Finite Group Databasebeta

← C3×#24644 #10643251

Subgroups — C3×#24644 #10643251

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic
4C3×#2356C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2014C212yesmaximal, complemented
7C3×#1982C212yesmaximal, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24644C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×C3×#121—22no—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C3×#460—24no—
21C3×#440—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#97C2211yescharacteristic
32C3×#392—21no—
33C3×#392—21no—
34C3×#391C2212yes—
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, complemented
45C2×#440C2×C311yescharacteristic, complemented
46#2541C2×C311yescharacteristic
47#2356C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2014C2×C312yescomplemented
50#1982C2×C312yescomplemented
51#1961C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1804C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16—24no—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C2×C3×SD16—42no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×#127—24no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#127—42no—
76C3×#121—48no—
77C3×C4×Q8—22no—
78C3×C4×Q8D414yescomplemented
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42—22no—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C3×#107—22no—
94C3×C4⋊C8—22no—
95C3×C4⋊C8—42no—
96C3×#103—22no—
97C3×#102—24no—
98C3×#102—22no—
99C3×#102—24no—
100C3×#102—42no—
101C3×#102—42no—
102C3×#97—22no—
103C3×#95—21no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95D411yescharacteristic
109C3×#95D411yescharacteristic
110C22×SD16C22×C311yescharacteristic
111C22×SD16C22×C312yes—
112C2×#127C22×C311yescharacteristic
113C2×#127C22×C312yes—
114C2×#121—22no—
115C2×C4×Q8C22×C311yescharacteristic
116C2×C4×D4C22×C311yescharacteristic
117#460—24no—
118#440—24no—
119C2×#107C22×C311yescharacteristic
120C2×C4⋊C8C22×C311yescharacteristic
121C2×C4⋊C8C22×C312yes—
122C2×#103C22×C311yescharacteristic
123C2×#102C22×C311yescharacteristic
124C2×#102C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×#97C22×C311yescharacteristic
129#392—21no—
130#392—21no—
131#391C22×C312yes—
132#390C22×C312yes—
133#387—22no—
134#383—21no—
135#382C22×C311yescharacteristic
136C4×C4⋊C4C22×C312yes—
137C4×C4⋊C4C22×C311yescharacteristic
138#376—21no—
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C24×C3—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8C2×D412yes—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—44no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—44no—
167C2×C3×D4—82no—
168C2×C3×D4—82no—
169C2×C3×D4—82no—
170C2×C3×D4#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—44no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C3×SD16—28no—
183C3×SD16—48no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4C2×D412yes—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—44no—
204C3×C42—24no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—44no—
208C3×C42C2×D412yes—
209C3×C42#4112yes—
210C22×Q8C23×C311yescharacteristic
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4C23×C312yes—
216C22×D4C23×C311yescharacteristic
217C23×C4—21no—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16—24no—
221C2×SD16—22no—
222C2×SD16—42no—
223C2×SD16—42no—
224C22×C8C23×C312yes—
225C22×C8C23×C311yescharacteristic
226C22×C8C23×C311yescharacteristic
227#127—24no—
228#127—22no—
229#127—22no—
230#127—42no—
231#121—48no—
232C4×Q8—22no—
233C4×Q8C3×D414yescomplemented
234C4×D4—22no—
235C4×D4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C3×D412yes—
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C312yes—
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42C23×C312yes—
247C2×C42C23×C311yescharacteristic
248#107—22no—
249C4⋊C8—22no—
250C4⋊C8—42no—
251#103—22no—
252#102—24no—
253#102—22no—
254#102—24no—
255#102—42no—
256#102—42no—
257#97—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95C3×D411yescharacteristic
264#95C3×D411yescharacteristic
265C23×C3—22no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3C22×D411yescharacteristic
278C3×Q8—22no—
279C3×Q8—22no—
280C3×Q8—41no—
281C3×Q8—41no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—42no—
285C3×D4—41no—
286C3×D4—42no—
287C3×D4—42no—
288C3×D4—41no—
289C3×D4—42no—
290C3×D4—88no—
291C3×D4—84no—
292C3×D4—84no—
293C3×D4—88no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—84no—
312C2×C3×C4#12112yes—
313C2×C3×C4#12112yes—
314C2×C3×C4AGL(1,ℤ/8)14yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C3×C8—22no—
322C3×C8—41no—
323C3×C8—42no—
324C24—21no—
325C24—22no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8C2×C3×D412yes—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4C3×#4112yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D412yes—
364C22×C4C24×C311yescharacteristic
365SD16—28no—
366SD16—48no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8C3×#4112yes—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4C2×C3×D412yes—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—44no—
391C42C2×C3×D412yes—
392C42C3×#4112yes—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—82no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3C2×#12111yescharacteristic
406C22×C3#54611yescharacteristic
407C22×C3#55011yescharacteristic
408C22×C3C2×AGL(1,ℤ/8)12yes—
409C22×C3#57712yes—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—24no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—82no—
417C3×C4—81no—
418C3×C4#52511yescharacteristic
419C3×C4#53511yescharacteristic
420C3×C4#55311yescharacteristic
421C3×C4#55511yescharacteristic
422C23—22no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23C22×C3×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439D4—22no—
440D4—22no—
441D4—42no—
442D4—41no—
443D4—42no—
444D4—42no—
445D4—41no—
446D4—42no—
447D4—88no—
448D4—84no—
449D4—84no—
450D4—88no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4C3×#12112yes—
470C2×C4C3×#12112yes—
471C2×C4C3×AGL(1,ℤ/8)14yes—
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C2×C3×#4111yescharacteristic
474C2×C4C2×C3×#4111yescharacteristic
475C2×C4C2×C3×#4111yescharacteristic
476C2×C4C3×H(5,𝔽2)11yescharacteristic
477C2×C4C3×H(5,𝔽2)11yescharacteristic
478C8—22no—
479C8—42no—
480C8—41no—
481C2×C3—82no—
482C2×C3—162no—
483C2×C3#241411yescharacteristic
484C2×C3#303811yescharacteristic
485C2×C3#304311yescharacteristic
486C2×C3#331012yes—
487C2×C3#339712yes—
488C22—42no—
489C22—44no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×C3×#12111yescharacteristic
501C22C3×#54611yescharacteristic
502C22C3×#55011yescharacteristic
503C22C2×C3×AGL(1,ℤ/8)12yes—
504C22C3×#57712yes—
505C4—22no—
506C4—22no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—81no—
513C4C3×#52511yescharacteristic
514C4C3×#53511yescharacteristic
515C4C3×#55311yescharacteristic
516C4C3×#55511yescharacteristic
517C3#2464411yescharacteristic, direct product factor, Sylow
518C2—82no—
519C2—162no—
520C2C3×#241411yescharacteristic
521C2C3×#303811yescharacteristic
522C2C3×#304311yescharacteristic
523C2C3×#331012yes—
524C2C3×#339712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2541C2
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
1. C2×C3×#460C2semidirect
2. C2×C3×#440C2semidirect
4. C3×#2356C2semidirect
5. C3×#2055C2semidirect
6. C3×#2014C2semidirect
7. C3×#1982C2semidirect
8. C3×#1961C2semidirect
9. C3×#1940C2semidirect
10. C3×#1925C2semidirect
11. C3×#1804C2semidirect
12. #24644C3direct (N × H)
44. C2×#460C2×C3semidirect
45. C2×#440C2×C3semidirect
47. #2356C2×C3semidirect
48. #2055C2×C3semidirect
49. #2014C2×C3semidirect
50. #1982C2×C3semidirect
51. #1961C2×C3semidirect
52. #1940C2×C3semidirect
53. #1925C2×C3semidirect
54. #1804C2×C3semidirect
78. C3×C4×Q8D4semidirect
233. C4×Q8C3×D4semidirect
517. C3#24644direct (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×#4402
C3×#25412
C3×#23562
C3×#20552
C3×#20142
C3×#19822
C3×#19612
C3×#19402
C3×#19252
C3×#18042
#246443