Finite Group Databasebeta

← C3×#24681 #10643288

Subgroups — C3×#24681 #10643288

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic
2C2×C3×C4×SD16C211yesmaximal, characteristic
3C3×#2568C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic
5C3×#2073C212yesmaximal
6C3×#2055C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1928C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24681C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×#117—21no—
18C3×#474—24no—
19C3×C4×SD16—24no—
20C2×C3×#107C2212yes—
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×C4×C8C2211yescharacteristic
29C3×#400—22no—
30C3×#391—21no—
31C3×#391—21no—
32C3×#391C2212yes—
33C3×#390C2212yes—
34C3×#388—22no—
35C3×#384C2211yescharacteristic
36C3×#382C2212yes—
37C3×#380—21no—
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#474C2×C311yescharacteristic
45C2×C4×SD16C2×C311yescharacteristic
46#2568C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic
48#2073C2×C312yes—
49#2055C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1928C2×C312yescomplemented
54#1803C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C3×#117—44no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—22no—
86C2×C3×C42—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#106—22no—
91C3×C4⋊C8—24no—
92C3×C4⋊C8—22no—
93C3×#103—24no—
94C3×#103—22no—
95C3×#102—22no—
96C3×#102—24no—
97C3×C4×C8—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95D412yes—
104C22×SD16C22×C311yescharacteristic
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C4×Q8C22×C311yescharacteristic
107C2×C4×D4C22×C311yescharacteristic
108C2×#117—21no—
109#474—24no—
110C4×SD16—24no—
111C2×#107C22×C312yes—
112C2×#106C22×C311yescharacteristic
113C2×C4⋊C8C22×C312yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#103C22×C312yes—
116C2×#103C22×C311yescharacteristic
117C2×#102C22×C312yes—
118C2×#102C22×C311yescharacteristic
119C2×C4×C8C22×C311yescharacteristic
120#400—22no—
121#391—21no—
122#391—21no—
123#391C22×C312yes—
124#390C22×C312yes—
125#388—22no—
126#384C22×C311yescharacteristic
127#382C22×C312yes—
128#380—21no—
129C4×C4⋊C4C22×C312yes—
130C4×C4⋊C4C22×C311yescharacteristic
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×Q8—42no—
144C2×C3×Q8#4112yes—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×SD16—28no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8#4112yes—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—81no—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—82no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—44no—
194C3×C42—42no—
195C3×C42C2×D412yes—
196C3×C42#4112yes—
197C22×Q8—21no—
198C22×Q8C23×C311yescharacteristic
199C22×D4C23×C311yescharacteristic
200C23×C4—21no—
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16—22no—
204C22×C8C23×C311yescharacteristic
205C22×C8C23×C312yes—
206C22×C8C23×C311yescharacteristic
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×D4—22no—
212C4×D4C3×D414yes—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C3×D411yescharacteristic
219C2×C4⋊C4C3×D411yescharacteristic
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42C23×C312yes—
230C2×C42C23×C311yescharacteristic
231#107—24no—
232#106—22no—
233C4⋊C8—24no—
234C4⋊C8—22no—
235#103—24no—
236#103—22no—
237#102—22no—
238#102—24no—
239C4×C8—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95C3×D412yes—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C22×D411yescharacteristic
254C3×Q8—22no—
255C3×Q8—22no—
256C3×Q8—41no—
257C3×Q8—41no—
258C3×Q8—84no—
259C3×D4—22no—
260C3×D4—22no—
261C3×D4—41no—
262C3×D4—41no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—48no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4#12112yes—
288C2×C3×C4#12112yes—
289C2×C3×C4#13512yes—
290C2×C3×C4AGL(1,ℤ/8)12yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C42−1+411yescharacteristic
296C2×C3×C42−1+411yescharacteristic
297C3×C8—22no—
298C3×C8—42no—
299C3×C8—41no—
300C24—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8C3×#4112yes—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4C2×C3×D412yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C3×D412yes—
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C24×C311yescharacteristic
329SD16—28no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8C3×#4112yes—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4C2×C3×D412yes—
353C22⋊C4—24no—
354C22⋊C4—44no—
355C22⋊C4—82no—
356C42—22no—
357C42—22no—
358C42—44no—
359C42—42no—
360C42C2×C3×D412yes—
361C42C3×#4112yes—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3C2×#12111yescharacteristic
370C22×C3#54711yescharacteristic
371C22×C3#54811yescharacteristic
372C22×C3C2×#13511yescharacteristic
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C3#57711yescharacteristic
375C22×C3#57811yescharacteristic
376C3×C4—22no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—82no—
382C3×C4—82no—
383C3×C4—81no—
384C3×C4#46314yes—
385C3×C4#52511yescharacteristic
386C3×C4#53611yescharacteristic
387C3×C4#55211yescharacteristic
388C3×C4#55611yescharacteristic
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C22×C3×D411yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—41no—
400Q8—41no—
401Q8—84no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4C3×#12112yes—
431C2×C4C3×#12112yes—
432C2×C4C3×#13512yes—
433C2×C4C3×AGL(1,ℤ/8)12yes—
434C2×C4C22×C3×D411yescharacteristic
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C2×C3×#4111yescharacteristic
437C2×C4C2×C3×#4111yescharacteristic
438C2×C4C3×2−1+411yescharacteristic
439C2×C4C3×2−1+411yescharacteristic
440C8—22no—
441C8—42no—
442C8—41no—
443C2×C3—82no—
444C2×C3#241511yescharacteristic
445C2×C3C2×#46311yescharacteristic
446C2×C3#304611yescharacteristic
447C2×C3#334011yescharacteristic
448C2×C3#335311yescharacteristic
449C2×C3#336411yescharacteristic
450C2×C3#336811yescharacteristic
451C22—42no—
452C22—42no—
453C22—42no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22C2×C3×#12111yescharacteristic
459C22C3×#54711yescharacteristic
460C22C3×#54811yescharacteristic
461C22C2×C3×#13511yescharacteristic
462C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
463C22C3×#57711yescharacteristic
464C22C3×#57811yescharacteristic
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4—82no—
473C4C3×#46314yes—
474C4C3×#52511yescharacteristic
475C4C3×#53611yescharacteristic
476C4C3×#55211yescharacteristic
477C4C3×#55611yescharacteristic
478C3#2468111yescharacteristic, direct product factor, Sylow
479C2—82no—
480C2C3×#241511yescharacteristic
481C2C2×C3×#46311yescharacteristic
482C2C3×#304611yescharacteristic
483C2C3×#334011yescharacteristic
484C2C3×#335311yescharacteristic
485C2C3×#336411yescharacteristic
486C2C3×#336811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
3. C3×#2568C2semidirect
7. C3×#1990C2semidirect
8. C3×#1949C2semidirect
9. C3×#1939C2semidirect
10. C3×#1928C2semidirect
11. C3×#1803C2semidirect
12. #24681C3direct (N × H)
46. #2568C2×C3semidirect
50. #1990C2×C3semidirect
51. #1949C2×C3semidirect
52. #1939C2×C3semidirect
53. #1928C2×C3semidirect
54. #1803C2×C3semidirect
478. C3#24681direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×C4×SD162
C3×#25682
C3×#23542
C3×#20732
C3×#20552
C3×#19902
C3×#19492
C3×#19392
C3×#19282
C3×#18032
#246813