Finite Group Databasebeta

← C3×#23409 #10642016

Subgroups — C3×#23409 #10642016

Order 768.

2014 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C212yesmaximal, complemented
2C2×C3×#537C211yesmaximal, characteristic, complemented
3C2×C3×#421C212yesmaximal, complemented
4C3×#2311C211yesmaximal, characteristic, complemented
5C3×#2007C214yesmaximal, complemented
6C3×#2006C214yesmaximal, complemented
7C2×C3×#343C211yesmaximal, characteristic, complemented
8#23409C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2212yes—
10C22×C3×#41C2211yescharacteristic
11C3×#575—28no—
12C3×#575—28no—
13C2×C3×#137—22no—
14C2×C3×#137C2214yescomplemented
15C2×C3×AGL(1,ℤ/8)—22no—
16C2×C3×AGL(1,ℤ/8)C2214yescomplemented
17C2×C3×#135—24no—
18C22×C3×M16C2212yes—
19C3×#537—24no—
20C3×#537—22no—
21C3×#537C2214yes—
22C2×C3×C22⋊Q8—22no—
23C2×C3×#121—22no—
24C2×C3×#121—22no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#421—28no—
28C2×C3×#104C2218yescomplemented
29C3×#391C2212yescomplemented
30C3×#390C2212yescomplemented
31C3×#386—22no—
32C3×#380—22no—
33C3×#343C2214yescomplemented
34C2×#575C2×C312yescomplemented
35C2×#537C2×C311yescharacteristic, complemented
36C2×#421C2×C312yescomplemented
37#2311C2×C311yescharacteristic, complemented
38#2007C2×C314yescomplemented
39#2006C2×C314yescomplemented
40C2×#343C2×C311yescharacteristic, complemented
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C2×C3×#41—24no—
47C2×C3×#41—24no—
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—24no—
51C2×C3×#41D414yes—
52C2×C3×#41D414yescomplemented
53C2×C3×#41C2314yes—
54C22×C3×Q8D412yes—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4D412yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C23×C3×C4D412yes—
64C23×C3×C4C2311yescharacteristic
65C3×#137—216no—
66C3×#137—28no—
67C3×#137—28no—
68C3×AGL(1,ℤ/8)—216no—
69C3×AGL(1,ℤ/8)—28no—
70C3×AGL(1,ℤ/8)—28no—
71C3×#135—232no—
72C2×C3×Q16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—24no—
75C2×C3×D8—24no—
76C2×C3×M16—22no—
77C2×C3×M16—24no—
78C2×C3×M16D418yescomplemented
79C2×C3×M16C2314yes—
80C22×C3×C8—22no—
81C3×C22⋊Q8—28no—
82C3×C22⋊Q8—28no—
83C3×#121—28no—
84C3×#121—28no—
85C3×#121—28no—
86C3×#121—28no—
87C3×#117—24no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D412yes—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—24no—
97C2×C3×C42C2312yes—
98C3×#104—216no—
99C3×#95—24no—
100C22×#41C22×C312yes—
101C22×#41C22×C311yescharacteristic
102#575—28no—
103#575—28no—
104C2×#137—22no—
105C2×#137C22×C314yescomplemented
106C2×AGL(1,ℤ/8)—22no—
107C2×AGL(1,ℤ/8)C22×C314yescomplemented
108C2×#135—24no—
109C22×M16C22×C312yes—
110#537—24no—
111#537—22no—
112#537C22×C314yes—
113C2×C22⋊Q8—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#117C22×C311yescharacteristic
117C22×C4⋊C4C22×C311yescharacteristic
118#421—28no—
119C2×#104C22×C318yescomplemented
120#391C22×C312yescomplemented
121#390C22×C312yescomplemented
122#386—22no—
123#380—22no—
124#343C22×C314yescomplemented
125C24×C3—22no—
126C24×C3—21no—
127C24×C3C2×D411yescharacteristic
128C3×#41—28no—
129C3×#41—28no—
130C3×#41—28no—
131C3×#41—28no—
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—24no—
135C3×#41—24no—
136C3×#41—44no—
137C3×#41—44no—
138C3×#41—44no—
139C3×#41—44no—
140C3×#41—44no—
141C3×#41—44no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—24no—
145C2×C3×Q8—24no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8C2×D414yes—
149C2×C3×Q8C2×D412yescomplemented
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—28no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—44no—
163C2×C3×D4—42no—
164C2×C3×D4—44no—
165C2×C3×D4—84no—
166C2×C3×D4C2×D414yes—
167C2×C3×D4C2×D412yescomplemented
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—42no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4—42no—
180C22×C3×C4—82no—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D414yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C3×Q16—216no—
190C3×SD16—216no—
191C3×SD16—216no—
192C3×D8—216no—
193C3×M16—28no—
194C3×M16—24no—
195C3×M16C2×D418yescomplemented
196C2×C3×C8—28no—
197C2×C3×C8—24no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—42no—
202C3×C22⋊C4—216no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—84no—
205C3×C42—24no—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41C3×D414yes—
217C2×#41C3×D414yescomplemented
218C2×#41C23×C314yes—
219C22×Q8C3×D412yes—
220C22×Q8C23×C311yescharacteristic
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4C3×D412yes—
225C22×D4C23×C311yescharacteristic
226C23×C4—22no—
227C23×C4—21no—
228C23×C4C3×D412yes—
229C23×C4C23×C311yescharacteristic
230#137—216no—
231#137—28no—
232#137—28no—
233AGL(1,ℤ/8)—216no—
234AGL(1,ℤ/8)—28no—
235AGL(1,ℤ/8)—28no—
236#135—232no—
237C2×Q16—24no—
238C2×SD16—24no—
239C2×SD16—24no—
240C2×D8—24no—
241C2×M16—22no—
242C2×M16—24no—
243C2×M16C3×D418yescomplemented
244C2×M16C23×C314yes—
245C22×C8—22no—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248#121—28no—
249#121—28no—
250#121—28no—
251#121—28no—
252#117—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C312yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—24no—
262C2×C42C23×C312yes—
263#104—216no—
264#95—24no—
265C23×C3—24no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—44no—
271C23×C3—42no—
272C23×C3—44no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—44no—
276C23×C3#12011yescharacteristic
277C23×C3#12011yescharacteristic
278C23×C3#12112yes—
279C23×C3#12712yes—
280C23×C3C22×D411yescharacteristic
281C3×Q8—24no—
282C3×Q8—24no—
283C3×Q8—24no—
284C3×Q8—42no—
285C3×Q8—44no—
286C3×D4—24no—
287C3×D4—24no—
288C3×D4—24no—
289C3×D4—44no—
290C3×D4—44no—
291C3×D4—44no—
292C3×D4—48no—
293C3×D4—44no—
294C3×D4—42no—
295C3×D4—44no—
296C3×D4—88no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—44no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4—48no—
307C2×C3×C4—48no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—82no—
313C2×C3×C4#12014yes—
314C2×C3×C4#12014yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#12114yes—
317C2×C3×C4#12114yes—
318C2×C3×C4#12112yes—
319C2×C3×C4#12712yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4C2×#4111yescharacteristic
324C3×C8—28no—
325C24—22no—
326C24—21no—
327C24C2×C3×D411yescharacteristic
328#41—28no—
329#41—28no—
330#41—28no—
331#41—28no—
332#41—24no—
333#41—24no—
334#41—24no—
335#41—24no—
336#41—44no—
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C2×C3×D414yes—
349C2×Q8C2×C3×D412yescomplemented
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—84no—
366C2×D4C2×C3×D414yes—
367C2×D4C2×C3×D412yescomplemented
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—82no—
381C22×C4C2×C3×D414yes—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D412yes—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D414yes—
387C22×C4C3×#4112yes—
388C22×C4C24×C311yescharacteristic
389Q16—216no—
390SD16—216no—
391SD16—216no—
392D8—216no—
393M16—28no—
394M16—24no—
395M16C2×C3×D418yescomplemented
396C2×C8—28no—
397C2×C8—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—42no—
402C22⋊C4—216no—
403C22⋊C4—42no—
404C22⋊C4—84no—
405C42—24no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—82no—
420C22×C3—88no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3#39212yes—
424C22×C3#39212yes—
425C22×C3C2×#12011yescharacteristic
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12711yescharacteristic
428C3×C4—44no—
429C3×C4—42no—
430C3×C4—44no—
431C3×C4—82no—
432C3×C4#39214yes—
433C3×C4C2×#12012yes—
434C3×C4C2×#12112yes—
435C23—24no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—41no—
445C23—41no—
446C23C3×#12011yescharacteristic
447C23C3×#12011yescharacteristic
448C23C3×#12112yes—
449C23C3×#12712yes—
450C23C22×C3×D411yescharacteristic
451Q8—24no—
452Q8—24no—
453Q8—24no—
454Q8—44no—
455Q8—42no—
456D4—24no—
457D4—24no—
458D4—24no—
459D4—44no—
460D4—48no—
461D4—44no—
462D4—44no—
463D4—44no—
464D4—44no—
465D4—42no—
466D4—88no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4C3×#12012yes—
484C2×C4C3×#12014yes—
485C2×C4C3×#12014yes—
486C2×C4C3×#12112yes—
487C2×C4C3×#12114yes—
488C2×C4C3×#12114yes—
489C2×C4C3×#12712yes—
490C2×C4C22×C3×D411yescharacteristic
491C2×C4C22×C3×D412yes—
492C2×C4C22×C3×D412yes—
493C2×C4C2×C3×#4111yescharacteristic
494C8—28no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—82no—
500C2×C3—84no—
501C2×C3C2×#39211yescharacteristic
502C2×C3#301911yescharacteristic
503C2×C3#302011yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—42no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—82no—
518C22—88no—
519C22—81no—
520C22—81no—
521C22C3×#39212yes—
522C22C3×#39212yes—
523C22C2×C3×#12011yescharacteristic
524C22C2×C3×#12111yescharacteristic
525C22C2×C3×#12711yescharacteristic
526C4—44no—
527C4—42no—
528C4—44no—
529C4—82no—
530C4C3×#39214yes—
531C4C2×C3×#12012yes—
532C4C2×C3×#12112yes—
533C3#2340911yescharacteristic, direct product factor, Sylow
534C2—21no—
535C2—21no—
536C2—22no—
537C2—22no—
538C2—82no—
539C2—84no—
540C2C2×C3×#39211yescharacteristic
541C2C3×#301911yescharacteristic
542C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#575C2semidirect
2. C2×C3×#537C2semidirect
3. C2×C3×#421C2semidirect
4. C3×#2311C2semidirect
5. C3×#2007C2semidirect
6. C3×#2006C2semidirect
7. C2×C3×#343C2semidirect
8. #23409C3direct (N × H)
14. C2×C3×#137C22semidirect
16. C2×C3×AGL(1,ℤ/8)C22semidirect
28. C2×C3×#104C22semidirect
29. C3×#391C22semidirect
30. C3×#390C22semidirect
33. C3×#343C22semidirect
34. C2×#575C2×C3semidirect
35. C2×#537C2×C3semidirect
36. C2×#421C2×C3semidirect
37. #2311C2×C3semidirect
38. #2007C2×C3semidirect
39. #2006C2×C3semidirect
40. C2×#343C2×C3semidirect
52. C2×C3×#41D4semidirect
78. C2×C3×M16D4semidirect
105. C2×#137C22×C3semidirect
107. C2×AGL(1,ℤ/8)C22×C3semidirect
119. C2×#104C22×C3semidirect
120. #391C22×C3semidirect
121. #390C22×C3semidirect
124. #343C22×C3semidirect
149. C2×C3×Q8C2×D4semidirect
167. C2×C3×D4C2×D4semidirect
195. C3×M16C2×D4semidirect
217. C2×#41C3×D4semidirect
243. C2×M16C3×D4semidirect
349. C2×Q8C2×C3×D4semidirect
367. C2×D4C2×C3×D4semidirect
395. M16C2×C3×D4semidirect
533. C3#23409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C2×C3×#5372
C2×C3×#4212
C3×#23112
C3×#20072
C3×#20062
C2×C3×#3432
#234093