Finite Group Databasebeta

← C3×#24375 #10642982

Subgroups — C3×#24375 #10642982

Order 768.

1422 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#449C211yesmaximal, characteristic
4C3×#2436C211yesmaximal, characteristic, complemented
5C3×#2341C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2012C212yesmaximal, complemented
8C3×#1895C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1851C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#24375C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C2×C3×#137—24no—
16C2×C3×#137—28no—
17C22×C3×Q16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2211yescharacteristic
20C22×C3×D8C2211yescharacteristic
21C22×C3×M16C2211yescharacteristic
22C2×C3×#124C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C3×#464—28no—
26C3×#449—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#393—21no—
35C3×#393—21no—
36C3×#391C2212yes—
37C3×#390—22no—
38C3×#390C2212yes—
39C3×#386—22no—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yes—
45C3×#336C2212yes—
46C22×#137C2×C311yescharacteristic, complemented
47C2×#464C2×C312yescomplemented
48C2×#449C2×C311yescharacteristic
49#2436C2×C311yescharacteristic, complemented
50#2341C2×C311yescharacteristic, complemented
51#2055C2×C312yescomplemented
52#2012C2×C312yescomplemented
53#1895C2×C311yescharacteristic, complemented
54#1878C2×C312yescomplemented
55#1851C2×C311yescharacteristic, complemented
56#1787C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—24no—
59C2×C3×#41—42no—
60C2×C3×#41D414yes—
61C22×C3×Q8—21no—
62C22×C3×Q8—24no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C3×#137—232no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16D418yescomplemented
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—22no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×D8—22no—
85C2×C3×D8—42no—
86C2×C3×M16—22no—
87C2×C3×M16—24no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2312yes—
91C3×#124—24no—
92C3×#124—42no—
93C3×#121—22no—
94C3×#121—22no—
95C3×C4×Q8—24no—
96C3×C4×Q8—42no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42—22no—
108C2×C3×C42C2312yes—
109C3×C4⋊C8—24no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#103—42no—
113C3×#102—22no—
114C3×#102—24no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C22×#41C22×C311yescharacteristic
122C23×Q8C22×C311yescharacteristic
123C2×#137—24no—
124C2×#137—28no—
125C22×Q16C22×C312yes—
126C22×SD16C22×C312yes—
127C22×SD16C22×C311yescharacteristic
128C22×D8C22×C311yescharacteristic
129C22×M16C22×C311yescharacteristic
130C2×#124C22×C312yes—
131C2×#121C22×C311yescharacteristic
132C2×C4×Q8C22×C312yes—
133#464—28no—
134#449—24no—
135C2×C4⋊C8C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C312yes—
141C2×C22⋊C8C22×C311yescharacteristic
142#393—21no—
143#393—21no—
144#391C22×C312yes—
145#390—22no—
146#390C22×C312yes—
147#386—22no—
148#380—21no—
149#380C22×C311yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4C22×C311yescharacteristic
152#340C22×C312yes—
153#336C22×C312yes—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C3×#41—28no—
158C3×#41—44no—
159C3×#41—48no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—28no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—44no—
169C2×C3×Q8—48no—
170C2×C3×Q8—44no—
171C2×C3×Q8—42no—
172C2×C3×Q8C2×D414yes—
173C2×C3×Q8C2×D412yes—
174C2×C3×Q8C2×D414yes—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—42no—
186C2×C3×D4C2×D412yes—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—44no—
195C22×C3×C4—44no—
196C22×C3×C4—42no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D412yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C3×Q16—216no—
204C3×SD16—216no—
205C3×SD16—44no—
206C3×D8—44no—
207C3×M16—28no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8C2×D414yes—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—44no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—44no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—82no—
226C3×C22⋊C4—81no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—44no—
230C2×#41—22no—
231C2×#41—24no—
232C2×#41—42no—
233C2×#41C3×D414yes—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C23×C311yescharacteristic
244C22×D4C23×C311yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C23×C311yescharacteristic
248#137—232no—
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16C3×D418yescomplemented
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—22no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×D8—22no—
258C2×D8—42no—
259C2×M16—22no—
260C2×M16—24no—
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C312yes—
264#124—24no—
265#124—42no—
266#121—22no—
267#121—22no—
268C4×Q8—24no—
269C4×Q8—42no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C3×D412yes—
280C2×C42—22no—
281C2×C42C23×C312yes—
282C4⋊C8—24no—
283#103—22no—
284#103—24no—
285#103—42no—
286#102—22no—
287#102—24no—
288C22⋊C8—22no—
289#95—21no—
290#95—22no—
291#95—22no—
292#95C3×D411yescharacteristic
293#95C3×D411yescharacteristic
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3#12112yes—
309C23×C3C22×D411yescharacteristic
310C3×Q8—22no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—24no—
314C3×Q8—48no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8#12014yes—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—48no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—84no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—48no—
340C2×C3×C4—48no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4#12112yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D412yes—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)12yes—
353C3×C8—24no—
354C3×C8—41no—
355C3×C8—41no—
356C24—21no—
357C24—21no—
358C24C2×C3×D411yescharacteristic
359#41—28no—
360#41—44no—
361#41—48no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—28no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8C2×C3×D414yes—
375C2×Q8C2×C3×D412yes—
376C2×Q8C2×C3×D414yes—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4C2×C3×D412yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D412yes—
403C22×C4C3×#4112yes—
404C22×C4C24×C311yescharacteristic
405Q16—216no—
406SD16—216no—
407SD16—44no—
408D8—44no—
409M16—28no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8C2×C3×D414yes—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C22⋊C4—42no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—82no—
428C22⋊C4—81no—
429C42—22no—
430C42—22no—
431C42—44no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—81no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3C2×#12111yescharacteristic
448C22×C3D4212yes—
449C22×C3#57512yes—
450C22×C3#57711yescharacteristic
451C22×C3#57811yescharacteristic
452C3×C4—22no—
453C3×C4—24no—
454C3×C4—44no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—82no—
458C3×C4—81no—
459C3×C4C2×#12011yescharacteristic
460C3×C4#53411yescharacteristic
461C3×C4#54612yes—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23C3×#12112yes—
477C23C22×C3×D411yescharacteristic
478Q8—22no—
479Q8—24no—
480Q8—24no—
481Q8—24no—
482Q8—48no—
483Q8—42no—
484Q8—42no—
485Q8C3×#12014yes—
486D4—22no—
487D4—22no—
488D4—48no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—84no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4C3×#12112yes—
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D412yes—
519C2×C4C2×C3×#4111yescharacteristic
520C2×C4C3×H(5,𝔽2)12yes—
521C8—24no—
522C8—41no—
523C8—41no—
524C2×C3—42no—
525C2×C3—82no—
526C2×C3—161no—
527C2×C3#241011yescharacteristic
528C2×C3#304011yescharacteristic
529C2×C3#304611yescharacteristic
530C2×C3#327112yes—
531C2×C3#327712yes—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22—81no—
546C22—81no—
547C22C2×C3×#12111yescharacteristic
548C22C3×D4212yes—
549C22C3×#57512yes—
550C22C3×#57711yescharacteristic
551C22C3×#57811yescharacteristic
552C4—22no—
553C4—24no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—81no—
559C4C2×C3×#12011yescharacteristic
560C4C3×#53411yescharacteristic
561C4C3×#54612yes—
562C3#2437511yescharacteristic, direct product factor, Sylow
563C2—42no—
564C2—82no—
565C2—161no—
566C2C3×#241011yescharacteristic
567C2C3×#304011yescharacteristic
568C2C3×#304611yescharacteristic
569C2C3×#327112yes—
570C2C3×#327712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#449C2
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. C22×C3×#137C2semidirect
2. C2×C3×#464C2semidirect
4. C3×#2436C2semidirect
5. C3×#2341C2semidirect
6. C3×#2055C2semidirect
7. C3×#2012C2semidirect
8. C3×#1895C2semidirect
9. C3×#1878C2semidirect
10. C3×#1851C2semidirect
11. C3×#1787C2semidirect
12. #24375C3direct (N × H)
46. C22×#137C2×C3semidirect
47. C2×#464C2×C3semidirect
49. #2436C2×C3semidirect
50. #2341C2×C3semidirect
51. #2055C2×C3semidirect
52. #2012C2×C3semidirect
53. #1895C2×C3semidirect
54. #1878C2×C3semidirect
55. #1851C2×C3semidirect
56. #1787C2×C3semidirect
78. C2×C3×Q16D4semidirect
251. C2×Q16C3×D4semidirect
562. C3#24375direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C2×C3×#4642
C2×C3×#4492
C3×#24362
C3×#23412
C3×#20552
C3×#20122
C3×#18952
C3×#18782
C3×#18512
C3×#17872
#243753