Finite Group Databasebeta

← C3×#71664 #11786675

Subgroups — C3×#71664 #11786675

Order 1152.

751 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#520C211yesmaximal, characteristic
2C2×C3×#837C211yesmaximal, characteristic
3C3×#5232C211yesmaximal, characteristic
4C3×#4995C211yesmaximal, characteristic, complemented
5C3×#4975C211yesmaximal, characteristic, complemented
6C3×#4532C211yesmaximal, characteristic, complemented
7C3×#4530C211yesmaximal, characteristic, complemented
8C3×#1859—31nomaximal
9#71664C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C32×C4×Q8—21no—
12C32×C4×D4—21no—
13C32×C4×D4—22no—
14C32×#117—21no—
15C32×#117C2211yescharacteristic
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×Dic3⋊C4—22no—
20C2×C3×C4×Dic3C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C3×#837C414yescomplemented
24C3×#835—21no—
25C3×#832—21no—
26C3×#808—21no—
27C3×#807—21no—
28C3×#520—21no—
29C3×#520D311yescharacteristic
30C3×#520C2×C311yescharacteristic
31C3×#428—31no—
32C2×C3×#104—31no—
33C3×#417—31no—
34C3×#384—31no—
35C2×#837C2×C311yescharacteristic
36#5232C2×C311yescharacteristic
37#4995C2×C311yescharacteristic, complemented
38#4975C2×C311yescharacteristic, complemented
39C3×#343—31no—
40C3×#339—31no—
41#4532C2×C311yescharacteristic, complemented
42#4530C2×C311yescharacteristic, complemented
43C32×#41—21no—
44C32×#41—21no—
45C32×#41C2×C412yes—
46C32×#41Q812yes—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4—21no—
49C2×C32×D4D411yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—22no—
52C22×C32×C4D412yes—
53C22×C32×C4C2311yescharacteristic
54C22×C3×Dic3—21no—
55C22×C3×Dic3—21no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4D411yescharacteristic
60C32×C4⋊C4D411yescharacteristic
61C32×C22⋊C4—22no—
62C32×C22⋊C4—21no—
63C32×C42—22no—
64C32×C42—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×Dic3⋊C4—44no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3C2×C412yescomplemented
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208C2×C412yescomplemented
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76#1859—31noSylow
77C2×C3×#41—21no—
78C2×C3×#41C2×D311yescharacteristic
79C2×C3×#41C22×C311yescharacteristic
80C2×C3×M16—31no—
81C2×C3×M16—31no—
82C3×C4×Q8—22no—
83C3×C4×Q8—21no—
84C3×C4×Q8—21no—
85C3×C4×D4—22no—
86C3×C4×D4—24no—
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×#117—22no—
92C3×#117—21no—
93C3×#117—21no—
94C3×#117—21no—
95C3×#117C2×D311yescharacteristic
96C3×#117C22×C311yescharacteristic
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—62no—
103C2×C3×C4⋊C4C2×D311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×C3×C42—31no—
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×Dic3⋊C4—22no—
108C2×C4×Dic3C22×C311yescharacteristic
109C2×#208C22×C311yescharacteristic
110C2×#208C22×C311yescharacteristic
111C3×#107—61no—
112C3×#106—61no—
113C3×#104—34no—
114C3×#103—61no—
115C3×#102—61no—
116#837C3×C414yescomplemented
117#835—21no—
118#832—21no—
119#808—21no—
120#807—21no—
121C23×C32—21no—
122C23×C32C2×D411yescharacteristic
123C32×Q8—21no—
124C32×Q8C4⋊C412yescomplemented
125C32×D4—22no—
126C32×D4—22no—
127C32×D4—21no—
128C32×D4C4⋊C412yescomplemented
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—42no—
136C2×C32×C4—82no—
137C2×C32×C4C22×C411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×Q811yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3—41no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—41no—
150#520C3×D311yescharacteristic
151#428—31no—
152C2×#104—31no—
153#417—31no—
154#384—31no—
155#343—31no—
156#339—31no—
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—21no—
160C3×#41—21no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41Dic612yes—
166C3×#41C4×D312yes—
167C3×#41C2×C3×C412yes—
168C3×#41C3×Q812yes—
169C2×C3×Q8—21no—
170C2×C3×Q8#6711yescharacteristic
171C2×C3×Q8C3×D411yescharacteristic
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4#6711yescharacteristic
177C2×C3×D4C3×D411yescharacteristic
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4#6712yes—
189C22×C3×C4C3×D412yes—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C23×C311yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C3×M16—32no—
195C3×M16—61no—
196C3×M16—61no—
197C3×M16—61no—
198C2×C3×C8—61no—
199C2×C3×C8—61no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—124no—
214C3×C4⋊C4#6711yescharacteristic
215C3×C4⋊C4#6711yescharacteristic
216C3×C4⋊C4C3×D411yescharacteristic
217C3×C4⋊C4C3×D411yescharacteristic
218C3×C22⋊C4—24no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—21no—
224C3×C42—24no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—32no—
231C3×C42—61no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234Dic3⋊C4—44no—
235C4×Dic3—21no—
236C4×Dic3C2×C3×C412yescomplemented
237#208—21no—
238#208—21no—
239#208—21no—
240#208C2×C3×C412yescomplemented
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32C4×D411yescharacteristic
249C22×C32#12011yescharacteristic
250C22×C32C22⋊Q811yescharacteristic
251C32×C4—22no—
252C32×C4—42no—
253C32×C4—41no—
254C32×C4—41no—
255C32×C4C2×C4⋊C411yescharacteristic
256C32×C4C4×D411yescharacteristic
257C32×C4C22⋊Q811yescharacteristic
258C32×C4#12311yescharacteristic
259C3×Dic3—24no—
260C3×Dic3—81no—
261C2×#41C2×C3×D311yescharacteristic
262C2×M16—31no—
263C2×M16—31no—
264C4×Q8—21no—
265C4×D4—21no—
266C4×D4—22no—
267#117—21no—
268#117C2×C3×D311yescharacteristic
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4C2×C3×D311yescharacteristic
273C2×C42—31no—
274#107—61no—
275#106—61no—
276#104—34no—
277#103—61no—
278#102—61no—
279C23×C3—22no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3C2×#6711yescharacteristic
284C23×C3C2×C3×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8—21no—
287C3×Q8—21no—
288C3×Q8—22no—
289C3×Q8Dic3⋊C412yescomplemented
290C3×Q8C3×C4⋊C412yescomplemented
291C3×D4—24no—
292C3×D4—24no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—21no—
299C3×D4—21no—
300C3×D4—22no—
301C3×D4Dic3⋊C412yescomplemented
302C3×D4C3×C4⋊C412yescomplemented
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—82no—
333C2×C3×C4—84no—
334C2×C3×C4—82no—
335C2×C3×C4—82no—
336C2×C3×C4—122no—
337C2×C3×C4—121no—
338C2×C3×C4C2×Dic611yescharacteristic
339C2×C3×C4C2×C4×D311yescharacteristic
340C2×C3×C4#23511yescharacteristic
341C2×C3×C4#23511yescharacteristic
342C2×C3×C4C2×#6711yescharacteristic
343C2×C3×C4C2×#6711yescharacteristic
344C2×C3×C4C22×C3×C411yescharacteristic
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×Q811yescharacteristic
348C2×C3×C4C3×#4111yescharacteristic
349C2×C3×C4C3×#4111yescharacteristic
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C3×C8—62no—
356C3×C8—121no—
357C3⋊C8—22no—
358C3⋊C8—41no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—42no—
362C2×C32#38511yescharacteristic
363C2×C32#45311yescharacteristic
364C2×C32#45611yescharacteristic
365#41—21no—
366#41—21no—
367#41C3×Dic612yes—
368#41C3×C4×D312yes—
369C2×Q8C3×#6711yescharacteristic
370C2×D4—21no—
371C2×D4C3×#6711yescharacteristic
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4C3×#6712yes—
377C22×C4C22×C3×D311yescharacteristic
378M16—32no—
379M16—61no—
380M16—61no—
381M16—61no—
382C2×C8—61no—
383C2×C8—61no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—124no—
390C4⋊C4C3×#6711yescharacteristic
391C4⋊C4C3×#6711yescharacteristic
392C22⋊C4—22no—
393C22⋊C4—21no—
394C42—22no—
395C42—21no—
396C42—32no—
397C42—61no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—24no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3#92411yescharacteristic
417C22×C3C4×#6711yescharacteristic
418C22×C3#95311yescharacteristic
419C22×C3C3×C4×D411yescharacteristic
420C22×C3C3×#12011yescharacteristic
421C22×C3C3×C22⋊Q811yescharacteristic
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—42no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—64no—
439C3×C4—241no—
440C3×C4C2×Dic3⋊C411yescharacteristic
441C3×C4#92411yescharacteristic
442C3×C4C4×#6711yescharacteristic
443C3×C4#92911yescharacteristic
444C3×C4C2×C3×C4⋊C411yescharacteristic
445C3×C4C3×C4×D411yescharacteristic
446C3×C4C3×C22⋊Q811yescharacteristic
447C3×C4C3×#12311yescharacteristic
448Dic3—24no—
449Dic3—81no—
450C32#185911yescharacteristic, complemented, Sylow
451C23—21no—
452C23C2×C3×#6711yescharacteristic
453Q8—21no—
454Q8C3×Dic3⋊C412yescomplemented
455D4—22no—
456D4—22no—
457D4—21no—
458D4C3×Dic3⋊C412yescomplemented
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—42no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—82no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4C2×C3×Dic611yescharacteristic
473C2×C4C2×C3×C4×D311yescharacteristic
474C2×C4C3×#23511yescharacteristic
475C2×C4C3×#23511yescharacteristic
476C2×C4C2×C3×#6711yescharacteristic
477C2×C4C2×C3×#6711yescharacteristic
478C8—62no—
479C8—121no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—44no—
490C2×C3—42no—
491C2×C3—42no—
492C2×C3#521011yescharacteristic
493C2×C3#524111yescharacteristic
494C2×C3#524611yescharacteristic
495C2×C3C3×#38511yescharacteristic
496C2×C3C3×#45311yescharacteristic
497C2×C3C3×#45611yescharacteristic
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22C3×#92411yescharacteristic
504C22C3×C4×#6711yescharacteristic
505C22C3×#95311yescharacteristic
506C4—22no—
507C4—42no—
508C4—41no—
509C4—41no—
510C4—64no—
511C4—241no—
512C4C2×C3×Dic3⋊C411yescharacteristic
513C4C3×#92411yescharacteristic
514C4C3×C4×#6711yescharacteristic
515C4C3×#92911yescharacteristic
516C3—21no—
517C3#7166411yescharacteristic, direct product factor
518C3C3×#185911yescharacteristic, complemented
519C2—21no—
520C2—21no—
521C2—42no—
522C2C3×#521011yescharacteristic
523C2C3×#524111yescharacteristic
524C2C3×#524611yescharacteristic
subgroupG/H
centerC22×C3#953
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#520C2
socleC22×C32#120
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#520D3
hypocenterC3C3×#1859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#520

Complements

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

NH ≅ G/Ntype
4. C3×#4995C2semidirect
5. C3×#4975C2semidirect
6. C3×#4532C2semidirect
7. C3×#4530C2semidirect
9. #71664C3direct (N × H)
23. C3×#837C4semidirect
37. #4995C2×C3semidirect
38. #4975C2×C3semidirect
41. #4532C2×C3semidirect
42. #4530C2×C3semidirect
69. C3×C4×Dic3C2×C4semidirect
73. C3×#208C2×C4semidirect
116. #837C3×C4semidirect
124. C32×Q8C4⋊C4semidirect
128. C32×D4C4⋊C4semidirect
236. C4×Dic3C2×C3×C4semidirect
240. #208C2×C3×C4semidirect
289. C3×Q8Dic3⋊C4semidirect
290. C3×Q8C3×C4⋊C4semidirect
301. C3×D4Dic3⋊C4semidirect
302. C3×D4C3×C4⋊C4semidirect
450. C32#1859semidirect
454. Q8C3×Dic3⋊C4semidirect
458. D4C3×Dic3⋊C4semidirect
517. C3#71664direct (N × H)
518. C3C3×#1859semidirect

Maximal subgroups

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

subgroupindex
C32×#5202
C2×C3×#8372
C3×#52322
C3×#49952
C3×#49752
C3×#45322
C3×#45302
C3×#18593
#716643