Finite Group Databasebeta

← C3×#24643 #10643250

Subgroups — C3×#24643 #10643250

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2012C212yesmaximal
7C3×#1983C212yesmaximal, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1927C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24643C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#461—24no—
20C3×#442—24no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×#97C2211yescharacteristic
31C3×#397—21no—
32C3×#397—21no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#388—22no—
36C3×#382C2211yescharacteristic
37C3×#381—21no—
38C3×#380—22no—
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#461C2×C311yescharacteristic, complemented
45C2×#442C2×C311yescharacteristic
46#2547C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2012C2×C312yes—
50#1983C2×C312yescomplemented
51#1960C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1927C2×C312yescomplemented
54#1804C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C2×C3×SD16—24no—
61C2×C3×SD16—42no—
62C2×C3×SD16—42no—
63C2×C3×D8—22no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#124—24no—
72C3×#124—42no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D414yes—
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—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—22no—
90C2×C3×C42C2312yes—
91C2×C3×C42C2311yescharacteristic
92C3×#106—22no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—42no—
95C3×#103—24no—
96C3×#103—24no—
97C3×#103—42no—
98C3×#102—22no—
99C3×#102—22no—
100C3×#102—42no—
101C3×#97—22no—
102C3×#95—21no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—21no—
107C3×#95D411yescharacteristic
108C3×#95D411yescharacteristic
109C22×SD16C22×C312yes—
110C22×D8C22×C311yescharacteristic
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×D4C22×C311yescharacteristic
114C2×C4×D4C22×C311yescharacteristic
115#461—24no—
116#442—24no—
117C2×#106C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C312yes—
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#102C22×C311yescharacteristic
124C2×#102C22×C311yescharacteristic
125C2×#102C22×C312yes—
126C2×#97C22×C311yescharacteristic
127#397—21no—
128#397—21no—
129#391C22×C312yes—
130#390C22×C312yes—
131#388—22no—
132#382C22×C311yescharacteristic
133#381—21no—
134#380—22no—
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C311yescharacteristic
137C4×C22⋊C4—21no—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—21no—
141C24×C3—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4C2×D412yes—
159C2×C3×D4#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—44no—
166C22×C3×C4—42no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C3×SD16—48no—
174C3×D8—28no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8#4112yes—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×D412yes—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—82no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—82no—
202C3×C42C2×D412yes—
203C3×C42#4112yes—
204C22×Q8C23×C312yes—
205C22×D4C23×C311yescharacteristic
206C22×D4C23×C311yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C2×SD16—24no—
210C2×SD16—42no—
211C2×SD16—42no—
212C2×D8—22no—
213C2×D8—22no—
214C2×D8—22no—
215C22×C8C23×C312yes—
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C311yescharacteristic
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220#124—24no—
221#124—42no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C3×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—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C42—22no—
239C2×C42C23×C312yes—
240C2×C42C23×C311yescharacteristic
241#106—22no—
242C4⋊C8—22no—
243C4⋊C8—42no—
244#103—24no—
245#103—24no—
246#103—42no—
247#102—22no—
248#102—22no—
249#102—42no—
250#97—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95C3×D411yescharacteristic
257#95C3×D411yescharacteristic
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3C22×D411yescharacteristic
269C3×Q8—42no—
270C3×Q8—42no—
271C3×Q8—42no—
272C3×Q8—42no—
273C3×D4—22no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—41no—
278C3×D4—41no—
279C3×D4—41no—
280C3×D4—41no—
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—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—84no—
300C2×C3×C4—84no—
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—21no—
313C24—21no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4C2×C3×D412yes—
331C2×D4C3×#4112yes—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345SD16—48no—
346D8—28no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8C3×#4112yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4C2×C3×D412yes—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C42—22no—
372C42—22no—
373C42—82no—
374C42C2×C3×D412yes—
375C42C3×#4112yes—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—44no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—82no—
386C22×C3C2×#12111yescharacteristic
387C22×C3#54611yescharacteristic
388C22×C3#55011yescharacteristic
389C22×C3#57512yes—
390C22×C3#57712yes—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—82no—
397C3×C4—82no—
398C3×C4—81no—
399C3×C4#52511yescharacteristic
400C3×C4#53511yescharacteristic
401C3×C4#55311yescharacteristic
402C3×C4#55511yescharacteristic
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C22×C3×D411yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418D4—22no—
419D4—22no—
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
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—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4C3×#12112yes—
447C2×C4C3×#12112yes—
448C2×C4C22×C3×D411yescharacteristic
449C2×C4C2×C3×#4111yescharacteristic
450C2×C4C2×C3×#4111yescharacteristic
451C2×C4C2×C3×#4111yescharacteristic
452C2×C4C3×H(5,𝔽2)11yescharacteristic
453C2×C4C3×H(5,𝔽2)11yescharacteristic
454C8—22no—
455C8—42no—
456C8—41no—
457C2×C3—82no—
458C2×C3—82no—
459C2×C3#241411yescharacteristic
460C2×C3#304011yescharacteristic
461C2×C3#304211yescharacteristic
462C2×C3#330712yes—
463C2×C3#340112yes—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—44no—
469C22—82no—
470C22—82no—
471C22—81no—
472C22—81no—
473C22—82no—
474C22C2×C3×#12111yescharacteristic
475C22C3×#54611yescharacteristic
476C22C3×#55011yescharacteristic
477C22C3×#57512yes—
478C22C3×#57712yes—
479C4—22no—
480C4—22no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—82no—
486C4—81no—
487C4C3×#52511yescharacteristic
488C4C3×#53511yescharacteristic
489C4C3×#55311yescharacteristic
490C4C3×#55511yescharacteristic
491C3#2464311yescharacteristic, direct product factor, Sylow
492C2—82no—
493C2—82no—
494C2C3×#241411yescharacteristic
495C2C3×#304011yescharacteristic
496C2C3×#304211yescharacteristic
497C2C3×#330712yes—
498C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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×#461C2semidirect
3. C3×#2547C2semidirect
4. C3×#2357C2semidirect
5. C3×#2055C2semidirect
7. C3×#1983C2semidirect
8. C3×#1960C2semidirect
9. C3×#1940C2semidirect
10. C3×#1927C2semidirect
11. C3×#1804C2semidirect
12. #24643C3direct (N × H)
44. C2×#461C2×C3semidirect
46. #2547C2×C3semidirect
47. #2357C2×C3semidirect
48. #2055C2×C3semidirect
50. #1983C2×C3semidirect
51. #1960C2×C3semidirect
52. #1940C2×C3semidirect
53. #1927C2×C3semidirect
54. #1804C2×C3semidirect
491. C3#24643direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×C3×#4422
C3×#25472
C3×#23572
C3×#20552
C3×#20122
C3×#19832
C3×#19602
C3×#19402
C3×#19272
C3×#18042
#246433