Finite Group Databasebeta

← C3×#23588 #10642195

Subgroups — C3×#23588 #10642195

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic, complemented
2C3×C4×AGL(1,ℤ/8)C212yesmaximal, complemented
3C3×C4×#124C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2040C211yesmaximal, characteristic, complemented
6C3×#2038C212yesmaximal, complemented
7C3×#2027C211yesmaximal, characteristic, complemented
8C3×#2006C211yesmaximal, characteristic
9C3×#2003C212yesmaximal, complemented
10C3×C4×#104C212yesmaximal, complemented
11C3×C4×#101C211yesmaximal, characteristic, complemented
12#23588C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#541—22no—
17C3×#539—22no—
18C3×#537C2211yescharacteristic
19C3×#536C2212yes—
20C3×#533—22no—
21C3×#529—22no—
22C3×#529—22no—
23C2×C3×#124C2211yescharacteristic
24C3×C4×#41C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#442—22no—
28C3×#440—22no—
29C3×C4×SD16—22no—
30C3×C4×D8—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#418C2212yes—
35C3×#417C2212yes—
36C2×C3×#101C2212yes—
37C3×C4×M16C2212yes—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—22no—
41C3×#382C2211yescharacteristic
42C3×C4×C22⋊C4—22no—
43C3×C43C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3461C2×C311yescharacteristic, complemented
49C4×AGL(1,ℤ/8)C2×C312yescomplemented
50C4×#124C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2040C2×C311yescharacteristic, complemented
53#2038C2×C312yescomplemented
54#2027C2×C311yescharacteristic, complemented
55#2006C2×C311yescharacteristic
56#2003C2×C312yescomplemented
57C4×#104C2×C312yescomplemented
58C4×#101C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C3×AGL(1,ℤ/8)—24no—
64C3×AGL(1,ℤ/8)—24no—
65C2×C3×SD16—22no—
66C2×C3×D8—22no—
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#126—24no—
72C3×#126—24no—
73C3×#125—22no—
74C3×#125—24no—
75C3×#125—22no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124—22no—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8D412yes—
93C3×C4×Q8D412yes—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D412yes—
100C3×C4×D4D412yes—
101C3×#117—24no—
102C3×#117—22no—
103C3×#117C2312yes—
104C2×C3×C4⋊C4D412yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—22no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42D412yes—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×#104—24no—
120C3×#104—24no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#101—24no—
126C3×#97—22no—
127C3×C4×C8—22no—
128C3×#95—22no—
129C3×#95—22no—
130C2×AGL(1,ℤ/8)C22×C312yes—
131#544C22×C311yescharacteristic
132#542—22no—
133#541—22no—
134#539—22no—
135#537C22×C311yescharacteristic
136#536C22×C312yes—
137#533—22no—
138#529—22no—
139#529—22no—
140C2×#124C22×C311yescharacteristic
141C4×#41C22×C312yes—
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144#442—22no—
145#440—22no—
146C4×SD16—22no—
147C4×D8—22no—
148#428C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#418C22×C312yes—
152#417C22×C312yes—
153C2×#101C22×C312yes—
154C4×M16C22×C312yes—
155#391C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#388—22no—
158#382C22×C311yescharacteristic
159C4×C22⋊C4—22no—
160C43C22×C311yescharacteristic
161#343C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C24×C3—21no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4#4112yes—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—44no—
190C22×C3×C4—41no—
191C22×C3×C4—41no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C3×SD16—28no—
198C3×D8—28no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16#4114yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—42no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—21no—
232C3×C42—22no—
233C3×C42—42no—
234C3×C42—42no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237C3×C42#4112yes—
238C2×#41C23×C312yes—
239C22×Q8C23×C311yescharacteristic
240C22×D4C23×C311yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—24no—
243AGL(1,ℤ/8)—24no—
244C2×SD16—22no—
245C2×D8—22no—
246C2×M16C23×C312yes—
247C2×M16C23×C312yes—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250#126—24no—
251#126—24no—
252#125—22no—
253#125—24no—
254#125—22no—
255#124—22no—
256#124—22no—
257#124—22no—
258#124—22no—
259#123—24no—
260#123—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8C3×D412yes—
272C4×Q8C3×D412yes—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4C3×D412yes—
279C4×D4C3×D412yes—
280#117—24no—
281#117—22no—
282#117C23×C312yes—
283C2×C4⋊C4C3×D412yes—
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42C3×D412yes—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296#107—22no—
297#106—22no—
298#104—24no—
299#104—24no—
300#103—22no—
301#102—22no—
302#102—22no—
303#102—22no—
304#101—24no—
305#97—22no—
306C4×C8—22no—
307#95—22no—
308#95—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3C22×D411yescharacteristic
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—42no—
319C3×Q8—42no—
320C3×Q8—82no—
321C3×D4—22no—
322C3×D4—24no—
323C3×D4—22no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—82no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—82no—
350C2×C3×C4—82no—
351C2×C3×C4—82no—
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4#12012yes—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4C22×D412yes—
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C4C2×#4111yescharacteristic
359C3×C8—24no—
360C3×C8—42no—
361C24—21no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8C2×C3×D412yes—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4C2×C3×D412yes—
378C2×D4C3×#4112yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C2×C3×D412yes—
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C24×C311yescharacteristic
393SD16—28no—
394D8—28no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16C3×#4114yes—
399C2×C8—22no—
400C2×C8—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C2×C3×D412yes—
413C22⋊C4—28no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—21no—
428C42—22no—
429C42—42no—
430C42—42no—
431C42C2×C3×D412yes—
432C42C2×C3×D412yes—
433C42C3×#4112yes—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—82no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#52511yescharacteristic
446C22×C3#53211yescharacteristic
447C3×C4—21no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—81no—
457C3×C4#45412yes—
458C3×C4#52512yes—
459C3×C4#53212yes—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23C22×C3×D411yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—42no—
470Q8—42no—
471Q8—82no—
472D4—22no—
473D4—24no—
474D4—22no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—82no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4C3×#12011yescharacteristic
504C2×C4C3×#12012yes—
505C2×C4C3×#12011yescharacteristic
506C2×C4C22×C3×D412yes—
507C2×C4C2×C3×#4111yescharacteristic
508C2×C4C2×C3×#4112yes—
509C2×C4C2×C3×#4111yescharacteristic
510C8—24no—
511C8—42no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—82no—
515C2×C3—82no—
516C2×C3#238811yescharacteristic
517C2×C3C2×#45411yescharacteristic
518C2×C3#301811yescharacteristic
519C22—21no—
520C22—21no—
521C22—44no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—82no—
527C22—81no—
528C22—81no—
529C22C2×C3×#12011yescharacteristic
530C22C3×#52511yescharacteristic
531C22C3×#53211yescharacteristic
532C4—21no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4C3×#45412yes—
543C4C3×#52512yes—
544C4C3×#53212yes—
545C3#2358811yescharacteristic, direct product factor, Sylow
546C2—21no—
547C2—21no—
548C2—82no—
549C2—82no—
550C2C3×#238811yescharacteristic
551C2C2×C3×#45411yescharacteristic
552C2C3×#301811yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC22×C4C24×C3
socleC22×C3C2×#120
prime radicalC3×#2006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3461C2semidirect
2. C3×C4×AGL(1,ℤ/8)C2semidirect
3. C3×C4×#124C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2038C2semidirect
7. C3×#2027C2semidirect
9. C3×#2003C2semidirect
10. C3×C4×#104C2semidirect
11. C3×C4×#101C2semidirect
12. #23588C3direct (N × H)
48. #3461C2×C3semidirect
49. C4×AGL(1,ℤ/8)C2×C3semidirect
50. C4×#124C2×C3semidirect
51. #2061C2×C3semidirect
52. #2040C2×C3semidirect
53. #2038C2×C3semidirect
54. #2027C2×C3semidirect
56. #2003C2×C3semidirect
57. C4×#104C2×C3semidirect
58. C4×#101C2×C3semidirect
545. C3#23588direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×C4×AGL(1,ℤ/8)2
C3×C4×#1242
C3×#20612
C3×#20402
C3×#20382
C3×#20272
C3×#20062
C3×#20032
C3×C4×#1042
C3×C4×#1012
#235883