Finite Group Databasebeta

← C3×#83416 #11865195

Subgroups — C3×#83416 #11865195

Order 1152.

1543 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C32×#536C211yesmaximal, characteristic, complemented
3C2×C3×#942C212yesmaximal, complemented
4C3×#5237C211yesmaximal, characteristic, complemented
5C3×#5095C214yesmaximal, complemented
6C3×#5061C214yesmaximal, complemented
7C3×#5005C212yesmaximal, complemented
8C3×#3022—31nomaximal
9#83416C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic, complemented
12C3×#1010—22no—
13C3×#1009—22no—
14C3×D3×#41—24no—
15C3×#1007C2212yes—
16C3×#1004—22no—
17C3×#1004C2212yes—
18C2×C3×#235—22no—
19C2×C3×#235C2212yes—
20C32×C4⋊Q8C2212yescomplemented
21C32×#124C2212yescomplemented
22C32×#123—22no—
23C32×C22⋊Q8—22no—
24C32×#117C2211yescharacteristic, complemented
25C2×C3×#216—22no—
26C3×#942—24no—
27C3×#942C2218yescomplemented
28C2×C3×#215—22no—
29C2×C3×#208C2212yescomplemented
30C3×#836C2212yescomplemented
31C3×#833C2212yescomplemented
32C3×#805C2218yescomplemented
33C3×#585—31no—
34#5964C2×C311yescharacteristic
35C2×C3×#137—32no—
36C3×#536—21no—
37C3×#536D311yescharacteristic, complemented
38C3×#536C2×C311yescharacteristic, complemented
39C2×#942C2×C312yescomplemented
40C3×#456—34no—
41C3×#455—34no—
42C3×#421—32no—
43C3×#413—31no—
44#5237C2×C311yescharacteristic, complemented
45#5095C2×C314yescomplemented
46#5061C2×C314yescomplemented
47#5005C2×C312yescomplemented
48C32×#41—22no—
49C32×#41—22no—
50C2×C32×Q8—21no—
51C2×C32×Q8—22no—
52C2×C32×Q8—22no—
53C2×C32×Q8C2312yes—
54C2×C32×Q8C2311yescharacteristic
55C2×C32×D4D412yes—
56C2×C32×D4C2311yescharacteristic
57C22×C32×C4—21no—
58C22×C32×C4D412yes—
59C22×C32×C4C2311yescharacteristic
60C2×C3×#67—24no—
61C2×C3×#67—22no—
62C3×#239—24no—
63C3×D3×Q8—24no—
64C3×#237—28no—
65C3×#237—24no—
66C3×D3×D4—28no—
67C3×D3×D4—24no—
68C3×#235—24no—
69C3×#235—24no—
70C3×#235—22no—
71C3×#235—24no—
72C3×#235—22no—
73C3×#235D414yes—
74C3×#235C2314yes—
75C2×C3×D12—22no—
76C2×C3×D12D412yes—
77C2×C3×C4×D3—24no—
78C2×C3×C4×D3—22no—
79C2×C3×Dic6—22no—
80C2×C3×Dic6D412yes—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—24no—
83C32×C22⋊C4—21no—
84C32×C22⋊C4—24no—
85C32×C42C2312yes—
86C3×#216—28no—
87C3×#215—28no—
88C3×#208—22no—
89C3×#208C2314yes—
90C2×C3×C3⋊C8—22no—
91#3022—31noSylow
92C3×2−1+4—32no—
93C3×2−1+4—62no—
94C3×H(5,𝔽2)—32no—
95C3×H(5,𝔽2)—62no—
96C3×H(5,𝔽2)—62no—
97C2×C3×#41—21no—
98C2×C3×#41—32no—
99C2×C3×#41—64no—
100C2×C3×#41—62no—
101C2×C3×#41C2×D311yescharacteristic
102C2×C3×#41C22×C311yescharacteristic
103C22×C3×Q8—21no—
104C22×C3×Q8C2×D311yescharacteristic, complemented
105C22×C3×Q8C22×C311yescharacteristic, complemented
106#1010—22no—
107#1009—22no—
108D3×#41—24no—
109#1007C22×C312yes—
110#1004—22no—
111#1004C22×C312yes—
112C2×#235—22no—
113C2×#235C22×C312yes—
114C3×#137—38no—
115C3×#137—64no—
116C2×C3×Q16—62no—
117C2×C3×SD16—62no—
118C2×C3×M16—32no—
119C3×C4⋊Q8—22no—
120C3×C4⋊Q8C2×D312yescomplemented
121C3×C4⋊Q8C22×C312yescomplemented
122C3×#124—22no—
123C3×#124C2×D312yescomplemented
124C3×#124C22×C312yescomplemented
125C3×#123—24no—
126C3×#123—22no—
127C3×#123—22no—
128C3×C22⋊Q8—24no—
129C3×C22⋊Q8—22no—
130C3×C22⋊Q8—22no—
131C3×#117—21no—
132C3×#117C2×D311yescharacteristic, complemented
133C3×#117C22×C311yescharacteristic, complemented
134C2×#216—22no—
135#942—24no—
136#942C22×C318yescomplemented
137C2×#215—22no—
138C2×#208C22×C312yescomplemented
139C3×#104—38no—
140C3×#101—32no—
141C3×#100—32no—
142#836C22×C312yescomplemented
143#833C22×C312yescomplemented
144#805C22×C318yescomplemented
145C23×C32C2×D412yes—
146C23×C32C2×D411yescharacteristic
147C22×C3×D3—24no—
148C22×C3×D3—22no—
149C32×Q8—22no—
150C32×Q8—24no—
151C32×Q8—41no—
152C32×Q8—41no—
153C32×D4—24no—
154C32×D4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—42no—
160C2×C32×C4C2×D412yes—
161C2×C32×C4C2×D412yes—
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2411yescharacteristic
165C3×#67—28no—
166C3×#67—24no—
167C3×#67—24no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3—22no—
170C3×D12—24no—
171C3×D12—22no—
172C3×D12C2×D414yes—
173C3×C4×D3—28no—
174C3×C4×D3—24no—
175C3×C4×D3—24no—
176C3×Dic6—24no—
177C3×Dic6—22no—
178C3×Dic6C2×D414yes—
179C3×C3⋊C8—24no—
180#585—31no—
181C2×#137—32no—
182#536C3×D311yescharacteristic, complemented
183#456—34no—
184#455—34no—
185#421—32no—
186#413—31no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—34no—
192C3×#41—34no—
193C3×#41—42no—
194C3×#41—42no—
195C3×#41—64no—
196C3×#41—64no—
197C3×#41—64no—
198C3×#41—68no—
199C3×#41—64no—
200C3×#41—62no—
201C3×#41—62no—
202C3×#41—64no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—22no—
210C2×C3×Q8—22no—
211C2×C3×Q8—21no—
212C2×C3×Q8—32no—
213C2×C3×Q8—42no—
214C2×C3×Q8—42no—
215C2×C3×Q8—62no—
216C2×C3×Q8—64no—
217C2×C3×Q8C22×D312yes—
218C2×C3×Q8C22×D311yescharacteristic
219C2×C3×Q8C23×C312yes—
220C2×C3×Q8C23×C311yescharacteristic
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4—32no—
224C2×C3×D4—62no—
225C2×C3×D4—64no—
226C2×C3×D4—68no—
227C2×C3×D4—62no—
228C2×C3×D4—64no—
229C2×C3×D4#6712yes—
230C2×C3×D4C3×D412yes—
231C2×C3×D4C22×D311yescharacteristic
232C2×C3×D4C23×C311yescharacteristic
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—22no—
237C22×C3×C4—21no—
238C22×C3×C4—64no—
239C22×C3×C4—62no—
240C22×C3×C4#6712yes—
241C22×C3×C4C3×D412yes—
242C22×C3×C4C22×D311yescharacteristic
243C22×C3×C4C23×C311yescharacteristic
244C2×#67—24no—
245C2×#67—22no—
246#239—24no—
247D3×Q8—24no—
248#237—28no—
249#237—24no—
250D3×D4—28no—
251D3×D4—24no—
252#235—24no—
253#235—24no—
254#235—22no—
255#235—24no—
256#235—22no—
257#235C3×D414yes—
258#235C23×C314yes—
259C2×D12—22no—
260C2×D12C3×D412yes—
261C2×C4×D3—24no—
262C2×C4×D3—22no—
263C2×Dic6—22no—
264C2×Dic6C3×D412yes—
265C3×Q16—68no—
266C3×SD16—68no—
267C3×M16—34no—
268C3×M16—62no—
269C2×C3×C8—62no—
270C3×C4⋊C4—22no—
271C3×C4⋊C4—28no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—24no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—28no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—24no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—24no—
282C3×C42—22no—
283C3×C42C22×D312yes—
284C3×C42C23×C312yes—
285#216—28no—
286#215—28no—
287#208—22no—
288#208C23×C314yes—
289C2×C3⋊C8—22no—
290C22×C32—22no—
291C22×C32—41no—
292C22×C32—42no—
293C22×C32#12012yes—
294C22×C32C22×D411yescharacteristic
295C2×C3×D3—24no—
296C2×C3×D3—82no—
297C2×C3×D3—84no—
298C32×C4—22no—
299C32×C4—42no—
300C32×C4—42no—
301C32×C4#12011yescharacteristic
302C32×C4#12011yescharacteristic
303C32×C4C22×D412yes—
304C3×Dic3—24no—
3052−1+4—32no—
3062−1+4—62no—
307H(5,𝔽2)—32no—
308H(5,𝔽2)—62no—
309H(5,𝔽2)—62no—
310C2×#41—32no—
311C2×#41—64no—
312C2×#41—62no—
313C2×#41C2×C3×D311yescharacteristic
314C22×Q8C2×C3×D311yescharacteristic, complemented
315#137—38no—
316#137—64no—
317C2×Q16—62no—
318C2×SD16—62no—
319C2×M16—32no—
320C4⋊Q8C2×C3×D312yescomplemented
321#124C2×C3×D312yescomplemented
322#123—22no—
323C22⋊Q8—22no—
324#117C2×C3×D311yescharacteristic, complemented
325#104—38no—
326#101—32no—
327#100—32no—
328C23×C3—21no—
329C23×C3—22no—
330C23×C3—64no—
331C23×C3—62no—
332C23×C3C2×#6712yes—
333C23×C3C2×#6711yescharacteristic
334C23×C3C2×C3×D412yes—
335C23×C3C2×C3×D411yescharacteristic
336C22×D3—24no—
337C22×D3—22no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—24no—
341C3×Q8—24no—
342C3×Q8—34no—
343C3×Q8—44no—
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—64no—
352C3×Q8—62no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—34no—
358C3×D4—42no—
359C3×D4—44no—
360C3×D4—64no—
361C3×D4—68no—
362C3×D4—64no—
363C3×D4—64no—
364C3×D4—62no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—44no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—68no—
386C2×C3×C4—64no—
387C2×C3×C4—62no—
388C2×C3×C4—64no—
389C2×C3×C4—64no—
390C2×C3×C4D3×D412yes—
391C2×C3×C4D3×D412yes—
392C2×C3×C4C2×#6712yes—
393C2×C3×C4C2×#6711yescharacteristic
394C2×C3×C4C2×C3×D412yes—
395C2×C3×C4C2×C3×D412yes—
396C2×C3×C4C2×C3×D412yes—
397C2×C3×C4C2×C3×D411yescharacteristic
398C2×C3×C4C23×D311yescharacteristic
399C2×C3×C4C24×C311yescharacteristic
400#67—28no—
401#67—24no—
402#67—24no—
403C2×Dic3—24no—
404C2×Dic3—22no—
405D12—24no—
406D12—22no—
407D12C2×C3×D414yes—
408C4×D3—28no—
409C4×D3—24no—
410C4×D3—24no—
411Dic6—24no—
412Dic6—22no—
413Dic6C2×C3×D414yes—
414C3×C8—64no—
415C3⋊C8—24no—
416C2×C32—21no—
417C2×C32—22no—
418C2×C32—42no—
419C2×C32C2×#12011yescharacteristic
420C3×D3—44no—
421#41—22no—
422#41—22no—
423#41—34no—
424#41—34no—
425#41—64no—
426#41—64no—
427#41—64no—
428#41—68no—
429#41—64no—
430#41—62no—
431#41—62no—
432#41—64no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—32no—
437C2×Q8—62no—
438C2×Q8—64no—
439C2×Q8C22×C3×D312yes—
440C2×Q8C22×C3×D311yescharacteristic
441C2×D4—32no—
442C2×D4—62no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4C3×#6712yes—
448C2×D4C22×C3×D311yescharacteristic
449C22×C4—21no—
450C22×C4—64no—
451C22×C4—62no—
452C22×C4C3×#6712yes—
453C22×C4C22×C3×D311yescharacteristic
454Q16—68no—
455SD16—68no—
456M16—34no—
457M16—62no—
458C2×C8—62no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C22⋊C4—21no—
462C22⋊C4—24no—
463C42C22×C3×D312yes—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—42no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—64no—
476C22×C3—242no—
477C22×C3—244no—
478C22×C3#93712yes—
479C22×C3C3×#12012yes—
480C22×C3C22×#6711yescharacteristic
481C22×C3C22×C3×D411yescharacteristic
482C2×D3—24no—
483C2×D3—82no—
484C2×D3—84no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—44no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—64no—
498C3×C4#93711yescharacteristic
499C3×C4#93711yescharacteristic
500C3×C4C3×#12011yescharacteristic
501C3×C4C3×#12011yescharacteristic
502C3×C4C2×D3×D412yes—
503C3×C4C22×C3×D412yes—
504Dic3—24no—
505C32#302211yescharacteristic, complemented, Sylow
506C23—64no—
507C23—62no—
508C23C2×C3×#6712yes—
509C23C2×C3×#6711yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—34no—
513Q8—41no—
514Q8—41no—
515Q8—64no—
516Q8—62no—
517D4—24no—
518D4—22no—
519D4—34no—
520D4—64no—
521D4—68no—
522D4—64no—
523D4—64no—
524D4—62no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—68no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4C3×D3×D412yes—
536C2×C4C3×D3×D412yes—
537C2×C4C2×C3×#6712yes—
538C2×C4C2×C3×#6711yescharacteristic
539C2×C4C23×C3×D311yescharacteristic
540C8—64no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—82no—
551C2×C3—124no—
552C2×C3C2×#93711yescharacteristic
553C2×C3C2×C3×#12011yescharacteristic
554D3—44no—
555C22—22no—
556C22—41no—
557C22—42no—
558C22—64no—
559C22—242no—
560C22—244no—
561C22C3×#93712yes—
562C22C22×C3×#6711yescharacteristic
563C4—22no—
564C4—42no—
565C4—42no—
566C4—64no—
567C4C3×#93711yescharacteristic
568C4C3×#93711yescharacteristic
569C4C2×C3×D3×D412yes—
570C3—21no—
571C3#8341611yescharacteristic, direct product factor
572C3C3×#302211yescharacteristic, complemented
573C2—21no—
574C2—22no—
575C2—42no—
576C2—124no—
577C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC2×C3C2×#937
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#536C2
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC3×#536D3
hypocenterC3C3×#3022
LCSG ▷ C2×C3×C4 ▷ 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. C2×C3×#942C2semidirect
4. C3×#5237C2semidirect
5. C3×#5095C2semidirect
6. C3×#5061C2semidirect
7. C3×#5005C2semidirect
9. #83416C3direct (N × H)
11. C22×C32×Q8C22semidirect
20. C32×C4⋊Q8C22semidirect
21. C32×#124C22semidirect
24. C32×#117C22semidirect
27. C3×#942C22semidirect
29. C2×C3×#208C22semidirect
30. C3×#836C22semidirect
31. C3×#833C22semidirect
32. C3×#805C22semidirect
37. C3×#536D3semidirect
38. C3×#536C2×C3semidirect
39. C2×#942C2×C3semidirect
44. #5237C2×C3semidirect
45. #5095C2×C3semidirect
46. #5061C2×C3semidirect
47. #5005C2×C3semidirect
104. C22×C3×Q8C2×D3semidirect
105. C22×C3×Q8C22×C3semidirect
120. C3×C4⋊Q8C2×D3semidirect
121. C3×C4⋊Q8C22×C3semidirect
123. C3×#124C2×D3semidirect
124. C3×#124C22×C3semidirect
132. C3×#117C2×D3semidirect
133. C3×#117C22×C3semidirect
136. #942C22×C3semidirect
138. C2×#208C22×C3semidirect
142. #836C22×C3semidirect
143. #833C22×C3semidirect
144. #805C22×C3semidirect
182. #536C3×D3semidirect
314. C22×Q8C2×C3×D3semidirect
320. C4⋊Q8C2×C3×D3semidirect
321. #124C2×C3×D3semidirect
324. #117C2×C3×D3semidirect
505. C32#3022semidirect
571. C3#83416direct (N × H)
572. C3C3×#3022semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C32×#5362
C2×C3×#9422
C3×#52372
C3×#50952
C3×#50612
C3×#50052
C3×#30223
#834163