Finite Group Databasebeta

← C3×#24647 #10643254

Subgroups — C3×#24647 #10643254

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic, complemented
5C3×#2054C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1983C212yesmaximal, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24647C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#459—24no—
20C3×#440—24no—
21C2×C3×#107C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#97C2211yescharacteristic
31C3×#394—21no—
32C3×#394—21no—
33C3×#390—22no—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#388—22no—
37C3×#382C2211yescharacteristic
38C3×#381—21no—
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#459C2×C311yescharacteristic
45C2×#440C2×C311yescharacteristic, complemented
46#2548C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic, complemented
48#2054C2×C312yescomplemented
49#2012C2×C312yescomplemented
50#1983C2×C312yescomplemented
51#1961C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1805C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C2×C3×SD16—22no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×D8—24no—
65C2×C3×D8—42no—
66C2×C3×D8—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#127—22no—
71C3×#127—22no—
72C3×#124—24no—
73C3×#124—42no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
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—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C42—21no—
91C2×C3×C42—22no—
92C2×C3×C42C2312yes—
93C2×C3×C42C2311yescharacteristic
94C3×#107—22no—
95C3×C4⋊C8—22no—
96C3×C4⋊C8—42no—
97C3×#103—22no—
98C3×#103—42no—
99C3×#102—22no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#102—42no—
103C3×#97—22no—
104C3×#95—21no—
105C3×#95—22no—
106C3×#95—21no—
107C3×#95D411yescharacteristic
108C3×#95D411yescharacteristic
109C22×SD16C22×C311yescharacteristic
110C22×D8C22×C312yes—
111C2×#127C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×D4C22×C311yescharacteristic
115#459—24no—
116#440—24no—
117C2×#107C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C311yescharacteristic
121C2×#103C22×C312yes—
122C2×#102C22×C311yescharacteristic
123C2×#102C22×C312yes—
124C2×#102C22×C312yes—
125C2×#102C22×C312yes—
126C2×#97C22×C311yescharacteristic
127#394—21no—
128#394—21no—
129#390—22no—
130#390C22×C312yes—
131#390C22×C312yes—
132#388—22no—
133#382C22×C311yescharacteristic
134#381—21no—
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C311yescharacteristic
137C4×C22⋊C4—21no—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—21no—
141C24×C3—22no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8#4112yes—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—44no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—82no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—82no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×SD16—28no—
176C3×D8—48no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8#4112yes—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—82no—
199C3×C22⋊C4—82no—
200C3×C22⋊C4—82no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—42no—
204C3×C42—82no—
205C3×C42C2×D412yes—
206C3×C42#4112yes—
207C22×Q8C23×C311yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C23×C312yes—
211C22×D4C23×C311yescharacteristic
212C23×C4—21no—
213C2×SD16—22no—
214C2×SD16—22no—
215C2×SD16—22no—
216C2×D8—24no—
217C2×D8—42no—
218C2×D8—42no—
219C22×C8C23×C312yes—
220C22×C8C23×C311yescharacteristic
221C22×C8C23×C311yescharacteristic
222#127—22no—
223#127—22no—
224#124—24no—
225#124—42no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×D4—22no—
229C4×D4C3×D414yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C3×D412yes—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42C23×C312yes—
245C2×C42C23×C311yescharacteristic
246#107—22no—
247C4⋊C8—22no—
248C4⋊C8—42no—
249#103—22no—
250#103—42no—
251#102—22no—
252#102—24no—
253#102—24no—
254#102—42no—
255#97—22no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95C3×D411yescharacteristic
260#95C3×D411yescharacteristic
261C23×C3—22no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C22×D411yescharacteristic
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—41no—
283C3×D4—42no—
284C3×D4—41no—
285C3×D4—42no—
286C3×D4—84no—
287C3×D4—84no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—84no—
307C2×C3×C4#12112yes—
308C2×C3×C4#12112yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C3×C8—22no—
316C3×C8—41no—
317C3×C8—42no—
318C24—21no—
319C24—22no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8C3×#4112yes—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—82no—
341C2×D4C2×C3×D412yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—82no—
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C24×C311yescharacteristic
353SD16—28no—
354D8—48no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8C3×#4112yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4C2×C3×D412yes—
372C22⋊C4—24no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—22no—
380C42—22no—
381C42—42no—
382C42—82no—
383C42C2×C3×D412yes—
384C42C3×#4112yes—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3C2×#12111yescharacteristic
398C22×C3#54611yescharacteristic
399C22×C3#55011yescharacteristic
400C22×C3#57512yes—
401C22×C3#57612yes—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4—81no—
410C3×C4#52511yescharacteristic
411C3×C4#53511yescharacteristic
412C3×C4#55311yescharacteristic
413C3×C4#55511yescharacteristic
414C23—22no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23C22×C3×D411yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431D4—22no—
432D4—22no—
433D4—42no—
434D4—42no—
435D4—41no—
436D4—42no—
437D4—41no—
438D4—42no—
439D4—84no—
440D4—84no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—84no—
460C2×C4C3×#12112yes—
461C2×C4C3×#12112yes—
462C2×C4C22×C3×D411yescharacteristic
463C2×C4C2×C3×#4111yescharacteristic
464C2×C4C2×C3×#4111yescharacteristic
465C2×C4C2×C3×#4111yescharacteristic
466C2×C4C3×H(5,𝔽2)11yescharacteristic
467C2×C4C3×H(5,𝔽2)11yescharacteristic
468C8—22no—
469C8—42no—
470C8—41no—
471C2×C3—82no—
472C2×C3—162no—
473C2×C3#241411yescharacteristic
474C2×C3#304011yescharacteristic
475C2×C3#304111yescharacteristic
476C2×C3#330912yes—
477C2×C3#340312yes—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×C3×#12111yescharacteristic
491C22C3×#54611yescharacteristic
492C22C3×#55011yescharacteristic
493C22C3×#57512yes—
494C22C3×#57612yes—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—81no—
503C4C3×#52511yescharacteristic
504C4C3×#53511yescharacteristic
505C4C3×#55311yescharacteristic
506C4C3×#55511yescharacteristic
507C3#2464711yescharacteristic, direct product factor, Sylow
508C2—82no—
509C2—162no—
510C2C3×#241411yescharacteristic
511C2C3×#304011yescharacteristic
512C2C3×#304111yescharacteristic
513C2C3×#330912yes—
514C2C3×#340312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
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×#440C2semidirect
3. C3×#2548C2semidirect
4. C3×#2357C2semidirect
5. C3×#2054C2semidirect
6. C3×#2012C2semidirect
7. C3×#1983C2semidirect
8. C3×#1961C2semidirect
9. C3×#1940C2semidirect
10. C3×#1925C2semidirect
11. C3×#1805C2semidirect
12. #24647C3direct (N × H)
45. C2×#440C2×C3semidirect
46. #2548C2×C3semidirect
47. #2357C2×C3semidirect
48. #2054C2×C3semidirect
49. #2012C2×C3semidirect
50. #1983C2×C3semidirect
51. #1961C2×C3semidirect
52. #1940C2×C3semidirect
53. #1925C2×C3semidirect
54. #1805C2×C3semidirect
507. C3#24647direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×C3×#4402
C3×#25482
C3×#23572
C3×#20542
C3×#20122
C3×#19832
C3×#19612
C3×#19402
C3×#19252
C3×#18052
#246473