Finite Group Databasebeta

← C3×#68775 #11787673

Subgroups — C3×#68775 #11787673

Order 1152.

723 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C3×#5523C211yesmaximal, characteristic
3C3×#5004C211yesmaximal, characteristic, complemented
4C3×#4929C211yesmaximal, characteristic, complemented
5C2×C3×#805C211yesmaximal, characteristic
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1882—31nomaximal
9#68775C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#909—22no—
18C3×#909C414yescomplemented
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C3×#892—21no—
22C3×D3×C4⋊C4—21no—
23C3×#67⋊C4—22no—
24C3×C4×D12—21no—
25C3×C4×Dic6—21no—
26C3×C3⋊#95—22no—
27C3×#810—21no—
28C3×#810—21no—
29C3×#805—22no—
30C2×C3×#137—31no—
31C3×#520—31no—
32C2×#909C2×C311yescharacteristic
33#5523C2×C311yescharacteristic
34C2×C3×#104—31no—
35C3×#419—31no—
36C3×#391—31no—
37#5004C2×C311yescharacteristic, complemented
38#4929C2×C311yescharacteristic, complemented
39C2×#805C2×C311yescharacteristic
40C3×#343—31no—
41C3×#339—21no—
42C3×#339D311yescharacteristic, complemented
43C3×#339C2×C311yescharacteristic, complemented
44#4554C2×C311yescharacteristic, complemented
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D412yes—
50C3×#235—21no—
51C3×#235—21no—
52C3×#235C2×C412yes—
53C3×#235D412yes—
54C2×C3×D12D411yescharacteristic
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—22no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2×C412yescomplemented
63C2×C3×Dic6D411yescharacteristic
64C32×M16—21no—
65C32×M16C2×C412yescomplemented
66C2×C32×C8—21no—
67C32×C4⋊C4D411yescharacteristic
68C32×C4⋊C4D411yescharacteristic
69C32×C22⋊C4—21no—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42—21no—
73C3×(C2×D3)⋊C4—22no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×Dic3⋊C4—22no—
77C3×C4×Dic3—22no—
78C3×#208—21no—
79C3×#208—21no—
80C2×C3×C3⋊C8—21no—
81C3×Dic12—24no—
82C3×#204—24no—
83#1882—31noSylow
84C2×C3×#41—31no—
85C22×C3×Q8—31no—
86C2×#235C22×C311yescharacteristic
87C22×Dic6C22×C311yescharacteristic
88C3×#137—34no—
89C3×#137—62no—
90C2×C3×Q16—61no—
91C2×C3×SD16—61no—
92C2×C3×M16—21no—
93C2×C3×M16—31no—
94C2×C3×M16C2×D311yescharacteristic
95C2×C3×M16C22×C311yescharacteristic
96C3×C4×Q8—61no—
97C3×C4×D4—61no—
98C3×C4×D4—62no—
99C3×#117—21no—
100C3×#117—61no—
101C3×#117C2×D311yescharacteristic
102C3×#117C22×C311yescharacteristic
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—61no—
105C2×C3×C42—21no—
106C2×C3×C42C2×D311yescharacteristic
107C2×C3×C42C22×C311yescharacteristic
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×#208C22×C311yescharacteristic
110#909—22no—
111#909C3×C414yescomplemented
112C2×Dic12—21no—
113C2×#204—21no—
114#892—21no—
115D3×C4⋊C4—21no—
116#67⋊C4—22no—
117C4×D12—21no—
118C4×Dic6—21no—
119C3×#104—62no—
120C3×#103—61no—
121C3×#103—61no—
122C3×#95—62no—
123C3⋊#95—22no—
124#810—21no—
125#810—21no—
126#805—22no—
127C23×C32C2×D411yescharacteristic
128C22×C3×D3—21no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
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—22no—
141C3×#67—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—42no—
148C3×D12—21no—
149C3×D12C22⋊C412yes—
150C3×C4×D3—22no—
151C3×C4×D3—22no—
152C3×C4×D3—82no—
153C3×Dic6—21no—
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—41no—
157C3×Dic6—41no—
158C3×Dic6C22⋊C412yes—
159C32×C8—22no—
160C3×C3⋊C8—41no—
161C2×#137—31no—
162#520—31no—
163C2×#104—31no—
164#419—31no—
165#391—31no—
166#343—31no—
167#339C3×D311yescharacteristic, complemented
168C3×#41—32no—
169C3×#41—32no—
170C3×#41—61no—
171C3×#41—61no—
172C2×C3×Q8—32no—
173C2×C3×Q8—31no—
174C2×C3×Q8—61no—
175C2×C3×Q8—61no—
176C2×C3×Q8—61no—
177C2×C3×Q8—61no—
178C2×C3×Q8—61no—
179C2×C3×D4—31no—
180C2×C3×D4—61no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—32no—
185C22×C3×C4—41no—
186C22×C3×C4—62no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C23×C311yescharacteristic
191C2×#67—21no—
192C22×Dic3—21no—
193C22×Dic3C3×D412yes—
194#235—21no—
195#235—21no—
196#235C2×C3×C412yes—
197#235C3×D412yes—
198C2×D12C3×D411yescharacteristic
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6C2×C3×C412yescomplemented
207C2×Dic6C3×D411yescharacteristic
208C3×Q16—64no—
209C3×SD16—64no—
210C3×M16—21no—
211C3×M16—21no—
212C3×M16—22no—
213C3×M16—41no—
214C3×M16—61no—
215C3×M16—61no—
216C3×M16C4×D312yescomplemented
217C3×M16C2×C3×C412yescomplemented
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—41no—
221C2×C3×C8—61no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—62no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4D1211yescharacteristic
228C3×C4⋊C4D1211yescharacteristic
229C3×C4⋊C4C3×D411yescharacteristic
230C3×C4⋊C4C3×D411yescharacteristic
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—41no—
234C3×C22⋊C4—62no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—22no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—41no—
243C3×C42—41no—
244C3×C42—62no—
245(C2×D3)⋊C4—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248Dic3⋊C4—22no—
249C4×Dic3—22no—
250#208—21no—
251#208—21no—
252C2×C3⋊C8—21no—
253Dic12—24no—
254#204—24no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32C4×D411yescharacteristic
258C22×C32#12011yescharacteristic
259C22×C32#12111yescharacteristic
260C2×C3×D3—22no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C32×C4—42no—
264C32×C4—41no—
265C32×C4—41no—
266C32×C4C2×C22⋊C411yescharacteristic
267C32×C4C4×D411yescharacteristic
268C32×C4#12111yescharacteristic
269C32×C4#12311yescharacteristic
270C3×Dic3—22no—
271C3×Dic3—42no—
272C3×Dic3—42no—
273C2×#41—31no—
274C22×Q8—31no—
275#137—34no—
276#137—62no—
277C2×Q16—61no—
278C2×SD16—61no—
279C2×M16—31no—
280C2×M16C2×C3×D311yescharacteristic
281C4×Q8—61no—
282C4×D4—61no—
283C4×D4—62no—
284#117—61no—
285#117C2×C3×D311yescharacteristic
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C42C2×C3×D311yescharacteristic
289#104—62no—
290#103—61no—
291#103—61no—
292#95—62no—
293C23×C3—21no—
294C23×C3—61no—
295C23×C3D3×D411yescharacteristic
296C23×C3C2×C3×D411yescharacteristic
297C22×D3—21no—
298C3×Q8—32no—
299C3×Q8—61no—
300C3×Q8—62no—
301C3×Q8—62no—
302C3×Q8—121no—
303C3×Q8—121no—
304C3×D4—32no—
305C3×D4—62no—
306C3×D4—62no—
307C3×D4—61no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—61no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—121no—
335C2×C3×C4—121no—
336C2×C3×C4—122no—
337C2×C3×C4—242no—
338C2×C3×C4C2×C4×D311yescharacteristic
339C2×C3×C4C2×D1211yescharacteristic
340C2×C3×C4#23511yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4#23911yescharacteristic
344C2×C3×C4C22×C3×C411yescharacteristic
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C3×#4111yescharacteristic
349C2×C3×C4C3×#4111yescharacteristic
350#67—22no—
351#67—22no—
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358D12—21no—
359D12C3×C22⋊C412yes—
360C4×D3—22no—
361C4×D3—22no—
362C4×D3—82no—
363Dic6—21no—
364Dic6—22no—
365Dic6—22no—
366Dic6—41no—
367Dic6—41no—
368Dic6C3×C22⋊C412yes—
369C3×C8—22no—
370C3×C8—22no—
371C3×C8—42no—
372C3×C8—121no—
373C3⋊C8—41no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32#38611yescharacteristic
377C2×C32#45511yescharacteristic
378C2×C32#45611yescharacteristic
379C3×D3—42no—
380#41—32no—
381#41—32no—
382#41—61no—
383#41—61no—
384C2×Q8—32no—
385C2×Q8—31no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×D4—31no—
392C2×D4—61no—
393C22×C4—21no—
394C22×C4—32no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4C22×C3×D311yescharacteristic
399Q16—64no—
400SD16—64no—
401M16—21no—
402M16—61no—
403M16—61no—
404M16C3×C4×D312yescomplemented
405C2×C8—21no—
406C2×C8—61no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4C3×D1211yescharacteristic
411C4⋊C4C3×D1211yescharacteristic
412C22⋊C4—21no—
413C22⋊C4—62no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—62no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—62no—
428C22×C3—121no—
429C22×C3—121no—
430C22×C3#88211yescharacteristic
431C22×C3#88411yescharacteristic
432C22×C3#89311yescharacteristic
433C22×C3C3×C4×D411yescharacteristic
434C22×C3C3×#12011yescharacteristic
435C22×C3C3×#12111yescharacteristic
436C2×D3—22no—
437C2×D3—41no—
438C2×D3—41no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—44no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—62no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4—122no—
454C3×C4—122no—
455C3×C4C4×D1211yescharacteristic
456C3×C4D3×C22⋊C411yescharacteristic
457C3×C4#89411yescharacteristic
458C3×C4#89511yescharacteristic
459C3×C4C2×C3×C22⋊C411yescharacteristic
460C3×C4C3×C4×D411yescharacteristic
461C3×C4C3×#12111yescharacteristic
462C3×C4C3×#12311yescharacteristic
463Dic3—22no—
464Dic3—42no—
465Dic3—42no—
466C32#188211yescharacteristic, complemented, Sylow
467C23—61no—
468C23C3×D3×D411yescharacteristic
469Q8—32no—
470Q8—61no—
471Q8—62no—
472Q8—62no—
473Q8—121no—
474Q8—121no—
475D4—32no—
476D4—62no—
477D4—62no—
478D4—61no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—122no—
492C2×C4—242no—
493C2×C4C2×C3×C4×D311yescharacteristic
494C2×C4C2×C3×D1211yescharacteristic
495C2×C4C3×#23511yescharacteristic
496C2×C4C3×D3×D411yescharacteristic
497C2×C4C3×D3×D411yescharacteristic
498C2×C4C3×#23911yescharacteristic
499C8—22no—
500C8—121no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—122no—
511C2×C3#466511yescharacteristic
512C2×C3#482511yescharacteristic
513C2×C3#482611yescharacteristic
514C2×C3C3×#38611yescharacteristic
515C2×C3C3×#45511yescharacteristic
516C2×C3C3×#45611yescharacteristic
517D3—42no—
518C22—21no—
519C22—21no—
520C22—62no—
521C22—121no—
522C22—121no—
523C22C3×#88211yescharacteristic
524C22C3×#88411yescharacteristic
525C22C3×#89311yescharacteristic
526C4—42no—
527C4—41no—
528C4—41no—
529C4—62no—
530C4—122no—
531C4—122no—
532C4C3×C4×D1211yescharacteristic
533C4C3×D3×C22⋊C411yescharacteristic
534C4C3×#89411yescharacteristic
535C4C3×#89511yescharacteristic
536C3—21no—
537C3#6877511yescharacteristic, direct product factor
538C3C3×#188211yescharacteristic, complemented
539C2—21no—
540C2—21no—
541C2—122no—
542C2C3×#466511yescharacteristic
543C2C3×#482511yescharacteristic
544C2C3×#482611yescharacteristic
subgroupG/H
centerC22×C3#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC22×C32#120
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1882
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
3. C3×#5004C2semidirect
4. C3×#4929C2semidirect
6. C32×#339C2semidirect
7. C3×#4554C2semidirect
9. #68775C3direct (N × H)
18. C3×#909C4semidirect
37. #5004C2×C3semidirect
38. #4929C2×C3semidirect
42. C3×#339D3semidirect
43. C3×#339C2×C3semidirect
44. #4554C2×C3semidirect
62. C2×C3×Dic6C2×C4semidirect
65. C32×M16C2×C4semidirect
111. #909C3×C4semidirect
167. #339C3×D3semidirect
206. C2×Dic6C2×C3×C4semidirect
216. C3×M16C4×D3semidirect
217. C3×M16C2×C3×C4semidirect
404. M16C3×C4×D3semidirect
466. C32#1882semidirect
537. C3#68775direct (N × H)
538. C3C3×#1882semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C3×#55232
C3×#50042
C3×#49292
C2×C3×#8052
C32×#3392
C3×#45542
C3×#18823
#687753