Finite Group Databasebeta

← C3×#83587 #11895913

Subgroups — C3×#83587 #11895913

Order 1152.

920 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#550C211yesmaximal, characteristic
2C3×#5567C211yesmaximal, characteristic, complemented
3C3×#5075C212yesmaximal, complemented
4C3×#5075C211yesmaximal, characteristic, complemented
5C3×#5071C212yesmaximal, complemented
6C3×#5070C211yesmaximal, characteristic, complemented
7C3×#5024C211yesmaximal, characteristic, complemented
8C3×#5018C212yesmaximal, complemented
9C3×C4×#214C211yesmaximal, characteristic, complemented
10C3×#5009C212yesmaximal, complemented
11C3×#5007C211yesmaximal, characteristic, complemented
12C3×#3368—31nomaximal
13#83587C311yesmaximal, characteristic, direct product factor
14C2×C32×#41—21no—
15C32×#127C2211yescharacteristic
16C32×#127C2212yes—
17C32×#121—21no—
18C32×#121—22no—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4C2211yescharacteristic
21C32×C4×D4C2212yes—
22C3×#944—21no—
23C2×C3×#214C2211yescharacteristic
24C2×C3×#214C2212yes—
25C3×#889—22no—
26C3×#887—21no—
27C3×#870C2212yes—
28C3×#869C2211yescharacteristic
29C3×C4×Dic6C2211yescharacteristic
30C3×C4×Dic6C2212yes—
31C3×#832C2212yes—
32C3×#832C2212yes—
33C3×#832C2212yes—
34C3×#832C2211yescharacteristic
35C3×#832C2212yes—
36C3×#810C2211yescharacteristic
37C3×#808C2211yescharacteristic
38C3×#807C2212yes—
39C3×C4⋊(C3⋊C8)C2211yescharacteristic
40C3×C4⋊(C3⋊C8)C2212yes—
41C3×#803C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#557—31no—
44C3×#550—21no—
45C3×#550D311yescharacteristic
46C3×#550C2×C311yescharacteristic
47#5567C2×C311yescharacteristic, complemented
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#5075C2×C311yescharacteristic, complemented
58#5075C2×C312yescomplemented
59#5071C2×C312yescomplemented
60#5070C2×C311yescharacteristic, complemented
61#5024C2×C311yescharacteristic, complemented
62#5018C2×C312yescomplemented
63C4×#214C2×C311yescharacteristic, complemented
64#5009C2×C312yescomplemented
65#5007C2×C311yescharacteristic, complemented
66C32×#41—44no—
67C2×C32×Q8D411yescharacteristic
68C2×C32×D4—22no—
69C2×C32×D4—21no—
70C2×C32×D4—22no—
71C2×C32×D4—21no—
72C2×C32×D4C2311yescharacteristic
73C2×C32×D4C2312yes—
74C22×C32×C4—22no—
75C22×C32×C4—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2311yescharacteristic
78C32×C4⋊C4D411yescharacteristic
79C32×C4⋊C4D412yes—
80C32×C4⋊C4C2311yescharacteristic
81C32×C22⋊C4—22no—
82C32×C22⋊C4—21no—
83C32×C42C2311yescharacteristic
84C32×C42C2312yes—
85C3×#214—22no—
86C3×#214—24no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2312yes—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4—22no—
94C3×C4×Dic3—22no—
95C3×C4×Dic3—21no—
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2312yes—
98C2×C3×C3⋊C8C2311yescharacteristic
99#3368—31noSylow
100C2×C3×#41—22no—
101C2×C3×#41—21no—
102C2×C3×#41—21no—
103C2×C3×SD16—31no—
104C2×C3×SD16—32no—
105C3×C4⋊Q8—31no—
106C3×C4⋊Q8—61no—
107C3×#127—21no—
108C3×#127—22no—
109C3×#127C2×D311yescharacteristic
110C3×#127C2×D312yes—
111C3×#127C22×C311yescharacteristic
112C3×#127C22×C312yes—
113C3×#125—32no—
114C3×#125—62no—
115C3×#121—22no—
116C3×#121—24no—
117C3×#121—21no—
118C3×#121—22no—
119C3×#121—21no—
120C3×#121—22no—
121C3×C4×Q8—21no—
122C3×C4×Q8—31no—
123C3×C4×Q8—32no—
124C3×C4×Q8—61no—
125C3×C4×Q8C2×D311yescharacteristic
126C3×C4×Q8C22×C311yescharacteristic
127C3×C4×D4—21no—
128C3×C4×D4—22no—
129C3×C4×D4C2×D311yescharacteristic
130C3×C4×D4C2×D312yes—
131C3×C4×D4C22×C311yescharacteristic
132C3×C4×D4C22×C312yes—
133#944—21no—
134C2×#214C22×C311yescharacteristic
135C2×#214C22×C312yes—
136#889—22no—
137#887—21no—
138#870C22×C312yes—
139#869C22×C311yescharacteristic
140C4×Dic6C22×C311yescharacteristic
141C4×Dic6C22×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#832C22×C312yes—
155#832C22×C312yes—
156#832C22×C312yes—
157#832C22×C311yescharacteristic
158#832C22×C312yes—
159#810C22×C311yescharacteristic
160#808C22×C311yescharacteristic
161#807C22×C312yes—
162C4⋊(C3⋊C8)C22×C311yescharacteristic
163C4⋊(C3⋊C8)C22×C312yes—
164#803C22×C312yes—
165C4×C3⋊C8C22×C311yescharacteristic
166C23×C32—22no—
167C23×C32—21no—
168C32×Q8—22no—
169C32×D4—21no—
170C32×D4—22no—
171C32×D4—21no—
172C32×D4—22no—
173C32×D4—44no—
174C32×D4—42no—
175C32×D4—82no—
176C32×D4—81no—
177C2×C32×C4—21no—
178C2×C32×C4—44no—
179C2×C32×C4—42no—
180C2×C32×C4C2×D412yes—
181C2×C32×C4C2×D412yes—
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4C2411yescharacteristic
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—22no—
188C3×Dic6—21no—
189C3×Dic6—42no—
190C3×Dic6#4112yescomplemented
191C3×C3⋊C8—21no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—21no—
194#557—31no—
195#550C3×D311yescharacteristic
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—48no—
206C3×#41—44no—
207C3×#41—44no—
208C2×C3×Q8—21no—
209C2×C3×Q8—31no—
210C2×C3×Q8—61no—
211C2×C3×Q8#6711yescharacteristic
212C2×C3×Q8C3×D411yescharacteristic
213C2×C3×D4—24no—
214C2×C3×D4—22no—
215C2×C3×D4—24no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—21no—
219C2×C3×D4—22no—
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4—22no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—22no—
227C2×C3×D4C22×D311yescharacteristic
228C2×C3×D4C22×D312yes—
229C2×C3×D4C23×C311yescharacteristic
230C2×C3×D4C23×C312yes—
231C22×C3×C4—24no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C2×Dic6—21no—
238C2×Dic6C23×C311yescharacteristic
239C3×SD16—62no—
240C3×SD16—64no—
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—32no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4#6711yescharacteristic
255C3×C4⋊C4#6712yes—
256C3×C4⋊C4C3×D411yescharacteristic
257C3×C4⋊C4C3×D412yes—
258C3×C4⋊C4C22×D311yescharacteristic
259C3×C4⋊C4C23×C311yescharacteristic
260C3×C22⋊C4—24no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—21no—
266C3×C42—21no—
267C3×C42—22no—
268C3×C42—62no—
269C3×C42—61no—
270C3×C42C22×D311yescharacteristic
271C3×C42C22×D312yes—
272C3×C42C23×C311yescharacteristic
273C3×C42C23×C312yes—
274#214—22no—
275#214—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3C23×C312yes—
278C4⋊Dic3C23×C311yescharacteristic
279Dic3⋊C4—21no—
280Dic3⋊C4—22no—
281Dic3⋊C4—22no—
282Dic3⋊C4—22no—
283C4×Dic3—22no—
284C4×Dic3—21no—
285C2×C3⋊C8C23×C311yescharacteristic
286C2×C3⋊C8C23×C311yescharacteristic
287C2×C3⋊C8C23×C312yes—
288C22×C32—42no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—41no—
292C22×C32—42no—
293C22×C32—41no—
294C22×C32C22×D411yescharacteristic
295C32×C4—22no—
296C32×C4—22no—
297C32×C4—21no—
298C32×C4—41no—
299C32×C4AGL(1,ℤ/8)12yes—
300C32×C4C2×#4111yescharacteristic
301C32×C42−1+411yescharacteristic
302C3×Dic3—22no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C2×#41—21no—
306C2×SD16—31no—
307C2×SD16—32no—
308C4⋊Q8—31no—
309C4⋊Q8—61no—
310#127C2×C3×D311yescharacteristic
311#127C2×C3×D312yes—
312#125—32no—
313#125—62no—
314#121—21no—
315#121—22no—
316C4×Q8—31no—
317C4×Q8—32no—
318C4×Q8—61no—
319C4×Q8C2×C3×D311yescharacteristic
320C4×D4C2×C3×D311yescharacteristic
321C4×D4C2×C3×D312yes—
322#107—32no—
323#106—31no—
324C4⋊C8—31no—
325C4⋊C8—32no—
326#103—31no—
327#102—31no—
328#102—32no—
329#102—32no—
330#102—32no—
331#102—32no—
332#97—32no—
333C4×C8—31no—
334C23×C3—24no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3—21no—
340C3×Q8—24no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—32no—
344C3×Q8—61no—
345C3×Q8—122no—
346C3×D4—21no—
347C3×D4—22no—
348C3×D4—21no—
349C3×D4—22no—
350C3×D4—21no—
351C3×D4—22no—
352C3×D4—21no—
353C3×D4—22no—
354C3×D4—41no—
355C3×D4—42no—
356C3×D4—41no—
357C3×D4—42no—
358C3×D4—48no—
359C3×D4—44no—
360C3×D4—44no—
361C3×D4—44no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4—82no—
365C3×D4—81no—
366C3×D4—82no—
367C3×D4—81no—
368C3×D4—84no—
369C3×D4—82no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—48no—
379C2×C3×C4—44no—
380C2×C3×C4—44no—
381C2×C3×C4—44no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—62no—
387C2×C3×C4C2×#6712yes—
388C2×C3×C4C2×#6711yescharacteristic
389C2×C3×C4C2×#6712yes—
390C2×C3×C4C2×#6711yescharacteristic
391C2×C3×C4C2×C3×D412yes—
392C2×C3×C4C2×C3×D412yes—
393C2×C3×C4C2×C3×D411yescharacteristic
394C2×C3×C4C2×C3×D411yescharacteristic
395C2×C3×C4C23×D311yescharacteristic
396C2×C3×C4C24×C311yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400Dic6—21no—
401Dic6—42no—
402Dic6C3×#4112yescomplemented
403C3×C8—61no—
404C3×C8—62no—
405C3×C8—61no—
406C3⋊C8—21no—
407C3⋊C8—22no—
408C3⋊C8—21no—
409C2×C32—82no—
410C2×C32—81no—
411C2×C32#54811yescharacteristic
412C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C32#57711yescharacteristic
414#41—44no—
415C2×Q8—31no—
416C2×Q8—61no—
417C2×Q8C3×#6711yescharacteristic
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4C22×C3×D311yescharacteristic
423C2×D4C22×C3×D312yes—
424C22×C4—22no—
425C22×C4—21no—
426SD16—62no—
427SD16—64no—
428C2×C8—31no—
429C2×C8—31no—
430C2×C8—32no—
431C4⋊C4—31no—
432C4⋊C4—32no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—61no—
438C4⋊C4C3×#6711yescharacteristic
439C4⋊C4C3×#6712yes—
440C4⋊C4C22×C3×D311yescharacteristic
441C22⋊C4—22no—
442C22⋊C4—21no—
443C42—62no—
444C42—61no—
445C42C22×C3×D311yescharacteristic
446C42C22×C3×D312yes—
447C22×C3—21no—
448C22×C3—44no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—42no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—41no—
466C22×C3C22×#6711yescharacteristic
467C22×C3C22×C3×D411yescharacteristic
468C3×C4—24no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—62no—
484C3×C4—122no—
485C3×C4—121no—
486C3×C4#93212yes—
487C3×C4C3×AGL(1,ℤ/8)12yes—
488C3×C4D3×#4111yescharacteristic
489C3×C4#101011yescharacteristic
490C3×C4C2×C3×#4111yescharacteristic
491C3×C4C3×2−1+411yescharacteristic
492Dic3—22no—
493Dic3—42no—
494Dic3—41no—
495C32#336811yescharacteristic, complemented, Sylow
496C23—22no—
497C23—21no—
498Q8—22no—
499Q8—32no—
500Q8—61no—
501Q8—122no—
502D4—21no—
503D4—22no—
504D4—21no—
505D4—22no—
506D4—44no—
507D4—42no—
508D4—82no—
509D4—81no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4C2×C3×#6712yes—
517C2×C4C2×C3×#6711yescharacteristic
518C2×C4C2×C3×#6712yes—
519C2×C4C2×C3×#6711yescharacteristic
520C2×C4C23×C3×D311yescharacteristic
521C8—61no—
522C8—62no—
523C8—61no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—84no—
528C2×C3—82no—
529C2×C3—82no—
530C2×C3—81no—
531C2×C3—82no—
532C2×C3—81no—
533C2×C3C2×#93211yescharacteristic
534C2×C3#583111yescharacteristic
535C2×C3#583611yescharacteristic
536C2×C3C3×#54811yescharacteristic
537C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
538C2×C3C3×#57711yescharacteristic
539C22—42no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22C22×C3×#6711yescharacteristic
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—62no—
551C4—122no—
552C4—121no—
553C4C3×#93212yes—
554C4C3×D3×#4111yescharacteristic
555C4C3×#101011yescharacteristic
556C3—21no—
557C3#8358711yescharacteristic, direct product factor
558C3C3×#336811yescharacteristic, complemented
559C2—82no—
560C2—81no—
561C2C2×C3×#93211yescharacteristic
562C2C3×#583111yescharacteristic
563C2C3×#583611yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#550C2
socleC22×C32C22×D4
hypercenterC3×#550D3
hypocenterC3C3×#3368
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#550

Complements

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

NH ≅ G/Ntype
2. C3×#5567C2semidirect
3. C3×#5075C2semidirect
4. C3×#5075C2semidirect
5. C3×#5071C2semidirect
6. C3×#5070C2semidirect
7. C3×#5024C2semidirect
8. C3×#5018C2semidirect
9. C3×C4×#214C2semidirect
10. C3×#5009C2semidirect
11. C3×#5007C2semidirect
13. #83587C3direct (N × H)
47. #5567C2×C3semidirect
57. #5075C2×C3semidirect
58. #5075C2×C3semidirect
59. #5071C2×C3semidirect
60. #5070C2×C3semidirect
61. #5024C2×C3semidirect
62. #5018C2×C3semidirect
63. C4×#214C2×C3semidirect
64. #5009C2×C3semidirect
65. #5007C2×C3semidirect
190. C3×Dic6#41semidirect
402. Dic6C3×#41semidirect
495. C32#3368semidirect
557. C3#83587direct (N × H)
558. C3C3×#3368semidirect

Maximal subgroups

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

subgroupindex
C32×#5502
C3×#55672
C3×#50752
C3×#50712
C3×#50702
C3×#50242
C3×#50182
C3×C4×#2142
C3×#50092
C3×#50072
C3×#33683
#835873