Finite Group Databasebeta

← C3×#83417 #11865395

Subgroups — C3×#83417 #11865395

Order 1152.

1567 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C32×#536C211yesmaximal, characteristic, complemented
3C3×#5816C212yesmaximal, complemented
4C3×#5240C211yesmaximal, characteristic, complemented
5C3×#5053C218yesmaximal, complemented
6C3×#5006C212yesmaximal, complemented
7C3×#3028—31nomaximal
8#83417C311yesmaximal, characteristic, direct product factor
9C2×C32×#41C2211yescharacteristic
10C22×C32×Q8C2211yescharacteristic, complemented
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007—21no—
15C3×#1004—21no—
16C3×#1004C2214yes—
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19C32×C4⋊Q8—21no—
20C32×#124—21no—
21C32×#123C2214yescomplemented
22C32×C22⋊Q8—22no—
23C32×#117C2211yescharacteristic, complemented
24C3×#947—22no—
25C3×#946—24no—
26C3×#944—22no—
27C3×#929C2218yescomplemented
28C3×#926C2212yescomplemented
29C3×#834C2214yescomplemented
30C3×#806C2218yescomplemented
31C3×#585—31no—
32#5964C2×C311yescharacteristic
33C3×#536—21no—
34C3×#536—32no—
35C3×#536D311yescharacteristic, complemented
36C3×#536C2×C311yescharacteristic, complemented
37#5816C2×C312yescomplemented
38C3×#458—38no—
39C3×#410—31no—
40C3×#410—32no—
41#5240C2×C311yescharacteristic, complemented
42#5053C2×C318yescomplemented
43#5006C2×C312yescomplemented
44C32×#41—22no—
45C32×#41—22no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—22no—
49C2×C32×Q8—22no—
50C2×C32×Q8D411yescharacteristic
51C2×C32×D4D411yescharacteristic
52C2×C32×D4C2312yes—
53C22×C32×C4D412yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C2×C3×#67—24no—
57C2×C3×#67C2314yes—
58C3×#239—24no—
59C3×D3×Q8—24no—
60C3×#237—24no—
61C3×#237—28no—
62C3×D3×D4—24no—
63C3×D3×D4—28no—
64C3×#235—24no—
65C3×#235—24no—
66C3×#235—22no—
67C3×#235—24no—
68C3×#235—22no—
69C3×#235—22no—
70C3×#235—22no—
71C2×C3×D12—22no—
72C2×C3×D12D412yes—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3D414yes—
75C2×C3×Dic6—22no—
76C2×C3×Dic6D412yes—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—24no—
79C32×C22⋊C4—24no—
80C32×C22⋊C4C2312yes—
81C32×C42—21no—
82C3×#217C2314yes—
83C3×(C2×D3)⋊C4—28no—
84C3×C4⋊Dic3—22no—
85C3×Dic3⋊C4—28no—
86C3×C4×Dic3—22no—
87#3028—31noSylow
88C3×2−1+4—62no—
89C3×2−1+4—61no—
90C3×H(5,𝔽2)—34no—
91C3×H(5,𝔽2)—62no—
92C3×H(5,𝔽2)—61no—
93C2×C3×#41—21no—
94C2×C3×#41—32no—
95C2×C3×#41—64no—
96C2×C3×#41—62no—
97C2×C3×#41C2×D311yescharacteristic
98C2×C3×#41C22×C311yescharacteristic
99C22×C3×Q8—21no—
100C22×C3×Q8C2×D311yescharacteristic, complemented
101C22×C3×Q8C22×C311yescharacteristic, complemented
102#1010—22no—
103#1009—22no—
104D3×#41—24no—
105#1007—21no—
106#1004—21no—
107#1004C22×C314yes—
108C2×#235—22no—
109C2×#235C22×C312yes—
110C3×C4⋊Q8—22no—
111C3×C4⋊Q8—21no—
112C3×C4⋊Q8—21no—
113C3×C4⋊Q8—62no—
114C3×#124—22no—
115C3×#124—21no—
116C3×#124—21no—
117C3×#124—62no—
118C3×#123—24no—
119C3×#123—38no—
120C3×#123C2×D314yescomplemented
121C3×#123C22×C314yescomplemented
122C3×C22⋊Q8—24no—
123C3×C22⋊Q8—22no—
124C3×C22⋊Q8—22no—
125C3×C22⋊Q8—64no—
126C3×#117—21no—
127C3×#117—32no—
128C3×#117C2×D311yescharacteristic, complemented
129C3×#117C22×C311yescharacteristic, complemented
130#947—22no—
131#946—24no—
132#944—22no—
133#929C22×C318yescomplemented
134#926C22×C312yescomplemented
135C3×#99—34no—
136C3×#99—38no—
137#834C22×C314yescomplemented
138#806C22×C318yescomplemented
139C23×C32C2×D412yes—
140C23×C32C2411yescharacteristic
141C22×C3×D3—24no—
142C22×C3×D3C2×D414yes—
143C32×Q8—22no—
144C32×Q8—41no—
145C32×Q8—42no—
146C32×Q8—41no—
147C32×D4—24no—
148C32×D4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—24no—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D412yes—
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C3×#67—28no—
160C3×#67—24no—
161C3×#67—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3C2×D414yes—
165C3×D12—24no—
166C3×D12—22no—
167C3×D12—22no—
168C3×C4×D3—28no—
169C3×C4×D3—24no—
170C3×C4×D3—24no—
171C3×Dic6—24no—
172C3×Dic6—22no—
173C3×Dic6—22no—
174#585—31no—
175#536—32no—
176#536C3×D311yescharacteristic, complemented
177#458—38no—
178#410—31no—
179#410—32no—
180C3×#41—22no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41—42no—
185C3×#41—42no—
186C3×#41—64no—
187C3×#41—64no—
188C3×#41—68no—
189C3×#41—64no—
190C3×#41—64no—
191C3×#41—62no—
192C3×#41—62no—
193C3×#41—62no—
194C3×#41—62no—
195C3×#41—64no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—22no—
199C2×C3×Q8—22no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—32no—
208C2×C3×Q8—42no—
209C2×C3×Q8—42no—
210C2×C3×Q8—62no—
211C2×C3×Q8—64no—
212C2×C3×Q8#6711yescharacteristic
213C2×C3×Q8C3×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—34no—
217C2×C3×D4—32no—
218C2×C3×D4—62no—
219C2×C3×D4—68no—
220C2×C3×D4—64no—
221C2×C3×D4—64no—
222C2×C3×D4#6711yescharacteristic
223C2×C3×D4C3×D411yescharacteristic
224C2×C3×D4C22×D312yes—
225C2×C3×D4C23×C312yes—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4—22no—
229C22×C3×C4—34no—
230C22×C3×C4—64no—
231C22×C3×C4#6712yes—
232C22×C3×C4C3×D412yes—
233C22×C3×C4C22×D312yes—
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C23×C312yes—
236C22×C3×C4C23×C311yescharacteristic
237C2×#67—24no—
238C2×#67C23×C314yes—
239#239—24no—
240D3×Q8—24no—
241#237—24no—
242#237—28no—
243D3×D4—24no—
244D3×D4—28no—
245#235—24no—
246#235—24no—
247#235—22no—
248#235—24no—
249#235—22no—
250#235—22no—
251#235—22no—
252C2×D12—22no—
253C2×D12C3×D412yes—
254C2×C4×D3—24no—
255C2×C4×D3C3×D414yes—
256C2×Dic6—22no—
257C2×Dic6C3×D412yes—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—28no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—24no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—68no—
265C3×C4⋊C4—62no—
266C3×C22⋊C4—28no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—34no—
271C3×C22⋊C4—68no—
272C3×C22⋊C4C22×D312yes—
273C3×C22⋊C4C23×C312yes—
274C3×C42—22no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—62no—
278#217C23×C314yes—
279(C2×D3)⋊C4—28no—
280C4⋊Dic3—22no—
281Dic3⋊C4—28no—
282C4×Dic3—22no—
283C22×C32—22no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32C22×D412yes—
287C22×C32C22×D411yescharacteristic
288C2×C3×D3—24no—
289C2×C3×D3—44no—
290C2×C3×D3—84no—
291C32×C4—22no—
292C32×C4—42no—
293C32×C4—42no—
294C32×C4#12011yescharacteristic
295C32×C4#12012yes—
296C32×C4#12011yescharacteristic
297C3×Dic3—24no—
298C3×Dic3—44no—
2992−1+4—62no—
3002−1+4—61no—
301H(5,𝔽2)—34no—
302H(5,𝔽2)—62no—
303H(5,𝔽2)—61no—
304C2×#41—32no—
305C2×#41—64no—
306C2×#41—62no—
307C2×#41C2×C3×D311yescharacteristic
308C22×Q8C2×C3×D311yescharacteristic, complemented
309C4⋊Q8—21no—
310C4⋊Q8—62no—
311#124—21no—
312#124—62no—
313#123—38no—
314#123C2×C3×D314yescomplemented
315C22⋊Q8—22no—
316C22⋊Q8—64no—
317#117—32no—
318#117C2×C3×D311yescharacteristic, complemented
319#99—34no—
320#99—38no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—34no—
324C23×C3—64no—
325C23×C3C2×#6712yes—
326C23×C3C2×C3×D412yes—
327C23×C3C23×D311yescharacteristic
328C23×C3C24×C311yescharacteristic
329C22×D3—24no—
330C22×D3C2×C3×D414yes—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—44no—
337C3×Q8—41no—
338C3×Q8—41no—
339C3×Q8—42no—
340C3×Q8—41no—
341C3×Q8—42no—
342C3×Q8—41no—
343C3×Q8—64no—
344C3×Q8—62no—
345C3×Q8—62no—
346C3×D4—24no—
347C3×D4—24no—
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—42no—
351C3×D4—44no—
352C3×D4—64no—
353C3×D4—68no—
354C3×D4—64no—
355C3×D4—64no—
356C3×D4—62no—
357C3×D4—62no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—34no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—44no—
379C2×C3×C4—68no—
380C2×C3×C4—64no—
381C2×C3×C4—64no—
382C2×C3×C4—64no—
383C2×C3×C4—64no—
384C2×C3×C4D3×D412yes—
385C2×C3×C4D3×D412yes—
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×D412yes—
392C2×C3×C4C2×C3×D411yescharacteristic
393C2×C3×C4C2×C3×D411yescharacteristic
394#67—28no—
395#67—24no—
396#67—24no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3C2×C3×D414yes—
400D12—24no—
401D12—22no—
402D12—22no—
403C4×D3—28no—
404C4×D3—24no—
405C4×D3—24no—
406Dic6—24no—
407Dic6—22no—
408Dic6—22no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—42no—
412C2×C32C2×#12011yescharacteristic
413C3×D3—44no—
414#41—22no—
415#41—22no—
416#41—64no—
417#41—64no—
418#41—68no—
419#41—64no—
420#41—64no—
421#41—62no—
422#41—62no—
423#41—62no—
424#41—62no—
425#41—64no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—32no—
431C2×Q8—62no—
432C2×Q8—64no—
433C2×Q8C3×#6711yescharacteristic
434C2×D4—34no—
435C2×D4—32no—
436C2×D4—62no—
437C2×D4—68no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4C3×#6711yescharacteristic
441C2×D4C22×C3×D312yes—
442C22×C4—34no—
443C22×C4—64no—
444C22×C4C3×#6712yes—
445C22×C4C22×C3×D312yes—
446C22×C4C22×C3×D311yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—68no—
450C4⋊C4—62no—
451C22⋊C4—24no—
452C22⋊C4—34no—
453C22⋊C4—68no—
454C22⋊C4C22×C3×D312yes—
455C42—21no—
456C42—62no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—44no—
468C22×C3—64no—
469C22×C3—124no—
470C22×C3—244no—
471C22×C3C2×D3×D412yes—
472C22×C3C22×#6711yescharacteristic
473C22×C3C22×C3×D412yes—
474C22×C3C22×C3×D411yescharacteristic
475C2×D3—24no—
476C2×D3—44no—
477C2×D3—84no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—22no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—44no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—64no—
491C3×C4—124no—
492C3×C4#93711yescharacteristic
493C3×C4#93711yescharacteristic
494C3×C4#93712yes—
495C3×C4C3×#12011yescharacteristic
496C3×C4C3×#12012yes—
497C3×C4C3×#12011yescharacteristic
498Dic3—24no—
499Dic3—44no—
500C32#302811yescharacteristic, complemented, Sylow
501C23—34no—
502C23—64no—
503C23C2×C3×#6712yes—
504C23C23×C3×D311yescharacteristic
505Q8—22no—
506Q8—41no—
507Q8—41no—
508Q8—42no—
509Q8—64no—
510Q8—62no—
511Q8—62no—
512D4—24no—
513D4—22no—
514D4—64no—
515D4—68no—
516D4—64no—
517D4—64no—
518D4—62no—
519D4—62no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—34no—
526C2×C4—68no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4C3×D3×D412yes—
532C2×C4C3×D3×D412yes—
533C2×C4C2×C3×#6712yes—
534C2×C4C2×C3×#6711yescharacteristic
535C2×C4C2×C3×#6711yescharacteristic
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—42no—
545C2×C3—82no—
546C2×C3—124no—
547C2×C3C2×#93711yescharacteristic
548C2×C3C2×C3×#12011yescharacteristic
549D3—44no—
550C22—22no—
551C22—41no—
552C22—42no—
553C22—64no—
554C22—124no—
555C22—244no—
556C22C2×C3×D3×D412yes—
557C22C22×C3×#6711yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—64no—
562C4—124no—
563C4C3×#93711yescharacteristic
564C4C3×#93711yescharacteristic
565C4C3×#93712yes—
566C3—21no—
567C3#8341711yescharacteristic, direct product factor
568C3C3×#302811yescharacteristic, complemented
569C2—21no—
570C2—22no—
571C2—42no—
572C2—124no—
573C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC2×C3C2×#937
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×#536C2
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC3×#536D3
hypocenterC3C3×#3028
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#536

Complements

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

NH ≅ G/Ntype
2. C32×#536C2semidirect
3. C3×#5816C2semidirect
4. C3×#5240C2semidirect
5. C3×#5053C2semidirect
6. C3×#5006C2semidirect
8. #83417C3direct (N × H)
10. C22×C32×Q8C22semidirect
21. C32×#123C22semidirect
23. C32×#117C22semidirect
27. C3×#929C22semidirect
28. C3×#926C22semidirect
29. C3×#834C22semidirect
30. C3×#806C22semidirect
35. C3×#536D3semidirect
36. C3×#536C2×C3semidirect
37. #5816C2×C3semidirect
41. #5240C2×C3semidirect
42. #5053C2×C3semidirect
43. #5006C2×C3semidirect
100. C22×C3×Q8C2×D3semidirect
101. C22×C3×Q8C22×C3semidirect
120. C3×#123C2×D3semidirect
121. C3×#123C22×C3semidirect
128. C3×#117C2×D3semidirect
129. C3×#117C22×C3semidirect
133. #929C22×C3semidirect
134. #926C22×C3semidirect
137. #834C22×C3semidirect
138. #806C22×C3semidirect
176. #536C3×D3semidirect
308. C22×Q8C2×C3×D3semidirect
314. #123C2×C3×D3semidirect
318. #117C2×C3×D3semidirect
500. C32#3028semidirect
567. C3#83417direct (N × H)
568. C3C3×#3028semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C32×#5362
C3×#58162
C3×#52402
C3×#50532
C3×#50062
C3×#30283
#834173