Finite Group Databasebeta

← C2×C3×#5623 #11901889

Subgroups — C2×C3×#5623 #11901889

Order 1152.

2321 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#944C212yesmaximal, complemented
3C2×C3×#924C211yesmaximal, characteristic
4C22×C3×Dic3⋊C4C211yesmaximal, characteristic
5C3×#5623C2116yesmaximal, direct product factor
6C2×C3×#889C212yesmaximal, complemented
7C2×C3×#889C212yesmaximal, complemented
8C2×C3×#888C212yesmaximal, complemented
9C2×C3×#878C212yesmaximal
10C2×C3×#877C212yesmaximal
11C2×C3×#544—31nomaximal
12C2×#5623C311yesmaximal, characteristic, direct product factor
13C22×C32×Q8C2211yescharacteristic
14C23×C32×C4C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C22×C3×Dic6C2211yescharacteristic
18C22×C3×Dic6C2212yes—
19C32×C22⋊Q8C2218yes—
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2212yes—
23C2×C3×#217C2212yes—
24C3×#944C22116yescomplemented
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C3×#924C2218yes—
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×Dic3⋊C4C2212yes—
30C2×C3×Dic3⋊C4C2212yes—
31C2×C3×Dic3⋊C4C2212yes—
32C2×C3×Dic3⋊C4C2212yes—
33C2×C3×Dic3⋊C4C2212yes—
34C2×C3×Dic3⋊C4C2212yes—
35C2×C3×Dic3⋊C4C2218yes—
36C2×C3×C4×Dic3C2212yes—
37C2×C3×C4×Dic3C2212yes—
38C3×#889C22116yescomplemented
39C3×#889C22116yescomplemented
40C3×#888C22116yescomplemented
41C3×#878C22116yes—
42C3×#877C22116yes—
43C3×#544—316no—
44C2×C3×C4⋊Q8—32no—
45C2×C3×C4⋊Q8—32no—
46C2×C3×#125—32no—
47C2×C3×#125—32no—
48C2×C3×C22⋊Q8—21no—
49C2×C3×C22⋊Q8—31no—
50C2×C3×C22⋊Q8—32no—
51C2×C3×C22⋊Q8D311yescharacteristic
52C2×C3×C22⋊Q8C2×C311yescharacteristic
53C2×C3×#117—32no—
54C22×C3×C4⋊C4—31no—
55C2×#944C2×C312yescomplemented
56C2×#924C2×C311yescharacteristic
57C22×Dic3⋊C4C2×C311yescharacteristic
58#5623C2×C3116yesdirect product factor
59C2×#889C2×C312yescomplemented
60C2×#889C2×C312yescomplemented
61C2×#888C2×C312yescomplemented
62C2×#878C2×C312yes—
63C2×#877C2×C312yes—
64C24×C32C2311yescharacteristic
65C2×C32×Q8—42no—
66C2×C32×Q8C2314yes—
67C22×C32×C4—42no—
68C22×C32×C4C2312yes—
69C22×C32×C4C2312yes—
70C22×C32×C4C2311yescharacteristic
71C22×C32×C4C2311yescharacteristic
72C22×C32×C4C2314yes—
73C22×C3×Dic3—42no—
74C22×C3×Dic3Q818yescomplemented
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2312yes—
77C22×C3×Dic3C2312yes—
78C22×C3×Dic3C2314yes—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2314yes—
81C2×C3×Dic6—42no—
82C2×C3×Dic6—44no—
83C2×C3×Dic6C2318yes—
84C2×C3×Dic6C2314yes—
85C32×C4⋊C4C2318yes—
86C32×C4⋊C4C2314yes—
87C32×C22⋊C4C2318yes—
88C3×#217C2318yes—
89C3×C4⋊Dic3C2318yes—
90C3×C4⋊Dic3C2314yes—
91C3×Dic3⋊C4C2318yes—
92C3×Dic3⋊C4C2318yes—
93C3×Dic3⋊C4C2318yes—
94C3×Dic3⋊C4C2318yes—
95C3×Dic3⋊C4C2318yes—
96C3×Dic3⋊C4C2318yes—
97C3×Dic3⋊C4C2318yes—
98C3×C4×Dic3C2318yes—
99C3×C4×Dic3C2318yes—
100C2×#544—31noSylow
101C22×C3×Q8—21no—
102C22×C3×Q8—31no—
103C22×C3×Q8—32no—
104C22×C3×Q8C2×D311yescharacteristic
105C22×C3×Q8C22×C311yescharacteristic
106C23×C3×C4—21no—
107C23×C3×C4—31no—
108C23×C3×C4—31no—
109C23×C3×C4C2×D311yescharacteristic
110C23×C3×C4C22×C311yescharacteristic
111C23×Dic3C22×C311yescharacteristic
112C23×Dic3C22×C311yescharacteristic
113C22×Dic6C22×C312yes—
114C22×Dic6C22×C311yescharacteristic
115C3×C4⋊Q8—316no—
116C3×C4⋊Q8—316no—
117C3×#125—316no—
118C3×#125—316no—
119C3×C22⋊Q8—28no—
120C3×C22⋊Q8—38no—
121C3×C22⋊Q8—316no—
122C3×C22⋊Q8C2×D318yes—
123C3×C22⋊Q8C22×C318yes—
124C3×#117—316no—
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4—22no—
127C2×C3×C4⋊C4—32no—
128C2×C3×C4⋊C4—32no—
129C2×C3×C4⋊C4—38no—
130C2×C3×C4⋊C4—32no—
131C2×C3×C4⋊C4—32no—
132C2×C3×C4⋊C4—32no—
133C2×C3×C4⋊C4—32no—
134C2×C3×C4⋊C4—32no—
135C2×C3×C4⋊C4—31no—
136C2×C3×C4⋊C4—32no—
137C2×C3×C4⋊C4C2×D311yescharacteristic
138C2×C3×C4⋊C4C2×D312yes—
139C2×C3×C4⋊C4C22×C312yes—
140C2×C3×C4⋊C4C22×C311yescharacteristic
141C2×C3×C22⋊C4—22no—
142C2×C3×C22⋊C4—32no—
143C2×C3×C22⋊C4C2×D312yes—
144C2×C3×C22⋊C4C22×C312yes—
145C2×C3×C42—32no—
146C2×C3×C42—32no—
147C2×#217C22×C312yes—
148#944C22×C3116yescomplemented
149C2×C4⋊Dic3C22×C312yes—
150C2×C4⋊Dic3C22×C311yescharacteristic
151#924C22×C318yes—
152C2×Dic3⋊C4C22×C312yes—
153C2×Dic3⋊C4C22×C312yes—
154C2×Dic3⋊C4C22×C312yes—
155C2×Dic3⋊C4C22×C312yes—
156C2×Dic3⋊C4C22×C312yes—
157C2×Dic3⋊C4C22×C312yes—
158C2×Dic3⋊C4C22×C312yes—
159C2×Dic3⋊C4C22×C318yes—
160C2×C4×Dic3C22×C312yes—
161C2×C4×Dic3C22×C312yes—
162#889C22×C3116yescomplemented
163#889C22×C3116yescomplemented
164#888C22×C3116yescomplemented
165#878C22×C3116yes—
166#877C22×C3116yes—
167C23×C32—22no—
168C23×C32—22no—
169C23×C32C2×Q814yes—
170C23×C32C2412yes—
171C23×C32C2411yescharacteristic
172C32×Q8—44no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—44no—
178C2×C32×C4C2414yes—
179C2×C32×C4C2414yes—
180C2×C32×C4C2412yes—
181C2×C32×C4C2412yes—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—44no—
186C2×C3×Dic3C2×Q818yes—
187C2×C3×Dic3C2×Q8116yescomplemented
188C2×C3×Dic3C2414yes—
189C2×C3×Dic3C2414yes—
190C2×C3×Dic3C2414yes—
191C2×C3×Dic3C2414yes—
192C3×Dic6—44no—
193C3×Dic6—48no—
194#544—316no—
195C2×C4⋊Q8—32no—
196C2×C4⋊Q8—32no—
197C2×#125—32no—
198C2×#125—32no—
199C2×C22⋊Q8—31no—
200C2×C22⋊Q8—32no—
201C2×C22⋊Q8C3×D311yescharacteristic
202C2×#117—32no—
203C22×C4⋊C4—31no—
204C24×C3—21no—
205C24×C3C22×D311yescharacteristic
206C24×C3C23×C311yescharacteristic
207C2×C3×Q8—24no—
208C2×C3×Q8—34no—
209C2×C3×Q8—38no—
210C2×C3×Q8—42no—
211C2×C3×Q8—42no—
212C2×C3×Q8—44no—
213C2×C3×Q8—122no—
214C2×C3×Q8—124no—
215C2×C3×Q8C22×D314yes—
216C2×C3×Q8C23×C314yes—
217C22×C3×C4—21no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—24no—
222C22×C3×C4—38no—
223C22×C3×C4—34no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—32no—
228C22×C3×C4—34no—
229C22×C3×C4—42no—
230C22×C3×C4—42no—
231C22×C3×C4—44no—
232C22×C3×C4—122no—
233C22×C3×C4C22×D312yes—
234C22×C3×C4C22×D312yes—
235C22×C3×C4C22×D311yescharacteristic
236C22×C3×C4C22×D311yescharacteristic
237C22×C3×C4C22×D314yes—
238C22×C3×C4C23×C312yes—
239C22×C3×C4C23×C312yes—
240C22×C3×C4C23×C311yescharacteristic
241C22×C3×C4C23×C311yescharacteristic
242C22×C3×C4C23×C314yes—
243C22×Dic3—42no—
244C22×Dic3C3×Q818yescomplemented
245C22×Dic3C23×C312yes—
246C22×Dic3C23×C312yes—
247C22×Dic3C23×C312yes—
248C22×Dic3C23×C314yes—
249C22×Dic3C23×C312yes—
250C22×Dic3C23×C314yes—
251C2×Dic6—42no—
252C2×Dic6—44no—
253C2×Dic6C23×C318yes—
254C2×Dic6C23×C314yes—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—28no—
257C3×C4⋊C4—38no—
258C3×C4⋊C4—38no—
259C3×C4⋊C4—38no—
260C3×C4⋊C4—38no—
261C3×C4⋊C4—38no—
262C3×C4⋊C4—38no—
263C3×C4⋊C4—38no—
264C3×C4⋊C4—34no—
265C3×C4⋊C4—38no—
266C3×C4⋊C4C22×D314yes—
267C3×C4⋊C4C22×D318yes—
268C3×C4⋊C4C23×C318yes—
269C3×C4⋊C4C23×C314yes—
270C3×C22⋊C4—28no—
271C3×C22⋊C4—38no—
272C3×C22⋊C4C22×D318yes—
273C3×C22⋊C4C23×C318yes—
274C3×C42—38no—
275C3×C42—38no—
276#217C23×C318yes—
277C4⋊Dic3C23×C318yes—
278C4⋊Dic3C23×C314yes—
279Dic3⋊C4C23×C318yes—
280Dic3⋊C4C23×C318yes—
281Dic3⋊C4C23×C318yes—
282Dic3⋊C4C23×C318yes—
283Dic3⋊C4C23×C318yes—
284Dic3⋊C4C23×C318yes—
285Dic3⋊C4C23×C318yes—
286C4×Dic3C23×C318yes—
287C4×Dic3C23×C318yes—
288C22×C32—28no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32C22×Q814yes—
292C22×C32C22×Q812yes—
293C22×C32H(5,𝔽2)12yes—
294C22×C322−1+412yes—
295C22×C32C2511yescharacteristic
296C32×C4—22no—
297C32×C4—22no—
298C32×C4—24no—
299C32×C4—24no—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—24no—
303C3×Dic3C22×Q818yes—
304C22×Q8—31no—
305C22×Q8—32no—
306C22×Q8C2×C3×D311yescharacteristic
307C23×C4—31no—
308C23×C4—31no—
309C23×C4C2×C3×D311yescharacteristic
310C4⋊Q8—316no—
311C4⋊Q8—316no—
312#125—316no—
313#125—316no—
314C22⋊Q8—38no—
315C22⋊Q8—316no—
316C22⋊Q8C2×C3×D318yes—
317#117—316no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—38no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4C2×C3×D311yescharacteristic
329C2×C4⋊C4C2×C3×D312yes—
330C2×C22⋊C4—32no—
331C2×C22⋊C4C2×C3×D312yes—
332C2×C42—32no—
333C2×C42—32no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—21no—
339C23×C3—22no—
340C23×C3—24no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3D3×Q814yes—
344C23×C3C2×C3×Q814yes—
345C23×C3C23×D312yes—
346C23×C3C23×D311yescharacteristic
347C23×C3C24×C312yes—
348C23×C3C24×C311yescharacteristic
349C3×Q8—44no—
350C3×Q8—44no—
351C3×Q8—48no—
352C3×Q8—124no—
353C3×Q8—128no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—24no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—38no—
368C2×C3×C4—316no—
369C2×C3×C4—34no—
370C2×C3×C4—34no—
371C2×C3×C4—34no—
372C2×C3×C4—34no—
373C2×C3×C4—44no—
374C2×C3×C4—42no—
375C2×C3×C4—44no—
376C2×C3×C4—44no—
377C2×C3×C4—44no—
378C2×C3×C4—48no—
379C2×C3×C4—64no—
380C2×C3×C4—64no—
381C2×C3×C4—64no—
382C2×C3×C4—124no—
383C2×C3×C4C23×D314yes—
384C2×C3×C4C23×D314yes—
385C2×C3×C4C23×D312yes—
386C2×C3×C4C23×D312yes—
387C2×C3×C4C24×C314yes—
388C2×C3×C4C24×C314yes—
389C2×C3×C4C24×C312yes—
390C2×C3×C4C24×C312yes—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—44no—
395C2×Dic3C2×C3×Q818yes—
396C2×Dic3C2×C3×Q8116yescomplemented
397C2×Dic3C24×C314yes—
398C2×Dic3C24×C314yes—
399C2×Dic3C24×C314yes—
400C2×Dic3C24×C314yes—
401Dic6—44no—
402Dic6—48no—
403C2×C32—24no—
404C2×C32#54414yescomplemented
405C2×C32C23×Q811yescharacteristic
406C2×C32C2×H(5,𝔽2)11yescharacteristic
407C2×C32C2×2−1+411yescharacteristic
408C24C22×C3×D311yescharacteristic
409C2×Q8—34no—
410C2×Q8—38no—
411C2×Q8—42no—
412C2×Q8—122no—
413C2×Q8—124no—
414C2×Q8C22×C3×D314yes—
415C22×C4—32no—
416C22×C4—34no—
417C22×C4—38no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—42no—
423C22×C4—122no—
424C22×C4C22×C3×D312yes—
425C22×C4C22×C3×D312yes—
426C22×C4C22×C3×D311yescharacteristic
427C22×C4C22×C3×D311yescharacteristic
428C22×C4C22×C3×D314yes—
429C4⋊C4—38no—
430C4⋊C4—38no—
431C4⋊C4—38no—
432C4⋊C4—38no—
433C4⋊C4—38no—
434C4⋊C4—34no—
435C4⋊C4—38no—
436C4⋊C4—38no—
437C4⋊C4—38no—
438C4⋊C4C22×C3×D314yes—
439C4⋊C4C22×C3×D318yes—
440C22⋊C4—38no—
441C22⋊C4C22×C3×D318yes—
442C42—38no—
443C42—38no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—28no—
449C22×C3—28no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—24no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—48no—
458C22×C3#100412yes—
459C22×C3C2×D3×Q814yes—
460C22×C3C2×D3×Q812yes—
461C22×C3#101012yes—
462C22×C3C22×C3×Q814yes—
463C22×C3C22×C3×Q812yes—
464C22×C3C3×H(5,𝔽2)12yes—
465C22×C3C3×2−1+412yes—
466C22×C3C24×D311yescharacteristic
467C22×C3C25×C311yescharacteristic
468C3×C4—24no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—24no—
472C3×C4—24no—
473C3×C4—22no—
474C3×C4—24no—
475C3×C4—22no—
476C3×C4—24no—
477C3×C4—38no—
478C3×C4—42no—
479C3×C4—44no—
480C3×C4—44no—
481C3×C4—64no—
482C3×C4—64no—
483C3×C4—64no—
484Dic3—24no—
485Dic3—24no—
486Dic3—24no—
487Dic3C22×C3×Q818yes—
488C32C2×#54411yescharacteristic, complemented, Sylow
489C23—22no—
490C23—22no—
491C23C3×D3×Q814yes—
492C23C23×C3×D312yes—
493C23C23×C3×D311yescharacteristic
494Q8—44no—
495Q8—124no—
496Q8—128no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—38no—
502C2×C4—316no—
503C2×C4—34no—
504C2×C4—34no—
505C2×C4—34no—
506C2×C4—34no—
507C2×C4—44no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—124no—
512C2×C4C23×C3×D314yes—
513C2×C4C23×C3×D314yes—
514C2×C4C23×C3×D312yes—
515C2×C4C23×C3×D312yes—
516C2×C3—24no—
517C2×C3—24no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—24no—
522C2×C3—44no—
523C2×C3#562314yesdirect product factor
524C2×C3C3×#54414yescomplemented
525C2×C3C2×#100411yescharacteristic
526C2×C3C22×D3×Q811yescharacteristic
527C2×C3C2×#101011yescharacteristic
528C2×C3C23×C3×Q811yescharacteristic
529C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
530C2×C3C2×C3×2−1+411yescharacteristic
531C22—22no—
532C22—22no—
533C22—28no—
534C22C3×#100412yes—
535C22C2×C3×D3×Q814yes—
536C22C2×C3×D3×Q812yes—
537C22C3×#101012yes—
538C22C24×C3×D311yescharacteristic
539C4—22no—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—38no—
544C4—64no—
545C4—64no—
546C4—64no—
547C3—21no—
548C3C2×#562311yescharacteristic, direct product factor
549C3C2×C3×#54411yescharacteristic, complemented
550C2—24no—
551C2C3×#562314yesdirect product factor
552C2C2×C3×#100411yescharacteristic
553C2C22×C3×D3×Q811yescharacteristic
554C2C2×C3×#101011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C24
prime radicalC24×C32C23
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C2×C3×#544
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#944C2semidirect
5. C3×#5623C2direct (N × H)
6. C2×C3×#889C2semidirect
7. C2×C3×#889C2semidirect
8. C2×C3×#888C2semidirect
12. C2×#5623C3direct (N × H)
24. C3×#944C22semidirect
38. C3×#889C22semidirect
39. C3×#889C22semidirect
40. C3×#888C22semidirect
55. C2×#944C2×C3semidirect
58. #5623C2×C3direct (N × H)
59. C2×#889C2×C3semidirect
60. C2×#889C2×C3semidirect
61. C2×#888C2×C3semidirect
74. C22×C3×Dic3Q8semidirect
148. #944C22×C3semidirect
162. #889C22×C3semidirect
163. #889C22×C3semidirect
164. #888C22×C3semidirect
187. C2×C3×Dic3C2×Q8semidirect
244. C22×Dic3C3×Q8semidirect
396. C2×Dic3C2×C3×Q8semidirect
404. C2×C32#544semidirect
488. C32C2×#544semidirect
523. C2×C3#5623direct (N × H)
524. C2×C3C3×#544semidirect
548. C3C2×#5623direct (N × H)
549. C3C2×C3×#544semidirect
551. C2C3×#5623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C2×C3×#9442
C2×C3×#9242
C22×C3×Dic3⋊C42
C3×#56232
C2×C3×#8892
C2×C3×#8882
C2×C3×#8782
C2×C3×#8772
C2×C3×#5443
C2×#56233