Finite Group Databasebeta

← C3×#29611 #10648218

Subgroups — C3×#29611 #10648218

Order 768.

1282 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2C3×#3094C211yesmaximal, characteristic, complemented
3C3×#3006C211yesmaximal, characteristic
4C3×#2436C211yesmaximal, characteristic, complemented
5C3×#2038C214yesmaximal, complemented
6C3×#2003C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1789C211yesmaximal, characteristic, complemented
9C3×#1784C211yesmaximal, characteristic, complemented
10#29611C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#536—22no—
15C3×#536—22no—
16C3×#536—22no—
17C3×#536C2214yes—
18C2×C3×C4⋊Q8—21no—
19C2×C3×#124—21no—
20C2×C3×#124C2212yescomplemented
21C2×C3×#123—22no—
22C2×C3×C22⋊Q8—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×#121C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C3×#482—22no—
27C3×#481—21no—
28C3×#480—21no—
29C3×#449—22no—
30C3×#448—21no—
31C3×#447—21no—
32C3×#428C2212yescomplemented
33C2×C3×#104C2214yes—
34C3×#418C2212yescomplemented
35C3×#417C2212yes—
36C2×C3×#101C2214yescomplemented
37C3×#407C2211yescharacteristic
38C3×#407C2211yescharacteristic
39C3×#393—22no—
40C3×#391C2212yescomplemented
41C3×#386—22no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#339C2214yescomplemented
44C2×#536C2×C311yescharacteristic, complemented
45#3094C2×C311yescharacteristic, complemented
46#3006C2×C311yescharacteristic
47#2436C2×C311yescharacteristic, complemented
48#2038C2×C314yescomplemented
49#2003C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1789C2×C311yescharacteristic, complemented
52#1784C2×C311yescharacteristic, complemented
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41—44no—
56C2×C3×#41D412yes—
57C2×C3×#41C2312yes—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C22×C3×Q8—44no—
63C22×C3×Q8D411yescharacteristic
64C22×C3×Q8D412yes—
65C22×C3×Q8C2312yes—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4D411yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4D412yes—
72C23×C3×C4C2311yescharacteristic
73C3×AGL(1,ℤ/8)—44no—
74C2×C3×SD16—22no—
75C2×C3×D8—22no—
76C2×C3×M16C2312yes—
77C2×C3×M16C2312yes—
78C3×C4⋊Q8—24no—
79C3×C4⋊Q8—24no—
80C3×#124—24no—
81C3×#124—24no—
82C3×#124—44no—
83C3×#123—28no—
84C3×#123—28no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—24no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#117—21no—
94C3×#117—21no—
95C3×#117D412yes—
96C3×#117C2312yes—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—24no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2312yes—
110C3×#107—22no—
111C3×#106—22no—
112C3×#104—44no—
113C3×#103—22no—
114C3×#102—22no—
115C3×#102—22no—
116C3×#102—22no—
117C3×#101—44no—
118C3×C22⋊C8—21no—
119C3×C22⋊C8—21no—
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×#95—21no—
123C3×#95—24no—
124C22×#41C22×C311yescharacteristic
125C23×Q8C22×C311yescharacteristic
126C2×AGL(1,ℤ/8)C22×C312yes—
127#536—22no—
128#536—22no—
129#536—22no—
130#536C22×C314yes—
131C2×C4⋊Q8—21no—
132C2×#124—21no—
133C2×#124C22×C312yescomplemented
134C2×#123—22no—
135C2×C22⋊Q8—21no—
136C2×C22⋊Q8—21no—
137C2×#121C22×C311yescharacteristic
138C2×#117C22×C311yescharacteristic
139#482—22no—
140#481—21no—
141#480—21no—
142#449—22no—
143#448—21no—
144#447—21no—
145#428C22×C312yescomplemented
146C2×#104C22×C314yes—
147#418C22×C312yescomplemented
148#417C22×C312yes—
149C2×#101C22×C314yescomplemented
150#407C22×C311yescharacteristic
151#407C22×C311yescharacteristic
152#393—22no—
153#391C22×C312yescomplemented
154#386—22no—
155C4×C22⋊C4C22×C311yescharacteristic
156#339C22×C314yescomplemented
157C24×C3—21no—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C3×#41—24no—
161C3×#41—24no—
162C3×#41—42no—
163C3×#41—42no—
164C3×#41—44no—
165C3×#41—44no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—24no—
172C2×C3×Q8—24no—
173C2×C3×Q8—44no—
174C2×C3×Q8—42no—
175C2×C3×Q8—44no—
176C2×C3×Q8—44no—
177C2×C3×Q8—44no—
178C2×C3×Q8—44no—
179C2×C3×Q8C2×D412yes—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—24no—
186C2×C3×D4—42no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4C2×D412yes—
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—44no—
203C22×C3×C4—44no—
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C3×SD16—44no—
212C3×D8—44no—
213C3×M16—24no—
214C3×M16—24no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4C2×D412yes—
224C3×C4⋊C4C2×D412yes—
225C3×C22⋊C4—28no—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—48no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—41no—
232C3×C22⋊C4—41no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—24no—
236C2×#41—21no—
237C2×#41—21no—
238C2×#41—44no—
239C2×#41C3×D412yes—
240C2×#41C23×C312yes—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—44no—
246C22×Q8C3×D411yescharacteristic
247C22×Q8C3×D412yes—
248C22×Q8C23×C312yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C3×D411yescharacteristic
252C22×D4C23×C311yescharacteristic
253C23×C4—21no—
254C23×C4C3×D412yes—
255C23×C4C23×C311yescharacteristic
256AGL(1,ℤ/8)—44no—
257C2×SD16—22no—
258C2×D8—22no—
259C2×M16C23×C312yes—
260C2×M16C23×C312yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263#124—24no—
264#124—24no—
265#124—44no—
266#123—28no—
267#123—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272#121—21no—
273#121—21no—
274#121—21no—
275#121—21no—
276#117—21no—
277#117—21no—
278#117C3×D412yes—
279#117C23×C312yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4C3×D411yescharacteristic
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—24no—
291C2×C42—21no—
292C2×C42C23×C312yes—
293#107—22no—
294#106—22no—
295#104—44no—
296#103—22no—
297#102—22no—
298#102—22no—
299#102—22no—
300#101—44no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305#95—21no—
306#95—24no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3#12011yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3C22×D411yescharacteristic
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—44no—
324C3×Q8—44no—
325C3×Q8—42no—
326C3×Q8—44no—
327C3×Q8—44no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—42no—
331C3×D4—44no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—82no—
336C3×D4—82no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—44no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4—42no—
354C2×C3×C4—44no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4H(5,𝔽2)14yes—
365C3×C8—42no—
366C3×C8—42no—
367C24—21no—
368C24—21no—
369C24C2×C3×D411yescharacteristic
370#41—24no—
371#41—24no—
372#41—42no—
373#41—42no—
374#41—44no—
375#41—44no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8C2×C3×D412yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—24no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4C2×C3×D412yes—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4C2×C3×D414yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C24×C311yescharacteristic
421SD16—44no—
422D8—44no—
423M16—24no—
424M16—24no—
425C2×C8—22no—
426C2×C8—22no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4C2×C3×D412yes—
434C4⋊C4C2×C3×D412yes—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—48no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C42—22no—
444C42—22no—
445C42—24no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—44no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—84no—
458C22×C3—84no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3#45412yes—
463C22×C3C2×#12011yescharacteristic
464C22×C3#53412yes—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—82no—
472C3×C4—81no—
473C3×C4#53412yes—
474C3×C4#53512yes—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23C3×#12011yescharacteristic
487C23C3×#12011yescharacteristic
488C23C22×C3×D411yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—44no—
495Q8—44no—
496D4—22no—
497D4—22no—
498D4—42no—
499D4—44no—
500D4—44no—
501D4—42no—
502D4—42no—
503D4—82no—
504D4—82no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4C3×#12011yescharacteristic
529C2×C4C3×#12011yescharacteristic
530C2×C4C22×C3×D411yescharacteristic
531C2×C4C22×C3×D411yescharacteristic
532C2×C4C3×H(5,𝔽2)14yes—
533C8—42no—
534C8—42no—
535C2×C3—22no—
536C2×C3—22no—
537C2×C3—41no—
538C2×C3—82no—
539C2×C3—161no—
540C2×C3#261311yescharacteristic
541C2×C3C2×#45411yescharacteristic
542C2×C3#302011yescharacteristic
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—84no—
555C22—84no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C3×#45412yes—
560C22C2×C3×#12011yescharacteristic
561C22C3×#53412yes—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4C3×#53412yes—
571C4C3×#53512yes—
572C3#2961111yescharacteristic, direct product factor, Sylow
573C2—22no—
574C2—22no—
575C2—41no—
576C2—82no—
577C2—161no—
578C2C3×#261311yescharacteristic
579C2C2×C3×#45411yescharacteristic
580C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3006C2
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×#536C2semidirect
2. C3×#3094C2semidirect
4. C3×#2436C2semidirect
5. C3×#2038C2semidirect
6. C3×#2003C2semidirect
7. C3×#1826C2semidirect
8. C3×#1789C2semidirect
9. C3×#1784C2semidirect
10. #29611C3direct (N × H)
20. C2×C3×#124C22semidirect
32. C3×#428C22semidirect
34. C3×#418C22semidirect
36. C2×C3×#101C22semidirect
40. C3×#391C22semidirect
43. C3×#339C22semidirect
44. C2×#536C2×C3semidirect
45. #3094C2×C3semidirect
47. #2436C2×C3semidirect
48. #2038C2×C3semidirect
49. #2003C2×C3semidirect
50. #1826C2×C3semidirect
51. #1789C2×C3semidirect
52. #1784C2×C3semidirect
133. C2×#124C22×C3semidirect
145. #428C22×C3semidirect
147. #418C22×C3semidirect
149. C2×#101C22×C3semidirect
153. #391C22×C3semidirect
156. #339C22×C3semidirect
572. C3#29611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C3×#30942
C3×#30062
C3×#24362
C3×#20382
C3×#20032
C3×#18262
C3×#17892
C3×#17842
#296113