Finite Group Databasebeta

← C3×#25788 #10644395

Subgroups — C3×#25788 #10644395

Order 768.

1892 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D8C211yesmaximal, characteristic, complemented
2C2×C3×#496C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C2×C3×#450C211yesmaximal, characteristic, complemented
5C2×C3×#447C211yesmaximal, characteristic, complemented
6C3×#2435C211yesmaximal, characteristic, complemented
7C3×#2433C211yesmaximal, characteristic, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#2011C212yesmaximal, complemented
10C3×#1972C212yesmaximal, complemented
11C3×#1752C211yesmaximal, characteristic, complemented
12#25788C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic, complemented
16C22×C3×SD16C2212yescomplemented
17C22×C3×D8—28no—
18C22×C3×D8—24no—
19C22×C3×D8C2212yes—
20C22×C3×D8C2212yes—
21C22×C3×D8C2212yescomplemented
22C23×C3×C8C2211yescharacteristic
23C2×C3×#127C2212yescomplemented
24C2×C3×#124C2212yescomplemented
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C2×C3×#120—22no—
28C3×#496—28no—
29C3×#465—24no—
30C3×#450—24no—
31C3×#447—24no—
32C2×C3×#106C2211yescharacteristic, complemented
33C2×C3×#103C2211yescharacteristic, complemented
34C2×C3×#102C2211yescharacteristic, complemented
35C2×C3×#102C2212yescomplemented
36C2×C3×#102C2212yescomplemented
37C2×C3×C22⋊C8C2211yescharacteristic
38C2×C3×#97C2212yescomplemented
39C3×#394—22no—
40C3×#392—22no—
41C3×#390C2212yescomplemented
42C3×#390C2212yescomplemented
43C3×#379—22no—
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×#336C2212yescomplemented
46C23×D8C2×C311yescharacteristic, complemented
47C2×#496C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic, complemented
49C2×#450C2×C311yescharacteristic, complemented
50C2×#447C2×C311yescharacteristic, complemented
51#2435C2×C311yescharacteristic, complemented
52#2433C2×C311yescharacteristic, complemented
53#2012C2×C312yescomplemented
54#2011C2×C312yescomplemented
55#1972C2×C312yescomplemented
56#1752C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C25×C3—21no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—24no—
61C22×C3×D4—24no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4—24no—
70C22×C3×D4—42no—
71C22×C3×D4—42no—
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2312yes—
74C22×C3×D4C2312yes—
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—22no—
79C2×C3×SD16—24no—
80C2×C3×SD16—42no—
81C2×C3×D8—232no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8—24no—
87C2×C3×D8—24no—
88C2×C3×D8—24no—
89C2×C3×D8—42no—
90C2×C3×D8D418yescomplemented
91C22×C3×C8—24no—
92C22×C3×C8D414yescomplemented
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2312yes—
95C22×C3×C8C2311yescharacteristic
96C3×#127—24no—
97C3×#127—42no—
98C3×#124—24no—
99C3×#124—42no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×#121—22no—
103C3×#121—22no—
104C3×#120—48no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—24no—
110C2×C3×C22⋊C4—24no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4D412yes—
114C2×C3×C42C2312yes—
115C3×#106—22no—
116C3×#103—22no—
117C3×#102—24no—
118C3×#102—24no—
119C3×#102—22no—
120C3×C22⋊C8D414yescomplemented
121C3×#97—24no—
122C3×#95D412yes—
123C23×D4C22×C311yescharacteristic
124C23×D4C22×C311yescharacteristic
125C22×SD16C22×C311yescharacteristic, complemented
126C22×SD16C22×C312yescomplemented
127C22×D8—28no—
128C22×D8—24no—
129C22×D8C22×C312yes—
130C22×D8C22×C312yes—
131C22×D8C22×C312yescomplemented
132C23×C8C22×C311yescharacteristic
133C2×#127C22×C312yescomplemented
134C2×#124C22×C312yescomplemented
135C2×C22⋊Q8C22×C311yescharacteristic
136C2×#121C22×C311yescharacteristic
137C2×#120—22no—
138#496—28no—
139#465—24no—
140#450—24no—
141#447—24no—
142C2×#106C22×C311yescharacteristic, complemented
143C2×#103C22×C311yescharacteristic, complemented
144C2×#102C22×C312yescomplemented
145C2×#102C22×C312yescomplemented
146C2×#102C22×C311yescharacteristic, complemented
147C2×C22⋊C8C22×C311yescharacteristic
148C2×#97C22×C312yescomplemented
149#394—22no—
150#392—22no—
151#390C22×C312yescomplemented
152#390C22×C312yescomplemented
153#379—22no—
154C4×C22⋊C4C22×C311yescharacteristic
155#336C22×C312yescomplemented
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—44no—
160C24×C3—44no—
161C24×C3—42no—
162C24×C3—42no—
163C24×C3—44no—
164C24×C3—44no—
165C24×C3—82no—
166C24×C3—82no—
167C24×C3C2×D411yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×D4—28no—
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—48no—
189C2×C3×D4—48no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—48no—
195C2×C3×D4—48no—
196C2×C3×D4—42no—
197C2×C3×D4—44no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—84no—
201C2×C3×D4C2×D414yes—
202C2×C3×D4C2×D414yes—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D414yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2411yescharacteristic
213C3×SD16—48no—
214C3×SD16—44no—
215C3×D8—216no—
216C3×D8—216no—
217C3×D8—48no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—28no—
222C2×C3×C8C2×D414yes—
223C2×C3×C8C2×D412yes—
224C2×C3×C8C2×D414yes—
225C2×C3×C8C2×D412yes—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—48no—
233C3×C22⋊C4—48no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—84no—
237C3×C42—22no—
238C3×C42—22no—
239C25—21no—
240C25—21no—
241C22×Q8C23×C311yescharacteristic
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—24no—
252C22×D4—42no—
253C22×D4—42no—
254C22×D4C23×C311yescharacteristic
255C22×D4C23×C312yes—
256C22×D4C23×C312yes—
257C23×C4C23×C311yescharacteristic
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×D8—232no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—42no—
272C2×D8C3×D418yescomplemented
273C22×C8—24no—
274C22×C8C3×D414yescomplemented
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C312yes—
277C22×C8C23×C311yescharacteristic
278#127—24no—
279#127—42no—
280#124—24no—
281#124—42no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284#121—22no—
285#121—22no—
286#120—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4C3×D412yes—
296C2×C42C23×C312yes—
297#106—22no—
298#103—22no—
299#102—24no—
300#102—24no—
301#102—22no—
302C22⋊C8C3×D414yescomplemented
303#97—24no—
304#95C3×D412yes—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—48no—
310C23×C3—48no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—816no—
324C23×C3—816no—
325C23×C3—84no—
326C23×C3—84no—
327C23×C3—84no—
328C23×C3—84no—
329C23×C3#12712yes—
330C23×C3AGL(1,ℤ/8)14yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—42no—
335C3×D4—24no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—48no—
341C3×D4—41no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—41no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—816no—
349C3×D4—84no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4#12712yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D412yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4H(5,𝔽2)12yes—
368C3×C8—22no—
369C3×C8—24no—
370C3×C8#12014yescomplemented
371C24—22no—
372C24—22no—
373C24—21no—
374C24—44no—
375C24—44no—
376C24—42no—
377C24—42no—
378C24—44no—
379C24—44no—
380C24—82no—
381C24—82no—
382C24C2×C3×D411yescharacteristic
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—84no—
416C2×D4C2×C3×D414yes—
417C2×D4C2×C3×D414yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4C2×C3×D414yes—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D412yes—
426C22×C4C2×C3×D412yes—
427C22×C4C24×C311yescharacteristic
428SD16—48no—
429SD16—44no—
430D8—216no—
431D8—216no—
432D8—48no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—28no—
437C2×C8C2×C3×D414yes—
438C2×C8C2×C3×D412yes—
439C2×C8C2×C3×D414yes—
440C2×C8C2×C3×D412yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C22⋊C4—24no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—84no—
452C42—22no—
453C42—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—28no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—48no—
460C22×C3—48no—
461C22×C3—816no—
462C22×C3—816no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—84no—
468C22×C3—84no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—84no—
473C22×C3#49614yescomplemented
474C22×C3C2×#12711yescharacteristic
475C22×C3D4212yes—
476C22×C3C2×AGL(1,ℤ/8)12yes—
477C22×C3#57612yes—
478C3×C4—22no—
479C3×C4—44no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×#12011yescharacteristic
484C3×C4#53511yescharacteristic
485C3×C4D4212yes—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—816no—
505C23—816no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23C3×#12712yes—
511C23C3×AGL(1,ℤ/8)14yes—
512C23C22×C3×D411yescharacteristic
513Q8—41no—
514Q8—41no—
515Q8—42no—
516D4—24no—
517D4—24no—
518D4—24no—
519D4—24no—
520D4—48no—
521D4—48no—
522D4—41no—
523D4—42no—
524D4—42no—
525D4—41no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—816no—
530D4—84no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4C3×#12712yes—
545C2×C4C22×C3×D411yescharacteristic
546C2×C4C22×C3×D412yes—
547C2×C4C22×C3×D411yescharacteristic
548C2×C4C3×H(5,𝔽2)12yes—
549C8—22no—
550C8—24no—
551C8C3×#12014yescomplemented
552C2×C3—24no—
553C2×C3—84no—
554C2×C3—84no—
555C2×C3—161no—
556C2×C3#243311yescharacteristic
557C2×C3C2×#49611yescharacteristic
558C2×C3#315511yescharacteristic
559C2×C3#326414yes—
560C22—22no—
561C22—22no—
562C22—28no—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—48no—
567C22—816no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—84no—
574C22—84no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—84no—
579C22C3×#49614yescomplemented
580C22C2×C3×#12711yescharacteristic
581C22C3×D4212yes—
582C22C2×C3×AGL(1,ℤ/8)12yes—
583C22C3×#57612yes—
584C4—22no—
585C4—44no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4C2×C3×#12011yescharacteristic
590C4C3×#53511yescharacteristic
591C4C3×D4212yes—
592C3#2578811yescharacteristic, direct product factor, Sylow
593C2—24no—
594C2—84no—
595C2—84no—
596C2—161no—
597C2C3×#243311yescharacteristic
598C2C2×C3×#49611yescharacteristic
599C2C3×#315511yescharacteristic
600C2C3×#326414yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×D8C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#465C2semidirect
4. C2×C3×#450C2semidirect
5. C2×C3×#447C2semidirect
6. C3×#2435C2semidirect
7. C3×#2433C2semidirect
8. C3×#2012C2semidirect
9. C3×#2011C2semidirect
10. C3×#1972C2semidirect
11. C3×#1752C2semidirect
12. #25788C3direct (N × H)
15. C22×C3×SD16C22semidirect
16. C22×C3×SD16C22semidirect
21. C22×C3×D8C22semidirect
23. C2×C3×#127C22semidirect
24. C2×C3×#124C22semidirect
32. C2×C3×#106C22semidirect
33. C2×C3×#103C22semidirect
34. C2×C3×#102C22semidirect
35. C2×C3×#102C22semidirect
36. C2×C3×#102C22semidirect
38. C2×C3×#97C22semidirect
41. C3×#390C22semidirect
42. C3×#390C22semidirect
45. C3×#336C22semidirect
46. C23×D8C2×C3semidirect
47. C2×#496C2×C3semidirect
48. C2×#465C2×C3semidirect
49. C2×#450C2×C3semidirect
50. C2×#447C2×C3semidirect
51. #2435C2×C3semidirect
52. #2433C2×C3semidirect
53. #2012C2×C3semidirect
54. #2011C2×C3semidirect
55. #1972C2×C3semidirect
56. #1752C2×C3semidirect
90. C2×C3×D8D4semidirect
92. C22×C3×C8D4semidirect
120. C3×C22⋊C8D4semidirect
125. C22×SD16C22×C3semidirect
126. C22×SD16C22×C3semidirect
131. C22×D8C22×C3semidirect
133. C2×#127C22×C3semidirect
134. C2×#124C22×C3semidirect
142. C2×#106C22×C3semidirect
143. C2×#103C22×C3semidirect
144. C2×#102C22×C3semidirect
145. C2×#102C22×C3semidirect
146. C2×#102C22×C3semidirect
148. C2×#97C22×C3semidirect
151. #390C22×C3semidirect
152. #390C22×C3semidirect
155. #336C22×C3semidirect
272. C2×D8C3×D4semidirect
274. C22×C8C3×D4semidirect
302. C22⋊C8C3×D4semidirect
370. C3×C8#120semidirect
473. C22×C3#496semidirect
551. C8C3×#120semidirect
579. C22C3×#496semidirect
592. C3#25788direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D82
C2×C3×#4962
C2×C3×#4652
C2×C3×#4502
C2×C3×#4472
C3×#24352
C3×#24332
C3×#20122
C3×#20112
C3×#19722
C3×#17522
#257883