Finite Group Databasebeta

← C3×#25784 #10644391

Subgroups — C3×#25784 #10644391

Order 768.

1618 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic
2C2×C3×#497C212yesmaximal, complemented
3C2×C3×#467C211yesmaximal, characteristic, complemented
4C2×C3×#451C211yesmaximal, characteristic, complemented
5C2×C3×#450C211yesmaximal, characteristic
6C3×#2435C211yesmaximal, characteristic
7C3×#2434C211yesmaximal, characteristic, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#2013C212yesmaximal, complemented
10C3×#1973C212yesmaximal, complemented
11C3×#1752C211yesmaximal, characteristic, complemented
12#25784C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2211yescharacteristic, complemented
16C22×C3×Q16C2212yescomplemented
17C22×C3×SD16—28no—
18C22×C3×SD16—24no—
19C22×C3×SD16C2212yes—
20C22×C3×SD16C2212yes—
21C22×C3×SD16C2212yes—
22C23×C3×C8C2211yescharacteristic
23C2×C3×C4⋊Q8C2212yescomplemented
24C2×C3×#124C2212yes—
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C3×#497—28no—
28C3×#467—24no—
29C3×#451—24no—
30C3×#450—24no—
31C2×C3×#107C2211yescharacteristic, complemented
32C2×C3×#103C2211yescharacteristic, complemented
33C2×C3×#103C2212yescomplemented
34C2×C3×#103C2211yescharacteristic, complemented
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×#97C2212yescomplemented
38C3×#395—22no—
39C3×#394—22no—
40C3×#391C2212yescomplemented
41C3×#390C2212yes—
42C3×#385—22no—
43C3×#379—22no—
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic
47C2×#497C2×C312yescomplemented
48C2×#467C2×C311yescharacteristic, complemented
49C2×#451C2×C311yescharacteristic, complemented
50C2×#450C2×C311yescharacteristic
51#2435C2×C311yescharacteristic
52#2434C2×C311yescharacteristic, complemented
53#2015C2×C312yescomplemented
54#2013C2×C312yescomplemented
55#1973C2×C312yescomplemented
56#1752C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—42no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—24no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—42no—
71C22×C3×D4C2312yes—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—24no—
76C2×C3×Q16—22no—
77C2×C3×Q16—24no—
78C2×C3×Q16—42no—
79C2×C3×SD16—232no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×SD16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—42no—
88C2×C3×SD16D418yes—
89C22×C3×C8—24no—
90C22×C3×C8D414yes—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2312yes—
93C22×C3×C8C2311yescharacteristic
94C3×C4⋊Q8—24no—
95C3×C4⋊Q8—42no—
96C3×#124—24no—
97C3×#124—42no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C2×C3×C4⋊C4—24no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—24no—
110C2×C3×C22⋊C4—24no—
111C2×C3×C22⋊C4D412yes—
112C2×C3×C42C2312yes—
113C3×#107—22no—
114C3×#103—24no—
115C3×#103—22no—
116C3×#103—22no—
117C3×#102—24no—
118C3×C22⋊C8D414yes—
119C3×#97—24no—
120C3×#95—24no—
121C3×#95D412yes—
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×Q16C22×C311yescharacteristic, complemented
125C22×Q16C22×C312yescomplemented
126C22×SD16—28no—
127C22×SD16—24no—
128C22×SD16C22×C312yes—
129C22×SD16C22×C312yes—
130C22×SD16C22×C312yes—
131C23×C8C22×C311yescharacteristic
132C2×C4⋊Q8C22×C312yescomplemented
133C2×#124C22×C312yes—
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×C22⋊Q8C22×C311yescharacteristic
136#497—28no—
137#467—24no—
138#451—24no—
139#450—24no—
140C2×#107C22×C311yescharacteristic, complemented
141C2×#103C22×C312yescomplemented
142C2×#103C22×C311yescharacteristic, complemented
143C2×#103C22×C311yescharacteristic, complemented
144C2×#102C22×C312yes—
145C2×C22⋊C8C22×C311yescharacteristic
146C2×#97C22×C312yescomplemented
147#395—22no—
148#394—22no—
149#391C22×C312yescomplemented
150#390C22×C312yes—
151#385—22no—
152#379—22no—
153C4×C22⋊C4C22×C311yescharacteristic
154#336C22×C312yescomplemented
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—42no—
158C24×C3—44no—
159C24×C3—82no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8C2×D414yes—
180C2×C3×D4—28no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—48no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D414yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—44no—
199C22×C3×C4—42no—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2411yescharacteristic
205C3×Q16—48no—
206C3×Q16—44no—
207C3×SD16—216no—
208C3×SD16—216no—
209C3×SD16—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—28no—
214C2×C3×C8C2×D414yes—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D414yes—
217C2×C3×C8C2×D412yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—48no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—48no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—84no—
230C3×C42—22no—
231C3×C42—22no—
232C25—21no—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C311yescharacteristic
240C22×Q8C23×C312yes—
241C22×D4—24no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—42no—
246C22×D4C23×C312yes—
247C23×C4—21no—
248C23×C4C23×C311yescharacteristic
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—22no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×SD16—232no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×SD16C3×D418yes—
264C22×C8—24no—
265C22×C8C3×D414yes—
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C312yes—
268C22×C8C23×C311yescharacteristic
269C4⋊Q8—24no—
270C4⋊Q8—42no—
271#124—24no—
272#124—42no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C3×D412yes—
287C2×C42C23×C312yes—
288#107—22no—
289#103—24no—
290#103—22no—
291#103—22no—
292#102—24no—
293C22⋊C8C3×D414yes—
294#97—24no—
295#95—24no—
296#95C3×D412yes—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—48no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—44no—
304C23×C3—44no—
305C23×C3—816no—
306C23×C3—84no—
307C23×C3—84no—
308C23×C3#12712yes—
309C23×C3#13714yes—
310C23×C3C22×D411yescharacteristic
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—48no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—41no—
317C3×Q8—41no—
318C3×Q8—41no—
319C3×Q8—41no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—48no—
325C3×D4—42no—
326C3×D4—42no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—48no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4#12712yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4H(5,𝔽2)12yes—
351C3×C8—22no—
352C3×C8—24no—
353C3×C8#12014yes—
354C24—22no—
355C24—44no—
356C24—42no—
357C24—44no—
358C24—82no—
359C24C2×C3×D411yescharacteristic
360C2×Q8—22no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8C2×C3×D414yes—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4C2×C3×D414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4C2×C3×D414yes—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D412yes—
402C22×C4C2×C3×D412yes—
403C22×C4C24×C311yescharacteristic
404Q16—48no—
405Q16—44no—
406SD16—216no—
407SD16—216no—
408SD16—48no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—28no—
413C2×C8C2×C3×D414yes—
414C2×C8C2×C3×D412yes—
415C2×C8C2×C3×D414yes—
416C2×C8C2×C3×D412yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C22⋊C4—24no—
425C22⋊C4—48no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—84no—
429C42—22no—
430C42—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—28no—
434C22×C3—44no—
435C22×C3—48no—
436C22×C3—816no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—84no—
440C22×C3#49714yescomplemented
441C22×C3C2×#12711yescharacteristic
442C22×C3D4212yes—
443C22×C3C2×#13712yes—
444C22×C3#57712yes—
445C3×C4—22no—
446C3×C4—44no—
447C3×C4—44no—
448C3×C4—81no—
449C3×C4—81no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4C2×#12011yescharacteristic
453C3×C4#53511yescharacteristic
454C3×C4D4212yes—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—48no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—816no—
464C23—84no—
465C23—84no—
466C23C3×#12712yes—
467C23C3×#13714yes—
468C23C22×C3×D411yescharacteristic
469Q8—24no—
470Q8—24no—
471Q8—48no—
472Q8—42no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—42no—
479Q8—42no—
480D4—24no—
481D4—24no—
482D4—48no—
483D4—42no—
484D4—42no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4C3×#12712yes—
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C22×C3×D412yes—
507C2×C4C22×C3×D411yescharacteristic
508C2×C4C3×H(5,𝔽2)12yes—
509C8—22no—
510C8—24no—
511C8C3×#12014yes—
512C2×C3—24no—
513C2×C3—84no—
514C2×C3#243311yescharacteristic
515C2×C3C2×#49711yescharacteristic
516C2×C3#315411yescharacteristic
517C2×C3#326614yes—
518C22—22no—
519C22—22no—
520C22—28no—
521C22—44no—
522C22—48no—
523C22—816no—
524C22—82no—
525C22—82no—
526C22—84no—
527C22C3×#49714yescomplemented
528C22C2×C3×#12711yescharacteristic
529C22C3×D4212yes—
530C22C2×C3×#13712yes—
531C22C3×#57712yes—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4C2×C3×#12011yescharacteristic
540C4C3×#53511yescharacteristic
541C4C3×D4212yes—
542C3#2578411yescharacteristic, direct product factor, Sylow
543C2—24no—
544C2—84no—
545C2C3×#243311yescharacteristic
546C2C2×C3×#49711yescharacteristic
547C2C3×#315411yescharacteristic
548C2C3×#326614yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#497C2semidirect
3. C2×C3×#467C2semidirect
4. C2×C3×#451C2semidirect
7. C3×#2434C2semidirect
8. C3×#2015C2semidirect
9. C3×#2013C2semidirect
10. C3×#1973C2semidirect
11. C3×#1752C2semidirect
12. #25784C3direct (N × H)
15. C22×C3×Q16C22semidirect
16. C22×C3×Q16C22semidirect
23. C2×C3×C4⋊Q8C22semidirect
31. C2×C3×#107C22semidirect
32. C2×C3×#103C22semidirect
33. C2×C3×#103C22semidirect
34. C2×C3×#103C22semidirect
37. C2×C3×#97C22semidirect
40. C3×#391C22semidirect
45. C3×#336C22semidirect
47. C2×#497C2×C3semidirect
48. C2×#467C2×C3semidirect
49. C2×#451C2×C3semidirect
52. #2434C2×C3semidirect
53. #2015C2×C3semidirect
54. #2013C2×C3semidirect
55. #1973C2×C3semidirect
56. #1752C2×C3semidirect
124. C22×Q16C22×C3semidirect
125. C22×Q16C22×C3semidirect
132. C2×C4⋊Q8C22×C3semidirect
140. C2×#107C22×C3semidirect
141. C2×#103C22×C3semidirect
142. C2×#103C22×C3semidirect
143. C2×#103C22×C3semidirect
146. C2×#97C22×C3semidirect
149. #391C22×C3semidirect
154. #336C22×C3semidirect
440. C22×C3#497semidirect
527. C22C3×#497semidirect
542. C3#25784direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×C3×#4972
C2×C3×#4672
C2×C3×#4512
C2×C3×#4502
C3×#24352
C3×#24342
C3×#20152
C3×#20132
C3×#19732
C3×#17522
#257843