Finite Group Databasebeta

← C3×#68877 #11786501

Subgroups — C3×#68877 #11786501

Order 1152.

689 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#892C211yesmaximal, characteristic
2C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
3C2×C3×#818C211yesmaximal, characteristic, complemented
4C3×#4987C211yesmaximal, characteristic
5C2×C3×#808C211yesmaximal, characteristic, complemented
6C32×#340C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1854—31nomaximal
9#68877C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×D3C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4×Dic3—21no—
19C22×C3×C3⋊C8C2211yescharacteristic
20C3×#892—22no—
21C3×#892—22no—
22C3×C3⋊#95—21no—
23C3×(C2×D3)⋊C8C414yescomplemented
24C3×#818—22no—
25C3×#808—22no—
26C2×C3×#117—31no—
27C2×#892C2×C311yescharacteristic
28C2×C3×#107—31no—
29C2×C3×#106—31no—
30C2×C3×C22⋊C8—31no—
31C3×#380—31no—
32C2×(C2×D3)⋊C8C2×C311yescharacteristic
33C2×#818C2×C311yescharacteristic, complemented
34#4987C2×C311yescharacteristic
35C2×#808C2×C311yescharacteristic, complemented
36C3×#340—21no—
37C3×#340—31no—
38C3×#340D311yescharacteristic, complemented
39C3×#340C2×C311yescharacteristic, complemented
40#4529C2×C311yescharacteristic, complemented
41C23×C3×D3D411yescharacteristic
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4C2311yescharacteristic
45C22×C3×Dic3—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3D411yescharacteristic
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—21no—
52C2×C3×C4×D3C2×C412yes—
53C2×C3×C4×D3Q812yes—
54C2×C32×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8C2×C412yes—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4D412yes—
61C3×(C2×D3)⋊C4—24no—
62C3×(C2×D3)⋊C4—42no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3D412yes—
67C3×C4×Dic3—24no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8C2×C412yes—
71#1854—31noSylow
72C23×C3×C4—31no—
73C22×C4×D3C22×C311yescharacteristic
74C22×C3×C8—21no—
75C22×C3×C8—31no—
76C22×C3×C8C2×D311yescharacteristic
77C22×C3×C8C22×C311yescharacteristic
78C3×#117—62no—
79C3×#117—62no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—31no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C4⋊C4C2×D311yescharacteristic
85C2×C3×C4⋊C4C2×D311yescharacteristic
86C2×C3×C4⋊C4C22×C311yescharacteristic
87C2×C3×C4⋊C4C22×C311yescharacteristic
88C2×C3×C22⋊C4—61no—
89C2×C3×C22⋊C4—61no—
90C2×C3×C42—61no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C22×C311yescharacteristic
94C2×C4⋊Dic3C22×C311yescharacteristic
95C2×C4×Dic3—21no—
96C22×C3⋊C8C22×C311yescharacteristic
97#892—22no—
98#892—22no—
99C3×#107—62no—
100C3×#106—62no—
101C3×C22⋊C8—34no—
102C3×#95—61no—
103C3⋊#95—21no—
104(C2×D3)⋊C8C3×C414yescomplemented
105#818—22no—
106#808—22no—
107C23×C32C2×D411yescharacteristic
108C22×C3×D3—21no—
109C22×C3×D3—21no—
110C22×C3×D3—21no—
111C22×C3×D3—21no—
112C22×C3×D3—21no—
113C22×C3×D3—21no—
114C22×C3×D3C4⋊C412yes—
115C2×C32×C4—22no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4C22×C411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×Q811yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3C4⋊C412yes—
136C3×C4×D3—22no—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×C4×D3—22no—
140C32×C8—22no—
141C3×C3⋊C8—22no—
142C2×#117—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×C22⋊C8—31no—
146#380—31no—
147#340—31no—
148#340C3×D311yescharacteristic, complemented
149C24×C3—31no—
150C23×D3C3×D411yescharacteristic
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—31no—
157C22×C3×C4—32no—
158C22×C3×C4—32no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C22×C3×C4—61no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C23×C311yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3C3×D411yescharacteristic
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3C2×C3×C412yes—
177C2×C4×D3C3×Q812yes—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—32no—
184C2×C3×C8—41no—
185C2×C3×C8—41no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C2×C3×C8C4×D312yes—
189C2×C3×C8C2×C3×C412yes—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—32no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4D1212yes—
205C3×C4⋊C4C3×D412yes—
206C3×C22⋊C4—64no—
207C3×C22⋊C4—122no—
208C3×C42—64no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—42no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C3×D412yes—
215C4×Dic3—24no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8C2×C3×C412yes—
219C22×C32C2×C4⋊C411yescharacteristic
220C22×C32#12011yescharacteristic
221C22×C32#12311yescharacteristic
222C22×C32#13511yescharacteristic
223C22×C32#13511yescharacteristic
224C22×C32AGL(1,ℤ/8)11yescharacteristic
225C22×C32#13711yescharacteristic
226C2×C3×D3—22no—
227C2×C3×D3—22no—
228C2×C3×D3—22no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C32×C4—42no—
234C32×C4—81no—
235C32×C4C4×D411yescharacteristic
236C32×C4C4×D411yescharacteristic
237C32×C4C22⋊Q811yescharacteristic
238C32×C4C22⋊Q811yescharacteristic
239C3×Dic3—22no—
240C3×Dic3—42no—
241C3×Dic3—81no—
242C23×C4—31no—
243C22×C8—31no—
244C22×C8C2×C3×D311yescharacteristic
245#117—62no—
246#117—62no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4C2×C3×D311yescharacteristic
250C2×C4⋊C4C2×C3×D311yescharacteristic
251C2×C22⋊C4—61no—
252C2×C22⋊C4—61no—
253C2×C42—61no—
254#107—62no—
255#106—62no—
256C22⋊C8—34no—
257#95—61no—
258C23×C3—21no—
259C23×C3—32no—
260C23×C3—61no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—61no—
264C23×C3—61no—
265C23×C3—61no—
266C23×C3D3×D411yescharacteristic
267C23×C3C2×C3×D411yescharacteristic
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3C3×C4⋊C412yes—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—32no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—81no—
304C2×C3×C4—81no—
305C2×C3×C4—81no—
306C2×C3×C4—81no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4—121no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4C2×C4×D311yescharacteristic
313C2×C3×C4C2×D1211yescharacteristic
314C2×C3×C4#23511yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4#23711yescharacteristic
317C2×C3×C4D3×Q811yescharacteristic
318C2×C3×C4C22×C3×C411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321C2×C3×C4C2×C3×Q811yescharacteristic
322C2×C3×C4C3×#4111yescharacteristic
323C2×C3×C4C3×#4111yescharacteristic
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3C3×C4⋊C412yes—
333C4×D3—22no—
334C4×D3—22no—
335C4×D3—22no—
336C4×D3—22no—
337C3×C8—22no—
338C3×C8—22no—
339C3×C8—42no—
340C3×C8—62no—
341C3⋊C8—22no—
342C2×C32#38511yescharacteristic
343C2×C32#42711yescharacteristic
344C2×C32#42811yescharacteristic
345C2×C32#44911yescharacteristic
346C2×C32#45211yescharacteristic
347C2×C32#48211yescharacteristic
348C2×C32#48511yescharacteristic
349C3×D3—42no—
350C24—31no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—31no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4C22×C3×D311yescharacteristic
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—32no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8C3×C4×D312yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—32no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4C3×D1212yes—
377C22⋊C4—64no—
378C22⋊C4—122no—
379C42—64no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3#88211yescharacteristic
395C22×C3#88311yescharacteristic
396C22×C3D3×C4⋊C411yescharacteristic
397C22×C3#91211yescharacteristic
398C22×C3#91411yescharacteristic
399C22×C3#91511yescharacteristic
400C22×C3#91911yescharacteristic
401C22×C3C2×C3×C4⋊C411yescharacteristic
402C22×C3C3×#12011yescharacteristic
403C22×C3C3×#12311yescharacteristic
404C22×C3C3×#13511yescharacteristic
405C22×C3C3×#13511yescharacteristic
406C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
407C22×C3C3×#13711yescharacteristic
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—22no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C2×D3—41no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—62no—
422C3×C4—82no—
423C3×C4—82no—
424C3×C4—81no—
425C3×C4—81no—
426C3×C4—122no—
427C3×C4—241no—
428C3×C4C4×D1211yescharacteristic
429C3×C4#67⋊C411yescharacteristic
430C3×C4#89611yescharacteristic
431C3×C4#89711yescharacteristic
432C3×C4C3×C4×D411yescharacteristic
433C3×C4C3×C4×D411yescharacteristic
434C3×C4C3×C22⋊Q811yescharacteristic
435C3×C4C3×C22⋊Q811yescharacteristic
436Dic3—22no—
437Dic3—42no—
438Dic3—81no—
439C32#185411yescharacteristic, complemented, Sylow
440C23—32no—
441C23—61no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23—61no—
446C23—61no—
447C23C3×D3×D411yescharacteristic
448C2×C4—22no—
449C2×C4—32no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4C2×C3×C4×D311yescharacteristic
468C2×C4C2×C3×D1211yescharacteristic
469C2×C4C3×#23511yescharacteristic
470C2×C4C3×D3×D411yescharacteristic
471C2×C4C3×#23711yescharacteristic
472C2×C4C3×D3×Q811yescharacteristic
473C8—22no—
474C8—62no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—122no—
483C2×C3#466811yescharacteristic
484C2×C3#478311yescharacteristic
485C2×C3#478411yescharacteristic
486C2×C3#481011yescharacteristic
487C2×C3#481411yescharacteristic
488C2×C3#486111yescharacteristic
489C2×C3#488011yescharacteristic
490C2×C3C3×#38511yescharacteristic
491C2×C3C3×#42711yescharacteristic
492C2×C3C3×#42811yescharacteristic
493C2×C3C3×#44911yescharacteristic
494C2×C3C3×#45211yescharacteristic
495C2×C3C3×#48211yescharacteristic
496C2×C3C3×#48511yescharacteristic
497D3—42no—
498C22—62no—
499C22—62no—
500C22—62no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22C3×#88211yescharacteristic
506C22C3×#88311yescharacteristic
507C22C3×D3×C4⋊C411yescharacteristic
508C22C3×#91211yescharacteristic
509C22C3×#91411yescharacteristic
510C22C3×#91511yescharacteristic
511C22C3×#91911yescharacteristic
512C4—42no—
513C4—62no—
514C4—81no—
515C4—122no—
516C4—241no—
517C4C3×C4×D1211yescharacteristic
518C4C3×#67⋊C411yescharacteristic
519C4C3×#89611yescharacteristic
520C4C3×#89711yescharacteristic
521C3—21no—
522C3#6887711yescharacteristic, direct product factor
523C3C3×#185411yescharacteristic, complemented
524C2—122no—
525C2C3×#466811yescharacteristic
526C2C3×#478311yescharacteristic
527C2C3×#478411yescharacteristic
528C2C3×#481011yescharacteristic
529C2C3×#481411yescharacteristic
530C2C3×#486111yescharacteristic
531C2C3×#488011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#340C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
cosocleC22×C3×C4C23×C3
hypercenterC3×#340D3
hypocenterC3C3×#1854
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
3. C2×C3×#818C2semidirect
5. C2×C3×#808C2semidirect
6. C32×#340C2semidirect
7. C3×#4529C2semidirect
9. #68877C3direct (N × H)
23. C3×(C2×D3)⋊C8C4semidirect
33. C2×#818C2×C3semidirect
35. C2×#808C2×C3semidirect
38. C3×#340D3semidirect
39. C3×#340C2×C3semidirect
40. #4529C2×C3semidirect
104. (C2×D3)⋊C8C3×C4semidirect
148. #340C3×D3semidirect
439. C32#1854semidirect
522. C3#68877direct (N × H)
523. C3C3×#1854semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8922
C2×C3×(C2×D3)⋊C82
C2×C3×#8182
C3×#49872
C2×C3×#8082
C32×#3402
C3×#45292
C3×#18543
#688773