Finite Group Databasebeta

← C3×#24190 #10642797

Subgroups — C3×#24190 #10642797

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic
3C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
4C3×#2582C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1957C211yesmaximal, characteristic
9C3×#1877C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24190C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C3×#559—24no—
16C3×#559—24no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#124C2212yes—
20C2×C3×#123—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2212yes—
26C3×#460—24no—
27C3×C4×Q16—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C3×#398—22no—
38C3×#391—21no—
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#559C2×C311yescharacteristic
47C2×#460C2×C311yescharacteristic
48C2×C4×Q16C2×C311yescharacteristic, complemented
49#2582C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2015C2×C312yescomplemented
52#1984C2×C311yescharacteristic, complemented
53#1957C2×C311yescharacteristic
54#1877C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C23×C3×C4—22no—
64C2×C3×Q16—22no—
65C2×C3×Q16—22no—
66C2×C3×Q16—22no—
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#127—22no—
74C3×#127D414yes—
75C3×#124—28no—
76C3×#124—24no—
77C3×#124—24no—
78C3×#123—216no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
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—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—22no—
106C3×C4⋊C8—22no—
107C3×#103—22no—
108C3×#103—22no—
109C3×#103—42no—
110C3×#102—24no—
111C3×#102—24no—
112C3×#102—42no—
113C3×C4×C8—22no—
114C3×#95—22no—
115C3×#95—22no—
116C3×#95—22no—
117C22×Q16C22×C311yescharacteristic
118C22×SD16C22×C312yes—
119#559—24no—
120#559—24no—
121C2×#127C22×C311yescharacteristic
122C2×#124—21no—
123C2×#124C22×C312yes—
124C2×#123—22no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C311yescharacteristic
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C22×C312yes—
130#460—24no—
131C4×Q16—24no—
132C2×#107C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C311yescharacteristic
135C2×#103C22×C311yescharacteristic
136C2×#103C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×C4×C8C22×C311yescharacteristic
141#398—22no—
142#391—21no—
143#391C22×C312yes—
144#389—21no—
145#389C22×C312yes—
146#384C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8C2×D412yes—
165C2×C3×Q8#4112yes—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—81no—
176C2×C3×D4—81no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—28no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C3×Q16—28no—
192C3×SD16—48no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8—21no—
201C2×C3×C8#4112yes—
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—42no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—81no—
213C3×C4⋊C4—82no—
214C3×C4⋊C4C2×D412yes—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—28no—
218C3×C42—24no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42—42no—
222C3×C42C2×D412yes—
223C3×C42C2×D412yes—
224C22×Q8C3×D411yescharacteristic
225C22×Q8C23×C311yescharacteristic
226C22×Q8C23×C311yescharacteristic
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C23×C312yes—
230C23×C4—22no—
231C2×Q16—22no—
232C2×Q16—22no—
233C2×Q16—22no—
234C2×SD16—24no—
235C2×SD16—24no—
236C2×SD16—42no—
237C22×C8C23×C312yes—
238C22×C8C23×C311yescharacteristic
239C22×C8C23×C311yescharacteristic
240#127—22no—
241#127C3×D414yes—
242#124—28no—
243#124—24no—
244#124—24no—
245#123—216no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8C3×D414yes—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C3×D411yescharacteristic
261C2×C4⋊C4C3×D412yes—
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C23×C312yes—
271C2×C42C23×C311yescharacteristic
272#107—22no—
273C4⋊C8—22no—
274#103—22no—
275#103—22no—
276#103—42no—
277#102—24no—
278#102—24no—
279#102—42no—
280C4×C8—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×Q8—44no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—48no—
306C3×D4—84no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—88no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12012yes—
333C2×C3×C4#13512yes—
334C2×C3×C4#13712yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341C3×C8—22no—
342C3×C8—41no—
343C3×C8—42no—
344C24—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8C2×C3×D412yes—
359C2×Q8C3×#4112yes—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C2×C3×D414yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C24×C311yescharacteristic
385Q16—28no—
386SD16—48no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8C3×#4112yes—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—81no—
407C4⋊C4—82no—
408C4⋊C4C2×C3×D412yes—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—42no—
416C42C2×C3×D412yes—
417C42C2×C3×D412yes—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57611yescharacteristic
431C22×C3#57711yescharacteristic
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—82no—
439C3×C4—81no—
440C3×C4#45214yes—
441C3×C4#52511yescharacteristic
442C3×C4#53711yescharacteristic
443C3×C4#54611yescharacteristic
444C3×C4#54711yescharacteristic
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C22×C3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—44no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—42no—
466D4—42no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—88no—
492C2×C4C3×#12012yes—
493C2×C4C3×#12012yes—
494C2×C4C3×#13512yes—
495C2×C4C3×#13712yes—
496C2×C4C22×C3×D411yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C2×C3×#4111yescharacteristic
499C2×C4C2×C3×#4111yescharacteristic
500C2×C4C3×H(5,𝔽2)11yescharacteristic
501C2×C4C3×2−1+411yescharacteristic
502C8—22no—
503C8—41no—
504C8—42no—
505C2×C3—84no—
506C2×C3#240711yescharacteristic
507C2×C3C2×#45211yescharacteristic
508C2×C3#301211yescharacteristic
509C2×C3#336511yescharacteristic
510C2×C3#337511yescharacteristic
511C2×C3#339611yescharacteristic
512C2×C3#340011yescharacteristic
513C22—44no—
514C22—44no—
515C22—44no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22C2×C3×#12011yescharacteristic
521C22C3×#54811yescharacteristic
522C22C3×#55011yescharacteristic
523C22C2×C3×#13511yescharacteristic
524C22C2×C3×#13711yescharacteristic
525C22C3×#57611yescharacteristic
526C22C3×#57711yescharacteristic
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—81no—
535C4C3×#45214yes—
536C4C3×#52511yescharacteristic
537C4C3×#53711yescharacteristic
538C4C3×#54611yescharacteristic
539C4C3×#54711yescharacteristic
540C3#2419011yescharacteristic, direct product factor, Sylow
541C2—84no—
542C2C3×#240711yescharacteristic
543C2C2×C3×#45211yescharacteristic
544C2C3×#301211yescharacteristic
545C2C3×#336511yescharacteristic
546C2C3×#337511yescharacteristic
547C2C3×#339611yescharacteristic
548C2C3×#340011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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
3. C2×C3×C4×Q16C2semidirect
4. C3×#2582C2semidirect
5. C3×#2042C2semidirect
6. C3×#2015C2semidirect
7. C3×#1984C2semidirect
9. C3×#1877C2semidirect
10. C3×#1862C2semidirect
11. C3×#1803C2semidirect
12. #24190C3direct (N × H)
48. C2×C4×Q16C2×C3semidirect
49. #2582C2×C3semidirect
50. #2042C2×C3semidirect
51. #2015C2×C3semidirect
52. #1984C2×C3semidirect
54. #1877C2×C3semidirect
55. #1862C2×C3semidirect
56. #1803C2×C3semidirect
540. C3#24190direct (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×#4602
C2×C3×C4×Q162
C3×#25822
C3×#20422
C3×#20152
C3×#19842
C3×#19572
C3×#18772
C3×#18622
C3×#18032
#241903