Finite Group Databasebeta

← C3×#68777 #11787765

Subgroups — C3×#68777 #11787765

Order 1152.

719 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic, complemented
2C3×#5477C211yesmaximal, characteristic
3C3×#5139C211yesmaximal, characteristic, complemented
4C3×#5006C211yesmaximal, characteristic, complemented
5C3×#5005C211yesmaximal, characteristic, complemented
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1884—31nomaximal
9#68777C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C3×#930—22no—
16C3×#926C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C3×#907—22no—
19C3×D3×M16—22no—
20C3×#901—22no—
21C3×#875C411yescharacteristic, complemented
22C3×#875C411yescharacteristic, complemented
23C3×#874C411yescharacteristic, complemented
24C3×C4×D12—22no—
25C3×#869C411yescharacteristic, complemented
26C3×#825—22no—
27C3×#806—22no—
28C3×#805—22no—
29C3×#568—31no—
30C3×#532—31no—
31#5745C2×C311yescharacteristic, complemented
32#5477C2×C311yescharacteristic
33C3×#421—31no—
34C3×#421—31no—
35C3×#410—31no—
36#5139C2×C311yescharacteristic, complemented
37#5006C2×C311yescharacteristic, complemented
38#5005C2×C311yescharacteristic, complemented
39C3×#339—21no—
40C3×#339—31no—
41C3×#339D311yescharacteristic, complemented
42C3×#339C2×C311yescharacteristic, complemented
43#4523C2×C311yescharacteristic, complemented
44C22×C32×C4—21no—
45C22×C32×C4C2311yescharacteristic
46C2×C3×#67—21no—
47C2×C3×#67D412yes—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12—22no—
57C2×C3×D12C2×C411yescharacteristic
58C2×C3×D12C2×C411yescharacteristic
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3D412yes—
61C2×C3×Dic6C2×C411yescharacteristic
62C2×C3×Dic6C2×C411yescharacteristic
63C32×M16—21no—
64C32×M16D412yescomplemented
65C2×C32×C8—21no—
66C32×C4⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C42—21no—
69C32×C42—21no—
70C32×C42C2×C411yescharacteristic
71C32×C42C2×C411yescharacteristic
72C3×#217—21no—
73C3×(C2×D3)⋊C4—22no—
74C3×(C2×D3)⋊C4—22no—
75C3×C4⋊Dic3—22no—
76C3×C4⋊Dic3—21no—
77C3×C4×Dic3—21no—
78C3×#208—21no—
79C3×#208D412yescomplemented
80C2×C3×C3⋊C8—21no—
81C3×ΓL(1,𝔽25)—24no—
82C3×D3×C8—24no—
83#1884—31noSylow
84C2×C3×#41—31no—
85C2×C3×#41—31no—
86C2×#235C22×C311yescharacteristic
87C2×#235C22×C311yescharacteristic
88C3×#131—62no—
89C3×#131—62no—
90C2×C3×M16—21no—
91C2×C3×M16—31no—
92C2×C3×M16—62no—
93C2×C3×M16C2×D311yescharacteristic
94C2×C3×M16C22×C311yescharacteristic
95C3×C4⋊Q8—31no—
96C3×#127—31no—
97C3×#124—31no—
98C3×#124—31no—
99C3×#121—62no—
100C3×C4×D4—62no—
101C3×#117—21no—
102C3×#117—31no—
103C3×#117C2×D311yescharacteristic
104C3×#117C22×C311yescharacteristic
105C2×C3×C42—21no—
106C2×C3×C42C2×D311yescharacteristic
107C2×C3×C42C22×C311yescharacteristic
108#930—22no—
109#926C22×C311yescharacteristic
110C2×#208C22×C311yescharacteristic
111#907—22no—
112D3×M16—22no—
113#901—22no—
114#875C3×C411yescharacteristic, complemented
115#875C3×C411yescharacteristic, complemented
116#874C3×C411yescharacteristic, complemented
117C4×D12—22no—
118#869C3×C411yescharacteristic, complemented
119C3×#104—62no—
120C3×#104—62no—
121C3×#99—62no—
122#825—22no—
123#806—22no—
124#805—22no—
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—21no—
127C22×C3×D3C22⋊C412yes—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—42no—
131C2×C32×C4—41no—
132C2×C32×C4C22×C411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4#4111yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C3×#67—22no—
139C3×#67—22no—
140C3×#67—41no—
141C3×#67—41no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3C22⋊C412yes—
145C3×D12—21no—
146C3×D12—21no—
147C3×D12—21no—
148C3×D12—21no—
149C3×D12—44no—
150C3×C4×D3—22no—
151C3×C4×D3—22no—
152C3×C4×D3—41no—
153C3×C4×D3—41no—
154C3×Dic6—21no—
155C3×Dic6—21no—
156C3×Dic6—21no—
157C3×Dic6—21no—
158C32×C8—22no—
159C3×C3⋊C8—22no—
160#568—31no—
161#532—31no—
162#421—31no—
163#421—31no—
164#410—31no—
165#339—31no—
166#339C3×D311yescharacteristic, complemented
167C3×#41—61no—
168C3×#41—61no—
169C3×#41—61no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—61no—
173C3×#41—61no—
174C3×#41—61no—
175C2×C3×Q8—31no—
176C2×C3×Q8—31no—
177C2×C3×D4—32no—
178C2×C3×D4—31no—
179C2×C3×D4—31no—
180C2×C3×D4—62no—
181C2×C3×D4—61no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—32no—
186C22×C3×C4—41no—
187C22×C3×C4—61no—
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C23×C311yescharacteristic
190C2×#67—21no—
191C2×#67C3×D412yes—
192#235—21no—
193#235—21no—
194#235—21no—
195#235—21no—
196#235—21no—
197#235—21no—
198#235—21no—
199#235—21no—
200C2×D12—22no—
201C2×D12C2×C3×C411yescharacteristic
202C2×D12C2×C3×C411yescharacteristic
203C2×C4×D3—21no—
204C2×C4×D3C3×D412yes—
205C2×Dic6C2×C3×C411yescharacteristic
206C2×Dic6C2×C3×C411yescharacteristic
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—22no—
210C3×M16—32no—
211C3×M16—41no—
212C3×M16—64no—
213C3×M16—61no—
214C3×M16D1212yescomplemented
215C3×M16C3×D412yescomplemented
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—41no—
219C2×C3×C8—64no—
220C2×C3×C8—61no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—62no—
225C3×C4⋊C4—61no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—41no—
229C3×C22⋊C4—62no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—61no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—41no—
239C3×C42—41no—
240C3×C42—61no—
241C3×C42C4×D311yescharacteristic
242C3×C42C4×D311yescharacteristic
243C3×C42C2×C3×C411yescharacteristic
244C3×C42C2×C3×C411yescharacteristic
245#217—21no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3—21no—
250C4×Dic3—21no—
251#208—21no—
252#208C3×D412yescomplemented
253C2×C3⋊C8—21no—
254ΓL(1,𝔽25)—24no—
255D3×C8—24no—
256C22×C32—41no—
257C22×C32C2×C22⋊C411yescharacteristic
258C22×C32#12111yescharacteristic
259C22×C32#12111yescharacteristic
260C2×C3×D3—22no—
261C2×C3×D3—42no—
262C2×C3×D3—42no—
263C2×C3×D3—41no—
264C32×C4—22no—
265C32×C4—41no—
266C32×C4—81no—
267C32×C4C4×D411yescharacteristic
268C32×C4C4×D411yescharacteristic
269C32×C4#12011yescharacteristic
270C32×C4#12311yescharacteristic
271C3×Dic3—22no—
272C3×Dic3—41no—
273C3×Dic3—81no—
274C2×#41—31no—
275C2×#41—31no—
276#131—62no—
277#131—62no—
278C2×M16—31no—
279C2×M16—62no—
280C2×M16C2×C3×D311yescharacteristic
281C4⋊Q8—31no—
282#127—31no—
283#124—31no—
284#124—31no—
285#121—62no—
286C4×D4—62no—
287#117—31no—
288#117C2×C3×D311yescharacteristic
289C2×C42C2×C3×D311yescharacteristic
290#104—62no—
291#104—62no—
292#99—62no—
293C23×C3—21no—
294C23×C3—32no—
295C23×C3—61no—
296C23×C3D3×D411yescharacteristic
297C23×C3C2×C3×D411yescharacteristic
298C22×D3—21no—
299C22×D3C3×C22⋊C412yes—
300C3×Q8—61no—
301C3×Q8—61no—
302C3×Q8—61no—
303C3×Q8—61no—
304C3×D4—62no—
305C3×D4—62no—
306C3×D4—61no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4—61no—
310C3×D4—124no—
311C3×D4—121no—
312C3×D4—121no—
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—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—32no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—61no—
335C2×C3×C4—121no—
336C2×C3×C4—121no—
337C2×C3×C4—121no—
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—
352#67—41no—
353#67—41no—
354C2×Dic3—21no—
355C2×Dic3—41no—
356C2×Dic3C3×C22⋊C412yes—
357D12—21no—
358D12—21no—
359D12—21no—
360D12—21no—
361D12—44no—
362C4×D3—22no—
363C4×D3—22no—
364C4×D3—41no—
365C4×D3—41no—
366Dic6—21no—
367Dic6—21no—
368Dic6—21no—
369Dic6—21no—
370C3×C8—22no—
371C3×C8—22no—
372C3×C8—42no—
373C3×C8—62no—
374C3⋊C8—22no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32#38611yescharacteristic
379C3×D3—42no—
380C3×D3—81no—
381#41—61no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388#41—61no—
389C2×Q8—31no—
390C2×Q8—31no—
391C2×D4—32no—
392C2×D4—31no—
393C2×D4—31no—
394C2×D4—62no—
395C2×D4—61no—
396C22×C4—21no—
397C22×C4—32no—
398C22×C4—61no—
399C22×C4C22×C3×D311yescharacteristic
400M16—21no—
401M16—32no—
402M16—64no—
403M16—61no—
404M16C3×D1212yescomplemented
405C2×C8—21no—
406C2×C8—64no—
407C2×C8—61no—
408C4⋊C4—21no—
409C4⋊C4—62no—
410C4⋊C4—61no—
411C22⋊C4—21no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—61no—
415C42—21no—
416C42—21no—
417C42—61no—
418C42C3×C4×D311yescharacteristic
419C42C3×C4×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—62no—
427C22×C3—122no—
428C22×C3—122no—
429C22×C3—121no—
430C22×C3D3×C22⋊C411yescharacteristic
431C22×C3#88411yescharacteristic
432C22×C3#89511yescharacteristic
433C22×C3C2×C3×C22⋊C411yescharacteristic
434C22×C3C3×#12111yescharacteristic
435C22×C3C3×#12111yescharacteristic
436C2×D3—22no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—41no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—62no—
451C3×C4—82no—
452C3×C4—81no—
453C3×C4—81no—
454C3×C4—121no—
455C3×C4—241no—
456C3×C4C4×D1211yescharacteristic
457C3×C4#88211yescharacteristic
458C3×C4#89311yescharacteristic
459C3×C4#89411yescharacteristic
460C3×C4C3×C4×D411yescharacteristic
461C3×C4C3×C4×D411yescharacteristic
462C3×C4C3×#12011yescharacteristic
463C3×C4C3×#12311yescharacteristic
464Dic3—22no—
465Dic3—41no—
466Dic3—81no—
467C32#188411yescharacteristic, complemented, Sylow
468C23—32no—
469C23—61no—
470C23C3×D3×D411yescharacteristic
471Q8—61no—
472Q8—61no—
473Q8—61no—
474Q8—61no—
475D4—62no—
476D4—62no—
477D4—61no—
478D4—61no—
479D4—61no—
480D4—61no—
481D4—124no—
482D4—121no—
483D4—121no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—32no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—61no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4C2×C3×C4×D311yescharacteristic
496C2×C4C2×C3×D1211yescharacteristic
497C2×C4C3×#23511yescharacteristic
498C2×C4C3×D3×D411yescharacteristic
499C2×C4C3×D3×D411yescharacteristic
500C2×C4C3×#23911yescharacteristic
501C8—22no—
502C8—62no—
503C2×C3—22no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—122no—
514C2×C3—241no—
515C2×C3#466511yescharacteristic
516C2×C3C3×#38611yescharacteristic
517D3—42no—
518D3—81no—
519C22—41no—
520C22—62no—
521C22—122no—
522C22—122no—
523C22—121no—
524C22C3×D3×C22⋊C411yescharacteristic
525C22C3×#88411yescharacteristic
526C22C3×#89511yescharacteristic
527C4—22no—
528C4—41no—
529C4—62no—
530C4—81no—
531C4—121no—
532C4—241no—
533C4C3×C4×D1211yescharacteristic
534C4C3×#88211yescharacteristic
535C4C3×#89311yescharacteristic
536C4C3×#89411yescharacteristic
537C3—21no—
538C3#6877711yescharacteristic, direct product factor
539C3C3×#188411yescharacteristic, complemented
540C2—21no—
541C2—21no—
542C2—21no—
543C2—122no—
544C2—241no—
545C2C3×#466511yescharacteristic
subgroupG/H
centerC3×C4#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC2×C32#386
prime radicalC3×#5477C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
1. C3×#5745C2semidirect
3. C3×#5139C2semidirect
4. C3×#5006C2semidirect
5. C3×#5005C2semidirect
6. C32×#339C2semidirect
7. C3×#4523C2semidirect
9. #68777C3direct (N × H)
21. C3×#875C4semidirect
22. C3×#875C4semidirect
23. C3×#874C4semidirect
25. C3×#869C4semidirect
31. #5745C2×C3semidirect
36. #5139C2×C3semidirect
37. #5006C2×C3semidirect
38. #5005C2×C3semidirect
41. C3×#339D3semidirect
42. C3×#339C2×C3semidirect
43. #4523C2×C3semidirect
64. C32×M16D4semidirect
79. C3×#208D4semidirect
114. #875C3×C4semidirect
115. #875C3×C4semidirect
116. #874C3×C4semidirect
118. #869C3×C4semidirect
166. #339C3×D3semidirect
214. C3×M16D12semidirect
215. C3×M16C3×D4semidirect
252. #208C3×D4semidirect
404. M16C3×D12semidirect
467. C32#1884semidirect
538. C3#68777direct (N × H)
539. C3C3×#1884semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
C3×#54772
C3×#51392
C3×#50062
C3×#50052
C32×#3392
C3×#45232
C3×#18843
#687773