Finite Group Databasebeta

← C3×#78677 #11878564

Subgroups — C3×#78677 #11878564

Order 1152.

883 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5833C212yesmaximal, complemented
2C32×#408C211yesmaximal, characteristic, complemented
3C3×#5137C211yesmaximal, characteristic, complemented
4C3×#5135C211yesmaximal, characteristic, complemented
5C3×#5114C212yesmaximal, complemented
6C3×#4735C212yesmaximal, complemented
7C3×#4734C212yesmaximal, complemented
8C3×#4733C212yesmaximal, complemented
9C3×#4721C212yesmaximal, complemented
10C3×#3204—31nomaximal
11#78677C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2212yes—
14C2×C32×M16C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C3×#946—22no—
17C3×#939—22no—
18C3×#939—22no—
19C3×#938—22no—
20C3×#930C2212yes—
21C3×#930C2212yes—
22C2×C3×C4⋊Dic3C2212yes—
23C3×#924C2212yes—
24C3×#924C2212yes—
25C2×C3×Dic12C2211yescharacteristic
26C2×C3×D24C2211yescharacteristic
27C2×C3×#204C2212yes—
28C32×C22⋊C8C2212yes—
29C32×C22⋊C8C2212yes—
30C3×#821C2212yes—
31C3×#821C2212yes—
32C3×#817C2211yescharacteristic
33C3×#817C2212yes—
34C3×#817C2211yescharacteristic
35C3×#816C2212yes—
36C3×#816C2212yes—
37C3×#537—32no—
38#5833C2×C312yescomplemented
39C3×#483—32no—
40C3×#481—32no—
41C3×#470—31no—
42C3×#469—31no—
43C3×#465—32no—
44C3×#452—32no—
45C3×#449—32no—
46C3×#408—21no—
47C3×#408D311yescharacteristic, complemented
48C3×#408C2×C311yescharacteristic, complemented
49#5137C2×C311yescharacteristic, complemented
50#5135C2×C311yescharacteristic, complemented
51#5114C2×C312yescomplemented
52#4735C2×C312yescomplemented
53#4734C2×C312yescomplemented
54#4733C2×C312yescomplemented
55#4721C2×C312yescomplemented
56C32×#41—41no—
57C32×#41—41no—
58C2×C32×Q8D411yescharacteristic
59C2×C32×D4D412yes—
60C2×C32×D4D411yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2312yes—
63C2×C3×#67—22no—
64C2×C3×#67—22no—
65C2×C3×#67—22no—
66C22×C3×Dic3—22no—
67C3×#235—42no—
68C3×#235—42no—
69C2×C3×D12C2312yes—
70C2×C3×C4×D3—22no—
71C2×C3×Dic6C2312yes—
72C32×M16—41no—
73C2×C32×C8—41no—
74C2×C32×C8C2312yes—
75C2×C32×C8C2311yescharacteristic
76C2×C32×C8C2311yescharacteristic
77C3×#217—22no—
78C3×#217—22no—
79C3×(C2×D3)⋊C4—22no—
80C3×(C2×D3)⋊C4—22no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3C2312yes—
83C3×C4⋊Dic3C2312yes—
84C3×Dic3⋊C4—22no—
85C3×Dic3⋊C4—22no—
86C3×Dic12—41no—
87C3×D24—41no—
88C3×#204—42no—
89#3204—31noSylow
90C2×C3×#41—21no—
91C2×C3×#41—32no—
92C2×C3×#41C2×D311yescharacteristic
93C2×C3×#41C22×C311yescharacteristic
94C2×#235C22×C312yes—
95C2×C3×Q16—31no—
96C2×C3×SD16—32no—
97C2×C3×D8—31no—
98C2×C3×M16—21no—
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C22×C311yescharacteristic
101C22×C3×C8—21no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C3×C22⋊Q8—32no—
105C3×C22⋊Q8—32no—
106C3×C22⋊Q8—62no—
107C3×#121—32no—
108C3×#121—32no—
109C3×#121—62no—
110C3×#121—62no—
111C3×#121—62no—
112C2×C3×C4⋊C4—32no—
113#946—22no—
114#939—22no—
115#939—22no—
116#938—22no—
117#930C22×C312yes—
118#930C22×C312yes—
119C2×C4⋊Dic3C22×C312yes—
120#924C22×C312yes—
121#924C22×C312yes—
122C2×Dic12C22×C311yescharacteristic
123C2×D24C22×C311yescharacteristic
124C2×#204C22×C312yes—
125C3×#106—32no—
126C3×#106—31no—
127C3×#106—31no—
128C3×#103—32no—
129C3×#103—32no—
130C3×#102—32no—
131C3×#102—32no—
132C3×C22⋊C8—22no—
133C3×C22⋊C8—22no—
134C3×C22⋊C8C2×D312yes—
135C3×C22⋊C8C2×D312yes—
136C3×C22⋊C8C22×C312yes—
137C3×C22⋊C8C22×C312yes—
138#821C22×C312yes—
139#821C22×C312yes—
140#817C22×C311yescharacteristic
141#817C22×C311yescharacteristic
142#817C22×C312yes—
143#816C22×C312yes—
144#816C22×C312yes—
145C23×C32C2×D411yescharacteristic
146C23×C32C2×D412yes—
147C22×C3×D3—22no—
148C32×Q8—41no—
149C32×D4—41no—
150C32×D4—42no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2411yescharacteristic
158C3×#67—44no—
159C3×#67—82no—
160C3×#67—82no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—82no—
165C3×D12—22no—
166C3×D12—22no—
167C3×C4×D3—44no—
168C3×Dic6—22no—
169C3×Dic6—22no—
170C32×C8—22no—
171C32×C8—21no—
172C32×C8—21no—
173#537—32no—
174#483—32no—
175#481—32no—
176#470—31no—
177#469—31no—
178#465—32no—
179#452—32no—
180#449—32no—
181#408C3×D311yescharacteristic, complemented
182C3×#41—41no—
183C3×#41—41no—
184C3×#41—41no—
185C3×#41—41no—
186C3×#41—42no—
187C3×#41—42no—
188C3×#41—122no—
189C3×#41—122no—
190C2×C3×Q8—21no—
191C2×C3×Q8—32no—
192C2×C3×Q8D1211yescharacteristic
193C2×C3×Q8C3×D411yescharacteristic
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—32no—
197C2×C3×D4—62no—
198C2×C3×D4—62no—
199C2×C3×D4—62no—
200C2×C3×D4D1211yescharacteristic
201C2×C3×D4D1212yes—
202C2×C3×D4C3×D412yes—
203C2×C3×D4C3×D411yescharacteristic
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—62no—
207C22×C3×C4—62no—
208C22×C3×C4C22×D312yes—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C23×C312yes—
211C22×C3×C4C23×C311yescharacteristic
212C2×#67—22no—
213C2×#67—22no—
214C2×#67—22no—
215C22×Dic3—22no—
216#235—42no—
217#235—42no—
218C2×D12C23×C312yes—
219C2×C4×D3—22no—
220C2×Dic6C23×C312yes—
221C3×Q16—121no—
222C3×SD16—122no—
223C3×D8—121no—
224C3×M16—41no—
225C3×M16—41no—
226C3×M16—42no—
227C2×C3×C8—22no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—41no—
231C2×C3×C8—41no—
232C2×C3×C8—42no—
233C2×C3×C8C22×D311yescharacteristic
234C2×C3×C8C22×D312yes—
235C2×C3×C8C22×D311yescharacteristic
236C2×C3×C8C23×C311yescharacteristic
237C2×C3×C8C23×C312yes—
238C2×C3×C8C23×C311yescharacteristic
239C3×C4⋊C4—32no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—62no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4—62no—
246C3×C22⋊C4—62no—
247C3×C22⋊C4—62no—
248#217—22no—
249#217—22no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3C23×C312yes—
254C4⋊Dic3C23×C312yes—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257Dic12—41no—
258D24—41no—
259#204—42no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32C22×D411yescharacteristic
266C2×C3×D3—42no—
267C2×C3×D3—42no—
268C2×C3×D3—42no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4H(5,𝔽2)11yescharacteristic
272C32×C4H(5,𝔽2)11yescharacteristic
273C3×Dic3—42no—
274C3×Dic3—42no—
275C3×Dic3—42no—
276C2×#41—32no—
277C2×#41C2×C3×D311yescharacteristic
278C2×Q16—31no—
279C2×SD16—32no—
280C2×D8—31no—
281C2×M16C2×C3×D311yescharacteristic
282C22×C8C2×C3×D311yescharacteristic
283C22⋊Q8—32no—
284C22⋊Q8—32no—
285C22⋊Q8—62no—
286#121—32no—
287#121—32no—
288#121—62no—
289#121—62no—
290#121—62no—
291C2×C4⋊C4—32no—
292#106—32no—
293#106—31no—
294#106—31no—
295#103—32no—
296#103—32no—
297#102—32no—
298#102—32no—
299C22⋊C8C2×C3×D312yes—
300C22⋊C8C2×C3×D312yes—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—62no—
304C23×C3C2×D1211yescharacteristic
305C23×C3C2×D1212yes—
306C23×C3C2×C3×D411yescharacteristic
307C23×C3C2×C3×D412yes—
308C22×D3—22no—
309C3×Q8—41no—
310C3×Q8—41no—
311C3×Q8—42no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—42no—
317C3×D4—41no—
318C3×D4—42no—
319C3×D4—44no—
320C3×D4—62no—
321C3×D4—62no—
322C3×D4—124no—
323C3×D4—242no—
324C3×D4—242no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—124no—
344C2×C3×C4—242no—
345C2×C3×C4C2×D1211yescharacteristic
346C2×C3×C4C2×D1212yes—
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D412yes—
349C2×C3×C4C23×D311yescharacteristic
350C2×C3×C4C24×C311yescharacteristic
351#67—44no—
352#67—82no—
353#67—82no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—82no—
358D12—22no—
359D12—22no—
360C4×D3—44no—
361Dic6—22no—
362Dic6—22no—
363C3×C8—22no—
364C3×C8—21no—
365C3×C8—21no—
366C3×C8—22no—
367C3×C8—21no—
368C3×C8—21no—
369C3×C8—42no—
370C3×C8—41no—
371C3×C8—41no—
372C2×C32—41no—
373C2×C32—42no—
374C2×C32#53411yescharacteristic
375C2×C32#57712yes—
376C3×D3—82no—
377#41—41no—
378#41—41no—
379#41—122no—
380#41—122no—
381C2×Q8—32no—
382C2×Q8C3×D1211yescharacteristic
383C2×D4—32no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4C3×D1211yescharacteristic
388C2×D4C3×D1212yes—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4C22×C3×D312yes—
392C22×C4C22×C3×D311yescharacteristic
393Q16—121no—
394SD16—122no—
395D8—121no—
396M16—41no—
397C2×C8—41no—
398C2×C8C22×C3×D311yescharacteristic
399C2×C8C22×C3×D312yes—
400C2×C8C22×C3×D311yescharacteristic
401C4⋊C4—32no—
402C4⋊C4—32no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—122no—
427C22×C3—122no—
428C22×C3—122no—
429C22×C3C22×D1211yescharacteristic
430C22×C3C22×C3×D411yescharacteristic
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—42no—
442C3×C4—122no—
443C3×C4—122no—
444C3×C4—122no—
445C3×C4#100411yescharacteristic
446C3×C4#100411yescharacteristic
447C3×C4C3×H(5,𝔽2)11yescharacteristic
448C3×C4C3×H(5,𝔽2)11yescharacteristic
449Dic3—42no—
450Dic3—42no—
451Dic3—42no—
452C32#320411yescharacteristic, complemented, Sylow
453C23—62no—
454C23C2×C3×D1211yescharacteristic
455C23C2×C3×D1212yes—
456Q8—41no—
457Q8—62no—
458Q8—62no—
459D4—42no—
460D4—41no—
461D4—62no—
462D4—62no—
463D4—124no—
464D4—242no—
465D4—242no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—124no—
474C2×C4—242no—
475C2×C4C2×C3×D1211yescharacteristic
476C2×C4C2×C3×D1212yes—
477C2×C4C23×C3×D311yescharacteristic
478C8—22no—
479C8—21no—
480C8—21no—
481C2×C3—21no—
482C2×C3—22no—
483C2×C3—42no—
484C2×C3—41no—
485C2×C3—42no—
486C2×C3—41no—
487C2×C3—42no—
488C2×C3—82no—
489C2×C3—242no—
490C2×C3#549311yescharacteristic
491C2×C3#575112yes—
492C2×C3C3×#53411yescharacteristic
493C2×C3C3×#57712yes—
494D3—82no—
495C22—22no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22C22×C3×D1211yescharacteristic
504C4—22no—
505C4—21no—
506C4—122no—
507C4—122no—
508C4—122no—
509C4C3×#100411yescharacteristic
510C4C3×#100411yescharacteristic
511C3—21no—
512C3#7867711yescharacteristic, direct product factor
513C3C3×#320411yescharacteristic, complemented
514C2—41no—
515C2—42no—
516C2—242no—
517C2C3×#549311yescharacteristic
518C2C3×#575112yes—
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#408C2
socleC22×C32C22×D4
hypercenterC3×#408D3
hypocenterC3C3×#3204
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
1. C3×#5833C2semidirect
2. C32×#408C2semidirect
3. C3×#5137C2semidirect
4. C3×#5135C2semidirect
5. C3×#5114C2semidirect
6. C3×#4735C2semidirect
7. C3×#4734C2semidirect
8. C3×#4733C2semidirect
9. C3×#4721C2semidirect
11. #78677C3direct (N × H)
38. #5833C2×C3semidirect
47. C3×#408D3semidirect
48. C3×#408C2×C3semidirect
49. #5137C2×C3semidirect
50. #5135C2×C3semidirect
51. #5114C2×C3semidirect
52. #4735C2×C3semidirect
53. #4734C2×C3semidirect
54. #4733C2×C3semidirect
55. #4721C2×C3semidirect
181. #408C3×D3semidirect
452. C32#3204semidirect
512. C3#78677direct (N × H)
513. C3C3×#3204semidirect

Maximal subgroups

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

subgroupindex
C3×#58332
C32×#4082
C3×#51372
C3×#51352
C3×#51142
C3×#47352
C3×#47342
C3×#47332
C3×#47212
C3×#32043
#786773