Finite Group Databasebeta

← C3×#25774 #10644381

Subgroups — C3×#25774 #10644381

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#492C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C2×C3×#448C211yesmaximal, characteristic, complemented
5C2×C3×#450C211yesmaximal, characteristic, complemented
6C3×#2434C211yesmaximal, characteristic, complemented
7C3×#2433C211yesmaximal, characteristic
8C3×#2014C212yesmaximal, complemented
9C3×#2013C212yesmaximal, complemented
10C3×#1968C212yesmaximal, complemented
11C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
12#25774C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16—28no—
16C22×C3×SD16—24no—
17C22×C3×SD16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2212yes—
20C22×C3×SD16C2211yescharacteristic, complemented
21C22×C3×SD16C2212yescomplemented
22C23×C3×C8C2211yescharacteristic
23C2×C3×C4⋊Q8C2212yescomplemented
24C2×C3×#127C2212yes—
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C2×C3×#120—22no—
28C3×#492—28no—
29C3×#465—24no—
30C3×#450—24no—
31C3×#448—24no—
32C2×C3×#106C2211yescharacteristic, complemented
33C2×C3×#103C2211yescharacteristic, complemented
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yes—
36C2×C3×#102C2211yescharacteristic, complemented
37C2×C3×C22⋊C8C2211yescharacteristic
38C2×C3×C4×C8C2212yescomplemented
39C3×#395—22no—
40C3×#392—22no—
41C3×#391C2212yescomplemented
42C3×#390C2212yes—
43C3×#385—22no—
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic, complemented
47C2×#492C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic, complemented
49C2×#448C2×C311yescharacteristic, complemented
50C2×#450C2×C311yescharacteristic, complemented
51#2434C2×C311yescharacteristic, complemented
52#2433C2×C311yescharacteristic
53#2014C2×C312yescomplemented
54#2013C2×C312yescomplemented
55#1968C2×C312yescomplemented
56C8×C22⋊C4C2×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×Q8C2312yes—
65C22×C3×D4—24no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—24no—
71C22×C3×D4—42no—
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2312yes—
74C23×C3×C4—21no—
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—232no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
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—22no—
87C2×C3×SD16—24no—
88C2×C3×SD16—42no—
89C2×C3×SD16—42no—
90C2×C3×SD16D418yescomplemented
91C22×C3×C8—24no—
92C22×C3×C8D414yescomplemented
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2312yes—
95C22×C3×C8C2311yescharacteristic
96C3×C4⋊Q8—24no—
97C3×C4⋊Q8—42no—
98C3×#127—24no—
99C3×#127—42no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×#121—22no—
103C3×#121—22no—
104C3×#120—48no—
105C2×C3×C4⋊C4—24no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—24no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4D412yes—
113C2×C3×C42C2312yes—
114C3×#106—22no—
115C3×#103—24no—
116C3×#103—22no—
117C3×#102—24no—
118C3×#102—22no—
119C3×C22⋊C8D414yescomplemented
120C3×C4×C8—24no—
121C3×#95—24no—
122C3×#95D412yes—
123C23×Q8C22×C311yescharacteristic
124C23×D4C22×C311yescharacteristic
125C22×SD16—28no—
126C22×SD16—24no—
127C22×SD16C22×C312yes—
128C22×SD16C22×C312yes—
129C22×SD16C22×C312yes—
130C22×SD16C22×C311yescharacteristic, complemented
131C22×SD16C22×C312yescomplemented
132C23×C8C22×C311yescharacteristic
133C2×C4⋊Q8C22×C312yescomplemented
134C2×#127C22×C312yes—
135C2×C22⋊Q8C22×C311yescharacteristic
136C2×#121C22×C311yescharacteristic
137C2×#120—22no—
138#492—28no—
139#465—24no—
140#450—24no—
141#448—24no—
142C2×#106C22×C311yescharacteristic, complemented
143C2×#103C22×C312yescomplemented
144C2×#103C22×C311yescharacteristic, complemented
145C2×#102C22×C312yes—
146C2×#102C22×C311yescharacteristic, complemented
147C2×C22⋊C8C22×C311yescharacteristic
148C2×C4×C8C22×C312yescomplemented
149#395—22no—
150#392—22no—
151#391C22×C312yescomplemented
152#390C22×C312yes—
153#385—22no—
154C4×C22⋊C4C22×C311yescharacteristic
155#336C22×C312yescomplemented
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—44no—
159C24×C3—42no—
160C24×C3—44no—
161C24×C3—82no—
162C24×C3C2×D411yescharacteristic
163C2×C3×Q8—22no—
164C2×C3×Q8—28no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8—48no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8C2×D414yes—
178C2×C3×D4—28no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—48no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—48no—
192C2×C3×D4—42no—
193C2×C3×D4—44no—
194C2×C3×D4—42no—
195C2×C3×D4—48no—
196C2×C3×D4—84no—
197C2×C3×D4C2×D414yes—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—44no—
205C22×C3×C4—42no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D414yes—
210C22×C3×C4C2411yescharacteristic
211C3×SD16—216no—
212C3×SD16—216no—
213C3×SD16—48no—
214C3×SD16—48no—
215C3×SD16—44no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—28no—
220C2×C3×C8C2×D414yes—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D414yes—
223C2×C3×C8C2×D412yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—48no—
229C3×C4⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—48no—
233C3×C22⋊C4SD1618yescomplemented
234C3×C42—22no—
235C3×C42—22no—
236C25—21no—
237C22×Q8—21no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8C23×C311yescharacteristic
243C22×Q8C23×C312yes—
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—24no—
250C22×D4—42no—
251C22×D4C23×C311yescharacteristic
252C22×D4C23×C312yes—
253C23×C4—21no—
254C23×C4C23×C311yescharacteristic
255C2×SD16—232no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—22no—
266C2×SD16—24no—
267C2×SD16—42no—
268C2×SD16—42no—
269C2×SD16C3×D418yescomplemented
270C22×C8—24no—
271C22×C8C3×D414yescomplemented
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C312yes—
274C22×C8C23×C311yescharacteristic
275C4⋊Q8—24no—
276C4⋊Q8—42no—
277#127—24no—
278#127—42no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—22no—
282#121—22no—
283#120—48no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4C3×D412yes—
292C2×C42C23×C312yes—
293#106—22no—
294#103—24no—
295#103—22no—
296#102—24no—
297#102—22no—
298C22⋊C8C3×D414yescomplemented
299C4×C8—24no—
300#95—24no—
301#95C3×D412yes—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—48no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—816no—
315C23×C3—84no—
316C23×C3—84no—
317C23×C3#12712yes—
318C23×C3C2×SD1614yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—24no—
321C3×Q8—24no—
322C3×Q8—48no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—42no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—48no—
331C3×D4—41no—
332C3×D4—42no—
333C3×D4—41no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—816no—
337C3×D4—84no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—48no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4#12712yes—
355C2×C3×C4C2×SD1618yescomplemented
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4H(5,𝔽2)12yes—
360C3×C8—22no—
361C3×C8—24no—
362C3×C8#12014yescomplemented
363C24—22no—
364C24—21no—
365C24—44no—
366C24—42no—
367C24—44no—
368C24—82no—
369C24C2×C3×D411yescharacteristic
370C2×Q8—22no—
371C2×Q8—28no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8C2×C3×D414yes—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—48no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—48no—
403C2×D4—84no—
404C2×D4C2×C3×D414yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D412yes—
416C22×C4C2×C3×D414yes—
417C22×C4C24×C311yescharacteristic
418SD16—216no—
419SD16—216no—
420SD16—48no—
421SD16—48no—
422SD16—44no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—28no—
427C2×C8C2×C3×D414yes—
428C2×C8C2×C3×D412yes—
429C2×C8C2×C3×D414yes—
430C2×C8C2×C3×D412yes—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—48no—
436C4⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—48no—
440C22⋊C4C3×SD1618yescomplemented
441C42—22no—
442C42—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—28no—
446C22×C3—44no—
447C22×C3—48no—
448C22×C3—816no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—84no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—84no—
456C22×C3#49214yescomplemented
457C22×C3C2×#12711yescharacteristic
458C22×C3D4212yes—
459C22×C3C22×SD1612yes—
460C22×C3#57712yes—
461C3×C4—22no—
462C3×C4—28no—
463C3×C4—44no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4C2×#12011yescharacteristic
468C3×C4#53511yescharacteristic
469C3×C4D4212yes—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—48no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—816no—
483C23—84no—
484C23—84no—
485C23C3×#12712yes—
486C23C2×C3×SD1614yes—
487C23C22×C3×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—48no—
491Q8—42no—
492Q8—42no—
493Q8—41no—
494Q8—41no—
495Q8—42no—
496D4—24no—
497D4—24no—
498D4—48no—
499D4—41no—
500D4—42no—
501D4—41no—
502D4—42no—
503D4—42no—
504D4—816no—
505D4—84no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4C3×#12712yes—
523C2×C4C2×C3×SD1618yescomplemented
524C2×C4C22×C3×D411yescharacteristic
525C2×C4C22×C3×D412yes—
526C2×C4C22×C3×D411yescharacteristic
527C2×C4C3×H(5,𝔽2)12yes—
528C8—22no—
529C8—24no—
530C8C3×#12014yescomplemented
531C2×C3—24no—
532C2×C3—84no—
533C2×C3—161no—
534C2×C3#243311yescharacteristic
535C2×C3C2×#49211yescharacteristic
536C2×C3#315411yescharacteristic
537C2×C3D4×SD1614yes—
538C22—22no—
539C22—22no—
540C22—28no—
541C22—44no—
542C22—48no—
543C22—816no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—84no—
551C22C3×#49214yescomplemented
552C22C2×C3×#12711yescharacteristic
553C22C3×D4212yes—
554C22C22×C3×SD1612yes—
555C22C3×#57712yes—
556C4—22no—
557C4—28no—
558C4—44no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4C2×C3×#12011yescharacteristic
563C4C3×#53511yescharacteristic
564C4C3×D4212yes—
565C3#2577411yescharacteristic, direct product factor, Sylow
566C2—24no—
567C2—84no—
568C2—161no—
569C2C3×#243311yescharacteristic
570C2C2×C3×#49211yescharacteristic
571C2C3×#315411yescharacteristic
572C2C3×D4×SD1614yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2433C2
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
1. C23×C3×SD16C2semidirect
2. C2×C3×#492C2semidirect
3. C2×C3×#465C2semidirect
4. C2×C3×#448C2semidirect
5. C2×C3×#450C2semidirect
6. C3×#2434C2semidirect
8. C3×#2014C2semidirect
9. C3×#2013C2semidirect
10. C3×#1968C2semidirect
11. C3×C8×C22⋊C4C2semidirect
12. #25774C3direct (N × H)
20. C22×C3×SD16C22semidirect
21. C22×C3×SD16C22semidirect
23. C2×C3×C4⋊Q8C22semidirect
32. C2×C3×#106C22semidirect
33. C2×C3×#103C22semidirect
34. C2×C3×#103C22semidirect
36. C2×C3×#102C22semidirect
38. C2×C3×C4×C8C22semidirect
41. C3×#391C22semidirect
45. C3×#336C22semidirect
46. C23×SD16C2×C3semidirect
47. C2×#492C2×C3semidirect
48. C2×#465C2×C3semidirect
49. C2×#448C2×C3semidirect
50. C2×#450C2×C3semidirect
51. #2434C2×C3semidirect
53. #2014C2×C3semidirect
54. #2013C2×C3semidirect
55. #1968C2×C3semidirect
56. C8×C22⋊C4C2×C3semidirect
90. C2×C3×SD16D4semidirect
92. C22×C3×C8D4semidirect
119. C3×C22⋊C8D4semidirect
130. C22×SD16C22×C3semidirect
131. C22×SD16C22×C3semidirect
133. C2×C4⋊Q8C22×C3semidirect
142. C2×#106C22×C3semidirect
143. C2×#103C22×C3semidirect
144. C2×#103C22×C3semidirect
146. C2×#102C22×C3semidirect
148. C2×C4×C8C22×C3semidirect
151. #391C22×C3semidirect
155. #336C22×C3semidirect
233. C3×C22⋊C4SD16semidirect
269. C2×SD16C3×D4semidirect
271. C22×C8C3×D4semidirect
298. C22⋊C8C3×D4semidirect
355. C2×C3×C4C2×SD16semidirect
362. C3×C8#120semidirect
440. C22⋊C4C3×SD16semidirect
456. C22×C3#492semidirect
523. C2×C4C2×C3×SD16semidirect
530. C8C3×#120semidirect
551. C22C3×#492semidirect
565. C3#25774direct (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×#4922
C2×C3×#4652
C2×C3×#4482
C2×C3×#4502
C3×#24342
C3×#24332
C3×#20142
C3×#20132
C3×#19682
C3×C8×C22⋊C42
#257743