Finite Group Databasebeta

← C3×#24202 #10642809

Subgroups — C3×#24202 #10642809

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 512 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×#441C211yesmaximal, characteristic, complemented
4C3×#2581C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic
9C3×#1877C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24202C311yesmaximal, 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×#441—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
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×#559C2×C311yescharacteristic
47C2×#460C2×C311yescharacteristic
48C2×#441C2×C311yescharacteristic, complemented
49#2581C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2015C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1961C2×C311yescharacteristic
54#1877C2×C312yescomplemented
55#1864C2×C312yescomplemented
56#1805C2×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—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D412yes—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C3×#106—22no—
104C3×C4⋊C8—22no—
105C3×#103—22no—
106C3×#103—22no—
107C3×#103—42no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—42no—
111C3×#97—22no—
112C3×#95—21no—
113C3×#95—21no—
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—
131#441—24no—
132C2×#106C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C311yescharacteristic
135C2×#103C22×C311yescharacteristic
136C2×#103C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#97C22×C311yescharacteristic
141#400—21no—
142#395—21no—
143#391C22×C312yes—
144#389C22×C312yes—
145#387—22no—
146#382C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
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—42no—
163C2×C3×Q8C2×D412yes—
164C2×C3×Q8#4112yes—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—44no—
173C2×C3×D4—82no—
174C2×C3×D4C2×D414yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—28no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—84no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C3×Q16—28no—
188C3×SD16—48no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8#4112yes—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—44no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C42—24no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—42no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C311yescharacteristic
220C22×Q8C23×C311yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C23×C312yes—
224C23×C4—22no—
225C2×Q16—22no—
226C2×Q16—22no—
227C2×Q16—22no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—42no—
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C311yescharacteristic
234#127—22no—
235#127C3×D414yes—
236#124—28no—
237#124—24no—
238#124—24no—
239#123—216no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8C3×D414yes—
246C4×D4—28no—
247C4×D4—24no—
248C4×D4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C3×D411yescharacteristic
253C2×C4⋊C4C3×D412yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C312yes—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C23×C312yes—
263C2×C42C23×C311yescharacteristic
264#106—22no—
265C4⋊C8—22no—
266#103—22no—
267#103—22no—
268#103—42no—
269#102—24no—
270#102—24no—
271#102—42no—
272#97—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3C22×D411yescharacteristic
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—44no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×Q8—41no—
293C3×D4—24no—
294C3×D4—24no—
295C3×D4—48no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—84no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—88no—
320C2×C3×C4#12012yes—
321C2×C3×C4#12012yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C3×C8—22no—
329C3×C8—41no—
330C3×C8—42no—
331C24—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—42no—
344C2×Q8C2×C3×D412yes—
345C2×Q8C3×#4112yes—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—44no—
354C2×D4—82no—
355C2×D4C2×C3×D414yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—84no—
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C24×C311yescharacteristic
368Q16—28no—
369SD16—48no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8C3×#4112yes—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4C2×C3×D412yes—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42C2×C3×D412yes—
398C42C2×C3×D412yes—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—84no—
403C22×C3—84no—
404C22×C3C2×#12011yescharacteristic
405C22×C3#54811yescharacteristic
406C22×C3#55011yescharacteristic
407C22×C3#57512yes—
408C22×C3#57712yes—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—82no—
417C3×C4—81no—
418C3×C4#52511yescharacteristic
419C3×C4#53711yescharacteristic
420C3×C4#54611yescharacteristic
421C3×C4#54711yescharacteristic
422C23—24no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23C22×C3×D411yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—22no—
431Q8—22no—
432Q8—44no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437D4—24no—
438D4—24no—
439D4—48no—
440D4—42no—
441D4—42no—
442D4—84no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—88no—
464C2×C4C3×#12012yes—
465C2×C4C3×#12012yes—
466C2×C4C22×C3×D411yescharacteristic
467C2×C4C22×C3×D411yescharacteristic
468C2×C4C2×C3×#4111yescharacteristic
469C2×C4C2×C3×#4111yescharacteristic
470C2×C4C3×H(5,𝔽2)11yescharacteristic
471C2×C4C3×2−1+411yescharacteristic
472C8—22no—
473C8—41no—
474C8—42no—
475C2×C3—84no—
476C2×C3#240711yescharacteristic
477C2×C3#300511yescharacteristic
478C2×C3#300911yescharacteristic
479C2×C3#337112yes—
480C2×C3#340112yes—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—84no—
485C22—84no—
486C22C2×C3×#12011yescharacteristic
487C22C3×#54811yescharacteristic
488C22C3×#55011yescharacteristic
489C22C3×#57512yes—
490C22C3×#57712yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—82no—
499C4—81no—
500C4C3×#52511yescharacteristic
501C4C3×#53711yescharacteristic
502C4C3×#54611yescharacteristic
503C4C3×#54711yescharacteristic
504C3#2420211yescharacteristic, direct product factor, Sylow
505C2—84no—
506C2C3×#240711yescharacteristic
507C2C3×#300511yescharacteristic
508C2C3×#300911yescharacteristic
509C2C3×#337112yes—
510C2C3×#340112yes—
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×#441C2semidirect
4. C3×#2581C2semidirect
5. C3×#2042C2semidirect
6. C3×#2015C2semidirect
7. C3×#1985C2semidirect
9. C3×#1877C2semidirect
10. C3×#1864C2semidirect
11. C3×#1805C2semidirect
12. #24202C3direct (N × H)
48. C2×#441C2×C3semidirect
49. #2581C2×C3semidirect
50. #2042C2×C3semidirect
51. #2015C2×C3semidirect
52. #1985C2×C3semidirect
54. #1877C2×C3semidirect
55. #1864C2×C3semidirect
56. #1805C2×C3semidirect
504. C3#24202direct (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×#4412
C3×#25812
C3×#20422
C3×#20152
C3×#19852
C3×#19612
C3×#18772
C3×#18642
C3×#18052
#242023