Finite Group Databasebeta

← C3×#83710 #11898406

Subgroups — C3×#83710 #11898406

Order 1152.

862 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C211yesmaximal, characteristic, complemented
2C3×#5584C211yesmaximal, characteristic, complemented
3C3×#5076C211yesmaximal, characteristic, complemented
4C3×#5072C211yesmaximal, characteristic
5C3×#5059C212yesmaximal, complemented
6C3×#5054C212yesmaximal, complemented
7C3×#5031C211yesmaximal, characteristic, complemented
8C3×C4×#215C211yesmaximal, characteristic, complemented
9C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
10C3×C4×#214C212yesmaximal, complemented
11C3×#5016C212yesmaximal, complemented
12C3×#3394—31nomaximal
13#83710C311yesmaximal, 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×#216C2212yes—
24C2×C3×#215C2211yescharacteristic
25C2×C3×#214C2212yes—
26C2×C3×#214C2212yes—
27C2×C3×#213C2212yes—
28C3×#898—22no—
29C3×#892—21no—
30C3×#875C2212yes—
31C3×C4×D12C2211yescharacteristic
32C3×#869C2211yescharacteristic
33C3×C4×Dic6C2212yes—
34C3×#835C2211yescharacteristic
35C3×#832C2212yes—
36C3×#810C2212yes—
37C3×#809C2211yescharacteristic
38C3×#808C2211yescharacteristic
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×#556—31no—
47#5584C2×C311yescharacteristic, complemented
48C3×#495—32no—
49C3×#492—31no—
50C3×#464—32no—
51C3×#463—32no—
52C3×#462—31no—
53C3×#459—31no—
54C3×C8×Q8—31no—
55C3×C4×SD16—31no—
56C3×C4×SD16—32no—
57#5076C2×C311yescharacteristic, complemented
58#5072C2×C311yescharacteristic
59#5059C2×C312yescomplemented
60#5054C2×C312yescomplemented
61#5031C2×C311yescharacteristic, complemented
62C4×#215C2×C311yescharacteristic, complemented
63Q8×C3⋊C8C2×C311yescharacteristic, complemented
64C4×#214C2×C312yescomplemented
65#5016C2×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×Dic6C2312yes—
75C32×C4⋊C4—21no—
76C32×C4⋊C4D411yescharacteristic
77C32×C4⋊C4D412yes—
78C32×C4⋊C4C2311yescharacteristic
79C32×C22⋊C4—22no—
80C32×C22⋊C4—22no—
81C32×C22⋊C4—22no—
82C32×C42—21no—
83C32×C42C2312yes—
84C32×C42C2311yescharacteristic
85C3×#216—42no—
86C3×#215—22no—
87C3×#214—24no—
88C3×#214—42no—
89C3×#213—42no—
90C3×(C2×D3)⋊C4—22no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—22no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—22no—
97C3×C4×Dic3—22no—
98C2×C3×C3⋊C8C2312yes—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101#3394—31noSylow
102C2×C3×Q16—32no—
103C2×C3×SD16—31no—
104C2×C3×SD16—32no—
105C2×C3×SD16—32no—
106C2×C3×D8—32no—
107C3×C4⋊Q8—31no—
108C3×#127—21no—
109C3×#127C2×D311yescharacteristic
110C3×#127C22×C311yescharacteristic
111C3×#126—62no—
112C3×#124—22no—
113C3×#124—21no—
114C3×#124—21no—
115C3×#124—22no—
116C3×#124—32no—
117C3×#124C2×D312yes—
118C3×#124C22×C312yes—
119C3×#123—24no—
120C3×#123—22no—
121C3×#123—22no—
122C3×C22⋊Q8—61no—
123C3×C4×Q8—21no—
124C3×C4×Q8—21no—
125C3×C4×Q8—32no—
126C3×C4×Q8C2×D311yescharacteristic
127C3×C4×Q8C2×D311yescharacteristic
128C3×C4×Q8C22×C311yescharacteristic
129C3×C4×Q8C22×C311yescharacteristic
130C3×C4×D4—22no—
131C3×C4×D4—21no—
132C3×C4×D4—21no—
133C3×C4×D4—22no—
134C3×C4×D4—31no—
135C3×C4×D4C2×D312yes—
136C3×C4×D4C22×C312yes—
137C3×#117—61no—
138#946—21no—
139C2×#216C22×C312yes—
140C2×#215C22×C311yescharacteristic
141C2×#214C22×C312yes—
142C2×#214C22×C312yes—
143C2×#213C22×C312yes—
144#898—22no—
145#892—21no—
146#875C22×C312yes—
147C4×D12C22×C311yescharacteristic
148#869C22×C311yescharacteristic
149C4×Dic6C22×C312yes—
150C3×#106—31no—
151C3×C4⋊C8—31no—
152C3×C4⋊C8—32no—
153C3×#103—31no—
154C3×#103—32no—
155C3×#102—32no—
156C3×#102—31no—
157C3×C4×C8—31no—
158C3×C4×C8—32no—
159#835C22×C311yescharacteristic
160#832C22×C312yes—
161#810C22×C312yes—
162#809C22×C311yescharacteristic
163#808C22×C311yescharacteristic
164C4⋊(C3⋊C8)C22×C312yes—
165C4⋊(C3⋊C8)C22×C311yescharacteristic
166C4×C3⋊C8C22×C312yes—
167C4×C3⋊C8C22×C311yescharacteristic
168C23×C32—22no—
169C22×C3×D3—21no—
170C32×Q8—22no—
171C32×Q8—21no—
172C32×Q8—21no—
173C32×D4—22no—
174C32×D4—22no—
175C32×D4—42no—
176C32×D4—81no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—82no—
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—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×C4×D3—42no—
190C3×Dic6—22no—
191C3×Dic6—22no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8#4112yes—
195#559C3×D311yescharacteristic, complemented
196#556—31no—
197#495—32no—
198#492—31no—
199#464—32no—
200#463—32no—
201#462—31no—
202#459—31no—
203C8×Q8—31no—
204C4×SD16—31no—
205C4×SD16—32no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—32no—
209C2×C3×Q8#6711yescharacteristic
210C2×C3×Q8C3×D411yescharacteristic
211C2×C3×Q8C22×D311yescharacteristic
212C2×C3×Q8C23×C311yescharacteristic
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—22no—
220C2×C3×D4—31no—
221C2×C3×D4C22×D312yes—
222C2×C3×D4C23×C312yes—
223C22×C3×C4—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—61no—
227C2×D12C23×C311yescharacteristic
228C2×C4×D3—21no—
229C2×Dic6C23×C312yes—
230C3×Q16—122no—
231C3×SD16—64no—
232C3×SD16—62no—
233C3×SD16—122no—
234C3×D8—122no—
235C2×C3×C8—31no—
236C2×C3×C8—32no—
237C2×C3×C8—31no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4#6711yescharacteristic
249C3×C4⋊C4#6712yes—
250C3×C4⋊C4C3×D411yescharacteristic
251C3×C4⋊C4C3×D412yes—
252C3×C4⋊C4C22×D311yescharacteristic
253C3×C4⋊C4C23×C311yescharacteristic
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—61no—
265C3×C22⋊C4—62no—
266C3×C42—22no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—22no—
270C3×C42—21no—
271C3×C42—62no—
272C3×C42C22×D311yescharacteristic
273C3×C42C22×D312yes—
274C3×C42C23×C312yes—
275C3×C42C23×C311yescharacteristic
276#216—42no—
277#215—22no—
278#214—24no—
279#214—42no—
280#213—42no—
281(C2×D3)⋊C4—21no—
282(C2×D3)⋊C4—22no—
283(C2×D3)⋊C4—22no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3C23×C311yescharacteristic
287Dic3⋊C4—22no—
288C4×Dic3—22no—
289C2×C3⋊C8C23×C311yescharacteristic
290C2×C3⋊C8C23×C312yes—
291C2×C3⋊C8C23×C311yescharacteristic
292C22×C32—42no—
293C22×C32—42no—
294C22×C32—42no—
295C22×C32C22×D411yescharacteristic
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4#13512yes—
305C32×C4C2×#4111yescharacteristic
306C32×C4H(5,𝔽2)11yescharacteristic
307C3×Dic3—42no—
308C3×Dic3—41no—
309C2×Q16—32no—
310C2×SD16—31no—
311C2×SD16—32no—
312C2×SD16—32no—
313C2×D8—32no—
314C4⋊Q8—31no—
315#127C2×C3×D311yescharacteristic
316#126—62no—
317#124—21no—
318#124—32no—
319#124C2×C3×D312yes—
320#123—22no—
321C22⋊Q8—61no—
322C4×Q8—32no—
323C4×Q8C2×C3×D311yescharacteristic
324C4×Q8C2×C3×D311yescharacteristic
325C4×D4—21no—
326C4×D4—31no—
327C4×D4C2×C3×D312yes—
328#117—61no—
329#106—31no—
330C4⋊C8—31no—
331C4⋊C8—32no—
332#103—31no—
333#103—32no—
334#102—32no—
335#102—31no—
336C4×C8—31no—
337C4×C8—32no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—61no—
342C22×D3—21no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—21no—
346C3×Q8—21no—
347C3×Q8—21no—
348C3×Q8—21no—
349C3×Q8—42no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—62no—
353C3×Q8—62no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—44no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—81no—
366C3×D4—81no—
367C3×D4—82no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—61no—
380C2×C3×C4—62no—
381C2×C3×C4—82no—
382C2×C3×C4—82no—
383C2×C3×C4—84no—
384C2×C3×C4—122no—
385C2×C3×C4C2×#6712yes—
386C2×C3×C4C2×#6712yes—
387C2×C3×C4C2×#6711yescharacteristic
388C2×C3×C4C2×#6711yescharacteristic
389C2×C3×C4C2×C3×D412yes—
390C2×C3×C4C2×C3×D412yes—
391C2×C3×C4C2×C3×D411yescharacteristic
392C2×C3×C4C2×C3×D411yescharacteristic
393C2×C3×C4C23×D311yescharacteristic
394C2×C3×C4C24×C311yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—22no—
397D12—21no—
398D12—21no—
399C4×D3—42no—
400Dic6—22no—
401Dic6—22no—
402C3×C8—32no—
403C3×C8—62no—
404C3×C8—61no—
405C3⋊C8—22no—
406C3⋊C8—21no—
407C3⋊C8C3×#4112yes—
408C2×C32—82no—
409C2×C32#55011yescharacteristic
410C2×C32C2×#13511yescharacteristic
411C2×C32#57711yescharacteristic
412C3×D3—81no—
413C2×Q8—32no—
414C2×Q8C3×#6711yescharacteristic
415C2×Q8C22×C3×D311yescharacteristic
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—31no—
419C2×D4C22×C3×D312yes—
420C22×C4—22no—
421C22×C4—61no—
422Q16—122no—
423SD16—64no—
424SD16—62no—
425SD16—122no—
426D8—122no—
427C2×C8—31no—
428C2×C8—32no—
429C2×C8—31no—
430C4⋊C4—21no—
431C4⋊C4—31no—
432C4⋊C4—61no—
433C4⋊C4—62no—
434C4⋊C4—61no—
435C4⋊C4C3×#6711yescharacteristic
436C4⋊C4C3×#6712yes—
437C4⋊C4C22×C3×D311yescharacteristic
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—62no—
442C22⋊C4—61no—
443C22⋊C4—62no—
444C42—21no—
445C42—62no—
446C42C22×C3×D311yescharacteristic
447C42C22×C3×D312yes—
448C22×C3—21no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3—121no—
461C22×C3C22×#6711yescharacteristic
462C22×C3C22×C3×D411yescharacteristic
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—22no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—121no—
485C3×C4—122no—
486C3×C4#95012yes—
487C3×C4C3×#13512yes—
488C3×C4D3×#4111yescharacteristic
489C3×C4#100911yescharacteristic
490C3×C4C2×C3×#4111yescharacteristic
491C3×C4C3×H(5,𝔽2)11yescharacteristic
492Dic3—41no—
493Dic3—42no—
494C32#339411yescharacteristic, complemented, Sylow
495C23—22no—
496C23—61no—
497Q8—22no—
498Q8—21no—
499Q8—21no—
500Q8—62no—
501Q8—62no—
502D4—22no—
503D4—22no—
504D4—42no—
505D4—61no—
506D4—61no—
507D4—81no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—82no—
513C2×C4—122no—
514C2×C4C2×C3×#6712yes—
515C2×C4C2×C3×#6712yes—
516C2×C4C2×C3×#6711yescharacteristic
517C2×C4C2×C3×#6711yescharacteristic
518C2×C4C23×C3×D311yescharacteristic
519C8—32no—
520C8—62no—
521C8—61no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—84no—
526C2×C3—82no—
527C2×C3—82no—
528C2×C3—241no—
529C2×C3C2×#95011yescharacteristic
530C2×C3#583411yescharacteristic
531C2×C3#583611yescharacteristic
532C2×C3C3×#55011yescharacteristic
533C2×C3C2×C3×#13511yescharacteristic
534C2×C3C3×#57711yescharacteristic
535D3—81no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—121no—
540C22—121no—
541C22—121no—
542C22C22×C3×#6711yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—121no—
549C4—122no—
550C4C3×#95012yes—
551C4C3×D3×#4111yescharacteristic
552C4C3×#100911yescharacteristic
553C3—21no—
554C3#8371011yescharacteristic, direct product factor
555C3C3×#339411yescharacteristic, complemented
556C2—82no—
557C2—241no—
558C2C2×C3×#95011yescharacteristic
559C2C3×#583411yescharacteristic
560C2C3×#583611yescharacteristic
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×#3394
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×#5584C2semidirect
3. C3×#5076C2semidirect
5. C3×#5059C2semidirect
6. C3×#5054C2semidirect
7. C3×#5031C2semidirect
8. C3×C4×#215C2semidirect
9. C3×Q8×C3⋊C8C2semidirect
10. C3×C4×#214C2semidirect
11. C3×#5016C2semidirect
13. #83710C3direct (N × H)
44. C3×#559D3semidirect
45. C3×#559C2×C3semidirect
47. #5584C2×C3semidirect
57. #5076C2×C3semidirect
59. #5059C2×C3semidirect
60. #5054C2×C3semidirect
61. #5031C2×C3semidirect
62. C4×#215C2×C3semidirect
63. Q8×C3⋊C8C2×C3semidirect
64. C4×#214C2×C3semidirect
65. #5016C2×C3semidirect
195. #559C3×D3semidirect
494. C32#3394semidirect
554. C3#83710direct (N × H)
555. C3C3×#3394semidirect

Maximal subgroups

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

subgroupindex
C32×#5592
C3×#55842
C3×#50762
C3×#50722
C3×#50592
C3×#50542
C3×#50312
C3×C4×#2152
C3×Q8×C3⋊C82
C3×C4×#2142
C3×#50162
C3×#33943
#837103