Finite Group Databasebeta

← C3×#24649 #10643256

Subgroups — C3×#24649 #10643256

Order 768.

884 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic
2C2×C3×#441C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic
4C3×#2356C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1983C212yesmaximal, complemented
8C3×#1961C211yesmaximal, characteristic
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24649C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C3×#460—24no—
20C3×#441—24no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#97C2211yescharacteristic
31C3×#394—21no—
32C3×#394—21no—
33C3×#391C2212yes—
34C3×#390—22no—
35C3×#390C2212yes—
36C3×#387—22no—
37C3×#383—21no—
38C3×#382C2211yescharacteristic
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#460C2×C311yescharacteristic
45C2×#441C2×C311yescharacteristic, complemented
46#2548C2×C311yescharacteristic
47#2356C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2015C2×C312yescomplemented
50#1983C2×C312yescomplemented
51#1961C2×C311yescharacteristic
52#1940C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1805C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C2×C3×Q16—22no—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×SD16—24no—
64C2×C3×SD16—42no—
65C2×C3×SD16—42no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×#127—22no—
70C3×#127—22no—
71C3×#124—24no—
72C3×#124—42no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8D414yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—21no—
88C2×C3×C42—22no—
89C2×C3×C42—21no—
90C2×C3×C42C2312yes—
91C2×C3×C42C2311yescharacteristic
92C3×#106—22no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—42no—
95C3×#103—22no—
96C3×#103—22no—
97C3×#103—42no—
98C3×#102—24no—
99C3×#102—24no—
100C3×#102—42no—
101C3×#97—22no—
102C3×#95—21no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—22no—
107C3×#95D411yescharacteristic
108C3×#95D411yescharacteristic
109C22×Q16C22×C311yescharacteristic
110C22×SD16C22×C312yes—
111C2×#127C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×Q8C22×C311yescharacteristic
115#460—24no—
116#441—24no—
117C2×#106C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C311yescharacteristic
121C2×#103C22×C311yescharacteristic
122C2×#103C22×C312yes—
123C2×#102C22×C312yes—
124C2×#102C22×C312yes—
125C2×#102C22×C312yes—
126C2×#97C22×C311yescharacteristic
127#394—21no—
128#394—21no—
129#391C22×C312yes—
130#390—22no—
131#390C22×C312yes—
132#387—22no—
133#383—21no—
134#382C22×C311yescharacteristic
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C311yescharacteristic
137#376—21no—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8C2×D412yes—
152C2×C3×Q8#4112yes—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—44no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—82no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C3×Q16—28no—
174C3×SD16—48no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8#4112yes—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×D412yes—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—82no—
196C3×C22⋊C4—82no—
197C3×C42—24no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—44no—
201C3×C42—42no—
202C3×C42C2×D412yes—
203C3×C42#4112yes—
204C22×Q8C23×C311yescharacteristic
205C22×Q8C23×C311yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C23×C312yes—
209C2×Q16—22no—
210C2×Q16—22no—
211C2×Q16—22no—
212C2×SD16—24no—
213C2×SD16—42no—
214C2×SD16—42no—
215C22×C8C23×C312yes—
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C311yescharacteristic
218#127—22no—
219#127—22no—
220#124—24no—
221#124—42no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8C3×D414yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C3×D412yes—
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42C23×C312yes—
240C2×C42C23×C311yescharacteristic
241#106—22no—
242C4⋊C8—22no—
243C4⋊C8—42no—
244#103—22no—
245#103—22no—
246#103—42no—
247#102—24no—
248#102—24no—
249#102—42no—
250#97—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95C3×D411yescharacteristic
257#95C3×D411yescharacteristic
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3C22×D411yescharacteristic
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—41no—
271C3×Q8—41no—
272C3×Q8—41no—
273C3×Q8—41no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—84no—
279C3×D4—84no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#12112yes—
302C2×C3×C4#12112yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4H(5,𝔽2)11yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C3×C8—22no—
310C3×C8—41no—
311C3×C8—42no—
312C24—22no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8C2×C3×D412yes—
324C2×Q8C3×#4112yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—82no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345Q16—28no—
346SD16—48no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8C3×#4112yes—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4C2×C3×D412yes—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—24no—
370C42—22no—
371C42—22no—
372C42—44no—
373C42—42no—
374C42C2×C3×D412yes—
375C42C3×#4112yes—
376C22×C3—82no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3C2×#12111yescharacteristic
384C22×C3#54611yescharacteristic
385C22×C3#55011yescharacteristic
386C22×C3#57512yes—
387C22×C3#57712yes—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—82no—
396C3×C4—81no—
397C3×C4#52511yescharacteristic
398C3×C4#53511yescharacteristic
399C3×C4#55311yescharacteristic
400C3×C4#55511yescharacteristic
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23C22×C3×D411yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—84no—
422D4—84no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4C3×#12112yes—
445C2×C4C3×#12112yes—
446C2×C4C22×C3×D411yescharacteristic
447C2×C4C2×C3×#4111yescharacteristic
448C2×C4C2×C3×#4111yescharacteristic
449C2×C4C2×C3×#4111yescharacteristic
450C2×C4C3×H(5,𝔽2)11yescharacteristic
451C2×C4C3×H(5,𝔽2)11yescharacteristic
452C8—22no—
453C8—42no—
454C8—41no—
455C2×C3—162no—
456C2×C3#241411yescharacteristic
457C2×C3#304011yescharacteristic
458C2×C3#304311yescharacteristic
459C2×C3#331712yes—
460C2×C3#340112yes—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22C2×C3×#12111yescharacteristic
469C22C3×#54611yescharacteristic
470C22C3×#55011yescharacteristic
471C22C3×#57512yes—
472C22C3×#57712yes—
473C4—22no—
474C4—22no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—82no—
481C4—81no—
482C4C3×#52511yescharacteristic
483C4C3×#53511yescharacteristic
484C4C3×#55311yescharacteristic
485C4C3×#55511yescharacteristic
486C3#2464911yescharacteristic, direct product factor, Sylow
487C2—162no—
488C2C3×#241411yescharacteristic
489C2C3×#304011yescharacteristic
490C2C3×#304311yescharacteristic
491C2C3×#331712yes—
492C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#441C2semidirect
4. C3×#2356C2semidirect
5. C3×#2055C2semidirect
6. C3×#2015C2semidirect
7. C3×#1983C2semidirect
9. C3×#1940C2semidirect
10. C3×#1926C2semidirect
11. C3×#1805C2semidirect
12. #24649C3direct (N × H)
45. C2×#441C2×C3semidirect
47. #2356C2×C3semidirect
48. #2055C2×C3semidirect
49. #2015C2×C3semidirect
50. #1983C2×C3semidirect
52. #1940C2×C3semidirect
53. #1926C2×C3semidirect
54. #1805C2×C3semidirect
486. C3#24649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×C3×#4412
C3×#25482
C3×#23562
C3×#20552
C3×#20152
C3×#19832
C3×#19612
C3×#19402
C3×#19262
C3×#18052
#246493