Finite Group Databasebeta

← C3×#24633 #10643240

Subgroups — C3×#24633 #10643240

Order 768.

898 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic
2C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic
4C3×#2355C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1983C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24633C311yesmaximal, 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×C4×Q16—24no—
21C2×C3×#107C2211yescharacteristic
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×C4×C8C2211yescharacteristic
31C3×#394—22no—
32C3×#391C2212yes—
33C3×#390—21no—
34C3×#390—21no—
35C3×#390C2212yes—
36C3×#387—22no—
37C3×#384C2211yescharacteristic
38C3×#382—21no—
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#460C2×C311yescharacteristic
45C2×C4×Q16C2×C311yescharacteristic, complemented
46#2548C2×C311yescharacteristic
47#2355C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2015C2×C312yescomplemented
50#1983C2×C312yescomplemented
51#1957C2×C311yescharacteristic
52#1939C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1803C2×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⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
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—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42—22no—
93C2×C3×C42C2312yes—
94C2×C3×C42C2311yescharacteristic
95C3×#107—22no—
96C3×C4⋊C8—22no—
97C3×C4⋊C8—42no—
98C3×#103—22no—
99C3×#103—22no—
100C3×#103—42no—
101C3×#102—24no—
102C3×#102—24no—
103C3×#102—42no—
104C3×C4×C8—22no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95D412yes—
109C22×Q16C22×C311yescharacteristic
110C22×SD16C22×C312yes—
111C2×#127C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×Q8C22×C311yescharacteristic
115#460—24no—
116C4×Q16—24no—
117C2×#107C22×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×C4×C8C22×C311yescharacteristic
127#394—22no—
128#391C22×C312yes—
129#390—21no—
130#390—21no—
131#390C22×C312yes—
132#387—22no—
133#384C22×C311yescharacteristic
134#382—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C22×C312yes—
137C4×C4⋊C4C22×C311yescharacteristic
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—42no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—81no—
164C2×C3×D4—81no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×Q16—28no—
176C3×SD16—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—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—81no—
197C3×C4⋊C4C2×D412yes—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—82no—
201C3×C22⋊C4—82no—
202C3×C42—24no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—42no—
206C3×C42—82no—
207C3×C42C2×D412yes—
208C3×C42#4112yes—
209C22×Q8C23×C311yescharacteristic
210C22×Q8C23×C311yescharacteristic
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C23×C312yes—
214C2×Q16—22no—
215C2×Q16—22no—
216C2×Q16—22no—
217C2×SD16—24no—
218C2×SD16—42no—
219C2×SD16—42no—
220C22×C8C23×C312yes—
221C22×C8C23×C311yescharacteristic
222C22×C8C23×C311yescharacteristic
223#127—22no—
224#127—22no—
225#124—24no—
226#124—42no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8C3×D414yes—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C3×D411yescharacteristic
236C2×C4⋊C4C3×D411yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42C23×C312yes—
248C2×C42C23×C311yescharacteristic
249#107—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252#103—22no—
253#103—22no—
254#103—42no—
255#102—24no—
256#102—24no—
257#102—42no—
258C4×C8—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95C3×D412yes—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—22no—
275C3×Q8—41no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—41no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—84no—
284C3×D4—84no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
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—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4#12112yes—
309C2×C3×C4#12112yes—
310C2×C3×C4#13512yes—
311C2×C3×C4#13712yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C3×C8—22no—
319C3×C8—41no—
320C3×C8—42no—
321C24—22no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C2×C3×D412yes—
333C2×Q8C3×#4112yes—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—81no—
345C2×D4—81no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C24×C311yescharacteristic
356Q16—28no—
357SD16—48no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8C3×#4112yes—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—81no—
378C4⋊C4C2×C3×D412yes—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—24no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42—82no—
388C42C2×C3×D412yes—
389C42C3×#4112yes—
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×C3C2×#13511yescharacteristic
401C22×C3C2×#13711yescharacteristic
402C22×C3#57611yescharacteristic
403C22×C3#57711yescharacteristic
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—82no—
411C3×C4—81no—
412C3×C4#46414yes—
413C3×C4#52511yescharacteristic
414C3×C4#53511yescharacteristic
415C3×C4#55311yescharacteristic
416C3×C4#55511yescharacteristic
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C22×C3×D411yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—84no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
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—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4C3×#12112yes—
463C2×C4C3×#12112yes—
464C2×C4C3×#13512yes—
465C2×C4C3×#13712yes—
466C2×C4C22×C3×D411yescharacteristic
467C2×C4C2×C3×#4111yescharacteristic
468C2×C4C2×C3×#4111yescharacteristic
469C2×C4C2×C3×#4111yescharacteristic
470C2×C4C3×H(5,𝔽2)11yescharacteristic
471C2×C4C3×H(5,𝔽2)11yescharacteristic
472C8—22no—
473C8—42no—
474C8—41no—
475C2×C3—162no—
476C2×C3#241411yescharacteristic
477C2×C3C2×#46411yescharacteristic
478C2×C3#304511yescharacteristic
479C2×C3#330411yescharacteristic
480C2×C3#333311yescharacteristic
481C2×C3#339611yescharacteristic
482C2×C3#340011yescharacteristic
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×C3×#12111yescharacteristic
491C22C3×#54611yescharacteristic
492C22C3×#55011yescharacteristic
493C22C2×C3×#13511yescharacteristic
494C22C2×C3×#13711yescharacteristic
495C22C3×#57611yescharacteristic
496C22C3×#57711yescharacteristic
497C4—22no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—81no—
505C4C3×#46414yes—
506C4C3×#52511yescharacteristic
507C4C3×#53511yescharacteristic
508C4C3×#55311yescharacteristic
509C4C3×#55511yescharacteristic
510C3#2463311yescharacteristic, direct product factor, Sylow
511C2—162no—
512C2C3×#241411yescharacteristic
513C2C2×C3×#46411yescharacteristic
514C2C3×#304511yescharacteristic
515C2C3×#330411yescharacteristic
516C2C3×#333311yescharacteristic
517C2C3×#339611yescharacteristic
518C2C3×#340011yescharacteristic
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×C4×Q16C2semidirect
4. C3×#2355C2semidirect
5. C3×#2055C2semidirect
6. C3×#2015C2semidirect
7. C3×#1983C2semidirect
9. C3×#1939C2semidirect
10. C3×#1925C2semidirect
11. C3×#1803C2semidirect
12. #24633C3direct (N × H)
45. C2×C4×Q16C2×C3semidirect
47. #2355C2×C3semidirect
48. #2055C2×C3semidirect
49. #2015C2×C3semidirect
50. #1983C2×C3semidirect
52. #1939C2×C3semidirect
53. #1925C2×C3semidirect
54. #1803C2×C3semidirect
510. C3#24633direct (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×C4×Q162
C3×#25482
C3×#23552
C3×#20552
C3×#20152
C3×#19832
C3×#19572
C3×#19392
C3×#19252
C3×#18032
#246333