Finite Group Databasebeta

← C2×C3×#5699 #11910298

Subgroups — C2×C3×#5699 #11910298

Order 1152.

2161 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#125C211yesmaximal, characteristic, complemented
2C3×#5699C2116yesmaximal, direct product factor
3C2×C3×#898C212yesmaximal, complemented
4C2×C3×#898C212yesmaximal, complemented
5C2×C3×#897C212yesmaximal, complemented
6C2×C3×#896C212yesmaximal, complemented
7C2×C3×#894C212yesmaximal
8C2×C3×#888C212yesmaximal, complemented
9C2×C3×#875C211yesmaximal, characteristic
10C2×C3×#870C211yesmaximal, characteristic, complemented
11C2×C3×#563—31nomaximal
12C2×#5699C311yesmaximal, characteristic, direct product factor
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×Dic6C2211yescharacteristic
16C22×C3×Dic6C2212yes—
17C32×#125C2218yescomplemented
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C42C2211yescharacteristic
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×(C2×D3)⋊C4C2212yes—
28C2×C3×C4⋊Dic3C2212yes—
29C2×C3×C4⋊Dic3C2211yescharacteristic
30C2×C3×C4⋊Dic3C2211yescharacteristic
31C2×C3×Dic3⋊C4C2212yes—
32C2×C3×Dic3⋊C4C2212yes—
33C2×C3×Dic3⋊C4C2212yes—
34C2×C3×C4×Dic3C2212yes—
35C3×#898C22116yescomplemented
36C3×#898C22116yescomplemented
37C3×#897C22116yescomplemented
38C3×#896C22116yescomplemented
39C3×#894C22116yes—
40C3×#888C22116yescomplemented
41C3×#875C2218yes—
42C3×#870C2218yescomplemented
43C3×#563—316no—
44C2×C3×C4⋊Q8—32no—
45C2×C3×#126—32no—
46C2×C3×#126—32no—
47C2×C3×#125—21no—
48C2×C3×#125—31no—
49C2×C3×#125D311yescharacteristic, complemented
50C2×C3×#125C2×C311yescharacteristic, complemented
51C2×C3×#124—31no—
52C2×C3×#123—32no—
53C2×C3×C22⋊Q8—32no—
54C2×C3×C22⋊Q8—32no—
55#5699C2×C3116yesdirect product factor
56C2×#898C2×C312yescomplemented
57C2×#898C2×C312yescomplemented
58C2×#897C2×C312yescomplemented
59C2×#896C2×C312yescomplemented
60C2×#894C2×C312yes—
61C2×#888C2×C312yescomplemented
62C2×#875C2×C311yescharacteristic
63C2×#870C2×C311yescharacteristic, complemented
64C23×C3×D3C2312yes—
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2312yes—
69C22×C32×C4C2311yescharacteristic
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2312yes—
73C2×C3×D12—42no—
74C2×C3×D12C2314yes—
75C2×C3×C4×D3—44no—
76C2×C3×C4×D3C2318yes—
77C2×C3×Dic6—42no—
78C2×C3×Dic6—44no—
79C2×C3×Dic6C2314yes—
80C2×C3×Dic6C2318yes—
81C32×C4⋊C4C2318yes—
82C32×C4⋊C4C2314yes—
83C32×C4⋊C4C2318yes—
84C32×C4⋊C4C2314yes—
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×C4⋊Dic3C2314yes—
93C3×C4⋊Dic3C2314yes—
94C3×Dic3⋊C4C2318yes—
95C3×Dic3⋊C4C2318yes—
96C3×Dic3⋊C4C2318yes—
97C3×C4×Dic3C2318yes—
98C2×#563—31noSylow
99C22×C3×Q8—32no—
100C22×C3×Q8—31no—
101C22×C3×D4—31no—
102C23×C3×C4—32no—
103C22×D12C22×C311yescharacteristic
104C22×C4×D3C22×C312yes—
105C22×Dic6C22×C311yescharacteristic
106C22×Dic6C22×C312yes—
107C3×C4⋊Q8—316no—
108C3×#126—316no—
109C3×#126—316no—
110C3×#125—28no—
111C3×#125—38no—
112C3×#125C2×D318yescomplemented
113C3×#125C22×C318yescomplemented
114C3×#124—38no—
115C3×#123—316no—
116C3×C22⋊Q8—316no—
117C3×C22⋊Q8—316no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4—22no—
120C2×C3×C4⋊C4—22no—
121C2×C3×C4⋊C4—21no—
122C2×C3×C4⋊C4—32no—
123C2×C3×C4⋊C4—32no—
124C2×C3×C4⋊C4—32no—
125C2×C3×C4⋊C4—31no—
126C2×C3×C4⋊C4—32no—
127C2×C3×C4⋊C4—31no—
128C2×C3×C4⋊C4C2×D311yescharacteristic
129C2×C3×C4⋊C4C2×D312yes—
130C2×C3×C4⋊C4C2×D312yes—
131C2×C3×C4⋊C4C2×D311yescharacteristic
132C2×C3×C4⋊C4C22×C312yes—
133C2×C3×C4⋊C4C22×C311yescharacteristic
134C2×C3×C4⋊C4C22×C312yes—
135C2×C3×C4⋊C4C22×C311yescharacteristic
136C2×C3×C22⋊C4—32no—
137C2×C3×C22⋊C4—32no—
138C2×C3×C22⋊C4—32no—
139C2×C3×C22⋊C4—32no—
140C2×C3×C22⋊C4—32no—
141C2×C3×C42—21no—
142C2×C3×C42—32no—
143C2×C3×C42C2×D311yescharacteristic
144C2×C3×C42C22×C311yescharacteristic
145C2×(C2×D3)⋊C4C22×C312yes—
146C2×(C2×D3)⋊C4C22×C312yes—
147C2×(C2×D3)⋊C4C22×C312yes—
148C2×(C2×D3)⋊C4C22×C312yes—
149C2×(C2×D3)⋊C4C22×C312yes—
150C2×C4⋊Dic3C22×C311yescharacteristic
151C2×C4⋊Dic3C22×C311yescharacteristic
152C2×C4⋊Dic3C22×C312yes—
153C2×Dic3⋊C4C22×C312yes—
154C2×Dic3⋊C4C22×C312yes—
155C2×Dic3⋊C4C22×C312yes—
156C2×C4×Dic3C22×C312yes—
157#898C22×C3116yescomplemented
158#898C22×C3116yescomplemented
159#897C22×C3116yescomplemented
160#896C22×C3116yescomplemented
161#894C22×C3116yes—
162#888C22×C3116yescomplemented
163#875C22×C318yes—
164#870C22×C318yescomplemented
165C23×C32C2411yescharacteristic
166C22×C3×D3—24no—
167C22×C3×D3—24no—
168C22×C3×D3—24no—
169C22×C3×D3C2414yes—
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4—22no—
173C2×C32×C4—24no—
174C2×C32×C4—22no—
175C2×C32×C4C2414yes—
176C2×C32×C4C2412yes—
177C2×C32×C4C2412yes—
178C2×C32×C4C2414yes—
179C2×C32×C4C2412yes—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3C2414yes—
184C2×C3×Dic3C2414yes—
185C2×C3×Dic3C2414yes—
186C3×D12—44no—
187C3×C4×D3—48no—
188C3×Dic6—44no—
189C3×Dic6—48no—
190#563—316no—
191C2×C4⋊Q8—32no—
192C2×#126—32no—
193C2×#126—32no—
194C2×#125—31no—
195C2×#125C3×D311yescharacteristic, complemented
196C2×#124—31no—
197C2×#123—32no—
198C2×C22⋊Q8—32no—
199C2×C22⋊Q8—32no—
200C24×C3—32no—
201C23×D3C23×C312yes—
202C2×C3×Q8—38no—
203C2×C3×Q8—34no—
204C2×C3×Q8—124no—
205C2×C3×Q8—122no—
206C2×C3×D4—34no—
207C2×C3×D4—122no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—32no—
214C22×C3×C4—32no—
215C22×C3×C4—32no—
216C22×C3×C4—38no—
217C22×C3×C4—124no—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C22×D312yes—
223C22×C3×C4C23×C312yes—
224C22×C3×C4C23×C311yescharacteristic
225C22×C3×C4C23×C311yescharacteristic
226C22×C3×C4C23×C312yes—
227C22×C3×C4C23×C311yescharacteristic
228C22×Dic3C23×C312yes—
229C22×Dic3C23×C312yes—
230C22×Dic3C23×C312yes—
231C2×D12—42no—
232C2×D12C23×C314yes—
233C2×C4×D3—44no—
234C2×C4×D3C23×C318yes—
235C2×Dic6—42no—
236C2×Dic6—44no—
237C2×Dic6C23×C314yes—
238C2×Dic6C23×C318yes—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—28no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—38no—
244C3×C4⋊C4—38no—
245C3×C4⋊C4—38no—
246C3×C4⋊C4—34no—
247C3×C4⋊C4—38no—
248C3×C4⋊C4—34no—
249C3×C4⋊C4C22×D314yes—
250C3×C4⋊C4C22×D318yes—
251C3×C4⋊C4C22×D318yes—
252C3×C4⋊C4C22×D314yes—
253C3×C4⋊C4C23×C318yes—
254C3×C4⋊C4C23×C314yes—
255C3×C4⋊C4C23×C318yes—
256C3×C4⋊C4C23×C314yes—
257C3×C22⋊C4—38no—
258C3×C22⋊C4—38no—
259C3×C22⋊C4—38no—
260C3×C22⋊C4—38no—
261C3×C22⋊C4—38no—
262C3×C42—24no—
263C3×C42—38no—
264C3×C42C22×D314yes—
265C3×C42C23×C314yes—
266(C2×D3)⋊C4C23×C318yes—
267(C2×D3)⋊C4C23×C318yes—
268(C2×D3)⋊C4C23×C318yes—
269(C2×D3)⋊C4C23×C318yes—
270(C2×D3)⋊C4C23×C318yes—
271C4⋊Dic3C23×C314yes—
272C4⋊Dic3C23×C314yes—
273C4⋊Dic3C23×C318yes—
274Dic3⋊C4C23×C318yes—
275Dic3⋊C4C23×C318yes—
276Dic3⋊C4C23×C318yes—
277C4×Dic3C23×C318yes—
278C22×C32H(5,𝔽2)12yes—
279C22×C322−1+412yes—
280C22×C322−1+412yes—
281C22×C32C2511yescharacteristic
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—48no—
286C32×C4—22no—
287C32×C4—24no—
288C32×C4—22no—
289C32×C4—24no—
290C32×C4—22no—
291C3×Dic3—24no—
292C3×Dic3—24no—
293C3×Dic3—24no—
294C22×Q8—32no—
295C22×Q8—31no—
296C22×D4—31no—
297C23×C4—32no—
298C4⋊Q8—316no—
299#126—316no—
300#126—316no—
301#125—38no—
302#125C2×C3×D318yescomplemented
303#124—38no—
304#123—316no—
305C22⋊Q8—316no—
306C22⋊Q8—316no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4C2×C3×D311yescharacteristic
314C2×C4⋊C4C2×C3×D312yes—
315C2×C4⋊C4C2×C3×D312yes—
316C2×C4⋊C4C2×C3×D311yescharacteristic
317C2×C22⋊C4—32no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C42—32no—
323C2×C42C2×C3×D311yescharacteristic
324C23×C3—21no—
325C23×C3—34no—
326C23×C3—64no—
327C23×C3—64no—
328C23×C3—64no—
329C23×C3C23×D311yescharacteristic
330C23×C3C24×C311yescharacteristic
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—24no—
334C22×D3C24×C314yes—
335C3×Q8—128no—
336C3×Q8—124no—
337C3×D4—124no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—34no—
355C2×C3×C4—34no—
356C2×C3×C4—34no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—64no—
362C2×C3×C4—64no—
363C2×C3×C4—64no—
364C2×C3×C4—128no—
365C2×C3×C4C23×D314yes—
366C2×C3×C4C23×D312yes—
367C2×C3×C4C23×D312yes—
368C2×C3×C4C23×D312yes—
369C2×C3×C4C23×D314yes—
370C2×C3×C4C24×C314yes—
371C2×C3×C4C24×C312yes—
372C2×C3×C4C24×C312yes—
373C2×C3×C4C24×C314yes—
374C2×C3×C4C24×C312yes—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3C24×C314yes—
379C2×Dic3C24×C314yes—
380C2×Dic3C24×C314yes—
381D12—44no—
382C4×D3—48no—
383Dic6—44no—
384Dic6—48no—
385C2×C32#56314yescomplemented
386C2×C32C2×H(5,𝔽2)11yescharacteristic
387C2×C32C2×2−1+411yescharacteristic
388C2×C32C2×2−1+411yescharacteristic
389C3×D3—44no—
390C24—32no—
391C2×Q8—38no—
392C2×Q8—34no—
393C2×Q8—124no—
394C2×Q8—122no—
395C2×D4—34no—
396C2×D4—122no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—38no—
401C22×C4—124no—
402C22×C4C22×C3×D312yes—
403C22×C4C22×C3×D311yescharacteristic
404C22×C4C22×C3×D311yescharacteristic
405C22×C4C22×C3×D311yescharacteristic
406C22×C4C22×C3×D312yes—
407C4⋊C4—34no—
408C4⋊C4—38no—
409C4⋊C4—38no—
410C4⋊C4—38no—
411C4⋊C4—38no—
412C4⋊C4—34no—
413C4⋊C4C22×C3×D314yes—
414C4⋊C4C22×C3×D318yes—
415C4⋊C4C22×C3×D318yes—
416C4⋊C4C22×C3×D314yes—
417C22⋊C4—38no—
418C22⋊C4—38no—
419C22⋊C4—38no—
420C22⋊C4—38no—
421C22⋊C4—38no—
422C42—38no—
423C42C22×C3×D314yes—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—64no—
431C22×C3—128no—
432C22×C3#100712yes—
433C22×C3#100912yes—
434C22×C3#101012yes—
435C22×C3C3×H(5,𝔽2)12yes—
436C22×C3C3×2−1+412yes—
437C22×C3C3×2−1+412yes—
438C22×C3C24×D311yescharacteristic
439C22×C3C25×C311yescharacteristic
440C2×D3—24no—
441C2×D3—24no—
442C2×D3—24no—
443C2×D3—48no—
444C3×C4—24no—
445C3×C4—22no—
446C3×C4—24no—
447C3×C4—22no—
448C3×C4—24no—
449C3×C4—22no—
450C3×C4—24no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—24no—
454C3×C4—22no—
455C3×C4—44no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—64no—
460C3×C4—64no—
461C3×C4—64no—
462Dic3—24no—
463Dic3—24no—
464Dic3—24no—
465C32C2×#56311yescharacteristic, complemented, Sylow
466C23—34no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23C23×C3×D311yescharacteristic
471Q8—128no—
472Q8—124no—
473D4—124no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—34no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—128no—
486C2×C4C23×C3×D314yes—
487C2×C4C23×C3×D312yes—
488C2×C4C23×C3×D312yes—
489C2×C4C23×C3×D312yes—
490C2×C4C23×C3×D314yes—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—24no—
495C2×C3—124no—
496C2×C3#569914yesdirect product factor
497C2×C3C3×#56314yescomplemented
498C2×C3C2×#100711yescharacteristic
499C2×C3C2×#100911yescharacteristic
500C2×C3C2×#101011yescharacteristic
501C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
502C2×C3C2×C3×2−1+411yescharacteristic
503C2×C3C2×C3×2−1+411yescharacteristic
504D3—44no—
505C22—64no—
506C22—64no—
507C22—64no—
508C22—128no—
509C22C3×#100712yes—
510C22C3×#100912yes—
511C22C3×#101012yes—
512C22C24×C3×D311yescharacteristic
513C4—22no—
514C4—24no—
515C4—22no—
516C4—24no—
517C4—22no—
518C4—64no—
519C4—64no—
520C4—64no—
521C3—21no—
522C3C2×#569911yescharacteristic, direct product factor
523C3C2×C3×#56311yescharacteristic, complemented
524C2—124no—
525C2C3×#569914yesdirect product factor
526C2C2×C3×#100711yescharacteristic
527C2C2×C3×#100911yescharacteristic
528C2C2×C3×#101011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#125C2
socleC23×C32C24
prime radicalC22×C3×D12C22
hypercenterC2×C3×#125D3
hypocenterC3C2×C3×#563
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#125

Complements

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

NH ≅ G/Ntype
1. C2×C32×#125C2semidirect
2. C3×#5699C2direct (N × H)
3. C2×C3×#898C2semidirect
4. C2×C3×#898C2semidirect
5. C2×C3×#897C2semidirect
6. C2×C3×#896C2semidirect
8. C2×C3×#888C2semidirect
10. C2×C3×#870C2semidirect
12. C2×#5699C3direct (N × H)
17. C32×#125C22semidirect
35. C3×#898C22semidirect
36. C3×#898C22semidirect
37. C3×#897C22semidirect
38. C3×#896C22semidirect
40. C3×#888C22semidirect
42. C3×#870C22semidirect
49. C2×C3×#125D3semidirect
50. C2×C3×#125C2×C3semidirect
55. #5699C2×C3direct (N × H)
56. C2×#898C2×C3semidirect
57. C2×#898C2×C3semidirect
58. C2×#897C2×C3semidirect
59. C2×#896C2×C3semidirect
61. C2×#888C2×C3semidirect
63. C2×#870C2×C3semidirect
112. C3×#125C2×D3semidirect
113. C3×#125C22×C3semidirect
157. #898C22×C3semidirect
158. #898C22×C3semidirect
159. #897C22×C3semidirect
160. #896C22×C3semidirect
162. #888C22×C3semidirect
164. #870C22×C3semidirect
195. C2×#125C3×D3semidirect
302. #125C2×C3×D3semidirect
385. C2×C32#563semidirect
465. C32C2×#563semidirect
496. C2×C3#5699direct (N × H)
497. C2×C3C3×#563semidirect
522. C3C2×#5699direct (N × H)
523. C3C2×C3×#563semidirect
525. C2C3×#5699direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1252
C3×#56992
C2×C3×#8982
C2×C3×#8972
C2×C3×#8962
C2×C3×#8942
C2×C3×#8882
C2×C3×#8752
C2×C3×#8702
C2×C3×#5633
C2×#56993