Finite Group Databasebeta

← C3×#24358 #10642965

Subgroups — C3×#24358 #10642965

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic
2C2×C3×#461C212yesmaximal, complemented
3C2×C3×#450C211yesmaximal, characteristic
4C3×#2434C211yesmaximal, characteristic, complemented
5C3×D4×C22⋊C4C211yesmaximal, characteristic
6C3×#2055C212yesmaximal, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#1892C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#24358C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16—24no—
16C22×C3×SD16—28no—
17C22×C3×SD16C2212yes—
18C22×C3×SD16C2211yescharacteristic
19C22×C3×SD16C2212yes—
20C22×C3×SD16C2211yescharacteristic
21C23×C3×C8C2211yescharacteristic
22C2×C3×C4⋊Q8C2212yescomplemented
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C22×C3×C22⋊C4—21no—
26C3×#461—28no—
27C3×#450—24no—
28C2×C3×C4⋊C8C2212yescomplemented
29C2×C3×#103C2212yescomplemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#103C2212yescomplemented
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C3×#395—22no—
36C3×#391C2212yescomplemented
37C3×#390C2212yes—
38C3×#386—22no—
39C3×#385—21no—
40C3×#385—21no—
41C3×#380C2211yescharacteristic
42C3×#379—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic
47C2×#461C2×C312yescomplemented
48C2×#450C2×C311yescharacteristic
49#2434C2×C311yescharacteristic, complemented
50D4×C22⋊C4C2×C311yescharacteristic
51#2055C2×C312yescomplemented
52#2013C2×C312yescomplemented
53#1892C2×C311yescharacteristic, complemented
54#1877C2×C312yescomplemented
55#1850C2×C311yescharacteristic, complemented
56#1787C2×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—42no—
69C22×C3×D4D414yes—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—22no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×SD16—232no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×SD16—42no—
85C2×C3×SD16D418yes—
86C22×C3×C8—22no—
87C22×C3×C8—24no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2312yes—
91C3×C4⋊Q8—24no—
92C3×C4⋊Q8—42no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C4×D4—24no—
96C3×C4×D4—42no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—44no—
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42C2312yes—
111C3×C4⋊C8—24no—
112C3×#103—24no—
113C3×#103—24no—
114C3×#103—42no—
115C3×#102—22no—
116C3×#102—22no—
117C3×C22⋊C8—22no—
118C3×#95—21no—
119C3×#95—22no—
120C3×#95—22no—
121C3×#95D412yes—
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×SD16—24no—
125C22×SD16—28no—
126C22×SD16C22×C312yes—
127C22×SD16C22×C311yescharacteristic
128C22×SD16C22×C312yes—
129C22×SD16C22×C311yescharacteristic
130C23×C8C22×C311yescharacteristic
131C2×C4⋊Q8C22×C312yescomplemented
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C4×D4C22×C312yes—
134C22×C22⋊C4—21no—
135#461—28no—
136#450—24no—
137C2×C4⋊C8C22×C312yescomplemented
138C2×#103C22×C312yescomplemented
139C2×#103C22×C312yescomplemented
140C2×#103C22×C312yescomplemented
141C2×#102C22×C311yescharacteristic
142C2×#102C22×C311yescharacteristic
143C2×C22⋊C8C22×C311yescharacteristic
144#395—22no—
145#391C22×C312yescomplemented
146#390C22×C312yes—
147#386—22no—
148#385—21no—
149#385—21no—
150#380C22×C311yescharacteristic
151#379—21no—
152C4×C22⋊C4C22×C311yescharacteristic
153#340C22×C312yescomplemented
154#336C22×C312yescomplemented
155C24×C3—21no—
156C24×C3—21no—
157C24×C3—24no—
158C24×C3—44no—
159C24×C3—42no—
160C24×C3—42no—
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—22no—
170C2×C3×Q8—21no—
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—24no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—44no—
186C2×C3×D4—48no—
187C2×C3×D4—44no—
188C2×C3×D4C2×D412yes—
189C2×C3×D4C2×D412yes—
190C2×C3×D4C2×D414yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—44no—
198C22×C3×C4—44no—
199C22×C3×C4—42no—
200C22×C3×C4—44no—
201C22×C3×C4—82no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4#4112yes—
207C22×C3×C4C2411yescharacteristic
208C3×SD16—216no—
209C3×SD16—216no—
210C3×SD16—44no—
211C3×SD16—44no—
212C2×C3×C8—28no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8C2×D414yes—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—44no—
225C3×C4⋊C4—44no—
226C3×C4⋊C4—42no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—44no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—44no—
232C3×C22⋊C4—82no—
233C3×C22⋊C4SD1614yescomplemented
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—42no—
248C22×D4C3×D414yes—
249C22×D4C23×C311yescharacteristic
250C22×D4C23×C311yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C23×C311yescharacteristic
254C2×SD16—232no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—22no—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×SD16C3×D418yes—
265C22×C8—22no—
266C22×C8—24no—
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C312yes—
270C4⋊Q8—24no—
271C4⋊Q8—42no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C4×D4—24no—
275C4×D4—42no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C23×C312yes—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C3×D411yescharacteristic
288C2×C22⋊C4C3×D411yescharacteristic
289C2×C42C23×C312yes—
290C4⋊C8—24no—
291#103—24no—
292#103—24no—
293#103—42no—
294#102—22no—
295#102—22no—
296C22⋊C8—22no—
297#95—21no—
298#95—22no—
299#95—22no—
300#95C3×D412yes—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—48no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—48no—
315C23×C3—84no—
316C23×C3—88no—
317C23×C3—88no—
318C23×C3#12112yes—
319C23×C3C2×SD1612yes—
320C23×C3#13712yes—
321C23×C3C22×D411yescharacteristic
322C3×Q8—24no—
323C3×Q8—24no—
324C3×Q8—48no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—42no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—42no—
336C3×D4#12014yes—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—48no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—88no—
358C2×C3×C4#12112yes—
359C2×C3×C4C2×SD1614yescomplemented
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D412yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)12yes—
364C3×C8—24no—
365C3×C8—41no—
366C3×C8—41no—
367C24—21no—
368C24—21no—
369C24—24no—
370C24—44no—
371C24—42no—
372C24—42no—
373C24—82no—
374C24C2×C3×D411yescharacteristic
375C2×Q8—22no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8C2×C3×D414yes—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—44no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4C2×C3×D412yes—
401C2×D4C2×C3×D412yes—
402C2×D4C2×C3×D414yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—82no—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D412yes—
418C22×C4C3×#4112yes—
419C22×C4C24×C311yescharacteristic
420SD16—216no—
421SD16—216no—
422SD16—44no—
423SD16—44no—
424C2×C8—28no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8C2×C3×D414yes—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C22⋊C4—22no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4C3×SD1614yescomplemented
446C42—22no—
447C42—22no—
448C22×C3—22no—
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—22no—
452C22×C3—24no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—816no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3#46114yescomplemented
464C22×C3C2×#12111yescharacteristic
465C22×C3D4212yes—
466C22×C3C22×SD1611yescharacteristic
467C22×C3C2×#13711yescharacteristic
468C22×C3#57712yes—
469C3×C4—22no—
470C3×C4—24no—
471C3×C4—44no—
472C3×C4—42no—
473C3×C4—81no—
474C3×C4—82no—
475C3×C4—81no—
476C3×C4C2×#12011yescharacteristic
477C3×C4#53411yescharacteristic
478C3×C4#54612yes—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—48no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—48no—
492C23—42no—
493C23—84no—
494C23—88no—
495C23—88no—
496C23C3×#12112yes—
497C23C2×C3×SD1612yes—
498C23C3×#13712yes—
499C23C22×C3×D411yescharacteristic
500Q8—24no—
501Q8—24no—
502Q8—48no—
503Q8—42no—
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—42no—
508D4—22no—
509D4—22no—
510D4—22no—
511D4—24no—
512D4—48no—
513D4—42no—
514D4C3×#12014yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—88no—
536C2×C4C3×#12112yes—
537C2×C4C2×C3×SD1614yescomplemented
538C2×C4C22×C3×D411yescharacteristic
539C2×C4C22×C3×D412yes—
540C2×C4C2×C3×#4111yescharacteristic
541C2×C4C3×H(5,𝔽2)12yes—
542C8—24no—
543C8—41no—
544C8—41no—
545C2×C3—24no—
546C2×C3—44no—
547C2×C3—82no—
548C2×C3#241011yescharacteristic
549C2×C3C2×#46111yescharacteristic
550C2×C3#304211yescharacteristic
551C2×C3#326612yes—
552C2×C3D4×SD1612yes—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—22no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—816no—
566C22—82no—
567C22—82no—
568C22C3×#46114yescomplemented
569C22C2×C3×#12111yescharacteristic
570C22C3×D4212yes—
571C22C22×C3×SD1611yescharacteristic
572C22C2×C3×#13711yescharacteristic
573C22C3×#57712yes—
574C4—22no—
575C4—24no—
576C4—44no—
577C4—42no—
578C4—81no—
579C4—82no—
580C4—81no—
581C4C2×C3×#12011yescharacteristic
582C4C3×#53411yescharacteristic
583C4C3×#54612yes—
584C3#2435811yescharacteristic, direct product factor, Sylow
585C2—24no—
586C2—44no—
587C2—82no—
588C2C3×#241011yescharacteristic
589C2C2×C3×#46111yescharacteristic
590C2C3×#304211yescharacteristic
591C2C3×#326612yes—
592C2C3×D4×SD1612yes—
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×#461C2semidirect
4. C3×#2434C2semidirect
6. C3×#2055C2semidirect
7. C3×#2013C2semidirect
8. C3×#1892C2semidirect
9. C3×#1877C2semidirect
10. C3×#1850C2semidirect
11. C3×#1787C2semidirect
12. #24358C3direct (N × H)
22. C2×C3×C4⋊Q8C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
31. C2×C3×#103C22semidirect
36. C3×#391C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
47. C2×#461C2×C3semidirect
49. #2434C2×C3semidirect
51. #2055C2×C3semidirect
52. #2013C2×C3semidirect
53. #1892C2×C3semidirect
54. #1877C2×C3semidirect
55. #1850C2×C3semidirect
56. #1787C2×C3semidirect
131. C2×C4⋊Q8C22×C3semidirect
137. C2×C4⋊C8C22×C3semidirect
138. C2×#103C22×C3semidirect
139. C2×#103C22×C3semidirect
140. C2×#103C22×C3semidirect
145. #391C22×C3semidirect
153. #340C22×C3semidirect
154. #336C22×C3semidirect
233. C3×C22⋊C4SD16semidirect
359. C2×C3×C4C2×SD16semidirect
445. C22⋊C4C3×SD16semidirect
463. C22×C3#461semidirect
537. C2×C4C2×C3×SD16semidirect
568. C22C3×#461semidirect
584. C3#24358direct (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×#4612
C2×C3×#4502
C3×#24342
C3×D4×C22⋊C42
C3×#20552
C3×#20132
C3×#18922
C3×#18772
C3×#18502
C3×#17872
#243583