Finite Group Databasebeta

← C3×#33014 #10651621

Subgroups — C3×#33014 #10651621

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C2×C3×#482C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2624C211yesmaximal, characteristic, complemented
5C3×#1879C212yesmaximal, complemented
6C3×#1879C212yesmaximal, complemented
7C3×#1864C212yesmaximal, complemented
8C3×#1862C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1818C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#33014C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#121C2211yescharacteristic
16C3×#518—24no—
17C3×#518—24no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C22×C3×C22⋊C4—22no—
25C3×#482—24no—
26C3×#447—24no—
27C2×C3×#107C2211yescharacteristic, complemented
28C2×C3×#106C2211yescharacteristic, complemented
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic, complemented
33C2×C3×#102C2211yescharacteristic, complemented
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#399—21no—
37C3×#397—21no—
38C3×#394—22no—
39C3×#390C2212yescomplemented
40C3×#389C2212yescomplemented
41C3×#380C2211yescharacteristic
42C3×#340C2212yescomplemented
43C3×#340C2212yescomplemented
44C3×#336C2212yescomplemented
45C3×#336C2212yescomplemented
46C2×#518C2×C311yescharacteristic, complemented
47C2×#482C2×C311yescharacteristic, complemented
48C2×#447C2×C311yescharacteristic
49#2624C2×C311yescharacteristic, complemented
50#1879C2×C312yescomplemented
51#1879C2×C312yescomplemented
52#1864C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1818C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×D4—24no—
59C22×C3×D4—24no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4D414yes—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2312yes—
66C23×C3×C4—22no—
67C23×C3×C4—22no—
68C23×C3×C4C2311yescharacteristic
69C2×C3×D8—24no—
70C2×C3×D8—24no—
71C2×C3×D8—24no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×#121—22no—
76C3×#121—22no—
77C3×C4×D4—28no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117—22no—
86C3×#117D414yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—216no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4D412yes—
103C2×C3×C42—21no—
104C2×C3×C42C2312yes—
105C3×#107—22no—
106C3×#106—22no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—22no—
111C3×#102—22no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3×#95—22no—
116C23×D4C22×C311yescharacteristic
117C22×D8C22×C312yes—
118C2×#121C22×C311yescharacteristic
119#518—24no—
120#518—24no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C22×C312yes—
124C2×C4×D4C22×C312yes—
125C2×#117C22×C311yescharacteristic
126C2×#117C22×C311yescharacteristic
127C22×C22⋊C4—22no—
128#482—24no—
129#447—24no—
130C2×#107C22×C311yescharacteristic, complemented
131C2×#106C22×C311yescharacteristic, complemented
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic, complemented
136C2×#102C22×C311yescharacteristic, complemented
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#399—21no—
140#397—21no—
141#394—22no—
142#390C22×C312yescomplemented
143#389C22×C312yescomplemented
144#380C22×C311yescharacteristic
145#340C22×C312yescomplemented
146#340C22×C312yescomplemented
147#336C22×C312yescomplemented
148#336C22×C312yescomplemented
149C24×C3—28no—
150C24×C3—22no—
151C24×C3—24no—
152C24×C3—21no—
153C24×C3—81no—
154C24×C3—82no—
155C24×C3—81no—
156C24×C3C2×D411yescharacteristic
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—28no—
160C2×C3×D4—216no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—82no—
171C2×C3×D4C2×D414yes—
172C2×C3×D4#4114yes—
173C2×C3×D4#4114yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—28no—
179C22×C3×C4—28no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—82no—
183C22×C3×C4—82no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2411yescharacteristic
192C3×D8—48no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—82no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—82no—
217C3×C22⋊C4—82no—
218C3×C22⋊C4—81no—
219C3×C42—24no—
220C3×C42—22no—
221C3×C42—22no—
222C25—21no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4C3×D414yes—
229C22×D4C23×C311yescharacteristic
230C22×D4C23×C312yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C23×C311yescharacteristic
234C2×D8—24no—
235C2×D8—24no—
236C2×D8—24no—
237C22×C8C23×C312yes—
238C22×C8C23×C311yescharacteristic
239C22×C8C23×C311yescharacteristic
240#121—22no—
241#121—22no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117C3×D414yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C3×D411yescharacteristic
254C2×C4⋊C4C3×D411yescharacteristic
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C22⋊C4—216no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C3×D412yes—
268C2×C42—21no—
269C2×C42C23×C312yes—
270#107—22no—
271#106—22no—
272#102—24no—
273#102—24no—
274#102—24no—
275#102—22no—
276#102—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279#95—21no—
280#95—22no—
281C23×C3—216no—
282C23×C3—24no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3—84no—
299C23×C3—84no—
300C23×C3—88no—
301C23×C3—88no—
302C23×C3#12012yes—
303C23×C3C22×D411yescharacteristic
304C3×D4—216no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—24no—
308C3×D4—42no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—84no—
312C3×D4#12318yes—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—88no—
334C2×C3×C4—88no—
335C2×C3×C4#12012yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4112yes—
339C2×C3×C4C2×#4112yes—
340C3×C8—42no—
341C3×C8—41no—
342C3×C8—41no—
343C24—28no—
344C24—22no—
345C24—24no—
346C24—21no—
347C24—81no—
348C24—82no—
349C24—81no—
350C24C2×C3×D411yescharacteristic
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—216no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—82no—
365C2×D4C2×C3×D414yes—
366C2×D4C3×#4114yes—
367C2×D4C3×#4114yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D412yes—
385C22×C4C24×C311yescharacteristic
386D8—48no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—82no—
403C4⋊C4C2×C3×D412yes—
404C4⋊C4C2×C3×D412yes—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—81no—
413C42—24no—
414C42—22no—
415C42—22no—
416C22×C3—22no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—816no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#12011yescharacteristic
435C22×C3#52512yes—
436C22×C3#57612yes—
437C22×C3#57611yescharacteristic
438C22×C3#57711yescharacteristic
439C3×C4—22no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—82no—
445C3×C4—81no—
446C3×C4C2×#12312yes—
447C3×C4#52511yescharacteristic
448C3×C4#52511yescharacteristic
449C23—216no—
450C23—21no—
451C23—22no—
452C23—21no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—84no—
467C23—84no—
468C23—88no—
469C23—88no—
470C23C3×#12012yes—
471C23C22×C3×D411yescharacteristic
472D4—216no—
473D4—24no—
474D4—24no—
475D4—24no—
476D4—42no—
477D4—41no—
478D4—41no—
479D4—84no—
480D4C3×#12318yes—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—88no—
503C2×C4C3×#12012yes—
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C2×C3×#4112yes—
507C2×C4C2×C3×#4112yes—
508C8—42no—
509C8—41no—
510C8—41no—
511C2×C3—42no—
512C2×C3—48no—
513C2×C3—161no—
514C2×C3#284911yescharacteristic
515C2×C3#300811yescharacteristic
516C2×C3#301211yescharacteristic
517C2×C3#311812yes—
518C2×C3#312212yes—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—28no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—44no—
530C22—44no—
531C22—816no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22C2×C3×#12011yescharacteristic
538C22C3×#52512yes—
539C22C3×#57612yes—
540C22C3×#57611yescharacteristic
541C22C3×#57711yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4—82no—
549C4C2×C3×#12312yes—
550C4C3×#52511yescharacteristic
551C4C3×#52511yescharacteristic
552C3#3301411yescharacteristic, direct product factor, Sylow
553C2—42no—
554C2—48no—
555C2—161no—
556C2C3×#284911yescharacteristic
557C2C3×#300811yescharacteristic
558C2C3×#301211yescharacteristic
559C2C3×#311812yes—
560C2C3×#312212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C2×C3×#482C2semidirect
4. C3×#2624C2semidirect
5. C3×#1879C2semidirect
6. C3×#1879C2semidirect
7. C3×#1864C2semidirect
8. C3×#1862C2semidirect
9. C3×#1825C2semidirect
10. C3×#1818C2semidirect
11. C3×#1786C2semidirect
12. #33014C3direct (N × H)
27. C2×C3×#107C22semidirect
28. C2×C3×#106C22semidirect
32. C2×C3×#102C22semidirect
33. C2×C3×#102C22semidirect
39. C3×#390C22semidirect
40. C3×#389C22semidirect
42. C3×#340C22semidirect
43. C3×#340C22semidirect
44. C3×#336C22semidirect
45. C3×#336C22semidirect
46. C2×#518C2×C3semidirect
47. C2×#482C2×C3semidirect
49. #2624C2×C3semidirect
50. #1879C2×C3semidirect
51. #1879C2×C3semidirect
52. #1864C2×C3semidirect
53. #1862C2×C3semidirect
54. #1825C2×C3semidirect
55. #1818C2×C3semidirect
56. #1786C2×C3semidirect
130. C2×#107C22×C3semidirect
131. C2×#106C22×C3semidirect
135. C2×#102C22×C3semidirect
136. C2×#102C22×C3semidirect
142. #390C22×C3semidirect
143. #389C22×C3semidirect
145. #340C22×C3semidirect
146. #340C22×C3semidirect
147. #336C22×C3semidirect
148. #336C22×C3semidirect
552. C3#33014direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×C3×#4822
C2×C3×#4472
C3×#26242
C3×#18792
C3×#18642
C3×#18622
C3×#18252
C3×#18182
C3×#17862
#330143