C3×#126—316no—
118C3×#125—316no—
119C3×#124—38no—
120C3×#123—316no—
121C3×C22⋊Q8—316no—
122C3×C22⋊Q8—316no—
123C2×C3×C4⋊C4—22no—
124C2×C3×C4⋊C4—22no—
125C2×C3×C4⋊C4—32no—
126C2×C3×C4⋊C4—32no—
127C2×C3×C4⋊C4—32no—
128C2×C3×C4⋊C4—32no—
129C2×C3×C4⋊C4—32no—
130C2×C3×C4⋊C4—32no—
131C2×C3×C4⋊C4C2×D312yes—
132C2×C3×C4⋊C4C2×D312yes—
133C2×C3×C4⋊C4C22×C312yes—
134C2×C3×C4⋊C4C22×C312yes—
135C2×C3×C22⋊C4—32no—
136C2×C3×C22⋊C4—32no—
137C2×C3×C22⋊C4—32no—
138C2×C3×C22⋊C4—32no—
139C2×C3×C22⋊C4—32no—
140C2×C3×C42—21no—
141C2×C3×C42—31no—
142C2×C3×C42—31no—
143C2×C3×C42C2×D311yescharacteristic
144C2×C3×C42C22×C311yescharacteristic
145#947C22×C318yes—
146#946C22×C3116yescomplemented
147#944C22×C318yescomplemented
148C2×(C2×D3)⋊C4C22×C312yes—
149C2×(C2×D3)⋊C4C22×C312yes—
150C2×(C2×D3)⋊C4C22×C312yes—
151C2×(C2×D3)⋊C4C22×C312yes—
152C2×(C2×D3)⋊C4C22×C312yes—
153C2×C4⋊Dic3C22×C312yes—
154C2×Dic3⋊C4C22×C312yes—
155C2×Dic3⋊C4C22×C312yes—
156C2×Dic3⋊C4C22×C312yes—
157C2×Dic3⋊C4C22×C312yes—
158C2×Dic3⋊C4C22×C312yes—
159C2×C4×Dic3C22×C311yescharacteristic
160C2×C4×Dic3C22×C311yescharacteristic
161#898C22×C3116yescomplemented
162#896C22×C3116yescomplemented
163#894C22×C3116yes—
164#889C22×C3116yescomplemented
165#876C22×C3116yescomplemented
166C23×C32C2411yescharacteristic
167C22×C3×D3—24no—
168C22×C3×D3—28no—
169C22×C3×D3C2414yes—
170C32×Q8—44no—
171C32×Q8—44no—
172C2×C32×C4—22no—
173C2×C32×C4—24no—
174C2×C32×C4—22no—
175C2×C32×C4—24no—
176C2×C32×C4—22no—
177C2×C32×C4C2412yes—
178C2×C32×C4C2412yes—
179C2×C32×C4C2414yes—
180C2×C32×C4C2412yes—
181C2×C32×C4C2414yes—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3C2414yes—
186C2×C3×Dic3C2414yes—
187C2×C3×Dic3C2414yes—
188C3×D12—44no—
189C3×C4×D3—48no—
190C3×Dic6—44no—
191#563—316no—
192C2×C4⋊Q8—31no—
193C2×C4⋊Q8C3×D311yescharacteristic, complemented
194C2×#126—32no—
195C2×#126—32no—
196C2×#125—32no—
197C2×#124—31no—
198C2×#123—32no—
199C2×C22⋊Q8—32no—
200C2×C22⋊Q8—32no—
201C24×C3—32no—
202C23×D3C23×C312yes—
203C2×C3×Q8—24no—
204C2×C3×Q8—24no—
205C2×C3×Q8—34no—
206C2×C3×Q8—42no—
207C2×C3×Q8—42no—
208C2×C3×Q8—42no—
209C2×C3×Q8—42no—
210C2×C3×Q8—44no—
211C2×C3×Q8—44no—
212C2×C3×Q8—122no—
213C2×C3×Q8C22×D314yes—
214C2×C3×Q8C22×D314yes—
215C2×C3×Q8C23×C314yes—
216C2×C3×Q8C23×C314yes—
217C2×C3×D4—34no—
218C2×C3×D4—122no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—22no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—38no—
228C22×C3×C4—124no—
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C22×D311yescharacteristic
231C22×C3×C4C22×D312yes—
232C22×C3×C4C22×D311yescharacteristic
233C22×C3×C4C22×D312yes—
234C22×C3×C4C23×C311yescharacteristic
235C22×C3×C4C23×C311yescharacteristic
236C22×C3×C4C23×C312yes—
237C22×C3×C4C23×C311yescharacteristic
238C22×C3×C4C23×C312yes—
239C22×Dic3C23×C312yes—
240C22×Dic3C23×C312yes—
241C22×Dic3C23×C312yes—
242C2×D12—42no—
243C2×D12C23×C314yes—
244C2×C4×D3—44no—
245C2×C4×D3C23×C318yes—
246C2×Dic6—42no—
247C2×Dic6C23×C314yes—
248C3×C4⋊C4—28no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—38no—
251C3×C4⋊C4—38no—
252C3×C4⋊C4—38no—
253C3×C4⋊C4—38no—
254C3×C4⋊C4—38no—
255C3×C4⋊C4—38no—
256C3×C4⋊C4C22×D318yes—
257C3×C4⋊C4C22×D318yes—
258C3×C4⋊C4C23×C318yes—
259C3×C4⋊C4C23×C318yes—
260C3×C22⋊C4—38no—
261C3×C22⋊C4—38no—
262C3×C22⋊C4—38no—
263C3×C22⋊C4—38no—
264C3×C22⋊C4—38no—
265C3×C42—24no—
266C3×C42—34no—
267C3×C42—34no—
268C3×C42C22×D314yes—
269C3×C42C23×C314yes—
270(C2×D3)⋊C4C23×C318yes—
271(C2×D3)⋊C4C23×C318yes—
272(C2×D3)⋊C4C23×C318yes—
273(C2×D3)⋊C4C23×C318yes—
274(C2×D3)⋊C4C23×C318yes—
275C4⋊Dic3C23×C318yes—
276Dic3⋊C4C23×C318yes—
277Dic3⋊C4C23×C318yes—
278Dic3⋊C4C23×C318yes—
279Dic3⋊C4C23×C318yes—
280Dic3⋊C4C23×C318yes—
281C4×Dic3C23×C314yes—
282C4×Dic3C23×C314yes—
283C22×C32H(5,𝔽2)12yes—
284C22×C322−1+414yes—
285C22×C32C2511yescharacteristic
286C2×C3×D3—24no—
287C2×C3×D3—28no—
288C2×C3×D3—48no—
289C32×C4—22no—
290C32×C4—24no—
291C32×C4—22no—
292C32×C4—24no—
293C32×C4—22no—
294C3×Dic3—24no—
295C3×Dic3—24no—
296C3×Dic3—24no—
297C22×Q8—31no—
298C22×Q8C2×C3×D311yescharacteristic
299C22×Q8C2×C3×D311yescharacteristic
300C22×D4—31no—
301C23×C4—32no—
302C4⋊Q8—38no—
303C4⋊Q8C2×C3×D318yescomplemented
304#126—316no—
305#126—316no—
306#125—316no—
307#124—38no—
308#123—316no—
309C22⋊Q8—316no—
310C22⋊Q8—316no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4C2×C3×D312yes—
318C2×C4⋊C4C2×C3×D312yes—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—32no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42C2×C3×D311yescharacteristic
327C23×C3—21no—
328C23×C3—34no—
329C23×C3—64no—
330C23×C3—68no—
331C23×C3C23×D311yescharacteristic
332C23×C3C24×C311yescharacteristic
333C22×D3—24no—
334C22×D3—28no—
335C22×D3C24×C314yes—
336C3×Q8—44no—
337C3×Q8—44no—
338C3×Q8—44no—
339C3×Q8—44no—
340C3×Q8—48no—
341C3×Q8—48no—
342C3×Q8—124no—
343C3×D4—124no—
344C2×C3×C4—28no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—34no—
362C2×C3×C4—34no—
363C2×C3×C4—34no—
364C2×C3×C4—44no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—64no—
368C2×C3×C4—64no—
369C2×C3×C4—64no—
370C2×C3×C4—128no—
371C2×C3×C4C23×D312yes—
372C2×C3×C4C23×D312yes—
373C2×C3×C4C23×D314yes—
374C2×C3×C4C23×D312yes—
375C2×C3×C4C23×D314yes—
376C2×C3×C4C24×C312yes—
377C2×C3×C4C24×C312yes—
378C2×C3×C4C24×C314yes—
379C2×C3×C4C24×C312yes—
380C2×C3×C4C24×C314yes—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3C24×C314yes—
385C2×Dic3C24×C314yes—
386C2×Dic3C24×C314yes—
387D12—44no—
388C4×D3—48no—
389Dic6—44no—
390C2×C32#56314yescomplemented
391C2×C32C2×H(5,𝔽2)11yescharacteristic
392C2×C32C2×2−1+412yes—
393C3×D3—44no—
394C24—32no—
395C2×Q8—34no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—122no—
399C2×Q8C22×C3×D314yes—
400C2×Q8C22×C3×D314yes—
401C2×D4—34no—
402C2×D4—122no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—38no—
407C22×C4—124no—
408C22×C4C22×C3×D311yescharacteristic
409C22×C4C22×C3×D311yescharacteristic
410C22×C4C22×C3×D312yes—
411C22×C4C22×C3×D311yescharacteristic
412C22×C4C22×C3×D312yes—
413C4⋊C4—38no—
414C4⋊C4—38no—
415C4⋊C4—38no—
416C4⋊C4—38no—
417C4⋊C4—38no—
418C4⋊C4—38no—
419C4⋊C4C22×C3×D318yes—
420C4⋊C4C22×C3×D318yes—
421C22⋊C4—38no—
422C22⋊C4—38no—
423C22⋊C4—38no—
424C22⋊C4—38no—
425C22⋊C4—38no—
426C42—34no—
427C42—34no—
428C42C22×C3×D314yes—
429C22×C3—22no—
430C22×C3—24no—
431C22×C3—21no—
432C22×C3—64no—
433C22×C3—68no—
434C22×C3—128no—
435C22×C3#100412yes—
436C22×C3#100714yes—
437C22×C3C3×H(5,𝔽2)12yes—
438C22×C3C3×2−1+414yes—
439C22×C3C24×D311yescharacteristic
440C22×C3C25×C311yescharacteristic
441C2×D3—24no—
442C2×D3—28no—
443C2×D3—48no—
444C3×C4—24no—
445C3×C4—28no—
446C3×C4—24no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—24no—
451C3×C4—22no—
452C3×C4—24no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—64no—
460C3×C4—64no—
461C3×C4—64no—
462Dic3—24no—
463Dic3—24no—
464Dic3—24no—
465C32C2×#56311yescharacteristic, complemented, Sylow
466C23—34no—
467C23—64no—
468C23—68no—
469C23C23×C3×D311yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—124no—
473D4—124no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—34no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—128no—
486C2×C4C23×C3×D312yes—
487C2×C4C23×C3×D312yes—
488C2×C4C23×C3×D314yes—
489C2×C4C23×C3×D312yes—
490C2×C4C23×C3×D314yes—
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—24no—
494C2×C3—124no—
495C2×C3#573514yesdirect product factor
496C2×C3C3×#56314yescomplemented
497C2×C3C2×#100411yescharacteristic
498C2×C3C2×#100712yes—
499C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
500C2×C3C2×C3×2−1+412yes—
501D3—44no—
502C22—64no—
503C22—68no—
504C22—128no—
505C22C3×#100412yes—
506C22C3×#100714yes—
507C22C24×C3×D311yescharacteristic
508C4—24no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—24no—
513C4—64no—
514C4—64no—
515C4—64no—
516C3—21no—
517C3C2×#573511yescharacteristic, direct product factor
518C3C2×C3×#56311yescharacteristic, complemented
519C2—124no—
520C2C3×#573514yesdirect product factor
521C2C2×C3×#100411yescharacteristic
522C2C2×C3×#100712yes—
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×#563
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×#946C2semidirect
4. C2×C3×#944C2semidirect
5. C3×#5735C2direct (N × H)
6. C2×C3×#898C2semidirect
7. C2×C3×#896C2semidirect
9. C2×C3×#889C2semidirect
10. C2×C3×#876C2semidirect
12. C2×#5735C3direct (N × H)
18. C32×C4⋊Q8C22semidirect
23. C3×#946C22semidirect
24. C3×#944C22semidirect
38. C3×#898C22semidirect
39. C3×#896C22semidirect
41. C3×#889C22semidirect
42. C3×#876C22semidirect
46. C2×C3×C4⋊Q8D3semidirect
47. C2×C3×C4⋊Q8C2×C3semidirect
56. C2×#946C2×C3semidirect
57. C2×#944C2×C3semidirect
58. #5735C2×C3direct (N × H)
59. C2×#898C2×C3semidirect
60. C2×#896C2×C3semidirect
62. C2×#889C2×C3semidirect
63. C2×#876C2×C3semidirect
114. C3×C4⋊Q8C2×D3semidirect
115. C3×C4⋊Q8C22×C3semidirect
146. #946C22×C3semidirect
147. #944C22×C3semidirect
161. #898C22×C3semidirect
162. #896C22×C3semidirect
164. #889C22×C3semidirect
165. #876C22×C3semidirect
193. C2×C4⋊Q8C3×D3semidirect
303. C4⋊Q8C2×C3×D3semidirect
390. C2×C32#563semidirect
465. C32C2×#563semidirect
495. C2×C3#5735direct (N × H)
496. C2×C3C3×#563semidirect
517. C3C2×#5735direct (N × H)
518. C3C2×C3×#563semidirect
520. C2C3×#5735direct (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×#9462
C2×C3×#9442
C3×#57352
C2×C3×#8982
C2×C3×#8962
C2×C3×#8942
C2×C3×#8892
C2×C3×#8762
C2×C3×#5633
C2×#57353