Finite Group Databasebeta

← C3×#24185 #10642792

Subgroups — C3×#24185 #10642792

Order 768.

1150 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
1C2×C3×#556C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×C3×C4×D8C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2012C212yesmaximal
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24185C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2211yescharacteristic
15C3×#556—24no—
16C3×#556—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C3×#461—24no—
28C3×C4×D8—24no—
29C2×C3×#107C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2211yescharacteristic
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×#556C2×C311yescharacteristic, complemented
47C2×#461C2×C311yescharacteristic, complemented
48C2×C4×D8C2×C311yescharacteristic
49#2538C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2012C2×C312yes—
52#1984C2×C311yescharacteristic, complemented
53#1956C2×C311yescharacteristic, complemented
54#1880C2×C312yescomplemented
55#1863C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C2×C3×D8—22no—
67C2×C3×D8—22no—
68C2×C3×D8—22no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D414yes—
74C3×#126—216no—
75C3×#124—24no—
76C3×#124—24no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—44no—
79C3×C4×Q8—24no—
80C3×C4×Q8—24no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yes—
87C3×#117—28no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—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—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C42—22no—
106C2×C3×C42C2312yes—
107C2×C3×C42C2311yescharacteristic
108C3×#107—22no—
109C3×C4⋊C8—22no—
110C3×#103—24no—
111C3×#103—24no—
112C3×#103—42no—
113C3×#102—22no—
114C3×#102—22no—
115C3×#102—42no—
116C3×C4×C8—22no—
117C3×#95—22no—
118C22×SD16C22×C312yes—
119C22×D8C22×C311yescharacteristic
120#556—24no—
121#556—24no—
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#126—22no—
124C2×#124C22×C312yes—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C4×Q8C22×C312yes—
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×D4C22×C311yescharacteristic
130C2×C4×D4C22×C311yescharacteristic
131C2×#117—21no—
132#461—24no—
133C4×D8—24no—
134C2×#107C22×C311yescharacteristic
135C2×C4⋊C8C22×C311yescharacteristic
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#103C22×C312yes—
139C2×#102C22×C311yescharacteristic
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—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8C2×D414yes—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4C2×D412yes—
173C2×C3×D4#4112yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C3×SD16—48no—
190C3×D8—28no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8#4112yes—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—81no—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—82no—
220C3×C42—28no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225C22×Q8C3×D411yescharacteristic
226C22×Q8C23×C312yes—
227C22×D4C23×C311yescharacteristic
228C22×D4C23×C311yescharacteristic
229C23×C4—21no—
230C23×C4—21no—
231C2×SD16—24no—
232C2×SD16—24no—
233C2×SD16—42no—
234C2×D8—22no—
235C2×D8—22no—
236C2×D8—22no—
237C22×C8C23×C312yes—
238C22×C8C23×C311yescharacteristic
239C22×C8C23×C311yescharacteristic
240C4⋊Q8—22no—
241C4⋊Q8C3×D414yes—
242#126—216no—
243#124—24no—
244#124—24no—
245C22⋊Q8—28no—
246C22⋊Q8—44no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4C3×D414yes—
255#117—28no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C3×D412yes—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C42—22no—
274C2×C42C23×C312yes—
275C2×C42C23×C311yescharacteristic
276#107—22no—
277C4⋊C8—22no—
278#103—24no—
279#103—24no—
280#103—42no—
281#102—22no—
282#102—22no—
283#102—42no—
284C4×C8—22no—
285#95—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3C22×D411yescharacteristic
301C3×Q8—24no—
302C3×Q8—24no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—44no—
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—48no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—84no—
336C2×C3×C4#12012yes—
337C2×C3×C4#12012yes—
338C2×C3×C4#13512yes—
339C2×C3×C4AGL(1,ℤ/8)12yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C42−1+411yescharacteristic
346C3×C8—22no—
347C3×C8—41no—
348C3×C8—42no—
349C24—21no—
350C24—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8C2×C3×D414yes—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4C2×C3×D412yes—
371C2×D4C3×#4112yes—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C24×C311yescharacteristic
387SD16—48no—
388D8—28no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8C3×#4112yes—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—81no—
411C4⋊C4C2×C3×D412yes—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—24no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—82no—
418C42—28no—
419C42—22no—
420C42—22no—
421C42C2×C3×D412yes—
422C42C2×C3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54811yescharacteristic
439C22×C3#55011yescharacteristic
440C22×C3C2×#13511yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3#57711yescharacteristic
443C22×C3#57811yescharacteristic
444C3×C4—22no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4—81no—
451C3×C4#44914yes—
452C3×C4#52511yescharacteristic
453C3×C4#53711yescharacteristic
454C3×C4#54611yescharacteristic
455C3×C4#54711yescharacteristic
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C22×C3×D411yescharacteristic
471Q8—24no—
472Q8—24no—
473Q8—44no—
474Q8—42no—
475Q8—42no—
476D4—22no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4C3×#12012yes—
507C2×C4C3×#12012yes—
508C2×C4C3×#13512yes—
509C2×C4C3×AGL(1,ℤ/8)12yes—
510C2×C4C22×C3×D411yescharacteristic
511C2×C4C22×C3×D411yescharacteristic
512C2×C4C2×C3×#4111yescharacteristic
513C2×C4C2×C3×#4111yescharacteristic
514C2×C4C3×H(5,𝔽2)11yescharacteristic
515C2×C4C3×2−1+411yescharacteristic
516C8—22no—
517C8—41no—
518C8—42no—
519C2×C3—82no—
520C2×C3—82no—
521C2×C3#240711yescharacteristic
522C2×C3C2×#44911yescharacteristic
523C2×C3#301311yescharacteristic
524C2×C3#336011yescharacteristic
525C2×C3#336711yescharacteristic
526C2×C3#339311yescharacteristic
527C2×C3#339711yescharacteristic
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×C3×#12011yescharacteristic
543C22C3×#54811yescharacteristic
544C22C3×#55011yescharacteristic
545C22C2×C3×#13511yescharacteristic
546C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
547C22C3×#57711yescharacteristic
548C22C3×#57811yescharacteristic
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—81no—
556C4C3×#44914yes—
557C4C3×#52511yescharacteristic
558C4C3×#53711yescharacteristic
559C4C3×#54611yescharacteristic
560C4C3×#54711yescharacteristic
561C3#2418511yescharacteristic, direct product factor, Sylow
562C2—82no—
563C2—82no—
564C2C3×#240711yescharacteristic
565C2C2×C3×#44911yescharacteristic
566C2C3×#301311yescharacteristic
567C2C3×#336011yescharacteristic
568C2C3×#336711yescharacteristic
569C2C3×#339311yescharacteristic
570C2C3×#339711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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×#556C2semidirect
2. C2×C3×#461C2semidirect
4. C3×#2538C2semidirect
5. C3×#2042C2semidirect
7. C3×#1984C2semidirect
8. C3×#1956C2semidirect
9. C3×#1880C2semidirect
10. C3×#1863C2semidirect
11. C3×#1802C2semidirect
12. #24185C3direct (N × H)
46. C2×#556C2×C3semidirect
47. C2×#461C2×C3semidirect
49. #2538C2×C3semidirect
50. #2042C2×C3semidirect
52. #1984C2×C3semidirect
53. #1956C2×C3semidirect
54. #1880C2×C3semidirect
55. #1863C2×C3semidirect
56. #1802C2×C3semidirect
561. C3#24185direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4612
C2×C3×C4×D82
C3×#25382
C3×#20422
C3×#20122
C3×#19842
C3×#19562
C3×#18802
C3×#18632
C3×#18022
#241853