Finite Group Databasebeta

← C3×#84116 #11865082

Subgroups — C3×#84116 #11865082

Order 1152.

1793 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C3×#5822C212yesmaximal, complemented
3C3×#5816C211yesmaximal, characteristic, complemented
4C3×#5246C214yesmaximal, complemented
5C3×#5245C214yesmaximal, complemented
6C3×#5236C212yesmaximal, complemented
7C2×C3×#836C211yesmaximal, characteristic, complemented
8C3×#3020—31nomaximal
9#84116C311yesmaximal, characteristic, direct product factor
10C32×2−1+4—24no—
11C32×2−1+4—22no—
12C32×2−1+4C2214yes—
13C2×C32×#41—22no—
14C2×C32×#41—22no—
15C2×C32×#41C2212yes—
16C22×C32×Q8—22no—
17C22×C32×Q8C2211yescharacteristic
18C2×C3×#235C2211yescharacteristic
19C3×#951C2214yescomplemented
20C3×#950—24no—
21C3×#949C2214yescomplemented
22C3×#947C2212yescomplemented
23C3×#946—22no—
24C3×#944C2212yescomplemented
25C3×#942—22no—
26C3×#932—22no—
27C3×#929—22no—
28C3×#926C2211yescharacteristic
29C2×C3×#208C2212yes—
30C3×#837C2218yescomplemented
31C3×#836C2214yescomplemented
32C2×C3×2−1+4—21no—
33C2×C3×2−1+4D311yescharacteristic, complemented
34C2×C3×2−1+4C2×C311yescharacteristic, complemented
35C3×#575—32no—
36C3×#536—31no—
37#5822C2×C312yescomplemented
38#5816C2×C311yescharacteristic, complemented
39C3×#456—34no—
40C3×#454—34no—
41C3×#421—32no—
42C2×C3×#101—31no—
43#5246C2×C314yescomplemented
44#5245C2×C314yescomplemented
45#5236C2×C312yescomplemented
46C2×#836C2×C311yescharacteristic, complemented
47C32×#41—28no—
48C32×#41—28no—
49C32×#41—28no—
50C32×#41—28no—
51C32×#41—22no—
52C32×#41—22no—
53C32×#41D414yes—
54C32×#41C2314yes—
55C2×C32×Q8—22no—
56C2×C32×Q8—28no—
57C2×C32×Q8—28no—
58C2×C32×Q8—22no—
59C2×C32×Q8—22no—
60C2×C32×Q8D412yes—
61C2×C32×Q8D412yes—
62C2×C32×Q8C2312yes—
63C2×C32×D4—22no—
64C2×C32×D4—22no—
65C2×C32×D4D412yes—
66C22×C32×C4—24no—
67C22×C32×C4—22no—
68C22×C32×C4D412yes—
69C22×C32×C4C2311yescharacteristic
70C2×C3×#67—21no—
71C3×#235—22no—
72C3×#235—22no—
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6C2311yescharacteristic
76C3×#217—21no—
77C3×#216—24no—
78C3×#215—24no—
79C3×#214—24no—
80C3×#213—24no—
81C3×(C2×D3)⋊C4—24no—
82C3×C4⋊Dic3—21no—
83C3×Dic3⋊C4—24no—
84C3×C4×Dic3C2312yes—
85C3×#208—22no—
86C3×#208C2314yes—
87C2×C3×C3⋊C8—22no—
88#3020—31noSylow
89C3×2−1+4—28no—
90C3×2−1+4—24no—
91C3×2−1+4—24no—
92C3×2−1+4—24no—
93C3×2−1+4—22no—
94C3×2−1+4—22no—
95C3×2−1+4—24no—
96C3×2−1+4C2×D314yes—
97C3×2−1+4C22×C314yes—
98C2×C3×#41—24no—
99C2×C3×#41—24no—
100C2×C3×#41—22no—
101C2×C3×#41—22no—
102C2×C3×#41—22no—
103C2×C3×#41—22no—
104C2×C3×#41—22no—
105C2×C3×#41—31no—
106C2×C3×#41C2×D312yes—
107C2×C3×#41C22×C312yes—
108C22×C3×Q8—24no—
109C22×C3×Q8—22no—
110C22×C3×Q8—22no—
111C22×C3×Q8—21no—
112C22×C3×Q8C2×D311yescharacteristic
113C22×C3×Q8C22×C311yescharacteristic
114C2×#235C22×C311yescharacteristic
115C3×#137—34no—
116C3×#137—62no—
117C3×AGL(1,ℤ/8)—34no—
118C3×AGL(1,ℤ/8)—62no—
119C3×#135—64no—
120C2×C3×M16—32no—
121C3×C4⋊Q8—32no—
122C3×#124—32no—
123C3×#123—62no—
124C3×C22⋊Q8—62no—
125C3×#117—31no—
126#951C22×C314yescomplemented
127#950—24no—
128#949C22×C314yescomplemented
129#947C22×C312yescomplemented
130#946—22no—
131#944C22×C312yescomplemented
132#942—22no—
133#932—22no—
134#929—22no—
135#926C22×C311yescharacteristic
136C2×#208C22×C312yes—
137C3×#104—38no—
138C3×#101—34no—
139#837C22×C318yescomplemented
140#836C22×C314yescomplemented
141C23×C32—22no—
142C23×C32C2×D411yescharacteristic
143C22×C3×D3—21no—
144C32×Q8—28no—
145C32×Q8—22no—
146C32×Q8—24no—
147C32×Q8—24no—
148C32×Q8C2×D414yes—
149C32×D4—28no—
150C32×D4—24no—
151C32×D4—24no—
152C32×D4—22no—
153C32×D4C2×D414yes—
154C2×C32×C4—216no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4C2×D414yes—
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2411yescharacteristic
165C3×#67—41no—
166C3×#67—41no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C3×D12—22no—
170C3×C4×D3—42no—
171C3×Dic6—22no—
172C3×C3⋊C8—24no—
173C2×2−1+4C3×D311yescharacteristic, complemented
174#575—32no—
175#536—31no—
176#456—34no—
177#454—34no—
178#421—32no—
179C2×#101—31no—
180C3×#41—216no—
181C3×#41—216no—
182C3×#41—28no—
183C3×#41—28no—
184C3×#41—28no—
185C3×#41—28no—
186C3×#41—216no—
187C3×#41—216no—
188C3×#41—24no—
189C3×#41—24no—
190C3×#41—28no—
191C3×#41—28no—
192C3×#41—28no—
193C3×#41—22no—
194C3×#41—22no—
195C3×#41—28no—
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—24no—
199C3×#41—24no—
200C3×#41—62no—
201C3×#41—62no—
202C3×#41#6714yes—
203C3×#41C3×D414yes—
204C3×#41C22×D314yes—
205C3×#41C23×C314yes—
206C2×C3×Q8—24no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—24no—
210C2×C3×Q8—216no—
211C2×C3×Q8—28no—
212C2×C3×Q8—28no—
213C2×C3×Q8—216no—
214C2×C3×Q8—24no—
215C2×C3×Q8—28no—
216C2×C3×Q8—28no—
217C2×C3×Q8—22no—
218C2×C3×Q8—22no—
219C2×C3×Q8—22no—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—22no—
223C2×C3×Q8—22no—
224C2×C3×Q8—31no—
225C2×C3×Q8#6712yes—
226C2×C3×Q8#6712yes—
227C2×C3×Q8C3×D412yes—
228C2×C3×Q8C3×D412yes—
229C2×C3×Q8C22×D312yes—
230C2×C3×Q8C23×C312yes—
231C2×C3×D4—24no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—24no—
235C2×C3×D4—22no—
236C2×C3×D4—22no—
237C2×C3×D4—22no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C2×C3×D4#6712yes—
241C2×C3×D4C3×D412yes—
242C22×C3×C4—28no—
243C22×C3×C4—24no—
244C22×C3×C4—24no—
245C22×C3×C4—24no—
246C22×C3×C4—22no—
247C22×C3×C4—22no—
248C22×C3×C4—22no—
249C22×C3×C4—21no—
250C22×C3×C4—61no—
251C22×C3×C4#6712yes—
252C22×C3×C4C3×D412yes—
253C22×C3×C4C22×D311yescharacteristic
254C22×C3×C4C23×C311yescharacteristic
255C2×#67—21no—
256#235—22no—
257#235—22no—
258C2×D12C23×C311yescharacteristic
259C2×C4×D3—21no—
260C2×Dic6C23×C311yescharacteristic
261C3×Q16—64no—
262C3×SD16—64no—
263C3×SD16—64no—
264C3×D8—64no—
265C3×M16—34no—
266C3×M16—62no—
267C2×C3×C8—62no—
268C3×C4⋊C4—64no—
269C3×C4⋊C4—61no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—64no—
272C3×C42—32no—
273#217—21no—
274#216—24no—
275#215—24no—
276#214—24no—
277#213—24no—
278(C2×D3)⋊C4—24no—
279C4⋊Dic3—21no—
280Dic3⋊C4—24no—
281C4×Dic3C23×C312yes—
282#208—22no—
283#208C23×C314yes—
284C2×C3⋊C8—22no—
285C22×C32—24no—
286C22×C32—41no—
287C22×C32—44no—
288C22×C32#12011yescharacteristic
289C22×C32#12011yescharacteristic
290C22×C32C22×D411yescharacteristic
291C2×C3×D3—42no—
292C2×C3×D3—41no—
293C32×C4—24no—
294C32×C4—24no—
295C32×C4#12012yes—
296C32×C4C22×D412yes—
297C3×Dic3—42no—
298C3×Dic3—41no—
2992−1+4—24no—
3002−1+4—22no—
3012−1+4C2×C3×D314yes—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—31no—
305C2×#41C2×C3×D312yes—
306C22×Q8—22no—
307C22×Q8C2×C3×D311yescharacteristic
308#137—34no—
309#137—62no—
310AGL(1,ℤ/8)—34no—
311AGL(1,ℤ/8)—62no—
312#135—64no—
313C2×M16—32no—
314C4⋊Q8—32no—
315#124—32no—
316#123—62no—
317C22⋊Q8—62no—
318#117—31no—
319#104—38no—
320#101—34no—
321C23×C3—24no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—61no—
326C23×C3C2×#6711yescharacteristic
327C23×C3C2×C3×D411yescharacteristic
328C22×D3—21no—
329C3×Q8—216no—
330C3×Q8—28no—
331C3×Q8—28no—
332C3×Q8—28no—
333C3×Q8—24no—
334C3×Q8—28no—
335C3×Q8—22no—
336C3×Q8—24no—
337C3×Q8—22no—
338C3×Q8—24no—
339C3×Q8—24no—
340C3×Q8—24no—
341C3×Q8—24no—
342C3×Q8—62no—
343C3×Q8C2×#6714yes—
344C3×Q8C2×C3×D414yes—
345C3×D4—216no—
346C3×D4—28no—
347C3×D4—28no—
348C3×D4—28no—
349C3×D4—28no—
350C3×D4—24no—
351C3×D4—24no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—24no—
358C3×D4—62no—
359C3×D4—121no—
360C3×D4—121no—
361C3×D4C2×#6714yes—
362C3×D4C2×C3×D414yes—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—232no—
366C2×C3×C4—216no—
367C2×C3×C4—216no—
368C2×C3×C4—24no—
369C2×C3×C4—28no—
370C2×C3×C4—28no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—62no—
387C2×C3×C4—61no—
388C2×C3×C4—122no—
389C2×C3×C4C2×#6714yes—
390C2×C3×C4C2×#6711yescharacteristic
391C2×C3×C4C2×#6712yes—
392C2×C3×C4C2×#6712yes—
393C2×C3×C4C2×C3×D414yes—
394C2×C3×C4C2×C3×D411yescharacteristic
395C2×C3×C4C2×C3×D412yes—
396C2×C3×C4C2×C3×D412yes—
397C2×C3×C4C23×D311yescharacteristic
398C2×C3×C4C24×C311yescharacteristic
399#67—41no—
400#67—41no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403D12—22no—
404C4×D3—42no—
405Dic6—22no—
406C3×C8—64no—
407C3⋊C8—24no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—44no—
412C2×C32C2×#12011yescharacteristic
413C3×D3—81no—
414#41—28no—
415#41—28no—
416#41—28no—
417#41—28no—
418#41—22no—
419#41—22no—
420#41—62no—
421#41—62no—
422#41C3×#6714yes—
423#41C22×C3×D314yes—
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—31no—
430C2×Q8C3×#6712yes—
431C2×Q8C3×#6712yes—
432C2×Q8C22×C3×D312yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—31no—
436C2×D4—61no—
437C2×D4C3×#6712yes—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—61no—
441C22×C4C3×#6712yes—
442C22×C4C22×C3×D311yescharacteristic
443Q16—64no—
444SD16—64no—
445SD16—64no—
446D8—64no—
447M16—34no—
448M16—62no—
449C2×C8—62no—
450C4⋊C4—61no—
451C4⋊C4—64no—
452C22⋊C4—61no—
453C22⋊C4—64no—
454C42—32no—
455C22×C3—28no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—48no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—122no—
468C22×C3—121no—
469C22×C3#95311yescharacteristic
470C22×C3#95311yescharacteristic
471C22×C3C3×#12011yescharacteristic
472C22×C3C3×#12011yescharacteristic
473C22×C3C22×#6711yescharacteristic
474C22×C3C22×C3×D411yescharacteristic
475C2×D3—42no—
476C2×D3—41no—
477C3×C4—28no—
478C3×C4—28no—
479C3×C4—24no—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—24no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—122no—
486C3×C4—121no—
487C3×C4#95312yes—
488C3×C4C3×#12012yes—
489C3×C4C22×#6712yes—
490C3×C4C22×C3×D412yes—
491Dic3—42no—
492Dic3—41no—
493C32#302011yescharacteristic, complemented, Sylow
494C23—22no—
495C23—61no—
496C23C2×C3×#6711yescharacteristic
497Q8—28no—
498Q8—22no—
499Q8—24no—
500Q8—24no—
501Q8—62no—
502Q8C2×C3×#6714yes—
503D4—28no—
504D4—24no—
505D4—24no—
506D4—22no—
507D4—62no—
508D4—121no—
509D4—121no—
510D4C2×C3×#6714yes—
511C2×C4—216no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—62no—
518C2×C4—61no—
519C2×C4—122no—
520C2×C4C2×C3×#6714yes—
521C2×C4C2×C3×#6711yescharacteristic
522C2×C4C2×C3×#6712yes—
523C2×C4C2×C3×#6712yes—
524C2×C4C23×C3×D311yescharacteristic
525C8—64no—
526C2×C3—22no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—48no—
537C2×C3—44no—
538C2×C3—44no—
539C2×C3—241no—
540C2×C3C2×#95311yescharacteristic
541C2×C3C2×C3×#12011yescharacteristic
542D3—81no—
543C22—24no—
544C22—41no—
545C22—44no—
546C22—122no—
547C22—121no—
548C22C3×#95311yescharacteristic
549C22C3×#95311yescharacteristic
550C22C22×C3×#6711yescharacteristic
551C4—24no—
552C4—24no—
553C4—122no—
554C4—121no—
555C4C3×#95312yes—
556C4C22×C3×#6712yes—
557C3—21no—
558C3#8411611yescharacteristic, direct product factor
559C3C3×#302011yescharacteristic, complemented
560C2—21no—
561C2—21no—
562C2—21no—
563C2—44no—
564C2—241no—
565C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC2×C3C2×#953
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×2−1+4C2
socleC2×C32C2×#120
hypercenterC2×C3×2−1+4D3
hypocenterC3C3×#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C3×#5822C2semidirect
3. C3×#5816C2semidirect
4. C3×#5246C2semidirect
5. C3×#5245C2semidirect
6. C3×#5236C2semidirect
7. C2×C3×#836C2semidirect
9. #84116C3direct (N × H)
19. C3×#951C22semidirect
21. C3×#949C22semidirect
22. C3×#947C22semidirect
24. C3×#944C22semidirect
30. C3×#837C22semidirect
31. C3×#836C22semidirect
33. C2×C3×2−1+4D3semidirect
34. C2×C3×2−1+4C2×C3semidirect
37. #5822C2×C3semidirect
38. #5816C2×C3semidirect
43. #5246C2×C3semidirect
44. #5245C2×C3semidirect
45. #5236C2×C3semidirect
46. C2×#836C2×C3semidirect
126. #951C22×C3semidirect
128. #949C22×C3semidirect
129. #947C22×C3semidirect
131. #944C22×C3semidirect
139. #837C22×C3semidirect
140. #836C22×C3semidirect
173. C2×2−1+4C3×D3semidirect
493. C32#3020semidirect
558. C3#84116direct (N × H)
559. C3C3×#3020semidirect

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C3×#58222
C3×#58162
C3×#52462
C3×#52452
C3×#52362
C2×C3×#8362
C3×#30203
#841163