Finite Group Databasebeta

← C3×#83690 #11895925

Subgroups — C3×#83690 #11895925

Order 1152.

864 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C3×#5577C211yesmaximal, characteristic
3C3×#5088C211yesmaximal, characteristic, complemented
4C3×#5087C211yesmaximal, characteristic, complemented
5C3×#5067C212yesmaximal, complemented
6C3×#5066C212yesmaximal, complemented
7C3×#5027C212yesmaximal, complemented
8C3×#5027C211yesmaximal, characteristic, complemented
9C3×#5026C212yesmaximal, complemented
10C3×C4×#215C211yesmaximal, characteristic, complemented
11C3×#5024C211yesmaximal, characteristic, complemented
12C3×#3368—31nomaximal
13#83690C311yesmaximal, characteristic, direct product factor
14C2×C3×#239—21no—
15C32×C4⋊Q8—21no—
16C32×C4⋊Q8C2211yescharacteristic
17C32×#125—22no—
18C32×#125C2212yes—
19C32×C4×Q8—21no—
20C32×C4×Q8C2211yescharacteristic
21C32×C4×Q8C2212yes—
22C32×C4×Q8C2211yescharacteristic
23C2×C3×#215C2211yescharacteristic
24C2×C3×#215C2212yes—
25C3×#895—21no—
26C3×#895—22no—
27C3×#874C2212yes—
28C3×#874C2211yescharacteristic
29C3×C4×D12C2211yescharacteristic
30C3×C4×D12C2212yes—
31C3×#835C2211yescharacteristic
32C3×#809C2211yescharacteristic
33C3×#809C2212yes—
34C3×#809C2212yes—
35C3×#809C2212yes—
36C3×#809C2212yes—
37C3×#808C2211yescharacteristic
38C3×#807C2212yes—
39C3×C4⋊(C3⋊C8)C2212yes—
40C3×C4⋊(C3⋊C8)C2211yescharacteristic
41C3×#803C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#557—21no—
44C3×#557D311yescharacteristic, complemented
45C3×#557C2×C311yescharacteristic, complemented
46C3×#550—31no—
47#5577C2×C311yescharacteristic
48C3×#490—32no—
49C3×#486—31no—
50C3×#478—32no—
51C3×#474—31no—
52C3×#460—32no—
53C3×#460—31no—
54C3×#446—31no—
55C3×#440—32no—
56C3×C4×SD16—31no—
57#5088C2×C311yescharacteristic, complemented
58#5087C2×C311yescharacteristic, complemented
59#5067C2×C312yescomplemented
60#5066C2×C312yescomplemented
61#5027C2×C312yescomplemented
62#5027C2×C311yescharacteristic, complemented
63#5026C2×C312yescomplemented
64C4×#215C2×C311yescharacteristic, complemented
65#5024C2×C311yescharacteristic, complemented
66C2×C32×Q8—21no—
67C2×C32×Q8D411yescharacteristic
68C2×C32×Q8C2311yescharacteristic
69C3×#239—44no—
70C2×C3×D12—22no—
71C2×C3×D12—21no—
72C2×C3×D12—22no—
73C2×C3×D12—21no—
74C2×C3×D12C2311yescharacteristic
75C2×C3×D12C2312yes—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—21no—
78C32×C4⋊C4—22no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—22no—
83C32×C4⋊C4D411yescharacteristic
84C32×C4⋊C4D412yes—
85C32×C4⋊C4C2312yes—
86C32×C4⋊C4C2311yescharacteristic
87C32×C42—22no—
88C32×C42—21no—
89C32×C42C2311yescharacteristic
90C32×C42C2312yes—
91C3×#215—22no—
92C3×#215—24no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2312yes—
98C2×C3×C3⋊C8C2311yescharacteristic
99#3368—31noSylow
100C2×C3×#41—61no—
101C2×#239—21no—
102C2×C3×SD16—31no—
103C2×C3×SD16—32no—
104C3×C4⋊Q8—22no—
105C3×C4⋊Q8—21no—
106C3×C4⋊Q8—21no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C2×D311yescharacteristic
109C3×C4⋊Q8C22×C311yescharacteristic
110C3×#127—31no—
111C3×#127—32no—
112C3×#125—24no—
113C3×#125—22no—
114C3×#125—22no—
115C3×#125—22no—
116C3×#125C2×D312yes—
117C3×#125C22×C312yes—
118C3×#121—61no—
119C3×#121—62no—
120C3×C4×Q8—22no—
121C3×C4×Q8—21no—
122C3×C4×Q8—21no—
123C3×C4×Q8—21no—
124C3×C4×Q8—22no—
125C3×C4×Q8—21no—
126C3×C4×Q8C2×D311yescharacteristic
127C3×C4×Q8C2×D312yes—
128C3×C4×Q8C2×D311yescharacteristic
129C3×C4×Q8C22×C311yescharacteristic
130C3×C4×Q8C22×C312yes—
131C3×C4×Q8C22×C311yescharacteristic
132C3×C4×D4—31no—
133C3×C4×D4—32no—
134C2×#215C22×C311yescharacteristic
135C2×#215C22×C312yes—
136#895—21no—
137#895—22no—
138#874C22×C312yes—
139#874C22×C311yescharacteristic
140C4×D12C22×C311yescharacteristic
141C4×D12C22×C312yes—
142C3×#107—32no—
143C3×#106—31no—
144C3×C4⋊C8—31no—
145C3×C4⋊C8—32no—
146C3×#103—31no—
147C3×#102—31no—
148C3×#102—32no—
149C3×#102—32no—
150C3×#102—32no—
151C3×#102—32no—
152C3×#97—32no—
153C3×C4×C8—31no—
154#835C22×C311yescharacteristic
155#809C22×C311yescharacteristic
156#809C22×C312yes—
157#809C22×C312yes—
158#809C22×C312yes—
159#809C22×C312yes—
160#808C22×C311yescharacteristic
161#807C22×C312yes—
162C4⋊(C3⋊C8)C22×C312yes—
163C4⋊(C3⋊C8)C22×C311yescharacteristic
164#803C22×C312yes—
165C4×C3⋊C8C22×C311yescharacteristic
166C22×C3×D3—22no—
167C22×C3×D3—21no—
168C32×Q8—21no—
169C32×Q8—22no—
170C32×Q8—42no—
171C32×Q8#4112yescomplemented
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—22no—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2411yescharacteristic
180C2×C3×Dic3—21no—
181C3×D12—21no—
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—22no—
185C3×D12—44no—
186C3×D12—42no—
187C3×D12—82no—
188C3×D12—81no—
189C3×C4×D3—44no—
190C3×C4×D3—42no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—21no—
194#557C3×D311yescharacteristic, complemented
195#550—31no—
196#490—32no—
197#486—31no—
198#478—32no—
199#474—31no—
200#460—32no—
201#460—31no—
202#446—31no—
203#440—32no—
204C4×SD16—31no—
205C3×#41—124no—
206C2×C3×Q8—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#6711yescharacteristic
212C2×C3×Q8C3×D411yescharacteristic
213C2×C3×Q8C22×D311yescharacteristic
214C2×C3×Q8C23×C311yescharacteristic
215C2×C3×D4—31no—
216C2×C3×D4—32no—
217C2×C3×D4—62no—
218C2×C3×D4—61no—
219C2×C3×D4—62no—
220C2×C3×D4—61no—
221C22×C3×C4—62no—
222C22×C3×C4—61no—
223#239—44no—
224C2×D12—22no—
225C2×D12—21no—
226C2×D12—22no—
227C2×D12—21no—
228C2×D12C23×C311yescharacteristic
229C2×D12C23×C312yes—
230C2×C4×D3—22no—
231C2×C4×D3—21no—
232C3×SD16—62no—
233C3×SD16—64no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—32no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4#6711yescharacteristic
258C3×C4⋊C4#6712yes—
259C3×C4⋊C4C3×D411yescharacteristic
260C3×C4⋊C4C3×D412yes—
261C3×C4⋊C4C22×D311yescharacteristic
262C3×C4⋊C4C22×D312yes—
263C3×C4⋊C4C23×C312yes—
264C3×C4⋊C4C23×C311yescharacteristic
265C3×C22⋊C4—62no—
266C3×C22⋊C4—61no—
267C3×C42—24no—
268C3×C42—22no—
269C3×C42—22no—
270C3×C42—21no—
271C3×C42—22no—
272C3×C42—21no—
273C3×C42—22no—
274C3×C42—21no—
275C3×C42C22×D311yescharacteristic
276C3×C42C22×D312yes—
277C3×C42C23×C311yescharacteristic
278C3×C42C23×C312yes—
279#215—22no—
280#215—24no—
281(C2×D3)⋊C4—22no—
282(C2×D3)⋊C4—21no—
283C4⋊Dic3C23×C311yescharacteristic
284C2×C3⋊C8C23×C311yescharacteristic
285C2×C3⋊C8C23×C311yescharacteristic
286C2×C3⋊C8C23×C312yes—
287C22×C32C22×D411yescharacteristic
288C2×C3×D3—42no—
289C2×C3×D3—41no—
290C2×C3×D3—42no—
291C2×C3×D3—41no—
292C2×C3×D3—42no—
293C2×C3×D3—41no—
294C32×C4—22no—
295C32×C4—22no—
296C32×C4—22no—
297C32×C4—21no—
298C32×C4—42no—
299C32×C4—41no—
300C32×C4AGL(1,ℤ/8)12yes—
301C32×C4C2×#4111yescharacteristic
302C32×C42−1+411yescharacteristic
303C3×Dic3—41no—
304C2×#41—61no—
305C2×SD16—31no—
306C2×SD16—32no—
307C4⋊Q8—21no—
308C4⋊Q8C2×C3×D311yescharacteristic
309#127—31no—
310#127—32no—
311#125—22no—
312#125C2×C3×D312yes—
313#121—61no—
314#121—62no—
315C4×Q8—21no—
316C4×Q8C2×C3×D311yescharacteristic
317C4×Q8C2×C3×D312yes—
318C4×Q8C2×C3×D311yescharacteristic
319C4×D4—31no—
320C4×D4—32no—
321#107—32no—
322#106—31no—
323C4⋊C8—31no—
324C4⋊C8—32no—
325#103—31no—
326#102—31no—
327#102—32no—
328#102—32no—
329#102—32no—
330#102—32no—
331#97—32no—
332C4×C8—31no—
333C23×C3—62no—
334C23×C3—61no—
335C22×D3—22no—
336C22×D3—21no—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—21no—
341C3×Q8—21no—
342C3×Q8—22no—
343C3×Q8—42no—
344C3×Q8—44no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×Q8#23912yescomplemented
348C3×Q8C3×#4112yescomplemented
349C3×D4—61no—
350C3×D4—62no—
351C3×D4—61no—
352C3×D4—62no—
353C3×D4—124no—
354C3×D4—122no—
355C3×D4—242no—
356C3×D4—241no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—61no—
372C2×C3×C4—124no—
373C2×C3×C4—122no—
374C2×C3×C4C2×#6712yes—
375C2×C3×C4C2×#6711yescharacteristic
376C2×C3×C4C2×#6712yes—
377C2×C3×C4C2×#6711yescharacteristic
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C2×C3×D412yes—
381C2×C3×C4C2×C3×D411yescharacteristic
382C2×C3×C4C23×D311yescharacteristic
383C2×C3×C4C24×C311yescharacteristic
384C2×Dic3—21no—
385D12—21no—
386D12—22no—
387D12—21no—
388D12—22no—
389D12—44no—
390D12—42no—
391D12—82no—
392D12—81no—
393C4×D3—44no—
394C4×D3—42no—
395C3×C8—61no—
396C3×C8—62no—
397C3×C8—61no—
398C3⋊C8—21no—
399C3⋊C8—22no—
400C3⋊C8—21no—
401C2×C32#54811yescharacteristic
402C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C32#57711yescharacteristic
404C3×D3—82no—
405C3×D3—81no—
406#41—124no—
407C2×Q8—21no—
408C2×Q8C3×#6711yescharacteristic
409C2×Q8C22×C3×D311yescharacteristic
410C2×D4—31no—
411C2×D4—32no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—62no—
415C2×D4—61no—
416C22×C4—62no—
417C22×C4—61no—
418SD16—62no—
419SD16—64no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—32no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—31no—
429C4⋊C4C3×#6711yescharacteristic
430C4⋊C4C3×#6712yes—
431C4⋊C4C22×C3×D311yescharacteristic
432C4⋊C4C22×C3×D312yes—
433C22⋊C4—62no—
434C22⋊C4—61no—
435C42—22no—
436C42—21no—
437C42C22×C3×D311yescharacteristic
438C42C22×C3×D312yes—
439C22×C3—21no—
440C22×C3—122no—
441C22×C3—121no—
442C22×C3—122no—
443C22×C3—121no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3C22×#6711yescharacteristic
447C22×C3C22×C3×D411yescharacteristic
448C2×D3—42no—
449C2×D3—41no—
450C2×D3—42no—
451C2×D3—41no—
452C2×D3—42no—
453C2×D3—41no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—21no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—121no—
476C3×C4#94912yes—
477C3×C4C3×AGL(1,ℤ/8)12yes—
478C3×C4C2×#23911yescharacteristic
479C3×C4#100711yescharacteristic
480C3×C4C2×C3×#4111yescharacteristic
481C3×C4C3×2−1+411yescharacteristic
482Dic3—41no—
483C32#336811yescharacteristic, complemented, Sylow
484C23—62no—
485C23—61no—
486Q8—22no—
487Q8—21no—
488Q8—42no—
489Q8C3×#23912yescomplemented
490D4—61no—
491D4—62no—
492D4—61no—
493D4—62no—
494D4—124no—
495D4—122no—
496D4—242no—
497D4—241no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—61no—
502C2×C4—124no—
503C2×C4—122no—
504C2×C4C2×C3×#6712yes—
505C2×C4C2×C3×#6711yescharacteristic
506C2×C4C2×C3×#6712yes—
507C2×C4C2×C3×#6711yescharacteristic
508C2×C4C23×C3×D311yescharacteristic
509C8—61no—
510C8—62no—
511C8—61no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—242no—
516C2×C3—241no—
517C2×C3#581711yescharacteristic
518C2×C3C2×#94911yescharacteristic
519C2×C3#583711yescharacteristic
520C2×C3C3×#54811yescharacteristic
521C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
522C2×C3C3×#57711yescharacteristic
523D3—82no—
524D3—81no—
525C22—122no—
526C22—121no—
527C22—122no—
528C22—121no—
529C22—122no—
530C22—121no—
531C22C22×C3×#6711yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—21no—
536C4—42no—
537C4—41no—
538C4—121no—
539C4C3×#94912yes—
540C4C2×C3×#23911yescharacteristic
541C4C3×#100711yescharacteristic
542C3—21no—
543C3#8369011yescharacteristic, direct product factor
544C3C3×#336811yescharacteristic, complemented
545C2—242no—
546C2—241no—
547C2C3×#581711yescharacteristic
548C2C2×C3×#94911yescharacteristic
549C2C3×#583711yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#557C2
socleC22×C32C22×D4
prime radicalC3×#5577C2
hypercenterC3×#557D3
hypocenterC3C3×#3368
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
3. C3×#5088C2semidirect
4. C3×#5087C2semidirect
5. C3×#5067C2semidirect
6. C3×#5066C2semidirect
7. C3×#5027C2semidirect
8. C3×#5027C2semidirect
9. C3×#5026C2semidirect
10. C3×C4×#215C2semidirect
11. C3×#5024C2semidirect
13. #83690C3direct (N × H)
44. C3×#557D3semidirect
45. C3×#557C2×C3semidirect
57. #5088C2×C3semidirect
58. #5087C2×C3semidirect
59. #5067C2×C3semidirect
60. #5066C2×C3semidirect
61. #5027C2×C3semidirect
62. #5027C2×C3semidirect
63. #5026C2×C3semidirect
64. C4×#215C2×C3semidirect
65. #5024C2×C3semidirect
171. C32×Q8#41semidirect
194. #557C3×D3semidirect
347. C3×Q8#239semidirect
348. C3×Q8C3×#41semidirect
483. C32#3368semidirect
489. Q8C3×#239semidirect
543. C3#83690direct (N × H)
544. C3C3×#3368semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
C3×#55772
C3×#50882
C3×#50872
C3×#50672
C3×#50662
C3×#50272
C3×#50262
C3×C4×#2152
C3×#50242
C3×#33683
#836903