Finite Group Databasebeta

← C3×#80320 #11895735

Subgroups — C3×#80320 #11895735

Order 1152.

862 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5718C211yesmaximal, characteristic, complemented
2C3×#5543C211yesmaximal, characteristic, complemented
3C32×#459C211yesmaximal, characteristic, complemented
4C3×#5152C212yesmaximal, complemented
5C3×#5150C212yesmaximal, complemented
6C3×#5070C211yesmaximal, characteristic, complemented
7C3×C4×#213C211yesmaximal, characteristic, complemented
8C3×#4842C211yesmaximal, characteristic, complemented
9C3×#4831C211yesmaximal, characteristic, complemented
10C3×#4764C212yesmaximal, complemented
11C3×#4763C212yesmaximal, complemented
12C3×#3366—31nomaximal
13#80320C311yesmaximal, characteristic, direct product factor
14C2×C3×#237—21no—
15C2×C32×D8C2212yes—
16C32×#127C2211yescharacteristic
17C32×C4×D4C2211yescharacteristic
18C3×#940C2212yes—
19C3×#939—22no—
20C3×#938—21no—
21C3×D4×Dic3C2212yes—
22C3×D4×Dic3C2212yes—
23C2×C3×#213C2211yescharacteristic
24C2×C3×C4⋊Dic3—21no—
25C3×#924—21no—
26C3×#890C2212yes—
27C3×#878—22no—
28C3×C4×D12C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C3×C4×Dic6C2211yescharacteristic
31C32×C4⋊C8C2211yescharacteristic
32C32×#102C2212yes—
33C3×#832C2211yescharacteristic
34C3×#832C2212yes—
35C3×#832C2212yes—
36C3×#821C2212yes—
37C3×#817C2212yes—
38C3×#815C2212yes—
39C3×Dic3⋊C8C2212yes—
40C3×#809C2211yescharacteristic
41C3×#807C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#554—31no—
44C3×#550—31no—
45#5718C2×C311yescharacteristic, complemented
46#5543C2×C311yescharacteristic, complemented
47C3×#490—32no—
48C3×#486—31no—
49C3×#478—32no—
50C3×#476—31no—
51C3×#459—21no—
52C3×#459—32no—
53C3×#459D311yescharacteristic, complemented
54C3×#459C2×C311yescharacteristic, complemented
55C3×#446—31no—
56C3×#442—32no—
57C3×C4×D8—31no—
58#5152C2×C312yescomplemented
59#5150C2×C312yescomplemented
60#5070C2×C311yescharacteristic, complemented
61C4×#213C2×C311yescharacteristic, complemented
62#4842C2×C311yescharacteristic, complemented
63#4831C2×C311yescharacteristic, complemented
64#4764C2×C312yescomplemented
65#4763C2×C312yescomplemented
66C2×C32×D4—21no—
67C2×C32×D4—21no—
68C2×C32×D4C2312yes—
69C2×C32×D4C2311yescharacteristic
70C22×C32×C4—21no—
71C2×C3×#67—22no—
72C2×C3×#67—22no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—22no—
75C3×#237—44no—
76C2×C3×D12C2311yescharacteristic
77C2×C3×C4×D3—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6D411yescharacteristic
80C32×D8—24no—
81C2×C32×C8C2312yes—
82C32×C4⋊C4C2311yescharacteristic
83C32×C22⋊C4—21no—
84C32×C42C2311yescharacteristic
85C3×#217—22no—
86C3×#217—22no—
87C3×#213—22no—
88C3×(C2×D3)⋊C4—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—22no—
91C3×C4⋊Dic3D411yescharacteristic
92C3×C4⋊Dic3C2312yes—
93C3×C4⋊Dic3C2311yescharacteristic
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4D412yes—
96C3×C4×Dic3C2312yes—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2311yescharacteristic
99#3366—31noSylow
100C2×C3×#41—61no—
101C2×#237—21no—
102C2×C3×D8—22no—
103C2×C3×D8—31no—
104C2×C3×D8C2×D312yes—
105C2×C3×D8C22×C312yes—
106C3×C4⋊Q8—31no—
107C3×#127—21no—
108C3×#127—32no—
109C3×#127C2×D311yescharacteristic
110C3×#127C22×C311yescharacteristic
111C3×#125—32no—
112C3×C22⋊Q8—62no—
113C3×C22⋊Q8—61no—
114C3×#121—61no—
115C3×#121—62no—
116C3×C4×Q8—31no—
117C3×C4×D4—21no—
118C3×C4×D4—32no—
119C3×C4×D4—32no—
120C3×C4×D4—31no—
121C3×C4×D4C2×D311yescharacteristic
122C3×C4×D4C22×C311yescharacteristic
123C2×C3×C4⋊C4—61no—
124#940C22×C312yes—
125#939—22no—
126#938—21no—
127D4×Dic3C22×C312yes—
128D4×Dic3C22×C312yes—
129C2×#213C22×C311yescharacteristic
130C2×C4⋊Dic3—21no—
131#924—21no—
132#890C22×C312yes—
133#878—22no—
134C4×D12C22×C311yescharacteristic
135#869C22×C311yescharacteristic
136C4×Dic6C22×C311yescharacteristic
137C3×#107—31no—
138C3×#106—32no—
139C3×C4⋊C8—21no—
140C3×C4⋊C8—32no—
141C3×C4⋊C8C2×D311yescharacteristic
142C3×C4⋊C8C22×C311yescharacteristic
143C3×#102—22no—
144C3×#102—31no—
145C3×#102—32no—
146C3×#102—32no—
147C3×#102—32no—
148C3×#102—31no—
149C3×#102C2×D312yes—
150C3×#102C22×C312yes—
151C3×#97—32no—
152C3×C4×C8—31no—
153#832C22×C312yes—
154#832C22×C311yescharacteristic
155#832C22×C312yes—
156#821C22×C312yes—
157#817C22×C312yes—
158#815C22×C312yes—
159Dic3⋊C8C22×C312yes—
160#809C22×C311yescharacteristic
161#807C22×C311yescharacteristic
162C4×C3⋊C8C22×C311yescharacteristic
163C23×C32—21no—
164C23×C32—22no—
165C22×C3×D3—21no—
166C32×D4—22no—
167C32×D4—21no—
168C32×D4—22no—
169C32×D4—42no—
170C32×D4—81no—
171C32×D4#4112yescomplemented
172C2×C32×C4—21no—
173C2×C32×C4—42no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C3×#67—44no—
178C3×#67—82no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—44no—
182C2×C3×Dic3—82no—
183C2×C3×Dic3C2×D412yes—
184C2×C3×Dic3C2×D412yes—
185C3×D12—21no—
186C3×D12—21no—
187C3×C4×D3—42no—
188C3×Dic6—22no—
189C3×Dic6—81no—
190C32×C8—22no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—21no—
193#554—31no—
194#550—31no—
195#490—32no—
196#486—31no—
197#478—32no—
198#476—31no—
199#459—32no—
200#459C3×D311yescharacteristic, complemented
201#446—31no—
202#442—32no—
203C4×D8—31no—
204C3×#41—124no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—22no—
213C2×C3×D4—31no—
214C2×C3×D4—41no—
215C2×C3×D4—41no—
216C2×C3×D4—62no—
217C2×C3×D4—62no—
218C2×C3×D4C22×D312yes—
219C2×C3×D4C22×D311yescharacteristic
220C2×C3×D4C23×C312yes—
221C2×C3×D4C23×C311yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C22×C3×C4—62no—
226C22×C3×C4—62no—
227C22×C3×C4—61no—
228C2×#67—22no—
229C2×#67—22no—
230C22×Dic3—22no—
231C22×Dic3—22no—
232#237—44no—
233C2×D12C23×C311yescharacteristic
234C2×C4×D3—21no—
235C2×Dic6—21no—
236C2×Dic6C3×D411yescharacteristic
237C3×D8—24no—
238C3×D8—24no—
239C3×D8—44no—
240C3×D8—62no—
241C2×C3×C8—22no—
242C2×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8C22×D312yes—
245C2×C3×C8C23×C312yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4C22×D311yescharacteristic
255C3×C4⋊C4C23×C311yescharacteristic
256C3×C22⋊C4—21no—
257C3×C22⋊C4—21no—
258C3×C22⋊C4—41no—
259C3×C22⋊C4—62no—
260C3×C22⋊C4—62no—
261C3×C22⋊C4—61no—
262C3×C42—21no—
263C3×C42—32no—
264C3×C42C22×D311yescharacteristic
265C3×C42C23×C311yescharacteristic
266#217—22no—
267#217—22no—
268#213—22no—
269(C2×D3)⋊C4—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—22no—
272C4⋊Dic3C3×D411yescharacteristic
273C4⋊Dic3C23×C311yescharacteristic
274C4⋊Dic3C23×C312yes—
275Dic3⋊C4—22no—
276Dic3⋊C4C3×D412yes—
277C4×Dic3C23×C312yes—
278C2×C3⋊C8C23×C311yescharacteristic
279C2×C3⋊C8C23×C311yescharacteristic
280C22×C32—22no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32C22×D411yescharacteristic
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C32×C4—21no—
291C32×C4—41no—
292C32×C4AGL(1,ℤ/8)12yes—
293C32×C4C2×#4111yescharacteristic
294C32×C42−1+411yescharacteristic
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—41no—
298C3×Dic3—42no—
299C2×#41—61no—
300C2×D8—31no—
301C2×D8C2×C3×D312yes—
302C4⋊Q8—31no—
303#127—32no—
304#127C2×C3×D311yescharacteristic
305#125—32no—
306C22⋊Q8—62no—
307C22⋊Q8—61no—
308#121—61no—
309#121—62no—
310C4×Q8—31no—
311C4×D4—32no—
312C4×D4—32no—
313C4×D4—31no—
314C4×D4C2×C3×D311yescharacteristic
315C2×C4⋊C4—61no—
316#107—31no—
317#106—32no—
318C4⋊C8—32no—
319C4⋊C8C2×C3×D311yescharacteristic
320#102—31no—
321#102—32no—
322#102—32no—
323#102—32no—
324#102—31no—
325#102C2×C3×D312yes—
326#97—32no—
327C4×C8—31no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—42no—
334C23×C3—61no—
335C22×D3—21no—
336C3×Q8—62no—
337C3×Q8—241no—
338C3×D4—22no—
339C3×D4—21no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—61no—
351C3×D4—61no—
352C3×D4—82no—
353C3×D4—81no—
354C3×D4—81no—
355C3×D4—82no—
356C3×D4—124no—
357C3×D4—242no—
358C3×D4#23712yescomplemented
359C3×D4C3×#4112yescomplemented
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—32no—
366C2×C3×C4—32no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—61no—
371C2×C3×C4—62no—
372C2×C3×C4—82no—
373C2×C3×C4—124no—
374C2×C3×C4—122no—
375C2×C3×C4—242no—
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4C2×C3×D411yescharacteristic
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C24×C311yescharacteristic
382#67—44no—
383#67—82no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—44no—
387C2×Dic3—82no—
388C2×Dic3C2×C3×D412yes—
389C2×Dic3C2×C3×D412yes—
390D12—21no—
391D12—21no—
392C4×D3—42no—
393Dic6—22no—
394Dic6—81no—
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—42no—
398C3×C8—61no—
399C3×C8—61no—
400C3⋊C8—21no—
401C3⋊C8—21no—
402C2×C32—42no—
403C2×C32—82no—
404C2×C32#54811yescharacteristic
405C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C32#57611yescharacteristic
407C3×D3—81no—
408#41—124no—
409C2×Q8—31no—
410C2×Q8—61no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—31no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4C22×C3×D312yes—
417C2×D4C22×C3×D311yescharacteristic
418C22×C4—21no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422D8—24no—
423D8—62no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8C22×C3×D312yes—
427C4⋊C4—31no—
428C4⋊C4—32no—
429C4⋊C4—32no—
430C4⋊C4—31no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—61no—
434C4⋊C4C22×C3×D311yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—62no—
437C22⋊C4—62no—
438C22⋊C4—61no—
439C42—32no—
440C42C22×C3×D311yescharacteristic
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—42no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—121no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3C2×D3×D411yescharacteristic
464C22×C3C22×C3×D411yescharacteristic
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—81no—
479C3×C4—121no—
480C3×C4—122no—
481C3×C4#91112yes—
482C3×C4C3×AGL(1,ℤ/8)12yes—
483C3×C4C2×#23711yescharacteristic
484C3×C4#101011yescharacteristic
485C3×C4C2×C3×#4111yescharacteristic
486C3×C4C3×2−1+411yescharacteristic
487Dic3—22no—
488Dic3—22no—
489Dic3—41no—
490Dic3—42no—
491C32#336611yescharacteristic, complemented, Sylow
492C23—21no—
493C23—22no—
494C23—61no—
495Q8—62no—
496Q8—241no—
497D4—22no—
498D4—21no—
499D4—22no—
500D4—42no—
501D4—61no—
502D4—61no—
503D4—81no—
504D4—124no—
505D4—242no—
506D4C3×#23712yescomplemented
507C2×C4—21no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—42no—
511C2×C4—61no—
512C2×C4—62no—
513C2×C4—124no—
514C2×C4—122no—
515C2×C4—242no—
516C2×C4C3×D3×D411yescharacteristic
517C2×C4C3×D3×D411yescharacteristic
518C2×C4C23×C3×D311yescharacteristic
519C8—22no—
520C8—61no—
521C8—61no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3—82no—
528C2×C3—82no—
529C2×C3—82no—
530C2×C3—162no—
531C2×C3—241no—
532C2×C3#559811yescharacteristic
533C2×C3C2×#91111yescharacteristic
534C2×C3#577411yescharacteristic
535C2×C3C3×#54811yescharacteristic
536C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
537C2×C3C3×#57611yescharacteristic
538D3—81no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22C2×C3×D3×D411yescharacteristic
549C4—21no—
550C4—41no—
551C4—62no—
552C4—62no—
553C4—121no—
554C4—122no—
555C4C3×#91112yes—
556C4C2×C3×#23711yescharacteristic
557C4C3×#101011yescharacteristic
558C3—21no—
559C3#8032011yescharacteristic, direct product factor
560C3C3×#336611yescharacteristic, complemented
561C2—42no—
562C2—82no—
563C2—241no—
564C2C3×#559811yescharacteristic
565C2C2×C3×#91111yescharacteristic
566C2C3×#577411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#459C2
socleC22×C32C22×D4
hypercenterC3×#459D3
hypocenterC3C3×#3366
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#459

Complements

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

NH ≅ G/Ntype
1. C3×#5718C2semidirect
2. C3×#5543C2semidirect
3. C32×#459C2semidirect
4. C3×#5152C2semidirect
5. C3×#5150C2semidirect
6. C3×#5070C2semidirect
7. C3×C4×#213C2semidirect
8. C3×#4842C2semidirect
9. C3×#4831C2semidirect
10. C3×#4764C2semidirect
11. C3×#4763C2semidirect
13. #80320C3direct (N × H)
45. #5718C2×C3semidirect
46. #5543C2×C3semidirect
53. C3×#459D3semidirect
54. C3×#459C2×C3semidirect
58. #5152C2×C3semidirect
59. #5150C2×C3semidirect
60. #5070C2×C3semidirect
61. C4×#213C2×C3semidirect
62. #4842C2×C3semidirect
63. #4831C2×C3semidirect
64. #4764C2×C3semidirect
65. #4763C2×C3semidirect
171. C32×D4#41semidirect
200. #459C3×D3semidirect
358. C3×D4#237semidirect
359. C3×D4C3×#41semidirect
491. C32#3366semidirect
506. D4C3×#237semidirect
559. C3#80320direct (N × H)
560. C3C3×#3366semidirect

Maximal subgroups

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

subgroupindex
C3×#57182
C3×#55432
C32×#4592
C3×#51522
C3×#51502
C3×#50702
C3×C4×#2132
C3×#48422
C3×#48312
C3×#47642
C3×#47632
C3×#33663
#803203