Finite Group Databasebeta

← C3×#80913 #11878549

Subgroups — C3×#80913 #11878549

Order 1152.

859 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5629C212yesmaximal, complemented
2C32×#470C211yesmaximal, characteristic, complemented
3C3×#5187C212yesmaximal, complemented
4C3×#5135C211yesmaximal, characteristic, complemented
5C3×#5129C211yesmaximal, characteristic, complemented
6C3×#5048C212yesmaximal, complemented
7C3×#4863C212yesmaximal, complemented
8C3×#4810C212yesmaximal, complemented
9C3×#4805C212yesmaximal, complemented
10C3×#3204—31nomaximal
11#80913C311yesmaximal, characteristic, direct product factor
12C2×C3×#239C2212yes—
13C2×C3×#235C2211yescharacteristic
14C2×C32×Q16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C32×C22⋊Q8C2212yes—
17C3×#946C2212yes—
18C2×C3×#215C2212yes—
19C3×#930C2212yes—
20C22×C3×C3⋊C8C2211yescharacteristic
21C2×C3×D24C2211yescharacteristic
22C3×#896—22no—
23C3×#895—22no—
24C3×#895C2212yes—
25C3×D3×C4⋊C4C2212yes—
26C3×#884—22no—
27C3×#884—22no—
28C32×#106C2211yescharacteristic
29C32×#103C2212yes—
30C3×#835C2212yes—
31C3×#821C2212yes—
32C3×(C2×D3)⋊C8C2212yes—
33C3×(C2×D3)⋊C8C2212yes—
34C3×#817C2211yescharacteristic
35C3×#809C2212yes—
36C3×#808C2212yes—
37C3×#537—32no—
38#5629C2×C312yescomplemented
39C3×#483—32no—
40C3×#481—32no—
41C3×#470—21no—
42C3×#470D311yescharacteristic, complemented
43C3×#470C2×C311yescharacteristic, complemented
44C3×#469—31no—
45C3×#465—32no—
46C3×#452—32no—
47C3×#449—32no—
48C3×#408—31no—
49#5187C2×C312yescomplemented
50#5135C2×C311yescharacteristic, complemented
51#5129C2×C311yescharacteristic, complemented
52#5048C2×C312yescomplemented
53#4863C2×C312yescomplemented
54#4810C2×C312yescomplemented
55#4805C2×C312yescomplemented
56C2×C32×Q8C2312yes—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—22no—
59C2×C3×#67D412yes—
60C3×#239—42no—
61C3×#239—42no—
62C3×#235—41no—
63C3×#235—41no—
64C2×C3×D12—22no—
65C2×C3×D12—22no—
66C2×C3×D12D411yescharacteristic
67C2×C3×D12C2312yes—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3C2312yes—
71C2×C3×Dic6D411yescharacteristic
72C32×Q16—41no—
73C32×M16—41no—
74C2×C32×C8C2311yescharacteristic
75C2×C32×C8C2311yescharacteristic
76C32×C4⋊C4—22no—
77C32×C4⋊C4C2312yes—
78C32×C22⋊C4—22no—
79C3×#215—42no—
80C3×(C2×D3)⋊C4—22no—
81C3×(C2×D3)⋊C4—22no—
82C3×(C2×D3)⋊C4—22no—
83C3×C4⋊Dic3C2312yes—
84C3×Dic3⋊C4—22no—
85C3×Dic3⋊C4—22no—
86C2×C3×C3⋊C8—41no—
87C2×C3×C3⋊C8C2312yes—
88C3×D24—41no—
89#3204—31noSylow
90C2×C3×#41—31no—
91C2×C3×#41—32no—
92C2×#239C22×C312yes—
93C2×#235C22×C311yescharacteristic
94C2×C3×Q16—21no—
95C2×C3×Q16C2×D311yescharacteristic
96C2×C3×Q16C22×C311yescharacteristic
97C2×C3×SD16—32no—
98C2×C3×D8—31no—
99C2×C3×M16—21no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C22×C311yescharacteristic
102C22×C3×C8—31no—
103C3×C22⋊Q8—22no—
104C3×C22⋊Q8—32no—
105C3×C22⋊Q8—62no—
106C3×C22⋊Q8C2×D312yes—
107C3×C22⋊Q8C22×C312yes—
108C3×#121—32no—
109C3×#121—32no—
110C3×#121—62no—
111C3×#121—62no—
112C3×#121—62no—
113C2×C3×C4⋊C4—32no—
114#946C22×C312yes—
115C2×#215C22×C312yes—
116#930C22×C312yes—
117C22×C3⋊C8C22×C311yescharacteristic
118C2×D24C22×C311yescharacteristic
119#896—22no—
120#895—22no—
121#895C22×C312yes—
122D3×C4⋊C4C22×C312yes—
123#884—22no—
124#884—22no—
125C3×#106—21no—
126C3×#106—31no—
127C3×#106—32no—
128C3×#106C2×D311yescharacteristic
129C3×#106C22×C311yescharacteristic
130C3×#103—22no—
131C3×#103—32no—
132C3×#103C2×D312yes—
133C3×#103C22×C312yes—
134C3×#102—32no—
135C3×#102—32no—
136C3×C22⋊C8—32no—
137C3×C22⋊C8—32no—
138#835C22×C312yes—
139#821C22×C312yes—
140(C2×D3)⋊C8C22×C312yes—
141(C2×D3)⋊C8C22×C312yes—
142#817C22×C311yescharacteristic
143#809C22×C312yes—
144#808C22×C312yes—
145C23×C32C2×D411yescharacteristic
146C22×C3×D3—22no—
147C22×C3×D3C2×D412yes—
148C32×Q8—22no—
149C32×Q8—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2411yescharacteristic
156C3×#67—42no—
157C3×#67—82no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3C2×D412yes—
160C3×D12—22no—
161C3×D12—22no—
162C3×D12—41no—
163C3×D12—44no—
164C3×D12—82no—
165C3×C4×D3—22no—
166C3×C4×D3—22no—
167C3×C4×D3—44no—
168C3×C4×D3—82no—
169C3×Dic6—41no—
170C32×C8—21no—
171C32×C8—21no—
172C3×C3⋊C8—22no—
173#537—32no—
174#483—32no—
175#481—32no—
176#470C3×D311yescharacteristic, complemented
177#469—31no—
178#465—32no—
179#452—32no—
180#449—32no—
181#408—31no—
182C3×#41—121no—
183C3×#41—121no—
184C3×#41—122no—
185C3×#41—122no—
186C2×C3×Q8—22no—
187C2×C3×Q8—31no—
188C2×C3×Q8C22×D312yes—
189C2×C3×Q8C23×C312yes—
190C2×C3×D4—32no—
191C2×C3×D4—31no—
192C2×C3×D4—32no—
193C2×C3×D4—62no—
194C2×C3×D4—62no—
195C2×C3×D4—62no—
196C22×C3×C4—21no—
197C22×C3×C4—32no—
198C22×C3×C4—62no—
199C22×C3×C4—62no—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C2×#67—22no—
203C2×#67C3×D412yes—
204#239—42no—
205#239—42no—
206#235—41no—
207#235—41no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12C3×D411yescharacteristic
211C2×D12C23×C312yes—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3C23×C312yes—
215C2×Dic6C3×D411yescharacteristic
216C3×Q16—41no—
217C3×Q16—41no—
218C3×Q16—42no—
219C3×SD16—122no—
220C3×D8—121no—
221C3×M16—41no—
222C3×M16—41no—
223C3×M16—42no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—32no—
227C2×C3×C8—121no—
228C2×C3×C8C22×D311yescharacteristic
229C2×C3×C8C22×D311yescharacteristic
230C2×C3×C8C23×C311yescharacteristic
231C2×C3×C8C23×C311yescharacteristic
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—32no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4C22×D312yes—
240C3×C4⋊C4C23×C312yes—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4—62no—
246C3×C22⋊C4—62no—
247#215—42no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—22no—
251C4⋊Dic3C23×C312yes—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254C2×C3⋊C8—41no—
255C2×C3⋊C8C23×C312yes—
256D24—41no—
257C22×C32—21no—
258C22×C32—22no—
259C22×C32C22×D411yescharacteristic
260C2×C3×D3—22no—
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—42no—
264C2×C3×D3—42no—
265C2×C3×D3—42no—
266C32×C4—21no—
267C32×C4—42no—
268C32×C4—42no—
269C32×C4H(5,𝔽2)11yescharacteristic
270C32×C4H(5,𝔽2)11yescharacteristic
271C3×Dic3—22no—
272C3×Dic3—42no—
273C2×#41—31no—
274C2×#41—32no—
275C2×Q16C2×C3×D311yescharacteristic
276C2×SD16—32no—
277C2×D8—31no—
278C2×M16C2×C3×D311yescharacteristic
279C22×C8—31no—
280C22⋊Q8—32no—
281C22⋊Q8—62no—
282C22⋊Q8C2×C3×D312yes—
283#121—32no—
284#121—32no—
285#121—62no—
286#121—62no—
287#121—62no—
288C2×C4⋊C4—32no—
289#106—31no—
290#106—32no—
291#106C2×C3×D311yescharacteristic
292#103—32no—
293#103C2×C3×D312yes—
294#102—32no—
295#102—32no—
296C22⋊C8—32no—
297C22⋊C8—32no—
298C23×C3—21no—
299C23×C3—32no—
300C23×C3—62no—
301C23×C3D3×D411yescharacteristic
302C23×C3C2×C3×D411yescharacteristic
303C22×D3—22no—
304C22×D3C2×C3×D412yes—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—121no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—121no—
315C3×D4—124no—
316C3×D4—122no—
317C3×D4—242no—
318C3×D4—242no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—32no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—124no—
338C2×C3×C4—242no—
339C2×C3×C4D3×D411yescharacteristic
340C2×C3×C4C2×C3×D411yescharacteristic
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C24×C311yescharacteristic
343#67—42no—
344#67—82no—
345C2×Dic3—22no—
346C2×Dic3C2×C3×D412yes—
347D12—22no—
348D12—22no—
349D12—41no—
350D12—44no—
351D12—82no—
352C4×D3—22no—
353C4×D3—22no—
354C4×D3—44no—
355C4×D3—82no—
356Dic6—41no—
357C3×C8—22no—
358C3×C8—21no—
359C3×C8—21no—
360C3×C8—21no—
361C3×C8—21no—
362C3×C8—41no—
363C3×C8—62no—
364C3⋊C8—22no—
365C2×C32—41no—
366C2×C32#53411yescharacteristic
367C2×C32#57712yes—
368C3×D3—42no—
369C3×D3—82no—
370#41—121no—
371#41—121no—
372#41—122no—
373#41—122no—
374C2×Q8—31no—
375C2×Q8C22×C3×D312yes—
376C2×D4—32no—
377C2×D4—32no—
378C2×D4—31no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C22×C4—32no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4C22×C3×D311yescharacteristic
386Q16—41no—
387SD16—122no—
388D8—121no—
389M16—41no—
390C2×C8—32no—
391C2×C8—121no—
392C2×C8C22×C3×D311yescharacteristic
393C2×C8C22×C3×D311yescharacteristic
394C4⋊C4—22no—
395C4⋊C4—32no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4C22×C3×D312yes—
399C22⋊C4—22no—
400C22⋊C4—62no—
401C22⋊C4—62no—
402C22⋊C4—62no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—42no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—122no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3C2×D3×D411yescharacteristic
417C22×C3C22×C3×D411yescharacteristic
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—62no—
434C3×C4—82no—
435C3×C4—82no—
436C3×C4—122no—
437C3×C4#100411yescharacteristic
438C3×C4#100911yescharacteristic
439C3×C4C3×H(5,𝔽2)11yescharacteristic
440C3×C4C3×H(5,𝔽2)11yescharacteristic
441Dic3—22no—
442Dic3—42no—
443C32#320411yescharacteristic, complemented, Sylow
444C23—32no—
445C23—62no—
446C23C3×D3×D411yescharacteristic
447Q8—22no—
448Q8—22no—
449Q8—121no—
450D4—62no—
451D4—62no—
452D4—121no—
453D4—124no—
454D4—122no—
455D4—242no—
456D4—242no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—124no—
466C2×C4—242no—
467C2×C4C3×D3×D411yescharacteristic
468C2×C4C23×C3×D311yescharacteristic
469C8—21no—
470C8—21no—
471C8—62no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—81no—
477C2×C3—122no—
478C2×C3—242no—
479C2×C3#560511yescharacteristic
480C2×C3#577812yes—
481C2×C3C3×#53411yescharacteristic
482C2×C3C3×#57712yes—
483D3—42no—
484D3—82no—
485C22—22no—
486C22—21no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22C2×C3×D3×D411yescharacteristic
494C4—21no—
495C4—42no—
496C4—42no—
497C4—62no—
498C4—122no—
499C4C3×#100411yescharacteristic
500C4C3×#100911yescharacteristic
501C3—21no—
502C3#8091311yescharacteristic, direct product factor
503C3C3×#320411yescharacteristic, complemented
504C2—41no—
505C2—122no—
506C2—242no—
507C2C3×#560511yescharacteristic
508C2C3×#577812yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#470C2
socleC22×C32C22×D4
hypercenterC3×#470D3
hypocenterC3C3×#3204
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#470

Complements

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

NH ≅ G/Ntype
1. C3×#5629C2semidirect
2. C32×#470C2semidirect
3. C3×#5187C2semidirect
4. C3×#5135C2semidirect
5. C3×#5129C2semidirect
6. C3×#5048C2semidirect
7. C3×#4863C2semidirect
8. C3×#4810C2semidirect
9. C3×#4805C2semidirect
11. #80913C3direct (N × H)
38. #5629C2×C3semidirect
42. C3×#470D3semidirect
43. C3×#470C2×C3semidirect
49. #5187C2×C3semidirect
50. #5135C2×C3semidirect
51. #5129C2×C3semidirect
52. #5048C2×C3semidirect
53. #4863C2×C3semidirect
54. #4810C2×C3semidirect
55. #4805C2×C3semidirect
176. #470C3×D3semidirect
443. C32#3204semidirect
502. C3#80913direct (N × H)
503. C3C3×#3204semidirect

Maximal subgroups

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

subgroupindex
C3×#56292
C32×#4702
C3×#51872
C3×#51352
C3×#51292
C3×#50482
C3×#48632
C3×#48102
C3×#48052
C3×#32043
#809133