Finite Group Databasebeta

← C3×#68757 #11788196

Subgroups — C3×#68757 #11788196

Order 1152.

771 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5524C211yesmaximal, characteristic
2C2×C3×#872C211yesmaximal, 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×#1897—31nomaximal
9#68757C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×Dic3⋊C4—21no—
17C2×C3×C4×Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#897C412yescomplemented
20C3×#895C412yescomplemented
21C3×#886—21no—
22C3×#882—21no—
23C3×C4×D12—22no—
24C3×#872—22no—
25C3×#872—22no—
26C3×#825—22no—
27C3×(C2×D3)⋊C8—22no—
28C3×#805—22no—
29C3×#525—31no—
30C2×C3×#117—31no—
31#5524C2×C311yescharacteristic
32C2×#872C2×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
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3C2×C412yes—
62C2×C3×C4×D3D412yes—
63C2×C3×Dic6C2×C411yescharacteristic
64C32×M16—21no—
65C32×M16—21no—
66C2×C32×C8—21no—
67C32×C4⋊C4C2×C412yes—
68C32×C22⋊C4—21no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42D412yescomplemented
72C3×(C2×D3)⋊C4—24no—
73C3×(C2×D3)⋊C4—22no—
74C3×(C2×D3)⋊C4—22no—
75C3×C4⋊Dic3—22no—
76C3×Dic3⋊C4—24no—
77C3×C4×Dic3—21no—
78C3×C4×Dic3D412yescomplemented
79C3×#208—21no—
80C3×#208—21no—
81C2×C3×C3⋊C8—21no—
82#1897—31noSylow
83C2×C3×#41—31no—
84C23×C3×C4—31no—
85C2×#235C22×C311yescharacteristic
86C22×C4×D3C22×C311yescharacteristic
87C2×C3×M16—21no—
88C2×C3×M16—31no—
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C22×C311yescharacteristic
91C3×#123—61no—
92C3×C22⋊Q8—32no—
93C3×#121—32no—
94C3×#120—61no—
95C3×C4×D4—62no—
96C3×#117—21no—
97C3×#117—62no—
98C3×#117—62no—
99C3×#117C2×D311yescharacteristic
100C3×#117C22×C311yescharacteristic
101C2×C3×C4⋊C4—61no—
102C2×C3×C22⋊C4—61no—
103C2×C3×C42—21no—
104C2×C3×C42—31no—
105C2×C3×C42C2×D311yescharacteristic
106C2×C3×C42C22×C311yescharacteristic
107C2×(C2×D3)⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3C22×C311yescharacteristic
110C2×#208C22×C311yescharacteristic
111#897C3×C412yescomplemented
112#895C3×C412yescomplemented
113#886—21no—
114#882—21no—
115C4×D12—22no—
116#872—22no—
117#872—22no—
118C3×#104—62no—
119C3×#104—62no—
120C3×C22⋊C8—62no—
121#825—22no—
122(C2×D3)⋊C8—22no—
123#805—22no—
124C23×C32C2×D411yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C22×C3×D3—21no—
128C22×C3×D3—42no—
129C22×C3×D3C22⋊C412yes—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—42no—
133C2×C32×C4C22×C411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C2×C32×C4#4111yescharacteristic
139C3×#67—41no—
140C3×#67—41no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3C22⋊C412yes—
147C3×D12—21no—
148C3×D12—21no—
149C3×D12—44no—
150C3×C4×D3—22no—
151C3×C4×D3—22no—
152C3×C4×D3—22no—
153C3×C4×D3—22no—
154C3×C4×D3—41no—
155C3×C4×D3—41no—
156C3×Dic6—21no—
157C3×Dic6—21no—
158C32×C8—41no—
159C3×C3⋊C8—41no—
160#525—31no—
161C2×#117—31no—
162C2×#104—31no—
163C2×#104—31no—
164#407—31no—
165#339—31no—
166#339C3×D311yescharacteristic, complemented
167C24×C3—31no—
168C23×D3C3×D411yescharacteristic
169C3×#41—61no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—61no—
173C2×C3×Q8—31no—
174C2×C3×D4—31no—
175C2×C3×D4—62no—
176C2×C3×D4—61no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—31no—
181C22×C3×C4—32no—
182C22×C3×C4—32no—
183C22×C3×C4—41no—
184C22×C3×C4—61no—
185C22×C3×C4—61no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C22×C3×C4—61no—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C23×C311yescharacteristic
192C2×#67—21no—
193C22×Dic3—21no—
194C22×Dic3C3×D411yescharacteristic
195#235—21no—
196#235—21no—
197#235—21no—
198#235—21no—
199C2×D12—22no—
200C2×D12C2×C3×C411yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3C2×C3×C412yes—
207C2×C4×D3C3×D412yes—
208C2×Dic6C2×C3×C411yescharacteristic
209C3×M16—22no—
210C3×M16—21no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—41no—
215C3×M16—61no—
216C3×M16—61no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—41no—
220C2×C3×C8—61no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—62no—
223C3×C4⋊C4—64no—
224C3×C4⋊C4C4×D312yes—
225C3×C4⋊C4C2×C3×C412yes—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—41no—
229C3×C22⋊C4—62no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—64no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42—32no—
238C3×C42—41no—
239C3×C42—41no—
240C3×C42—61no—
241C3×C42D1212yescomplemented
242C3×C42C3×D412yescomplemented
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246C4⋊Dic3—22no—
247Dic3⋊C4—24no—
248C4×Dic3—21no—
249C4×Dic3C3×D412yescomplemented
250#208—21no—
251#208—21no—
252C2×C3⋊C8—21no—
253C22×C32—22no—
254C22×C32C2×C22⋊C411yescharacteristic
255C22×C32#12011yescharacteristic
256C22×C32#12311yescharacteristic
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—42no—
261C2×C3×D3—42no—
262C2×C3×D3—41no—
263C2×C3×D3—42no—
264C32×C4—42no—
265C32×C4—42no—
266C32×C4C4×D411yescharacteristic
267C32×C4C4×D411yescharacteristic
268C32×C4#12111yescharacteristic
269C32×C4#12111yescharacteristic
270C3×Dic3—22no—
271C3×Dic3—42no—
272C3×Dic3—41no—
273C2×#41—31no—
274C23×C4—31no—
275C2×M16—31no—
276C2×M16C2×C3×D311yescharacteristic
277#123—61no—
278C22⋊Q8—32no—
279#121—32no—
280#120—61no—
281C4×D4—62no—
282#117—62no—
283#117—62no—
284#117C2×C3×D311yescharacteristic
285C2×C4⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—31no—
288C2×C42C2×C3×D311yescharacteristic
289#104—62no—
290#104—62no—
291C22⋊C8—62no—
292C23×C3—21no—
293C23×C3—32no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—122no—
298C23×C3D3×D411yescharacteristic
299C23×C3C2×C3×D411yescharacteristic
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—42no—
304C22×D3C3×C22⋊C412yes—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4—124no—
310C3×D4—121no—
311C3×D4—121no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—32no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—44no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—122no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4C2×C4×D311yescharacteristic
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4#23511yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4#23911yescharacteristic
345C2×C3×C4C22×C3×C411yescharacteristic
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C3×#4111yescharacteristic
350C2×C3×C4C3×#4111yescharacteristic
351#67—41no—
352#67—41no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—42no—
358C2×Dic3C3×C22⋊C412yes—
359D12—21no—
360D12—21no—
361D12—44no—
362C4×D3—22no—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—41no—
367C4×D3—41no—
368Dic6—21no—
369Dic6—21no—
370C3×C8—42no—
371C3×C8—41no—
372C3×C8—41no—
373C3×C8—121no—
374C3⋊C8—41no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32#38611yescharacteristic
378C2×C32#42112yes—
379C3×D3—42no—
380C3×D3—81no—
381C24—31no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41—61no—
386C2×Q8—31no—
387C2×D4—31no—
388C2×D4—62no—
389C2×D4—61no—
390C22×C4—21no—
391C22×C4—31no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4C22×C3×D311yescharacteristic
401M16—21no—
402M16—21no—
403M16—61no—
404M16—61no—
405C2×C8—21no—
406C2×C8—61no—
407C4⋊C4—62no—
408C4⋊C4—64no—
409C4⋊C4C3×C4×D312yes—
410C22⋊C4—21no—
411C22⋊C4—62no—
412C22⋊C4—62no—
413C22⋊C4—64no—
414C42—21no—
415C42—21no—
416C42—32no—
417C42—61no—
418C42C3×D1212yescomplemented
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—122no—
429C22×C3—122no—
430C22×C3—121no—
431C22×C3—122no—
432C22×C3D3×C22⋊C411yescharacteristic
433C22×C3#88211yescharacteristic
434C22×C3#89411yescharacteristic
435C22×C3C2×C3×C22⋊C411yescharacteristic
436C22×C3C3×#12011yescharacteristic
437C22×C3C3×#12311yescharacteristic
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—62no—
455C3×C4—82no—
456C3×C4—122no—
457C3×C4—121no—
458C3×C4C4×D1211yescharacteristic
459C3×C4#88411yescharacteristic
460C3×C4#89311yescharacteristic
461C3×C4#89511yescharacteristic
462C3×C4C3×C4×D411yescharacteristic
463C3×C4C3×C4×D411yescharacteristic
464C3×C4C3×#12111yescharacteristic
465C3×C4C3×#12111yescharacteristic
466Dic3—22no—
467Dic3—42no—
468Dic3—41no—
469C32#189711yescharacteristic, complemented, Sylow
470C23—32no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—122no—
475C23C3×D3×D411yescharacteristic
476Q8—61no—
477Q8—61no—
478D4—61no—
479D4—61no—
480D4—124no—
481D4—121no—
482D4—121no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—32no—
486C2×C4—42no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—62no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4C2×C3×C4×D311yescharacteristic
499C2×C4C2×C3×D1211yescharacteristic
500C2×C4C3×#23511yescharacteristic
501C2×C4C3×D3×D411yescharacteristic
502C2×C4C3×D3×D411yescharacteristic
503C2×C4C3×#23911yescharacteristic
504C8—41no—
505C8—121no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—122no—
515C2×C3—241no—
516C2×C3#466511yescharacteristic
517C2×C3#482112yes—
518C2×C3C3×#38611yescharacteristic
519C2×C3C3×#42112yes—
520D3—42no—
521D3—81no—
522C22—22no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—122no—
527C22—122no—
528C22—121no—
529C22—122no—
530C22C3×D3×C22⋊C411yescharacteristic
531C22C3×#88211yescharacteristic
532C22C3×#89411yescharacteristic
533C4—42no—
534C4—42no—
535C4—62no—
536C4—122no—
537C4—121no—
538C4C3×C4×D1211yescharacteristic
539C4C3×#88411yescharacteristic
540C4C3×#89311yescharacteristic
541C4C3×#89511yescharacteristic
542C3—21no—
543C3#6875711yescharacteristic, direct product factor
544C3C3×#189711yescharacteristic, complemented
545C2—21no—
546C2—21no—
547C2—122no—
548C2—241no—
549C2C3×#466511yescharacteristic
550C2C3×#482112yes—
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×#1897
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×#872C2semidirect
3. C2×C3×#825C2semidirect
4. C3×#5126C2semidirect
5. C2×C3×#805C2semidirect
6. C32×#339C2semidirect
7. C3×#4532C2semidirect
9. #68757C3direct (N × H)
19. C3×#897C4semidirect
20. C3×#895C4semidirect
32. C2×#872C2×C3semidirect
36. C2×#825C2×C3semidirect
37. #5126C2×C3semidirect
38. C2×#805C2×C3semidirect
41. C3×#339D3semidirect
42. C3×#339C2×C3semidirect
43. #4532C2×C3semidirect
71. C32×C42D4semidirect
78. C3×C4×Dic3D4semidirect
111. #897C3×C4semidirect
112. #895C3×C4semidirect
166. #339C3×D3semidirect
241. C3×C42D12semidirect
242. C3×C42C3×D4semidirect
249. C4×Dic3C3×D4semidirect
418. C42C3×D12semidirect
469. C32#1897semidirect
543. C3#68757direct (N × H)
544. C3C3×#1897semidirect

Maximal subgroups

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

subgroupindex
C3×#55242
C2×C3×#8722
C2×C3×#8252
C3×#51262
C2×C3×#8052
C32×#3392
C3×#45322
C3×#18973
#687573