Finite Group Databasebeta

← C3×#81786 #11878743

Subgroups — C3×#81786 #11878743

Order 1152.

957 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
1C3×#5716C211yesmaximal, characteristic, complemented
2C3×#5716C211yesmaximal, characteristic, complemented
3C32×#493C211yesmaximal, characteristic, complemented
4C3×#5158C214yesmaximal, complemented
5C3×#5153C214yesmaximal, complemented
6C3×#5076C211yesmaximal, characteristic, complemented
7C3×#5070C211yesmaximal, characteristic, complemented
8C3×#4685C211yesmaximal, characteristic, complemented
9C3×#4681C211yesmaximal, characteristic, complemented
10C3×#3207—31nomaximal
11#81786C311yesmaximal, characteristic, direct product factor
12C2×C3×#237—21no—
13C2×C3×#237C2212yes—
14C2×C32×D8C2214yes—
15C32×#127C2211yescharacteristic
16C32×#127C2211yescharacteristic
17C3×#938—21no—
18C3×#938—21no—
19C3×#938—21no—
20C3×#938C2212yes—
21C3×#936—21no—
22C3×#936C2212yes—
23C3×D4×Dic3—21no—
24C3×D4×Dic3—21no—
25C3×D4×Dic3—21no—
26C3×D4×Dic3C2212yes—
27C2×C3×#214C2214yes—
28C3×C42×D3C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C3×#869C2211yescharacteristic
31C32×C4×C8C2211yescharacteristic
32C3×#832C2214yes—
33C3×(C2×D3)⋊C8C2212yes—
34C3×#816C2214yes—
35C3×Dic3⋊C8C2212yes—
36C3×C4×C3⋊C8C2211yescharacteristic
37C3×#532—31no—
38C3×#532—31no—
39#5716C2×C311yescharacteristic, complemented
40#5716C2×C311yescharacteristic, complemented
41C3×#493—21no—
42C3×#493D311yescharacteristic, complemented
43C3×#493C2×C311yescharacteristic, complemented
44C3×#492—31no—
45C3×#487—31no—
46C3×#486—31no—
47C3×#463—34no—
48C3×#449—34no—
49C3×#431—31no—
50#5158C2×C314yescomplemented
51#5153C2×C314yescomplemented
52#5076C2×C311yescharacteristic, complemented
53#5070C2×C311yescharacteristic, complemented
54#4685C2×C311yescharacteristic, complemented
55#4681C2×C311yescharacteristic, complemented
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4C2312yes—
61C2×C32×D4C2312yes—
62C2×C3×#67—22no—
63C2×C3×#67—22no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—22no—
66C3×#237—44no—
67C3×#237—44no—
68C2×C3×C4×D3D411yescharacteristic
69C2×C3×C4×D3D411yescharacteristic
70C2×C3×C4×D3C2311yescharacteristic
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2312yes—
73C32×D8—44no—
74C2×C32×C8C2312yes—
75C32×C42C2311yescharacteristic
76C3×#217—22no—
77C3×#217—22no—
78C3×#217—22no—
79C3×#217—22no—
80C3×#214—44no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3C2312yes—
85C3×C4×Dic3D411yescharacteristic
86C3×C4×Dic3D411yescharacteristic
87C3×C4×Dic3C2311yescharacteristic
88C2×C3×C3⋊C8C2312yes—
89#3207—31noSylow
90C2×C3×#41—32no—
91C2×C3×#41—61no—
92C2×#237—21no—
93C2×#237C22×C312yes—
94C2×C3×SD16—34no—
95C2×C3×D8—24no—
96C2×C3×D8C2×D314yes—
97C2×C3×D8C22×C314yes—
98C3×C4⋊Q8—31no—
99C3×C4⋊Q8—31no—
100C3×#127—21no—
101C3×#127—21no—
102C3×#127C2×D311yescharacteristic
103C3×#127C2×D311yescharacteristic
104C3×#127C22×C311yescharacteristic
105C3×#127C22×C311yescharacteristic
106C3×#124—32no—
107C3×#124—61no—
108C3×#121—32no—
109C3×#121—61no—
110C3×#121—61no—
111C3×#121—61no—
112C3×C4×D4—32no—
113C3×C4×D4—61no—
114C3×C4×D4—61no—
115C3×C4×D4—61no—
116C2×C3×C42—31no—
117#938—21no—
118#938—21no—
119#938—21no—
120#938C22×C312yes—
121#936—21no—
122#936C22×C312yes—
123D4×Dic3—21no—
124D4×Dic3—21no—
125D4×Dic3—21no—
126D4×Dic3C22×C312yes—
127C2×#214C22×C314yes—
128C42×D3C22×C311yescharacteristic
129#869C22×C311yescharacteristic
130#869C22×C311yescharacteristic
131C3×C4⋊C8—32no—
132C3×#103—34no—
133C3×#102—34no—
134C3×C22⋊C8—32no—
135C3×C4×C8—21no—
136C3×C4×C8—31no—
137C3×C4×C8C2×D311yescharacteristic
138C3×C4×C8C22×C311yescharacteristic
139#832C22×C314yes—
140(C2×D3)⋊C8C22×C312yes—
141#816C22×C314yes—
142Dic3⋊C8C22×C312yes—
143C4×C3⋊C8C22×C311yescharacteristic
144C23×C32—22no—
145C23×C32—22no—
146C22×C3×D3C2×D411yescharacteristic
147C32×D4—24no—
148C32×D4—24no—
149C32×D4—42no—
150C32×D4—42no—
151C32×D4—42no—
152C32×D4—42no—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2411yescharacteristic
156C3×#67—44no—
157C3×#67—44no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—44no—
161C2×C3×Dic3—44no—
162C2×C3×Dic3C2×D411yescharacteristic
163C2×C3×Dic3C2×D411yescharacteristic
164C2×C3×Dic3C2×D411yescharacteristic
165C3×C4×D3—22no—
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C3×Dic6—24no—
169C3×Dic6—42no—
170C32×C8—22no—
171C3×C3⋊C8—22no—
172#532—31no—
173#532—31no—
174#493C3×D311yescharacteristic, complemented
175#492—31no—
176#487—31no—
177#486—31no—
178#463—34no—
179#449—34no—
180#431—31no—
181C3×#41—124no—
182C3×#41—124no—
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—41no—
196C2×C3×D4—41no—
197C2×C3×D4—41no—
198C2×C3×D4—41no—
199C2×C3×D4—62no—
200C2×C3×D4—62no—
201C2×C3×D4C22×D312yes—
202C2×C3×D4C22×D312yes—
203C2×C3×D4C23×C312yes—
204C2×C3×D4C23×C312yes—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—62no—
209C22×C3×C4—62no—
210C2×#67—22no—
211C2×#67—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214#237—44no—
215#237—44no—
216C2×C4×D3C3×D411yescharacteristic
217C2×C4×D3C3×D411yescharacteristic
218C2×C4×D3C23×C311yescharacteristic
219C2×Dic6—21no—
220C2×Dic6C23×C312yes—
221C3×SD16—124no—
222C3×D8—44no—
223C3×D8—44no—
224C3×D8—48no—
225C2×C3×C8—22no—
226C2×C3×C8—32no—
227C2×C3×C8C22×D312yes—
228C2×C3×C8C23×C312yes—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C22⋊C4—62no—
234C3×C22⋊C4—62no—
235C3×C22⋊C4—62no—
236C3×C22⋊C4—62no—
237C3×C42—21no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—31no—
241C3×C42C22×D311yescharacteristic
242C3×C42C23×C311yescharacteristic
243#217—22no—
244#217—22no—
245#217—22no—
246#217—22no—
247#214—44no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C23×C312yes—
252C4×Dic3C3×D411yescharacteristic
253C4×Dic3C3×D411yescharacteristic
254C4×Dic3C23×C311yescharacteristic
255C2×C3⋊C8C23×C312yes—
256C22×C32—42no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32C22×D411yescharacteristic
263C2×C3×D3—21no—
264C2×C3×D3—21no—
265C2×C3×D3—21no—
266C32×C4#13512yes—
267C32×C4AGL(1,ℤ/8)12yes—
268C32×C4H(5,𝔽2)12yes—
269C3×Dic3—21no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—42no—
273C3×Dic3—42no—
274C2×#41—32no—
275C2×#41—61no—
276C2×SD16—34no—
277C2×D8C2×C3×D314yes—
278C4⋊Q8—31no—
279C4⋊Q8—31no—
280#127C2×C3×D311yescharacteristic
281#127C2×C3×D311yescharacteristic
282#124—32no—
283#124—61no—
284#121—32no—
285#121—61no—
286#121—61no—
287#121—61no—
288C4×D4—32no—
289C4×D4—61no—
290C4×D4—61no—
291C4×D4—61no—
292C2×C42—31no—
293C4⋊C8—32no—
294#103—34no—
295#102—34no—
296C22⋊C8—32no—
297C4×C8—31no—
298C4×C8C2×C3×D311yescharacteristic
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—31no—
304C23×C3—42no—
305C23×C3—42no—
306C22×D3C2×C3×D411yescharacteristic
307C3×Q8—64no—
308C3×Q8—122no—
309C3×D4—24no—
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—24no—
313C3×D4—44no—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—82no—
324C3×D4—82no—
325C3×D4—82no—
326C3×D4—82no—
327C3×D4—124no—
328C3×D4—124no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—124no—
341C2×C3×C4—124no—
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4C2×C3×D411yescharacteristic
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C23×D311yescharacteristic
347C2×C3×C4C24×C311yescharacteristic
348#67—44no—
349#67—44no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—44no—
353C2×Dic3—44no—
354C2×Dic3C2×C3×D411yescharacteristic
355C2×Dic3C2×C3×D411yescharacteristic
356C2×Dic3C2×C3×D411yescharacteristic
357C4×D3—22no—
358C4×D3—22no—
359C4×D3—22no—
360Dic6—24no—
361Dic6—42no—
362C3×C8—24no—
363C3×C8—22no—
364C3×C8—22no—
365C3×C8—62no—
366C3⋊C8—22no—
367C2×C32—82no—
368C2×C32—82no—
369C2×C32#53511yescharacteristic
370C2×C32C2×#13511yescharacteristic
371C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
372C3×D3—41no—
373#41—124no—
374#41—124no—
375C2×Q8—32no—
376C2×Q8—61no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4C22×C3×D312yes—
384C2×D4C22×C3×D312yes—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—62no—
389C22×C4—62no—
390SD16—124no—
391D8—44no—
392C2×C8—32no—
393C2×C8C22×C3×D312yes—
394C4⋊C4—32no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C22⋊C4—62no—
399C22⋊C4—62no—
400C22⋊C4—62no—
401C22⋊C4—62no—
402C42—31no—
403C42—31no—
404C42—31no—
405C42C22×C3×D311yescharacteristic
406C22×C3—21no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—61no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3C2×D3×D411yescharacteristic
429C22×C3C22×C3×D411yescharacteristic
430C2×D3—21no—
431C2×D3—21no—
432C2×D3—21no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—122no—
440C3×C4—122no—
441C3×C4#91112yes—
442C3×C4#91212yes—
443C3×C4C3×#13512yes—
444C3×C4C3×AGL(1,ℤ/8)12yes—
445C3×C4#100412yes—
446C3×C4C3×H(5,𝔽2)12yes—
447Dic3—21no—
448Dic3—21no—
449Dic3—21no—
450Dic3—42no—
451Dic3—42no—
452C32#320711yescharacteristic, complemented, Sylow
453C23—22no—
454C23—22no—
455C23—31no—
456Q8—64no—
457Q8—122no—
458D4—24no—
459D4—24no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—124no—
465D4—124no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—124no—
475C2×C4—124no—
476C2×C4C3×D3×D411yescharacteristic
477C2×C4C3×D3×D411yescharacteristic
478C2×C4C23×C3×D311yescharacteristic
479C8—22no—
480C8—62no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—82no—
485C2×C3—82no—
486C2×C3—82no—
487C2×C3—82no—
488C2×C3—121no—
489C2×C3—162no—
490C2×C3—162no—
491C2×C3#571511yescharacteristic
492C2×C3C2×#91111yescharacteristic
493C2×C3C2×#91211yescharacteristic
494C2×C3C3×#53511yescharacteristic
495C2×C3C2×C3×#13511yescharacteristic
496C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
497D3—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—61no—
505C22—61no—
506C22—61no—
507C22C2×C3×D3×D411yescharacteristic
508C4—61no—
509C4—61no—
510C4—61no—
511C4—122no—
512C4—122no—
513C4C3×#91112yes—
514C4C3×#91212yes—
515C4C3×#100412yes—
516C3—21no—
517C3#8178611yescharacteristic, direct product factor
518C3C3×#320711yescharacteristic, complemented
519C2—82no—
520C2—82no—
521C2—121no—
522C2C3×#571511yescharacteristic
523C2C2×C3×#91111yescharacteristic
524C2C2×C3×#91211yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#493C2
socleC22×C32C22×D4
hypercenterC3×#493D3
hypocenterC3C3×#3207
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#493

Complements

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

NH ≅ G/Ntype
1. C3×#5716C2semidirect
2. C3×#5716C2semidirect
3. C32×#493C2semidirect
4. C3×#5158C2semidirect
5. C3×#5153C2semidirect
6. C3×#5076C2semidirect
7. C3×#5070C2semidirect
8. C3×#4685C2semidirect
9. C3×#4681C2semidirect
11. #81786C3direct (N × H)
39. #5716C2×C3semidirect
40. #5716C2×C3semidirect
42. C3×#493D3semidirect
43. C3×#493C2×C3semidirect
50. #5158C2×C3semidirect
51. #5153C2×C3semidirect
52. #5076C2×C3semidirect
53. #5070C2×C3semidirect
54. #4685C2×C3semidirect
55. #4681C2×C3semidirect
174. #493C3×D3semidirect
452. C32#3207semidirect
517. C3#81786direct (N × H)
518. C3C3×#3207semidirect

Maximal subgroups

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

subgroupindex
C3×#57162
C32×#4932
C3×#51582
C3×#51532
C3×#50762
C3×#50702
C3×#46852
C3×#46812
C3×#32073
#817863