Finite Group Databasebeta

← C3×#80492 #11888027

Subgroups — C3×#80492 #11888027

Order 1152.

826 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5726C211yesmaximal, characteristic
2C3×#5584C211yesmaximal, characteristic, complemented
3C32×#462C211yesmaximal, characteristic, complemented
4C3×#5188C212yesmaximal, complemented
5C3×#5187C212yesmaximal, complemented
6C3×#5079C211yesmaximal, characteristic, complemented
7C3×C4×#215C211yesmaximal, characteristic, complemented
8C3×#4846C211yesmaximal, characteristic, complemented
9C3×#4834C211yesmaximal, characteristic, complemented
10C3×#4814C212yesmaximal, complemented
11C3×#4804C212yesmaximal, complemented
12C3×#3302—31nomaximal
13#80492C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2212yes—
15C2×C32×Q16C2212yes—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18C3×#946C2212yes—
19C3×#946C2212yes—
20C2×C3×#215C2211yescharacteristic
21C2×C3×D3×C8C2212yes—
22C3×#898—22no—
23C3×#895—21no—
24C3×#894—22no—
25C3×#892C2212yes—
26C3×D3×C4⋊C4—21no—
27C3×#875—21no—
28C3×C4×D12C2211yescharacteristic
29C3×C4×D12C2211yescharacteristic
30C3×#869C2211yescharacteristic
31C3×C4×Dic6—21no—
32C32×C4⋊C8C2211yescharacteristic
33C32×#103C2212yes—
34C3×#835C2212yes—
35C3×#835C2211yescharacteristic
36C3×#835C2212yes—
37C3×#821C2212yes—
38C3×(C2×D3)⋊C8C2212yes—
39C3×#818C2212yes—
40C3×#809C2211yescharacteristic
41C3×#808C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#556—31no—
44C3×#547—31no—
45#5726C2×C311yescharacteristic
46#5584C2×C311yescharacteristic, complemented
47C3×#487—31no—
48C3×#485—32no—
49C3×#474—31no—
50C3×#467—32no—
51C3×#462—21no—
52C3×#462D311yescharacteristic, complemented
53C3×#462C2×C311yescharacteristic, complemented
54C3×#452—32no—
55C3×C4×SD16—31no—
56C3×C8×D4—31no—
57C3×#416—32no—
58#5188C2×C312yescomplemented
59#5187C2×C312yescomplemented
60#5079C2×C311yescharacteristic, complemented
61C4×#215C2×C311yescharacteristic, complemented
62#4846C2×C311yescharacteristic, complemented
63#4834C2×C311yescharacteristic, complemented
64#4814C2×C312yescomplemented
65#4804C2×C312yescomplemented
66C2×C32×Q8C2311yescharacteristic
67C2×C32×Q8C2312yes—
68C3×#239—24no—
69C3×#239—42no—
70C2×C3×D12—22no—
71C2×C3×D12D411yescharacteristic
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3C2312yes—
76C2×C3×Dic6—21no—
77C32×Q16—42no—
78C2×C32×C8C2312yes—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2311yescharacteristic
83C32×C42—21no—
84C32×C42C2311yescharacteristic
85C3×#215—22no—
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—22no—
89C3×(C2×D3)⋊C4—21no—
90C3×(C2×D3)⋊C4D412yes—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3D411yescharacteristic
93C3×C4⋊Dic3C2312yes—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×Dic3⋊C4—22no—
96C3×Dic3⋊C4—22no—
97C3×C4×Dic3—22no—
98C2×C3×C3⋊C8C2311yescharacteristic
99C2×C3×C3⋊C8C2311yescharacteristic
100C3×D3×C8—42no—
101#3302—31noSylow
102C2×C3×#41—32no—
103C2×#239C22×C312yes—
104C2×C3×Q16—22no—
105C2×C3×Q16C2×D312yes—
106C2×C3×Q16C22×C312yes—
107C2×C3×SD16—31no—
108C22×C3×C8—32no—
109C3×C4⋊Q8—21no—
110C3×C4⋊Q8—31no—
111C3×C4⋊Q8C2×D311yescharacteristic
112C3×C4⋊Q8C22×C311yescharacteristic
113C3×#126—62no—
114C3×#124—61no—
115C3×#123—62no—
116C3×C22⋊Q8—32no—
117C3×C22⋊Q8—32no—
118C3×#121—61no—
119C3×C4×Q8—21no—
120C3×C4×Q8—61no—
121C3×C4×Q8C2×D311yescharacteristic
122C3×C4×Q8C22×C311yescharacteristic
123C3×C4×D4—31no—
124C3×C4×D4—31no—
125C3×#117—32no—
126C2×C3×C4⋊C4—61no—
127#946C22×C312yes—
128#946C22×C312yes—
129C2×#215C22×C311yescharacteristic
130C2×D3×C8C22×C312yes—
131#898—22no—
132#895—21no—
133#894—22no—
134#892C22×C312yes—
135D3×C4⋊C4—21no—
136#875—21no—
137C4×D12C22×C311yescharacteristic
138C4×D12C22×C311yescharacteristic
139#869C22×C311yescharacteristic
140C4×Dic6—21no—
141C3×#107—32no—
142C3×#106—31no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8C2×D311yescharacteristic
145C3×C4⋊C8C22×C311yescharacteristic
146C3×#103—22no—
147C3×#103—32no—
148C3×#103—31no—
149C3×#103—32no—
150C3×#103C2×D312yes—
151C3×#103C22×C312yes—
152C3×#102—32no—
153C3×#102—31no—
154C3×C22⋊C8—32no—
155C3×C4×C8—31no—
156#835C22×C312yes—
157#835C22×C311yescharacteristic
158#835C22×C312yes—
159#821C22×C312yes—
160(C2×D3)⋊C8C22×C312yes—
161#818C22×C312yes—
162#809C22×C311yescharacteristic
163#808C22×C311yescharacteristic
164C4×C3⋊C8C22×C311yescharacteristic
165C22×C3×D3—21no—
166C22×C3×D3C2×D412yes—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—22no—
170C32×Q8—22no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C2×D412yes—
180C3×D12—22no—
181C3×D12—21no—
182C3×D12—44no—
183C3×D12#4112yes—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—44no—
187C3×C4×D3—42no—
188C3×Dic6—42no—
189C32×C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192#556—31no—
193#547—31no—
194#487—31no—
195#485—32no—
196#474—31no—
197#467—32no—
198#462C3×D311yescharacteristic, complemented
199#452—32no—
200C4×SD16—31no—
201C8×D4—31no—
202#416—32no—
203C3×#41—64no—
204C3×#41—122no—
205C2×C3×Q8—21no—
206C2×C3×Q8—22no—
207C2×C3×Q8—61no—
208C2×C3×Q8C22×D312yes—
209C2×C3×Q8C22×D311yescharacteristic
210C2×C3×Q8C23×C312yes—
211C2×C3×Q8C23×C311yescharacteristic
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—62no—
215C22×C3×C4—32no—
216C22×C3×C4—62no—
217C22×C3×C4—61no—
218#239—24no—
219#239—42no—
220C2×D12—22no—
221C2×D12C3×D411yescharacteristic
222C2×D12C23×C311yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—22no—
225C2×C4×D3C23×C312yes—
226C2×Dic6—21no—
227C3×Q16—42no—
228C3×Q16—42no—
229C3×Q16—44no—
230C3×SD16—62no—
231C2×C3×C8—22no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—122no—
235C2×C3×C8C22×D312yes—
236C2×C3×C8C23×C312yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4C22×D311yescharacteristic
254C3×C4⋊C4C23×C311yescharacteristic
255C3×C22⋊C4—32no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—62no—
258C3×C22⋊C4—62no—
259C3×C22⋊C4—61no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—41no—
264C3×C42—62no—
265C3×C42C22×D311yescharacteristic
266C3×C42C23×C311yescharacteristic
267#215—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—21no—
272(C2×D3)⋊C4C3×D412yes—
273C4⋊Dic3—21no—
274C4⋊Dic3C3×D411yescharacteristic
275C4⋊Dic3C23×C311yescharacteristic
276C4⋊Dic3C23×C312yes—
277Dic3⋊C4—22no—
278Dic3⋊C4—22no—
279C4×Dic3—22no—
280C2×C3⋊C8C23×C311yescharacteristic
281C2×C3⋊C8C23×C311yescharacteristic
282D3×C8—42no—
283C22×C32C22×D411yescharacteristic
284C2×C3×D3—22no—
285C2×C3×D3—22no—
286C2×C3×D3—22no—
287C2×C3×D3—22no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C32×C4—21no—
291C32×C4—41no—
292C32×C4—42no—
293C32×C4—41no—
294C32×C4#13512yes—
295C32×C4C2×#4111yescharacteristic
296C32×C4H(5,𝔽2)11yescharacteristic
297C3×Dic3—22no—
298C3×Dic3—42no—
299C3×Dic3—41no—
300C2×#41—32no—
301C2×Q16C2×C3×D312yes—
302C2×SD16—31no—
303C22×C8—32no—
304C4⋊Q8—31no—
305C4⋊Q8C2×C3×D311yescharacteristic
306#126—62no—
307#124—61no—
308#123—62no—
309C22⋊Q8—32no—
310C22⋊Q8—32no—
311#121—61no—
312C4×Q8—61no—
313C4×Q8C2×C3×D311yescharacteristic
314C4×D4—31no—
315C4×D4—31no—
316#117—32no—
317C2×C4⋊C4—61no—
318#107—32no—
319#106—31no—
320C4⋊C8C2×C3×D311yescharacteristic
321#103—32no—
322#103—31no—
323#103—32no—
324#103C2×C3×D312yes—
325#102—32no—
326#102—31no—
327C22⋊C8—32no—
328C4×C8—31no—
329C23×C3—32no—
330C23×C3—61no—
331C22×D3—21no—
332C22×D3C2×C3×D412yes—
333C3×Q8—22no—
334C3×Q8—21no—
335C3×Q8—22no—
336C3×Q8—21no—
337C3×Q8—22no—
338C3×Q8—21no—
339C3×Q8—22no—
340C3×Q8—21no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—42no—
344C3×Q8—42no—
345C3×Q8—122no—
346C3×D4—32no—
347C3×D4—62no—
348C3×D4—61no—
349C3×D4—124no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—32no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—124no—
368C2×C3×C4—122no—
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C2×C3×D411yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C24×C311yescharacteristic
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3C2×C3×D412yes—
378D12—22no—
379D12—21no—
380D12—44no—
381D12C3×#4112yes—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—44no—
385C4×D3—42no—
386Dic6—42no—
387C3×C8—22no—
388C3×C8—22no—
389C3×C8—42no—
390C3×C8—61no—
391C3×C8—61no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C2×C32#54611yescharacteristic
395C2×C32C2×#13511yescharacteristic
396C2×C32#57811yescharacteristic
397C3×D3—42no—
398C3×D3—42no—
399#41—64no—
400#41—122no—
401C2×Q8—61no—
402C2×Q8C22×C3×D312yes—
403C2×Q8C22×C3×D311yescharacteristic
404C2×D4—31no—
405C2×D4—31no—
406C2×D4—62no—
407C22×C4—32no—
408C22×C4—62no—
409C22×C4—61no—
410Q16—42no—
411SD16—62no—
412C2×C8—31no—
413C2×C8—31no—
414C2×C8—122no—
415C2×C8C22×C3×D312yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—31no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4C22×C3×D311yescharacteristic
426C22⋊C4—32no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C22⋊C4—62no—
430C22⋊C4—61no—
431C42—21no—
432C42—62no—
433C42C22×C3×D311yescharacteristic
434C22×C3—21no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3C2×D3×D411yescharacteristic
442C22×C3C22×C3×D411yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—41no—
448C2×D3—41no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—42no—
459C3×C4—41no—
460C3×C4—42no—
461C3×C4—41no—
462C3×C4—62no—
463C3×C4—82no—
464C3×C4—81no—
465C3×C4—121no—
466C3×C4—122no—
467C3×C4#91912yes—
468C3×C4C3×#13512yes—
469C3×C4#100411yescharacteristic
470C3×C4D3×#4111yescharacteristic
471C3×C4C2×C3×#4111yescharacteristic
472C3×C4C3×H(5,𝔽2)11yescharacteristic
473Dic3—22no—
474Dic3—42no—
475Dic3—41no—
476C32#330211yescharacteristic, complemented, Sylow
477C23—32no—
478C23—61no—
479Q8—21no—
480Q8—22no—
481Q8—21no—
482Q8—22no—
483Q8—122no—
484D4—32no—
485D4—62no—
486D4—61no—
487D4—124no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—32no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—61no—
496C2×C4—124no—
497C2×C4—122no—
498C2×C4C3×D3×D411yescharacteristic
499C2×C4C3×D3×D411yescharacteristic
500C2×C4C23×C3×D311yescharacteristic
501C8—22no—
502C8—61no—
503C8—61no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—122no—
508C2×C3—122no—
509C2×C3#561211yescharacteristic
510C2×C3C2×#91911yescharacteristic
511C2×C3#577911yescharacteristic
512C2×C3C3×#54611yescharacteristic
513C2×C3C2×C3×#13511yescharacteristic
514C2×C3C3×#57811yescharacteristic
515D3—42no—
516D3—42no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—121no—
522C22—121no—
523C22C2×C3×D3×D411yescharacteristic
524C4—21no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—62no—
529C4—121no—
530C4—122no—
531C4C3×#91912yes—
532C4C3×#100411yescharacteristic
533C4C3×D3×#4111yescharacteristic
534C3—21no—
535C3#8049211yescharacteristic, direct product factor
536C3C3×#330211yescharacteristic, complemented
537C2—122no—
538C2—122no—
539C2C3×#561211yescharacteristic
540C2C2×C3×#91911yescharacteristic
541C2C3×#577911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#462C2
socleC22×C32C22×D4
prime radicalC3×#5726C2
hypercenterC3×#462D3
hypocenterC3C3×#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#462

Complements

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

NH ≅ G/Ntype
2. C3×#5584C2semidirect
3. C32×#462C2semidirect
4. C3×#5188C2semidirect
5. C3×#5187C2semidirect
6. C3×#5079C2semidirect
7. C3×C4×#215C2semidirect
8. C3×#4846C2semidirect
9. C3×#4834C2semidirect
10. C3×#4814C2semidirect
11. C3×#4804C2semidirect
13. #80492C3direct (N × H)
46. #5584C2×C3semidirect
52. C3×#462D3semidirect
53. C3×#462C2×C3semidirect
58. #5188C2×C3semidirect
59. #5187C2×C3semidirect
60. #5079C2×C3semidirect
61. C4×#215C2×C3semidirect
62. #4846C2×C3semidirect
63. #4834C2×C3semidirect
64. #4814C2×C3semidirect
65. #4804C2×C3semidirect
198. #462C3×D3semidirect
476. C32#3302semidirect
535. C3#80492direct (N × H)
536. C3C3×#3302semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
C3×#55842
C32×#4622
C3×#51882
C3×#51872
C3×#50792
C3×C4×#2152
C3×#48462
C3×#48342
C3×#48142
C3×#48042
C3×#33023
#804923