Finite Group Databasebeta

← C3×#24187 #10642794

Subgroups — C3×#24187 #10642794

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2538C211yesmaximal, characteristic
5C3×#2043C212yesmaximal, complemented
6C3×#2013C212yesmaximal
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1865C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24187C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C3×#557—24no—
16C3×#557—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—21no—
20C2×C3×#125—22no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#461—24no—
28C3×C4×SD16—24no—
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38C3×#396—22no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#384C2211yescharacteristic
42C3×#380—21no—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#557C2×C311yescharacteristic, complemented
47C2×#461C2×C311yescharacteristic
48C2×C4×SD16C2×C311yescharacteristic
49#2538C2×C311yescharacteristic
50#2043C2×C312yescomplemented
51#2013C2×C312yes—
52#1984C2×C311yescharacteristic, complemented
53#1956C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1865C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yes—
75C3×#125—216no—
76C3×#125—28no—
77C3×C22⋊Q8—44no—
78C3×C4×Q8—28no—
79C3×C4×Q8—24no—
80C3×C4×Q8—24no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D414yes—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C42—22no—
105C2×C3×C42—21no—
106C2×C3×C42C2312yes—
107C2×C3×C42C2311yescharacteristic
108C3×#106—22no—
109C3×C4⋊C8—22no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#103—24no—
113C3×#103—42no—
114C3×#103—42no—
115C3×#102—22no—
116C3×C4×C8—22no—
117C3×#95—22no—
118C22×SD16C22×C311yescharacteristic
119C22×SD16C22×C312yes—
120#557—24no—
121#557—24no—
122C2×C4⋊Q8C22×C312yes—
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×#125—21no—
125C2×#125—22no—
126C2×C22⋊Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C22×C312yes—
129C2×C4×Q8C22×C311yescharacteristic
130C2×C4×Q8C22×C311yescharacteristic
131C2×C4×D4C22×C311yescharacteristic
132#461—24no—
133C4×SD16—24no—
134C2×#106C22×C311yescharacteristic
135C2×C4⋊C8C22×C311yescharacteristic
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#103C22×C311yescharacteristic
139C2×#103C22×C312yes—
140C2×#103C22×C312yes—
141C2×#102C22×C311yescharacteristic
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—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8SD1614yescomplemented
165C2×C3×Q8C2×D414yes—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4#4112yes—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C3×SD16—28no—
187C3×SD16—48no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—21no—
196C2×C3×C8#4112yes—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—44no—
212C3×C4⋊C4—81no—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—82no—
217C3×C42—28no—
218C3×C42—24no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42C2×D412yes—
222C3×C42C2×D412yes—
223C22×Q8C3×D411yescharacteristic
224C22×Q8C23×C312yes—
225C22×Q8C23×C311yescharacteristic
226C22×D4C23×C311yescharacteristic
227C23×C4—21no—
228C2×SD16—22no—
229C2×SD16—22no—
230C2×SD16—24no—
231C2×SD16—24no—
232C2×SD16—22no—
233C2×SD16—42no—
234C22×C8C23×C312yes—
235C22×C8C23×C311yescharacteristic
236C22×C8C23×C311yescharacteristic
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8C3×D414yes—
241#125—216no—
242#125—28no—
243C22⋊Q8—44no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8C3×D414yes—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
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—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42C23×C312yes—
273C2×C42C23×C311yescharacteristic
274#106—22no—
275C4⋊C8—22no—
276#103—22no—
277#103—24no—
278#103—24no—
279#103—42no—
280#103—42no—
281#102—22no—
282C4×C8—22no—
283#95—22no—
284C23×C3—22no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—28no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—24no—
296C3×Q8—24no—
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—41no—
300C3×Q8—41no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—84no—
328C2×C3×C4#12012yes—
329C2×C3×C4#12012yes—
330C2×C3×C4C2×SD1614yes—
331C2×C3×C4C2×SD1612yes—
332C2×C3×C4#13712yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4H(5,𝔽2)11yescharacteristic
338C2×C3×C42−1+411yescharacteristic
339C3×C8—22no—
340C3×C8—41no—
341C3×C8—42no—
342C24—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8C3×SD1614yescomplemented
356C2×Q8C2×C3×D414yes—
357C2×Q8C2×C3×D412yes—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4C3×#4112yes—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D412yes—
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C24×C311yescharacteristic
377SD16—28no—
378SD16—48no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8C3×#4112yes—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—81no—
404C4⋊C4C2×C3×D412yes—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—82no—
408C42—28no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42C2×C3×D412yes—
413C42C2×C3×D412yes—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54811yescharacteristic
423C22×C3#55011yescharacteristic
424C22×C3C22×SD1611yescharacteristic
425C22×C3C2×#13711yescharacteristic
426C22×C3#57711yescharacteristic
427C22×C3#57811yescharacteristic
428C3×C4—24no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4—81no—
436C3×C4#44814yes—
437C3×C4#52511yescharacteristic
438C3×C4#53711yescharacteristic
439C3×C4#54611yescharacteristic
440C3×C4#54711yescharacteristic
441C23—22no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C22×C3×D411yescharacteristic
449Q8—28no—
450Q8—22no—
451Q8—22no—
452Q8—24no—
453Q8—24no—
454Q8—42no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458D4—22no—
459D4—22no—
460D4—41no—
461D4—41no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4C3×#12012yes—
486C2×C4C3×#12012yes—
487C2×C4C2×C3×SD1614yes—
488C2×C4C2×C3×SD1612yes—
489C2×C4C3×#13712yes—
490C2×C4C22×C3×D411yescharacteristic
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C2×C3×#4111yescharacteristic
493C2×C4C2×C3×#4111yescharacteristic
494C2×C4C3×H(5,𝔽2)11yescharacteristic
495C2×C4C3×2−1+411yescharacteristic
496C8—22no—
497C8—41no—
498C8—42no—
499C2×C3—82no—
500C2×C3#240711yescharacteristic
501C2×C3C2×#44811yescharacteristic
502C2×C3#301311yescharacteristic
503C2×C3#335911yescharacteristic
504C2×C3#337211yescharacteristic
505C2×C3#339211yescharacteristic
506C2×C3#339811yescharacteristic
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22C2×C3×#12011yescharacteristic
515C22C3×#54811yescharacteristic
516C22C3×#55011yescharacteristic
517C22C22×C3×SD1611yescharacteristic
518C22C2×C3×#13711yescharacteristic
519C22C3×#57711yescharacteristic
520C22C3×#57811yescharacteristic
521C4—24no—
522C4—22no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—81no—
529C4C3×#44814yes—
530C4C3×#52511yescharacteristic
531C4C3×#53711yescharacteristic
532C4C3×#54611yescharacteristic
533C4C3×#54711yescharacteristic
534C3#2418711yescharacteristic, direct product factor, Sylow
535C2—82no—
536C2C3×#240711yescharacteristic
537C2C2×C3×#44811yescharacteristic
538C2C3×#301311yescharacteristic
539C2C3×#335911yescharacteristic
540C2C3×#337211yescharacteristic
541C2C3×#339211yescharacteristic
542C2C3×#339811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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×#557C2semidirect
5. C3×#2043C2semidirect
7. C3×#1984C2semidirect
8. C3×#1956C2semidirect
10. C3×#1865C2semidirect
11. C3×#1803C2semidirect
12. #24187C3direct (N × H)
46. C2×#557C2×C3semidirect
50. #2043C2×C3semidirect
52. #1984C2×C3semidirect
53. #1956C2×C3semidirect
55. #1865C2×C3semidirect
56. #1803C2×C3semidirect
164. C2×C3×Q8SD16semidirect
355. C2×Q8C3×SD16semidirect
534. C3#24187direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4612
C2×C3×C4×SD162
C3×#25382
C3×#20432
C3×#20132
C3×#19842
C3×#19562
C3×#18802
C3×#18652
C3×#18032
#241873