Finite Group Databasebeta

← C3×#68756 #11788162

Subgroups — C3×#68756 #11788162

Order 1152.

939 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5524C211yesmaximal, characteristic
2C2×C3×C42×D3C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic, complemented
4C3×#5126C211yesmaximal, characteristic, complemented
5C2×C3×#805C211yesmaximal, characteristic, complemented
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4532C211yesmaximal, characteristic, complemented
8C3×#1896—31nomaximal
9#68756C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×C4×D3—21no—
12C22×C3×C4×D3C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C32×#117C2211yescharacteristic
15C2×C32×C42C2211yescharacteristic
16C2×C3×C4×Dic3—21no—
17C2×C3×C4×Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#897C412yescomplemented
20C3×#895C412yescomplemented
21C3×#886—21no—
22C3×#882—21no—
23C3×C4×D12—22no—
24C3×C42×D3—22no—
25C3×C42×D3—22no—
26C3×#825—22no—
27C3×(C2×D3)⋊C8—22no—
28C3×#805—22no—
29C3×#525—31no—
30C22×C3×C42—31no—
31#5524C2×C311yescharacteristic
32C2×C42×D3C2×C311yescharacteristic, complemented
33C2×C3×#104—31no—
34C2×C3×#104—31no—
35C3×#407—31no—
36C2×#825C2×C311yescharacteristic, complemented
37#5126C2×C311yescharacteristic, complemented
38C2×#805C2×C311yescharacteristic, complemented
39C3×#339—21no—
40C3×#339—31no—
41C3×#339D311yescharacteristic, complemented
42C3×#339C2×C311yescharacteristic, complemented
43#4532C2×C311yescharacteristic, complemented
44C23×C3×D3D411yescharacteristic
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C2×C3×D12—22no—
55C2×C3×D12C2×C411yescharacteristic, complemented
56C2×C3×C4×D3—28no—
57C2×C3×C4×D3—24no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3—21no—
63C2×C3×C4×D3C2×C412yes—
64C2×C3×C4×D3D412yes—
65C2×C3×Dic6C2×C411yescharacteristic, complemented
66C32×M16—21no—
67C32×M16—21no—
68C2×C32×C8—21no—
69C32×C4⋊C4C2×C412yescomplemented
70C32×C22⋊C4—21no—
71C32×C42—21no—
72C32×C42—21no—
73C32×C42D412yescomplemented
74C3×(C2×D3)⋊C4—22no—
75C3×(C2×D3)⋊C4—22no—
76C3×C4⋊Dic3—22no—
77C3×C4×Dic3—24no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3D412yescomplemented
80C3×#208—21no—
81C3×#208—21no—
82C2×C3×C3⋊C8—21no—
83#1896—31noSylow
84C2×C3×#41—31no—
85C23×C3×C4—31no—
86C23×C3×C4—61no—
87C2×#235C22×C311yescharacteristic
88C22×C4×D3—21no—
89C22×C4×D3C22×C311yescharacteristic
90C2×C3×M16—21no—
91C2×C3×M16—31no—
92C2×C3×M16C2×D311yescharacteristic
93C2×C3×M16C22×C311yescharacteristic
94C3×#123—61no—
95C3×C22⋊Q8—32no—
96C3×#121—32no—
97C3×#120—61no—
98C3×C4×D4—62no—
99C3×#117—21no—
100C3×#117C2×D311yescharacteristic
101C3×#117C22×C311yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42—31no—
104C2×C3×C42—62no—
105C2×C3×C42—61no—
106C2×C3×C42—62no—
107C2×C3×C42C2×D311yescharacteristic
108C2×C3×C42C22×C311yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3C22×C311yescharacteristic
111C2×#208C22×C311yescharacteristic
112#897C3×C412yescomplemented
113#895C3×C412yescomplemented
114#886—21no—
115#882—21no—
116C4×D12—22no—
117C42×D3—22no—
118C42×D3—22no—
119C3×#104—62no—
120C3×#104—62no—
121C3×C22⋊C8—62no—
122#825—22no—
123(C2×D3)⋊C8—22no—
124#805—22no—
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—21no—
127C22×C3×D3—21no—
128C22×C3×D3—24no—
129C22×C3×D3—21no—
130C22×C3×D3C22⋊C412yes—
131C2×C32×C4—24no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4C22×C411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4#4111yescharacteristic
139C2×C32×C4#4111yescharacteristic
140C3×#67—41no—
141C3×#67—41no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3C22⋊C412yes—
148C3×D12—21no—
149C3×D12—21no—
150C3×D12—44no—
151C3×C4×D3—216no—
152C3×C4×D3—22no—
153C3×C4×D3—22no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C3×C4×D3—41no—
157C3×C4×D3—41no—
158C3×Dic6—21no—
159C3×Dic6—21no—
160C32×C8—41no—
161C3×C3⋊C8—41no—
162#525—31no—
163C22×C42—31no—
164C2×#104—31no—
165C2×#104—31no—
166#407—31no—
167#339—31no—
168#339C3×D311yescharacteristic, complemented
169C24×C3—31no—
170C23×D3C3×D411yescharacteristic
171C3×#41—61no—
172C3×#41—61no—
173C3×#41—61no—
174C3×#41—61no—
175C2×C3×Q8—31no—
176C2×C3×D4—31no—
177C2×C3×D4—62no—
178C2×C3×D4—61no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—31no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—41no—
186C22×C3×C4—64no—
187C22×C3×C4—68no—
188C22×C3×C4—61no—
189C22×C3×C4—61no—
190C22×C3×C4—61no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C22×C3×C4—61no—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C23×C311yescharacteristic
196C2×#67—21no—
197C22×Dic3—21no—
198C22×Dic3C3×D411yescharacteristic
199#235—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203C2×D12—22no—
204C2×D12C2×C3×C411yescharacteristic, complemented
205C2×C4×D3—24no—
206C2×C4×D3—28no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3C2×C3×C412yes—
213C2×C4×D3C3×D412yes—
214C2×Dic6C2×C3×C411yescharacteristic, complemented
215C3×M16—22no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—41no—
221C3×M16—61no—
222C3×M16—61no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—41no—
226C2×C3×C8—61no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4C4×D312yescomplemented
230C3×C4⋊C4C2×C3×C412yescomplemented
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—41no—
234C3×C22⋊C4—62no—
235C3×C22⋊C4—62no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—22no—
241C3×C42—32no—
242C3×C42—41no—
243C3×C42—41no—
244C3×C42—64no—
245C3×C42—61no—
246C3×C42D1212yescomplemented
247C3×C42C3×D412yescomplemented
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250C4⋊Dic3—22no—
251C4×Dic3—24no—
252C4×Dic3—21no—
253C4×Dic3C3×D412yescomplemented
254#208—21no—
255#208—21no—
256C2×C3⋊C8—21no—
257C22×C32—22no—
258C22×C32C2×C22⋊C411yescharacteristic
259C22×C32#12011yescharacteristic
260C22×C32#12311yescharacteristic
261C2×C3×D3—24no—
262C2×C3×D3—24no—
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3—41no—
266C2×C3×D3—42no—
267C2×C3×D3#10414yescomplemented
268C32×C4—24no—
269C32×C4—42no—
270C32×C4C4×D411yescharacteristic
271C32×C4C4×D411yescharacteristic
272C32×C4#12111yescharacteristic
273C32×C4#12111yescharacteristic
274C3×Dic3—24no—
275C3×Dic3—22no—
276C3×Dic3—41no—
277C2×#41—31no—
278C23×C4—31no—
279C23×C4—61no—
280C2×M16—31no—
281C2×M16C2×C3×D311yescharacteristic
282#123—61no—
283C22⋊Q8—32no—
284#121—32no—
285#120—61no—
286C4×D4—62no—
287#117C2×C3×D311yescharacteristic
288C2×C42—31no—
289C2×C42—62no—
290C2×C42—61no—
291C2×C42—62no—
292C2×C42C2×C3×D311yescharacteristic
293#104—62no—
294#104—62no—
295C22⋊C8—62no—
296C23×C3—21no—
297C23×C3—32no—
298C23×C3—64no—
299C23×C3—61no—
300C23×C3—61no—
301C23×C3—61no—
302C23×C3D3×D411yescharacteristic
303C23×C3C2×C3×D411yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—24no—
307C22×D3—21no—
308C22×D3C3×C22⋊C412yes—
309C3×Q8—61no—
310C3×Q8—61no—
311C3×D4—61no—
312C3×D4—61no—
313C3×D4—124no—
314C3×D4—121no—
315C3×D4—121no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—616no—
333C2×C3×C4—64no—
334C2×C3×C4—62no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—61no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4C2×C4×D311yescharacteristic
345C2×C3×C4C2×D1211yescharacteristic
346C2×C3×C4#23511yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4#23911yescharacteristic
350C2×C3×C4C22×C3×C411yescharacteristic
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C3×#4111yescharacteristic
355C2×C3×C4C3×#4111yescharacteristic
356#67—41no—
357#67—41no—
358C2×Dic3—24no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3C3×C22⋊C412yes—
364D12—21no—
365D12—21no—
366D12—44no—
367C4×D3—216no—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371C4×D3—22no—
372C4×D3—41no—
373C4×D3—41no—
374Dic6—21no—
375Dic6—21no—
376C3×C8—42no—
377C3×C8—41no—
378C3×C8—41no—
379C3×C8—121no—
380C3⋊C8—41no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32#38611yescharacteristic
384C2×C32C2×#10412yes—
385C3×D3—24no—
386C3×D3—81no—
387C24—31no—
388#41—61no—
389#41—61no—
390#41—61no—
391#41—61no—
392C2×Q8—31no—
393C2×D4—31no—
394C2×D4—62no—
395C2×D4—61no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—64no—
401C22×C4—68no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4C22×C3×D311yescharacteristic
409M16—21no—
410M16—21no—
411M16—61no—
412M16—61no—
413C2×C8—21no—
414C2×C8—61no—
415C4⋊C4—62no—
416C4⋊C4C3×C4×D312yescomplemented
417C22⋊C4—21no—
418C22⋊C4—62no—
419C22⋊C4—62no—
420C42—21no—
421C42—21no—
422C42—32no—
423C42—64no—
424C42—61no—
425C42C3×D1212yescomplemented
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—34no—
432C22×C3—42no—
433C22×C3—64no—
434C22×C3—64no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—121no—
438C22×C3—122no—
439C22×C3D3×C22⋊C411yescharacteristic
440C22×C3#88211yescharacteristic
441C22×C3#89411yescharacteristic
442C22×C3C2×C3×C22⋊C411yescharacteristic
443C22×C3C3×#12011yescharacteristic
444C22×C3C3×#12311yescharacteristic
445C2×D3—24no—
446C2×D3—24no—
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—41no—
450C2×D3—42no—
451C2×D3C3×#10414yescomplemented
452C3×C4—24no—
453C3×C4—24no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—64no—
462C3×C4—62no—
463C3×C4—82no—
464C3×C4—121no—
465C3×C4C4×D1211yescharacteristic
466C3×C4#88411yescharacteristic
467C3×C4#89311yescharacteristic
468C3×C4#89511yescharacteristic
469C3×C4C3×C4×D411yescharacteristic
470C3×C4C3×C4×D411yescharacteristic
471C3×C4C3×#12111yescharacteristic
472C3×C4C3×#12111yescharacteristic
473Dic3—24no—
474Dic3—22no—
475Dic3—41no—
476C32#189611yescharacteristic, complemented, Sylow
477C23—32no—
478C23—64no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23C3×D3×D411yescharacteristic
483Q8—61no—
484Q8—61no—
485D4—61no—
486D4—61no—
487D4—124no—
488D4—121no—
489D4—121no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—32no—
494C2×C4—616no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—61no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4C2×C3×C4×D311yescharacteristic
507C2×C4C2×C3×D1211yescharacteristic
508C2×C4C3×#23511yescharacteristic
509C2×C4C3×D3×D411yescharacteristic
510C2×C4C3×D3×D411yescharacteristic
511C2×C4C3×#23911yescharacteristic
512C8—41no—
513C8—121no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—64no—
523C2×C3—241no—
524C2×C3#466511yescharacteristic
525C2×C3D3×#10412yes—
526C2×C3C3×#38611yescharacteristic
527C2×C3C2×C3×#10412yes—
528D3—24no—
529D3—81no—
530C22—22no—
531C22—34no—
532C22—64no—
533C22—64no—
534C22—62no—
535C22—62no—
536C22—121no—
537C22—122no—
538C22C3×D3×C22⋊C411yescharacteristic
539C22C3×#88211yescharacteristic
540C22C3×#89411yescharacteristic
541C4—24no—
542C4—42no—
543C4—64no—
544C4—62no—
545C4—121no—
546C4C3×C4×D1211yescharacteristic
547C4C3×#88411yescharacteristic
548C4C3×#89311yescharacteristic
549C4C3×#89511yescharacteristic
550C3—21no—
551C3#6875611yescharacteristic, direct product factor
552C3C3×#189611yescharacteristic, complemented
553C2—21no—
554C2—21no—
555C2—64no—
556C2—241no—
557C2C3×#466511yescharacteristic
558C2C3×D3×#10412yes—
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC22×C32C2×C22⋊C4
prime radicalC3×#5524C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
2. C2×C3×C42×D3C2semidirect
3. C2×C3×#825C2semidirect
4. C3×#5126C2semidirect
5. C2×C3×#805C2semidirect
6. C32×#339C2semidirect
7. C3×#4532C2semidirect
9. #68756C3direct (N × H)
19. C3×#897C4semidirect
20. C3×#895C4semidirect
32. C2×C42×D3C2×C3semidirect
36. C2×#825C2×C3semidirect
37. #5126C2×C3semidirect
38. C2×#805C2×C3semidirect
41. C3×#339D3semidirect
42. C3×#339C2×C3semidirect
43. #4532C2×C3semidirect
55. C2×C3×D12C2×C4semidirect
65. C2×C3×Dic6C2×C4semidirect
69. C32×C4⋊C4C2×C4semidirect
73. C32×C42D4semidirect
79. C3×C4×Dic3D4semidirect
112. #897C3×C4semidirect
113. #895C3×C4semidirect
168. #339C3×D3semidirect
204. C2×D12C2×C3×C4semidirect
214. C2×Dic6C2×C3×C4semidirect
229. C3×C4⋊C4C4×D3semidirect
230. C3×C4⋊C4C2×C3×C4semidirect
246. C3×C42D12semidirect
247. C3×C42C3×D4semidirect
253. C4×Dic3C3×D4semidirect
267. C2×C3×D3#104semidirect
416. C4⋊C4C3×C4×D3semidirect
425. C42C3×D12semidirect
451. C2×D3C3×#104semidirect
476. C32#1896semidirect
551. C3#68756direct (N × H)
552. C3C3×#1896semidirect

Maximal subgroups

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

subgroupindex
C3×#55242
C2×C3×C42×D32
C2×C3×#8252
C3×#51262
C2×C3×#8052
C32×#3392
C3×#45322
C3×#18963
#687563