Finite Group Databasebeta

← C3×#83713 #11895299

Subgroups — C3×#83713 #11895299

Order 1152.

842 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C211yesmaximal, characteristic, complemented
2C3×#5579C211yesmaximal, characteristic, complemented
3C3×#5072C211yesmaximal, characteristic
4C3×#5070C211yesmaximal, characteristic, complemented
5C3×#5057C212yesmaximal, complemented
6C3×#5056C212yesmaximal, complemented
7C3×C4×#215C211yesmaximal, characteristic, complemented
8C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
9C3×#5021C211yesmaximal, characteristic, complemented
10C3×C4×#213C212yesmaximal, complemented
11C3×#5009C212yesmaximal, complemented
12C3×#3361—31nomaximal
13#83713C311yesmaximal, characteristic, direct product factor
14C32×#127C2211yescharacteristic
15C32×#124—21no—
16C32×#124C2212yes—
17C32×#123—22no—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4—21no—
21C32×C4×D4C2212yes—
22C3×#946—21no—
23C2×C3×#215C2211yescharacteristic
24C2×C3×#213C2212yes—
25C3×#896—22no—
26C3×D3×C4⋊C4—21no—
27C3×C4×D12C2212yes—
28C3×C4×D12C2211yescharacteristic
29C3×#870C2212yes—
30C3×#869C2211yescharacteristic
31C3×#835C2212yes—
32C3×#835C2211yescharacteristic
33C3×#832C2212yes—
34C3×#832C2212yes—
35C3×#832C2212yes—
36C3×#809C2211yescharacteristic
37C3×#808C2211yescharacteristic
38C3×#807C2212yes—
39C3×C4⋊(C3⋊C8)C2212yes—
40C3×C4⋊(C3⋊C8)C2211yescharacteristic
41C3×C4×C3⋊C8C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#559—21no—
44C3×#559D311yescharacteristic, complemented
45C3×#559C2×C311yescharacteristic, complemented
46C3×#554—31no—
47#5579C2×C311yescharacteristic, complemented
48C3×#488—32no—
49C3×#486—31no—
50C3×#478—32no—
51C3×#477—31no—
52C3×#463—32no—
53C3×#459—31no—
54C3×C8×Q8—31no—
55C3×C4×SD16—31no—
56C3×C4×D8—32no—
57#5072C2×C311yescharacteristic
58#5070C2×C311yescharacteristic, complemented
59#5057C2×C312yescomplemented
60#5056C2×C312yescomplemented
61C4×#215C2×C311yescharacteristic, complemented
62Q8×C3⋊C8C2×C311yescharacteristic, complemented
63#5021C2×C311yescharacteristic, complemented
64C4×#213C2×C312yescomplemented
65#5009C2×C312yescomplemented
66C2×C32×Q8D411yescharacteristic
67C2×C32×Q8C2311yescharacteristic
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4C2312yes—
71C22×C32×C4—22no—
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—22no—
75C2×C3×Dic6—21no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4D411yescharacteristic
78C32×C4⋊C4D412yes—
79C32×C4⋊C4C2311yescharacteristic
80C32×C22⋊C4—22no—
81C32×C22⋊C4—22no—
82C32×C22⋊C4—22no—
83C32×C42—21no—
84C32×C42C2312yes—
85C32×C42C2311yescharacteristic
86C3×#215—22no—
87C3×#213—24no—
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4—22no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2312yes—
92C3×C4⋊Dic3C2311yescharacteristic
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4—22no—
95C2×C3×C3⋊C8C2312yes—
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2311yescharacteristic
98#3361—31noSylow
99C2×C3×SD16—31no—
100C2×C3×D8—32no—
101C3×C4⋊Q8—31no—
102C3×#127—21no—
103C3×#127C2×D311yescharacteristic
104C3×#127C22×C311yescharacteristic
105C3×#125—32no—
106C3×#124—22no—
107C3×#124—21no—
108C3×#124—21no—
109C3×#124—22no—
110C3×#124C2×D312yes—
111C3×#124C22×C312yes—
112C3×#123—24no—
113C3×#123—22no—
114C3×#123—22no—
115C3×C22⋊Q8—61no—
116C3×C22⋊Q8—62no—
117C3×C4×Q8—21no—
118C3×C4×Q8—21no—
119C3×C4×Q8C2×D311yescharacteristic
120C3×C4×Q8C2×D311yescharacteristic
121C3×C4×Q8C22×C311yescharacteristic
122C3×C4×Q8C22×C311yescharacteristic
123C3×C4×D4—22no—
124C3×C4×D4—21no—
125C3×C4×D4—21no—
126C3×C4×D4—22no—
127C3×C4×D4—32no—
128C3×C4×D4—31no—
129C3×C4×D4C2×D312yes—
130C3×C4×D4C22×C312yes—
131C2×C3×C4⋊C4—61no—
132#946—21no—
133C2×#215C22×C311yescharacteristic
134C2×#213C22×C312yes—
135#896—22no—
136D3×C4⋊C4—21no—
137C4×D12C22×C312yes—
138C4×D12C22×C311yescharacteristic
139#870C22×C312yes—
140#869C22×C311yescharacteristic
141C3×#107—32no—
142C3×#106—31no—
143C3×C4⋊C8—31no—
144C3×C4⋊C8—32no—
145C3×#103—31no—
146C3×#103—32no—
147C3×#102—32no—
148C3×#102—32no—
149C3×#102—32no—
150C3×#102—31no—
151C3×C4×C8—32no—
152C3×C4×C8—31no—
153#835C22×C312yes—
154#835C22×C311yescharacteristic
155#832C22×C312yes—
156#832C22×C312yes—
157#832C22×C312yes—
158#809C22×C311yescharacteristic
159#808C22×C311yescharacteristic
160#807C22×C312yes—
161C4⋊(C3⋊C8)C22×C312yes—
162C4⋊(C3⋊C8)C22×C311yescharacteristic
163C4×C3⋊C8C22×C312yes—
164C4×C3⋊C8C22×C311yescharacteristic
165C23×C32—22no—
166C22×C3×D3—21no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—42no—
173C32×D4—81no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—82no—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2411yescharacteristic
182C2×C3×Dic3—21no—
183C2×C3×Dic3—22no—
184C3×D12—21no—
185C3×D12#4112yes—
186C3×C4×D3—42no—
187C3×C4×D3—82no—
188C3×Dic6—81no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192#559C3×D311yescharacteristic, complemented
193#554—31no—
194#488—32no—
195#486—31no—
196#478—32no—
197#477—31no—
198#463—32no—
199#459—31no—
200C8×Q8—31no—
201C4×SD16—31no—
202C4×D8—32no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—61no—
206C2×C3×Q8#6711yescharacteristic
207C2×C3×Q8C3×D411yescharacteristic
208C2×C3×Q8C22×D311yescharacteristic
209C2×C3×Q8C23×C311yescharacteristic
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—22no—
217C2×C3×D4—31no—
218C2×C3×D4C22×D312yes—
219C2×C3×D4C23×C312yes—
220C22×C3×C4—24no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—61no—
224C22×C3×C4—62no—
225C2×D12C23×C311yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—22no—
228C2×Dic6—21no—
229C3×SD16—62no—
230C3×D8—64no—
231C2×C3×C8—31no—
232C2×C3×C8—31no—
233C2×C3×C8—32no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4#6711yescharacteristic
246C3×C4⋊C4#6712yes—
247C3×C4⋊C4C3×D411yescharacteristic
248C3×C4⋊C4C3×D412yes—
249C3×C4⋊C4C22×D311yescharacteristic
250C3×C4⋊C4C23×C311yescharacteristic
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—61no—
261C3×C22⋊C4—62no—
262C3×C42—22no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—22no—
266C3×C42—21no—
267C3×C42C22×D312yes—
268C3×C42C22×D311yescharacteristic
269C3×C42C23×C311yescharacteristic
270C3×C42C23×C312yes—
271#215—22no—
272#213—24no—
273(C2×D3)⋊C4—21no—
274(C2×D3)⋊C4—22no—
275C4⋊Dic3—21no—
276C4⋊Dic3C23×C312yes—
277C4⋊Dic3C23×C311yescharacteristic
278Dic3⋊C4—22no—
279Dic3⋊C4—22no—
280C2×C3⋊C8C23×C311yescharacteristic
281C2×C3⋊C8C23×C312yes—
282C2×C3⋊C8C23×C311yescharacteristic
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32C22×D411yescharacteristic
287C2×C3×D3—22no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C32×C4—22no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4#13512yes—
296C32×C4C2×#4111yescharacteristic
297C32×C42−1+411yescharacteristic
298C3×Dic3—41no—
299C3×Dic3—42no—
300C2×SD16—31no—
301C2×D8—32no—
302C4⋊Q8—31no—
303#127C2×C3×D311yescharacteristic
304#125—32no—
305#124—21no—
306#124C2×C3×D312yes—
307#123—22no—
308C22⋊Q8—61no—
309C22⋊Q8—62no—
310C4×Q8C2×C3×D311yescharacteristic
311C4×Q8C2×C3×D311yescharacteristic
312C4×D4—21no—
313C4×D4—32no—
314C4×D4—31no—
315C4×D4C2×C3×D312yes—
316C2×C4⋊C4—61no—
317#107—32no—
318#106—31no—
319C4⋊C8—31no—
320C4⋊C8—32no—
321#103—31no—
322#103—32no—
323#102—32no—
324#102—32no—
325#102—32no—
326#102—31no—
327C4×C8—32no—
328C4×C8—31no—
329C23×C3—24no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—61no—
333C22×D3—21no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—42no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8—241no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—32no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—44no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—61no—
355C3×D4—81no—
356C3×D4—81no—
357C3×D4—82no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—82no—
372C2×C3×C4—82no—
373C2×C3×C4—84no—
374C2×C3×C4—122no—
375C2×C3×C4—242no—
376C2×C3×C4C2×#6712yes—
377C2×C3×C4C2×#6712yes—
378C2×C3×C4C2×#6711yescharacteristic
379C2×C3×C4C2×#6711yescharacteristic
380C2×C3×C4C2×C3×D412yes—
381C2×C3×C4C2×C3×D412yes—
382C2×C3×C4C2×C3×D411yescharacteristic
383C2×C3×C4C2×C3×D411yescharacteristic
384C2×C3×C4C23×D311yescharacteristic
385C2×C3×C4C24×C311yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—22no—
388D12—21no—
389D12C3×#4112yes—
390C4×D3—42no—
391C4×D3—82no—
392Dic6—81no—
393C3×C8—62no—
394C3×C8—61no—
395C3×C8—61no—
396C3⋊C8—22no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C2×C32—82no—
400C2×C32#54811yescharacteristic
401C2×C32C2×#13511yescharacteristic
402C2×C32#57611yescharacteristic
403C3×D3—42no—
404C2×Q8—61no—
405C2×Q8C3×#6711yescharacteristic
406C2×Q8C22×C3×D311yescharacteristic
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—31no—
410C2×D4C22×C3×D312yes—
411C22×C4—22no—
412C22×C4—61no—
413C22×C4—62no—
414SD16—62no—
415D8—64no—
416C2×C8—31no—
417C2×C8—31no—
418C2×C8—32no—
419C4⋊C4—21no—
420C4⋊C4—31no—
421C4⋊C4—32no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—62no—
425C4⋊C4C3×#6711yescharacteristic
426C4⋊C4C3×#6712yes—
427C4⋊C4C22×C3×D311yescharacteristic
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—61no—
432C22⋊C4—62no—
433C42—21no—
434C42C22×C3×D312yes—
435C42C22×C3×D311yescharacteristic
436C22×C3—21no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—62no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3C22×#6711yescharacteristic
450C22×C3C22×C3×D411yescharacteristic
451C2×D3—22no—
452C2×D3—41no—
453C2×D3—41no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—22no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—121no—
473C3×C4—122no—
474C3×C4#95012yes—
475C3×C4C3×#13512yes—
476C3×C4D3×#4111yescharacteristic
477C3×C4#101011yescharacteristic
478C3×C4C2×C3×#4111yescharacteristic
479C3×C4C3×2−1+411yescharacteristic
480Dic3—41no—
481Dic3—42no—
482C32#336111yescharacteristic, complemented, Sylow
483C23—22no—
484C23—61no—
485Q8—22no—
486Q8—21no—
487Q8—21no—
488Q8—241no—
489D4—22no—
490D4—22no—
491D4—32no—
492D4—42no—
493D4—61no—
494D4—81no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—82no—
500C2×C4—122no—
501C2×C4—242no—
502C2×C4C2×C3×#6712yes—
503C2×C4C2×C3×#6712yes—
504C2×C4C2×C3×#6711yescharacteristic
505C2×C4C2×C3×#6711yescharacteristic
506C2×C4C23×C3×D311yescharacteristic
507C8—62no—
508C8—61no—
509C8—61no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—84no—
514C2×C3—82no—
515C2×C3—82no—
516C2×C3—122no—
517C2×C3C2×#95011yescharacteristic
518C2×C3#583111yescharacteristic
519C2×C3#583511yescharacteristic
520C2×C3C3×#54811yescharacteristic
521C2×C3C2×C3×#13511yescharacteristic
522C2×C3C3×#57611yescharacteristic
523D3—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—62no—
528C22—121no—
529C22—121no—
530C22C22×C3×#6711yescharacteristic
531C4—22no—
532C4—22no—
533C4—21no—
534C4—41no—
535C4—41no—
536C4—121no—
537C4—122no—
538C4C3×#95012yes—
539C4C3×D3×#4111yescharacteristic
540C4C3×#101011yescharacteristic
541C3—21no—
542C3#8371311yescharacteristic, direct product factor
543C3C3×#336111yescharacteristic, complemented
544C2—82no—
545C2—122no—
546C2C2×C3×#95011yescharacteristic
547C2C3×#583111yescharacteristic
548C2C3×#583511yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#559C2
socleC22×C32C22×D4
prime radicalC3×#5072C2
hypercenterC3×#559D3
hypocenterC3C3×#3361
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#559

Complements

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

NH ≅ G/Ntype
1. C32×#559C2semidirect
2. C3×#5579C2semidirect
4. C3×#5070C2semidirect
5. C3×#5057C2semidirect
6. C3×#5056C2semidirect
7. C3×C4×#215C2semidirect
8. C3×Q8×C3⋊C8C2semidirect
9. C3×#5021C2semidirect
10. C3×C4×#213C2semidirect
11. C3×#5009C2semidirect
13. #83713C3direct (N × H)
44. C3×#559D3semidirect
45. C3×#559C2×C3semidirect
47. #5579C2×C3semidirect
58. #5070C2×C3semidirect
59. #5057C2×C3semidirect
60. #5056C2×C3semidirect
61. C4×#215C2×C3semidirect
62. Q8×C3⋊C8C2×C3semidirect
63. #5021C2×C3semidirect
64. C4×#213C2×C3semidirect
65. #5009C2×C3semidirect
192. #559C3×D3semidirect
482. C32#3361semidirect
542. C3#83713direct (N × H)
543. C3C3×#3361semidirect

Maximal subgroups

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

subgroupindex
C32×#5592
C3×#55792
C3×#50722
C3×#50702
C3×#50572
C3×#50562
C3×C4×#2152
C3×Q8×C3⋊C82
C3×#50212
C3×C4×#2132
C3×#50092
C3×#33613
#837133