Finite Group Databasebeta

← C3×#24200 #10642807

Subgroups — C3×#24200 #10642807

Order 768.

1380 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×C3×#440C211yesmaximal, characteristic, complemented
4C3×#2581C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2014C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24200C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—22no—
21C2×C3×#121—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C3×#460—24no—
27C3×#440—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#400—21no—
38C3×#395—21no—
39C3×#391C2212yes—
40C3×#389C2212yes—
41C3×#387—22no—
42C3×#382C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic
47C2×#460C2×C311yescharacteristic, complemented
48C2×#440C2×C311yescharacteristic, complemented
49#2581C2×C311yescharacteristic, complemented
50#2041C2×C312yescomplemented
51#2014C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1961C2×C311yescharacteristic, complemented
54#1877C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C2×C3×SD16—22no—
70C2×C3×SD16—22no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C2×C3×SD16—22no—
74C2×C3×SD16—42no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×#127—24no—
79C3×#127—24no—
80C3×#127—22no—
81C3×#127D414yes—
82C3×#121—28no—
83C3×#121—216no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×D4—24no—
89C3×C4×D4—24no—
90C3×C4×D4—22no—
91C3×C4×D4D414yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2312yes—
103C2×C3×C42C2311yescharacteristic
104C3×#107—22no—
105C3×C4⋊C8—22no—
106C3×#103—22no—
107C3×#102—22no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—42no—
111C3×#102—42no—
112C3×#97—22no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—22no—
116C3×#95—22no—
117C3×#95—22no—
118C22×#41—21no—
119C22×SD16C22×C311yescharacteristic
120C22×SD16C22×C312yes—
121#550—24no—
122#550—24no—
123C2×#127C22×C312yes—
124C2×#127C22×C311yescharacteristic
125C2×#121—22no—
126C2×#121—21no—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×D4C22×C312yes—
130C2×C4×D4C22×C311yescharacteristic
131#460—24no—
132#440—24no—
133C2×#107C22×C311yescharacteristic
134C2×C4⋊C8C22×C311yescharacteristic
135C2×#103C22×C311yescharacteristic
136C2×#102C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×#97C22×C311yescharacteristic
142#400—21no—
143#395—21no—
144#391C22×C312yes—
145#389C22×C312yes—
146#387—22no—
147#382C22×C311yescharacteristic
148#340C22×C312yes—
149#336C22×C312yes—
150#336C22×C312yes—
151C24×C3—22no—
152C24×C3—21no—
153C3×#41—432no—
154C2×C3×Q8—22no—
155C2×C3×Q8—24no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8#4112yescomplemented
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—44no—
179C2×C3×D4—82no—
180C2×C3×D4—82no—
181C2×C3×D4—82no—
182C2×C3×D4C2×D414yes—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—28no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—44no—
191C22×C3×C4—44no—
192C22×C3×C4—44no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2411yescharacteristic
198C3×SD16—28no—
199C3×SD16—48no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—21no—
206C2×C3×C8—22no—
207C2×C3×C8—21no—
208C2×C3×C8#4112yescomplemented
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—44no—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—24no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—42no—
224C3×C42C2×D412yes—
225C3×C42C2×D412yes—
226C2×#41—28no—
227C2×#41—48no—
228C22×Q8C3×D411yescharacteristic
229C22×Q8C23×C311yescharacteristic
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4C23×C312yes—
235C22×D4C23×C311yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C2×SD16—22no—
239C2×SD16—22no—
240C2×SD16—24no—
241C2×SD16—24no—
242C2×SD16—22no—
243C2×SD16—42no—
244C22×C8C23×C312yes—
245C22×C8C23×C311yescharacteristic
246C22×C8C23×C311yescharacteristic
247#127—24no—
248#127—24no—
249#127—22no—
250#127C3×D414yes—
251#121—28no—
252#121—216no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4C3×D414yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C3×D412yes—
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C42—21no—
271C2×C42C23×C312yes—
272C2×C42C23×C311yescharacteristic
273#107—22no—
274C4⋊C8—22no—
275#103—22no—
276#102—22no—
277#102—24no—
278#102—24no—
279#102—42no—
280#102—42no—
281#97—22no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—28no—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×Q8—41no—
304C3×Q8—41no—
305C3×D4—22no—
306C3×D4—22no—
307C3×D4—24no—
308C3×D4—24no—
309C3×D4—48no—
310C3×D4—416no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—84no—
316C3×D4—88no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—48no—
327C2×C3×C4—416no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4#12012yes—
337C2×C3×C4#12012yes—
338C2×C3×C4AGL(1,ℤ/8)14yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)11yescharacteristic
344C2×C3×C42−1+411yescharacteristic
345C3×C8—22no—
346C3×C8—41no—
347C3×C8—42no—
348C24—22no—
349C24—21no—
350#41—432no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8C3×#4112yescomplemented
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×C3×D414yes—
380C2×D4C2×C3×D412yes—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D412yes—
394C22×C4C24×C311yescharacteristic
395SD16—28no—
396SD16—48no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8C3×#4112yescomplemented
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4C2×C3×D412yes—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C42—22no—
419C42—22no—
420C42—42no—
421C42C2×C3×D412yes—
422C42C2×C3×D412yes—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3C2×#12011yescharacteristic
436C22×C3#54811yescharacteristic
437C22×C3#55011yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)12yes—
439C22×C3#57712yes—
440C3×C4—22no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—82no—
447C3×C4—81no—
448C3×C4#52511yescharacteristic
449C3×C4#53711yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#54711yescharacteristic
452C23—24no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23C22×C3×D411yescharacteristic
465Q8—28no—
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—22no—
472D4—24no—
473D4—24no—
474D4—48no—
475D4—416no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—84no—
481D4—88no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—416no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4C3×#12012yes—
502C2×C4C3×#12012yes—
503C2×C4C3×AGL(1,ℤ/8)14yes—
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C2×C3×#4111yescharacteristic
507C2×C4C2×C3×#4111yescharacteristic
508C2×C4C3×H(5,𝔽2)11yescharacteristic
509C2×C4C3×2−1+411yescharacteristic
510C8—22no—
511C8—41no—
512C8—42no—
513C2×C3—82no—
514C2×C3—84no—
515C2×C3#240711yescharacteristic
516C2×C3#300311yescharacteristic
517C2×C3#300911yescharacteristic
518C2×C3#336812yes—
519C2×C3#339712yes—
520C22—42no—
521C22—44no—
522C22—42no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×C3×#12011yescharacteristic
533C22C3×#54811yescharacteristic
534C22C3×#55011yescharacteristic
535C22C2×C3×AGL(1,ℤ/8)12yes—
536C22C3×#57712yes—
537C4—22no—
538C4—24no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4C3×#52511yescharacteristic
546C4C3×#53711yescharacteristic
547C4C3×#54611yescharacteristic
548C4C3×#54711yescharacteristic
549C3#2420011yescharacteristic, direct product factor, Sylow
550C2—82no—
551C2—84no—
552C2C3×#240711yescharacteristic
553C2C3×#300311yescharacteristic
554C2C3×#300911yescharacteristic
555C2C3×#336812yes—
556C2C3×#339712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
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
2. C2×C3×#460C2semidirect
3. C2×C3×#440C2semidirect
4. C3×#2581C2semidirect
5. C3×#2041C2semidirect
6. C3×#2014C2semidirect
7. C3×#1985C2semidirect
8. C3×#1961C2semidirect
9. C3×#1877C2semidirect
10. C3×#1862C2semidirect
11. C3×#1804C2semidirect
12. #24200C3direct (N × H)
47. C2×#460C2×C3semidirect
48. C2×#440C2×C3semidirect
49. #2581C2×C3semidirect
50. #2041C2×C3semidirect
51. #2014C2×C3semidirect
52. #1985C2×C3semidirect
53. #1961C2×C3semidirect
54. #1877C2×C3semidirect
55. #1862C2×C3semidirect
56. #1804C2×C3semidirect
161. C2×C3×Q8#41semidirect
208. C2×C3×C8#41semidirect
358. C2×Q8C3×#41semidirect
405. C2×C8C3×#41semidirect
549. C3#24200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4602
C2×C3×#4402
C3×#25812
C3×#20412
C3×#20142
C3×#19852
C3×#19612
C3×#18772
C3×#18622
C3×#18042
#242003