Finite Group Databasebeta

← C3×#81860 #11886573

Subgroups — C3×#81860 #11886573

Order 1152.

1000 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5762C212yesmaximal, complemented
2C3×#5757C212yesmaximal, complemented
3C32×#496C211yesmaximal, characteristic, complemented
4C3×#5177C212yesmaximal, complemented
5C3×#5160C212yesmaximal, complemented
6C3×#5077C212yesmaximal, complemented
7C3×#5062C211yesmaximal, characteristic, complemented
8C3×#5061C211yesmaximal, characteristic, complemented
9C3×#4717C212yesmaximal, complemented
10C3×#3288—31nomaximal
11#81860C311yesmaximal, characteristic, direct product factor
12C3×#1007C2211yescharacteristic, complemented
13C3×#1004C2211yescharacteristic
14C3×#1004C2212yescomplemented
15C2×C32×SD16C2212yescomplemented
16C2×C32×D8C2212yescomplemented
17C32×#127C2211yescharacteristic, complemented
18C32×#124C2211yescharacteristic, complemented
19C3×#942C2212yescomplemented
20C3×#932C2212yes—
21C3×#932C2212yescomplemented
22C3×#932C2212yescomplemented
23C3×#916—22no—
24C3×#915—22no—
25C3×#914—22no—
26C3×D3×SD16—22no—
27C3×#912—22no—
28C3×#911—22no—
29C3×#911—22no—
30C3×D3×D8—22no—
31C3×#904C2212yes—
32C3×#901C2212yescomplemented
33C3×#901C2212yescomplemented
34C32×#97C2211yescharacteristic, complemented
35C3×#836C2211yescharacteristic, complemented
36C3×#833C2211yescharacteristic, complemented
37C3×#833C2212yescomplemented
38C3×#819C2212yescomplemented
39C3×#805C2212yescomplemented
40C3×#805C2212yescomplemented
41C3×#577—32no—
42C3×#576—32no—
43#5762C2×C312yescomplemented
44#5757C2×C312yescomplemented
45C3×#496—21no—
46C3×#496D311yescharacteristic, complemented
47C3×#496C2×C311yescharacteristic, complemented
48C3×#472—32no—
49C3×#471—32no—
50C3×#455—31no—
51C3×#454—31no—
52C3×#453—32no—
53C3×#444—32no—
54#5177C2×C312yescomplemented
55#5160C2×C312yescomplemented
56#5077C2×C312yescomplemented
57#5062C2×C311yescharacteristic, complemented
58#5061C2×C311yescharacteristic, complemented
59#4717C2×C312yescomplemented
60C2×C32×Q8C2311yescharacteristic
61C2×C32×D4—41no—
62C2×C32×D4C2312yes—
63C2×C32×D4C2311yescharacteristic
64C2×C3×#67—41no—
65C2×C3×#67—42no—
66C3×#239—22no—
67C3×D3×Q8—22no—
68C3×#237—22no—
69C3×#237—22no—
70C3×#237—22no—
71C3×D3×D4—22no—
72C3×D3×D4—22no—
73C3×D3×D4—22no—
74C3×#235—41no—
75C3×#235C2312yes—
76C3×#235C2312yes—
77C32×SD16—22no—
78C32×SD16—22no—
79C32×D8—22no—
80C32×D8—22no—
81C2×C32×C8C2312yes—
82C32×C22⋊C4—41no—
83C32×C42C2311yescharacteristic
84C3×#216—22no—
85C3×#215—22no—
86C3×#214—22no—
87C3×#214—22no—
88C3×#214—22no—
89C3×#213—22no—
90C3×#213—22no—
91C3×#213—22no—
92C3×#208C2312yes—
93C3×#208C2312yes—
94C3×Dic12D412yescomplemented
95C3×D24D412yescomplemented
96C3×#204D412yescomplemented
97C3×#204D412yescomplemented
98C3×ΓL(1,𝔽25)—22no—
99C3×ΓL(1,𝔽25)—22no—
100C3×D3×C8—22no—
101C3×D3×C8—22no—
102#3288—31noSylow
103C3×2−1+4—31no—
104C3×H(5,𝔽2)—31no—
105C3×H(5,𝔽2)—32no—
106#1007C22×C311yescharacteristic, complemented
107#1004C22×C311yescharacteristic
108#1004C22×C312yescomplemented
109C3×#137—32no—
110C3×#137—62no—
111C3×AGL(1,ℤ/8)—32no—
112C3×AGL(1,ℤ/8)—32no—
113C3×AGL(1,ℤ/8)—32no—
114C3×AGL(1,ℤ/8)—62no—
115C3×AGL(1,ℤ/8)—62no—
116C3×AGL(1,ℤ/8)—62no—
117C3×#135—32no—
118C3×#135—62no—
119C3×#135—62no—
120C2×C3×SD16—22no—
121C2×C3×SD16—62no—
122C2×C3×SD16C2×D312yescomplemented
123C2×C3×SD16C22×C312yescomplemented
124C2×C3×D8—22no—
125C2×C3×D8—62no—
126C2×C3×D8C2×D312yescomplemented
127C2×C3×D8C22×C312yescomplemented
128C3×#131—32no—
129C3×#131—32no—
130C3×#127—21no—
131C3×#127C2×D311yescharacteristic, complemented
132C3×#127C22×C311yescharacteristic, complemented
133C3×#124—21no—
134C3×#124C2×D311yescharacteristic, complemented
135C3×#124C22×C311yescharacteristic, complemented
136#942C22×C312yescomplemented
137#932C22×C312yes—
138#932C22×C312yescomplemented
139#932C22×C312yescomplemented
140#916—22no—
141#915—22no—
142#914—22no—
143D3×SD16—22no—
144#912—22no—
145#911—22no—
146#911—22no—
147D3×D8—22no—
148#904C22×C312yes—
149#901C22×C312yescomplemented
150#901C22×C312yescomplemented
151C3×#108—32no—
152C3×#104—32no—
153C3×#104—32no—
154C3×#101—31no—
155C3×#100—32no—
156C3×#100—31no—
157C3×#97—21no—
158C3×#97C2×D311yescharacteristic, complemented
159C3×#97C22×C311yescharacteristic, complemented
160#836C22×C311yescharacteristic, complemented
161#833C22×C311yescharacteristic, complemented
162#833C22×C312yescomplemented
163#819C22×C312yescomplemented
164#805C22×C312yescomplemented
165#805C22×C312yescomplemented
166C23×C32—21no—
167C23×C32—22no—
168C22×C3×D3—42no—
169C22×C3×D3—41no—
170C32×Q8—22no—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—82no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4C2411yescharacteristic
178C3×#67—22no—
179C3×#67—22no—
180C3×#67—42no—
181C3×#67—42no—
182C3×#67—42no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—41no—
185C3×D12—41no—
186C3×D12C2×D412yes—
187C3×D12C2×D412yes—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3—42no—
191C3×Dic6—41no—
192C3×Dic6C2×D412yes—
193C3×Dic6C2×D412yes—
194C32×C8C2×D412yes—
195C32×C8C2×D412yes—
196C3×C3⋊C8—22no—
197C3×C3⋊C8—22no—
198#577—32no—
199#576—32no—
200#496C3×D311yescharacteristic, complemented
201#472—32no—
202#471—32no—
203#455—31no—
204#454—31no—
205#453—32no—
206#444—32no—
207C3×#41—32no—
208C3×#41—32no—
209C3×#41—62no—
210C3×#41—62no—
211C3×#41—62no—
212C3×#41—62no—
213C3×#41—121no—
214C2×C3×Q8—21no—
215C2×C3×Q8—62no—
216C2×C3×Q8C22×D311yescharacteristic
217C2×C3×Q8C23×C311yescharacteristic
218C2×C3×D4—21no—
219C2×C3×D4—22no—
220C2×C3×D4—41no—
221C2×C3×D4—41no—
222C2×C3×D4—42no—
223C2×C3×D4—62no—
224C2×C3×D4—62no—
225C2×C3×D4—62no—
226C2×C3×D4—121no—
227C2×C3×D4—122no—
228C2×C3×D4C22×D311yescharacteristic
229C2×C3×D4C22×D312yes—
230C2×C3×D4C23×C312yes—
231C2×C3×D4C23×C311yescharacteristic
232C2×#67—41no—
233C2×#67—42no—
234#239—22no—
235D3×Q8—22no—
236#237—22no—
237#237—22no—
238#237—22no—
239D3×D4—22no—
240D3×D4—22no—
241D3×D4—22no—
242#235—41no—
243#235C23×C312yes—
244#235C23×C312yes—
245C3×Q16—32no—
246C3×Q16—62no—
247C3×SD16—22no—
248C3×SD16—22no—
249C3×SD16—22no—
250C3×SD16—22no—
251C3×SD16—32no—
252C3×SD16—32no—
253C3×SD16—42no—
254C3×SD16—42no—
255C3×SD16—62no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×D8—22no—
260C3×D8—22no—
261C3×D8—22no—
262C3×D8—22no—
263C3×D8—32no—
264C3×D8—42no—
265C3×D8—42no—
266C3×D8—62no—
267C3×D8—62no—
268C3×D8—62no—
269C3×M16—32no—
270C3×M16—32no—
271C3×M16—62no—
272C3×M16—62no—
273C2×C3×C8—22no—
274C2×C3×C8—62no—
275C2×C3×C8—62no—
276C2×C3×C8C22×D312yes—
277C2×C3×C8C23×C312yes—
278C3×C22⋊C4—41no—
279C3×C22⋊C4—41no—
280C3×C22⋊C4—42no—
281C3×C42—21no—
282C3×C42C22×D311yescharacteristic
283C3×C42C23×C311yescharacteristic
284#216—22no—
285#215—22no—
286#214—22no—
287#214—22no—
288#214—22no—
289#213—22no—
290#213—22no—
291#213—22no—
292#208C23×C312yes—
293#208C23×C312yes—
294Dic12C3×D412yescomplemented
295D24C3×D412yescomplemented
296#204C3×D412yescomplemented
297#204C3×D412yescomplemented
298ΓL(1,𝔽25)—22no—
299ΓL(1,𝔽25)—22no—
300D3×C8—22no—
301D3×C8—22no—
302C22×C32—41no—
303C22×C32—42no—
304C22×C32—82no—
305C22×C32—81no—
306C22×C32H(5,𝔽2)11yescharacteristic
307C2×C3×D3—22no—
308C2×C3×D3—22no—
309C2×C3×D3—162no—
310C2×C3×D3—161no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4C22×D412yes—
314C3×Dic3—22no—
315C3×Dic3—22no—
3162−1+4—31no—
317H(5,𝔽2)—31no—
318H(5,𝔽2)—32no—
319#137—32no—
320#137—62no—
321AGL(1,ℤ/8)—32no—
322AGL(1,ℤ/8)—32no—
323AGL(1,ℤ/8)—32no—
324AGL(1,ℤ/8)—62no—
325AGL(1,ℤ/8)—62no—
326AGL(1,ℤ/8)—62no—
327#135—32no—
328#135—62no—
329#135—62no—
330C2×SD16—62no—
331C2×SD16C2×C3×D312yescomplemented
332C2×D8—62no—
333C2×D8C2×C3×D312yescomplemented
334#131—32no—
335#131—32no—
336#127C2×C3×D311yescharacteristic, complemented
337#124C2×C3×D311yescharacteristic, complemented
338#108—32no—
339#104—32no—
340#104—32no—
341#101—31no—
342#100—32no—
343#100—31no—
344#97C2×C3×D311yescharacteristic, complemented
345C23×C3—21no—
346C23×C3—22no—
347C23×C3—21no—
348C23×C3—22no—
349C23×C3—41no—
350C23×C3—42no—
351C23×C3—122no—
352C23×C3—121no—
353C22×D3—42no—
354C22×D3—41no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—32no—
358C3×Q8—32no—
359C3×Q8—42no—
360C3×Q8—121no—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—32no—
368C3×D4—32no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—62no—
373C3×D4—62no—
374C3×D4—82no—
375C3×D4—82no—
376C3×D4—84no—
377C3×D4—121no—
378C3×D4—122no—
379C3×D4—122no—
380C3×D4—122no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4—122no—
391C2×C3×C4—121no—
392C2×C3×C4—122no—
393C2×C3×C4C23×D311yescharacteristic
394C2×C3×C4C24×C311yescharacteristic
395#67—22no—
396#67—22no—
397#67—42no—
398#67—42no—
399#67—42no—
400C2×Dic3—42no—
401C2×Dic3—41no—
402D12—41no—
403D12C2×C3×D412yes—
404D12C2×C3×D412yes—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—42no—
408Dic6—41no—
409Dic6C2×C3×D412yes—
410Dic6C2×C3×D412yes—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—62no—
414C3×C8—62no—
415C3×C8D3×D412yes—
416C3×C8D3×D412yes—
417C3×C8C2×C3×D412yes—
418C3×C8C2×C3×D412yes—
419C3⋊C8—22no—
420C3⋊C8—22no—
421C2×C32—21no—
422C2×C32—82no—
423C2×C32—81no—
424C2×C32D4211yescharacteristic
425C3×D3—42no—
426C3×D3—42no—
427#41—32no—
428#41—32no—
429#41—62no—
430#41—62no—
431#41—62no—
432#41—62no—
433#41—121no—
434C2×Q8—62no—
435C2×Q8C22×C3×D311yescharacteristic
436C2×D4—41no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—121no—
441C2×D4—122no—
442C2×D4C22×C3×D311yescharacteristic
443C2×D4C22×C3×D312yes—
444Q16—32no—
445Q16—62no—
446SD16—22no—
447SD16—22no—
448SD16—32no—
449SD16—32no—
450SD16—62no—
451SD16—62no—
452SD16—62no—
453SD16—62no—
454D8—22no—
455D8—22no—
456D8—32no—
457D8—62no—
458D8—62no—
459D8—62no—
460M16—32no—
461M16—32no—
462M16—62no—
463M16—62no—
464C2×C8—62no—
465C2×C8—62no—
466C2×C8C22×C3×D312yes—
467C22⋊C4—41no—
468C42C22×C3×D311yescharacteristic
469C22×C3—21no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—41no—
473C22×C3—42no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—81no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—84no—
483C22×C3—82no—
484C22×C3—481no—
485C22×C3—482no—
486C22×C3#100411yescharacteristic
487C22×C3C3×H(5,𝔽2)11yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—161no—
491C2×D3—162no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—81no—
501C3×C4C2×D3×D412yes—
502C3×C4C22×C3×D412yes—
503Dic3—22no—
504Dic3—22no—
505C32#328811yescharacteristic, complemented, Sylow
506C23—21no—
507C23—22no—
508C23—122no—
509C23—121no—
510Q8—22no—
511Q8—32no—
512Q8—32no—
513Q8—121no—
514D4—22no—
515D4—22no—
516D4—22no—
517D4—32no—
518D4—32no—
519D4—62no—
520D4—62no—
521D4—82no—
522D4—121no—
523D4—122no—
524D4—122no—
525D4—122no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—122no—
533C2×C4C23×C3×D311yescharacteristic
534C8—62no—
535C8—62no—
536C8C3×D3×D412yes—
537C8C3×D3×D412yes—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—82no—
543C2×C3—81no—
544C2×C3—82no—
545C2×C3—81no—
546C2×C3—122no—
547C2×C3—122no—
548C2×C3—162no—
549C2×C3—161no—
550C2×C3#571711yescharacteristic
551C2×C3C3×D4211yescharacteristic
552D3—42no—
553D3—42no—
554C22—41no—
555C22—42no—
556C22—62no—
557C22—62no—
558C22—82no—
559C22—81no—
560C22—481no—
561C22—482no—
562C22C3×#100411yescharacteristic
563C4—41no—
564C4—41no—
565C4—62no—
566C4—62no—
567C4C2×C3×D3×D412yes—
568C3—21no—
569C3#8186011yescharacteristic, direct product factor
570C3C3×#328811yescharacteristic, complemented
571C2—21no—
572C2—82no—
573C2—81no—
574C2—122no—
575C2—122no—
576C2C3×#571711yescharacteristic
subgroupG/H
centerC2×C3#5717
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#496C2
socleC2×C32D42
hypercenterC3×#496D3
hypocenterC3C3×#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#496

Complements

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

NH ≅ G/Ntype
1. C3×#5762C2semidirect
2. C3×#5757C2semidirect
3. C32×#496C2semidirect
4. C3×#5177C2semidirect
5. C3×#5160C2semidirect
6. C3×#5077C2semidirect
7. C3×#5062C2semidirect
8. C3×#5061C2semidirect
9. C3×#4717C2semidirect
11. #81860C3direct (N × H)
12. C3×#1007C22semidirect
14. C3×#1004C22semidirect
15. C2×C32×SD16C22semidirect
16. C2×C32×D8C22semidirect
17. C32×#127C22semidirect
18. C32×#124C22semidirect
19. C3×#942C22semidirect
21. C3×#932C22semidirect
22. C3×#932C22semidirect
32. C3×#901C22semidirect
33. C3×#901C22semidirect
34. C32×#97C22semidirect
35. C3×#836C22semidirect
36. C3×#833C22semidirect
37. C3×#833C22semidirect
38. C3×#819C22semidirect
39. C3×#805C22semidirect
40. C3×#805C22semidirect
43. #5762C2×C3semidirect
44. #5757C2×C3semidirect
46. C3×#496D3semidirect
47. C3×#496C2×C3semidirect
54. #5177C2×C3semidirect
55. #5160C2×C3semidirect
56. #5077C2×C3semidirect
57. #5062C2×C3semidirect
58. #5061C2×C3semidirect
59. #4717C2×C3semidirect
94. C3×Dic12D4semidirect
95. C3×D24D4semidirect
96. C3×#204D4semidirect
97. C3×#204D4semidirect
106. #1007C22×C3semidirect
108. #1004C22×C3semidirect
122. C2×C3×SD16C2×D3semidirect
123. C2×C3×SD16C22×C3semidirect
126. C2×C3×D8C2×D3semidirect
127. C2×C3×D8C22×C3semidirect
131. C3×#127C2×D3semidirect
132. C3×#127C22×C3semidirect
134. C3×#124C2×D3semidirect
135. C3×#124C22×C3semidirect
136. #942C22×C3semidirect
138. #932C22×C3semidirect
139. #932C22×C3semidirect
149. #901C22×C3semidirect
150. #901C22×C3semidirect
158. C3×#97C2×D3semidirect
159. C3×#97C22×C3semidirect
160. #836C22×C3semidirect
161. #833C22×C3semidirect
162. #833C22×C3semidirect
163. #819C22×C3semidirect
164. #805C22×C3semidirect
165. #805C22×C3semidirect
200. #496C3×D3semidirect
294. Dic12C3×D4semidirect
295. D24C3×D4semidirect
296. #204C3×D4semidirect
297. #204C3×D4semidirect
331. C2×SD16C2×C3×D3semidirect
333. C2×D8C2×C3×D3semidirect
336. #127C2×C3×D3semidirect
337. #124C2×C3×D3semidirect
344. #97C2×C3×D3semidirect
505. C32#3288semidirect
569. C3#81860direct (N × H)
570. C3C3×#3288semidirect

Maximal subgroups

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

subgroupindex
C3×#57622
C3×#57572
C32×#4962
C3×#51772
C3×#51602
C3×#50772
C3×#50622
C3×#50612
C3×#47172
C3×#32883
#818603