Finite Group Databasebeta

← C3×#24186 #10642793

Subgroups — C3×#24186 #10642793

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
4C3×#2538C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24186C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2212yes—
15C3×#559—24no—
16C3×#559—24no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#124C2212yes—
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2212yes—
27C3×#459—24no—
28C3×C4×SD16—24no—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C3×#396—22no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#384C2211yescharacteristic
42C3×#380—21no—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#559C2×C311yescharacteristic, complemented
47C2×#459C2×C311yescharacteristic
48C2×C4×SD16C2×C311yescharacteristic, complemented
49#2538C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2012C2×C312yescomplemented
52#1984C2×C311yescharacteristic, complemented
53#1957C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1864C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C2×C3×D8—24no—
69C2×C3×D8—24no—
70C2×C3×D8—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#127—22no—
75C3×#127D414yes—
76C3×#124—28no—
77C3×#124—24no—
78C3×#124—24no—
79C3×#123—216no—
80C3×C22⋊Q8—44no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D414yes—
85C3×C4×D4—28no—
86C3×C4×D4—24no—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D412yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42—21no—
106C2×C3×C42C2312yes—
107C2×C3×C42C2311yescharacteristic
108C3×#106—22no—
109C3×C4⋊C8—22no—
110C3×#103—22no—
111C3×#103—42no—
112C3×#102—24no—
113C3×#102—24no—
114C3×#102—22no—
115C3×#102—42no—
116C3×C4×C8—22no—
117C3×#95—22no—
118C22×SD16C22×C311yescharacteristic
119C22×D8C22×C312yes—
120#559—24no—
121#559—24no—
122C2×#127C22×C311yescharacteristic
123C2×#124—21no—
124C2×#124C22×C312yes—
125C2×#123—22no—
126C2×C22⋊Q8—21no—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4C22×C312yes—
131C2×C4×D4C22×C311yescharacteristic
132#459—24no—
133C4×SD16—24no—
134C2×#106C22×C311yescharacteristic
135C2×C4⋊C8C22×C311yescharacteristic
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#102C22×C311yescharacteristic
141C2×#102C22×C312yes—
142C2×C4×C8C22×C311yescharacteristic
143#396—22no—
144#390C22×C312yes—
145#389C22×C312yes—
146#384C22×C311yescharacteristic
147#380—21no—
148#340C22×C312yes—
149#336C22×C312yes—
150#336C22×C312yes—
151C24×C3—21no—
152C24×C3—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D412yes—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—81no—
177C2×C3×D4—81no—
178C2×C3×D4C2×D414yes—
179C2×C3×D4#4112yes—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—28no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—82no—
189C22×C3×C4—82no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C3×SD16—28no—
196C3×D8—48no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—21no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C2×C3×C8#4112yes—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—44no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—81no—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—44no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—82no—
224C3×C42—24no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C22×Q8C3×D411yescharacteristic
230C22×Q8C23×C311yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C23×C312yes—
234C22×D4C23×C311yescharacteristic
235C23×C4—22no—
236C23×C4—21no—
237C2×SD16—22no—
238C2×SD16—22no—
239C2×SD16—22no—
240C2×D8—24no—
241C2×D8—24no—
242C2×D8—42no—
243C22×C8C23×C312yes—
244C22×C8C23×C311yescharacteristic
245C22×C8C23×C311yescharacteristic
246#127—22no—
247#127C3×D414yes—
248#124—28no—
249#124—24no—
250#124—24no—
251#123—216no—
252C22⋊Q8—44no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8C3×D414yes—
257C4×D4—28no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—22no—
261C4×D4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C3×D412yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C42—21no—
278C2×C42C23×C312yes—
279C2×C42C23×C311yescharacteristic
280#106—22no—
281C4⋊C8—22no—
282#103—22no—
283#103—42no—
284#102—24no—
285#102—24no—
286#102—22no—
287#102—42no—
288C4×C8—22no—
289#95—22no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×Q8—44no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—24no—
313C3×D4—24no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—42no—
317C3×D4—41no—
318C3×D4—48no—
319C3×D4—84no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—88no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4#13512yes—
345C2×C3×C4AGL(1,ℤ/8)12yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C3×C8—22no—
353C3×C8—41no—
354C3×C8—42no—
355C24—21no—
356C24—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C3×D412yes—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4C2×C3×D414yes—
383C2×D4C3×#4112yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—28no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C24×C311yescharacteristic
399SD16—28no—
400D8—48no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8C3×#4112yes—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—81no—
421C4⋊C4C2×C3×D412yes—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—82no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42C2×C3×D412yes—
432C42C2×C3×D412yes—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—81no—
444C22×C3—82no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3C2×#12011yescharacteristic
448C22×C3#54811yescharacteristic
449C22×C3#55011yescharacteristic
450C22×C3C2×#13511yescharacteristic
451C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
452C22×C3#57611yescharacteristic
453C22×C3#57711yescharacteristic
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—82no—
460C3×C4—81no—
461C3×C4#44914yes—
462C3×C4#52511yescharacteristic
463C3×C4#53711yescharacteristic
464C3×C4#54611yescharacteristic
465C3×C4#54711yescharacteristic
466C23—24no—
467C23—22no—
468C23—41no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23C22×C3×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—44no—
484Q8—41no—
485Q8—41no—
486D4—22no—
487D4—22no—
488D4—24no—
489D4—24no—
490D4—48no—
491D4—42no—
492D4—41no—
493D4—42no—
494D4—41no—
495D4—84no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—88no—
517C2×C4—84no—
518C2×C4C3×#12012yes—
519C2×C4C3×#12012yes—
520C2×C4C3×#13512yes—
521C2×C4C3×AGL(1,ℤ/8)12yes—
522C2×C4C22×C3×D411yescharacteristic
523C2×C4C22×C3×D411yescharacteristic
524C2×C4C2×C3×#4111yescharacteristic
525C2×C4C2×C3×#4111yescharacteristic
526C2×C4C3×H(5,𝔽2)11yescharacteristic
527C2×C4C3×2−1+411yescharacteristic
528C8—22no—
529C8—41no—
530C8—42no—
531C2×C3—82no—
532C2×C3—84no—
533C2×C3#240711yescharacteristic
534C2×C3C2×#44911yescharacteristic
535C2×C3#301211yescharacteristic
536C2×C3#336111yescharacteristic
537C2×C3#337311yescharacteristic
538C2×C3#339411yescharacteristic
539C2×C3#339911yescharacteristic
540C22—42no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—82no—
548C22—81no—
549C22—82no—
550C22—81no—
551C22—81no—
552C22—82no—
553C22—82no—
554C22C2×C3×#12011yescharacteristic
555C22C3×#54811yescharacteristic
556C22C3×#55011yescharacteristic
557C22C2×C3×#13511yescharacteristic
558C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
559C22C3×#57611yescharacteristic
560C22C3×#57711yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—81no—
568C4C3×#44914yes—
569C4C3×#52511yescharacteristic
570C4C3×#53711yescharacteristic
571C4C3×#54611yescharacteristic
572C4C3×#54711yescharacteristic
573C3#2418611yescharacteristic, direct product factor, Sylow
574C2—84no—
575C2—82no—
576C2C3×#240711yescharacteristic
577C2C2×C3×#44911yescharacteristic
578C2C3×#301211yescharacteristic
579C2C3×#336111yescharacteristic
580C2C3×#337311yescharacteristic
581C2C3×#339411yescharacteristic
582C2C3×#339911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#559C2semidirect
3. C2×C3×C4×SD16C2semidirect
4. C3×#2538C2semidirect
5. C3×#2042C2semidirect
6. C3×#2012C2semidirect
7. C3×#1984C2semidirect
8. C3×#1957C2semidirect
9. C3×#1879C2semidirect
10. C3×#1864C2semidirect
11. C3×#1803C2semidirect
12. #24186C3direct (N × H)
46. C2×#559C2×C3semidirect
48. C2×C4×SD16C2×C3semidirect
49. #2538C2×C3semidirect
50. #2042C2×C3semidirect
51. #2012C2×C3semidirect
52. #1984C2×C3semidirect
53. #1957C2×C3semidirect
54. #1879C2×C3semidirect
55. #1864C2×C3semidirect
56. #1803C2×C3semidirect
573. C3#24186direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4592
C2×C3×C4×SD162
C3×#25382
C3×#20422
C3×#20122
C3×#19842
C3×#19572
C3×#18792
C3×#18642
C3×#18032
#241863