Finite Group Databasebeta

← C3×#27644 #10646251

Subgroups — C3×#27644 #10646251

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#480C212yesmaximal
2C3×C4×#121C211yesmaximal, characteristic
3C2×C3×#384C211yesmaximal, characteristic, complemented
4C3×#2083C212yesmaximal, complemented
5C3×#2071C212yesmaximal, complemented
6C3×#1947C212yesmaximal, complemented
7C3×#1823C212yesmaximal, complemented
8C3×#1802C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic, complemented
10#27644C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#121C2211yescharacteristic
12C2×C3×C4×D4—21no—
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4—21no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#480—28no—
19C2×C3×#107C2214yes—
20C2×C3×C4⋊C8C2212yes—
21C2×C3×#102C2214yes—
22C2×C3×C22⋊C8C2212yes—
23C3×#390C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#388—21no—
26C3×#386—21no—
27C3×#384—21no—
28C3×#384—21no—
29C3×#384C2212yes—
30C3×#384C2212yes—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C3×#340C2214yes—
35C3×#340C2214yes—
36C3×#336C2212yes—
37C2×#480C2×C312yes—
38C4×#121C2×C311yescharacteristic
39C2×#384C2×C311yescharacteristic, complemented
40#2083C2×C312yescomplemented
41#2071C2×C312yescomplemented
42#1947C2×C312yescomplemented
43#1823C2×C312yescomplemented
44#1802C2×C312yescomplemented
45#1797C2×C311yescharacteristic, complemented
46C22×C3×D4—21no—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—22no—
49C23×C3×C4—21no—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×C3×C8C2314yes—
54C3×#121—22no—
55C3×#121D414yes—
56C3×C4×D4—24no—
57C3×C4×D4—44no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—44no—
67C2×C3×C4⋊C4—44no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2314yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—24no—
78C2×C3×C42—21no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#107—28no—
84C3×C4⋊C8—24no—
85C3×#102—28no—
86C3×C22⋊C8—24no—
87C3×#95—21no—
88C3×#95—21no—
89C3×#95—21no—
90C2×#121C22×C311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4C22×C311yescharacteristic
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C22×C312yes—
96C22×C42C22×C311yescharacteristic
97#480—28no—
98C2×#107C22×C314yes—
99C2×C4⋊C8C22×C312yes—
100C2×#102C22×C314yes—
101C2×C22⋊C8C22×C312yes—
102#390C22×C311yescharacteristic
103#389C22×C311yescharacteristic
104#388—21no—
105#386—21no—
106#384—21no—
107#384—21no—
108#384C22×C312yes—
109#384C22×C312yes—
110#380C22×C311yescharacteristic
111C4×C4⋊C4C22×C311yescharacteristic
112C4×C22⋊C4—21no—
113#340C22×C314yes—
114#340C22×C314yes—
115#336C22×C312yes—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C2×C3×D4—24no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—22no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C2×C3×D4C2×D412yes—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—24no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4—44no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4D814yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4112yes—
150C22×C3×C4C2411yescharacteristic
151C2×C3×C8—24no—
152C2×C3×C8—24no—
153C2×C3×C8—24no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—44no—
160C3×C4⋊C4—44no—
161C3×C4⋊C4—44no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4C2×D412yes—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—42no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—42no—
175C3×C42#4114yes—
176C22×D4—21no—
177C22×D4C23×C311yescharacteristic
178C23×C4—22no—
179C23×C4—21no—
180C23×C4C3×D411yescharacteristic
181C23×C4C3×D411yescharacteristic
182C23×C4C23×C311yescharacteristic
183C22×C8C23×C314yes—
184#121—22no—
185#121C3×D414yes—
186C4×D4—24no—
187C4×D4—44no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C23×C314yes—
199C2×C4⋊C4C23×C311yescharacteristic
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42C3×D411yescharacteristic
210C2×C42C3×D411yescharacteristic
211C2×C42C23×C311yescharacteristic
212C2×C42C23×C311yescharacteristic
213#107—28no—
214C4⋊C8—24no—
215#102—28no—
216C22⋊C8—24no—
217#95—21no—
218#95—21no—
219#95—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—22no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3#12112yes—
232C23×C3C2×D812yes—
233C23×C3AGL(1,ℤ/8)12yes—
234C23×C3C22×D411yescharacteristic
235C3×D4—24no—
236C3×D4—48no—
237C3×D4—42no—
238C2×C3×C4—28no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—28no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—44no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—88no—
263C2×C3×C4—84no—
264C2×C3×C4#12112yes—
265C2×C3×C4#12312yes—
266C2×C3×C4#12312yes—
267C2×C3×C4C2×D812yes—
268C2×C3×C4C2×D812yes—
269C2×C3×C4#13712yes—
270C2×C3×C4#13712yes—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C42−1+412yes—
275C3×C8—44no—
276C24—21no—
277C24C2×C3×D411yescharacteristic
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4C2×C3×D412yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C3×D814yes—
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D412yes—
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C3×#4112yes—
310C22×C4C24×C311yescharacteristic
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C4⋊C4—21no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4C2×C3×D412yes—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—42no—
335C42C3×#4114yes—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#45914yes—
350C22×C3C2×#12111yescharacteristic
351C22×C3C2×#12311yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3C22×D811yescharacteristic
354C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C3C2×#13711yescharacteristic
356C22×C3C2×#13711yescharacteristic
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—42no—
362C3×C4—81no—
363C3×C4—81no—
364C3×C4—84no—
365C3×C4#48014yescomplemented
366C3×C4#54712yes—
367C3×C4#55212yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C3×#12112yes—
380C23C2×C3×D812yes—
381C23C3×AGL(1,ℤ/8)12yes—
382C23C22×C3×D411yescharacteristic
383D4—24no—
384D4—48no—
385D4—42no—
386C2×C4—28no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—88no—
411C2×C4—84no—
412C2×C4C3×#12112yes—
413C2×C4C3×#12312yes—
414C2×C4C3×#12312yes—
415C2×C4C2×C3×D812yes—
416C2×C4C2×C3×D812yes—
417C2×C4C3×#13712yes—
418C2×C4C3×#13712yes—
419C2×C4C22×C3×D411yescharacteristic
420C2×C4C2×C3×#4112yes—
421C2×C4C2×C3×#4111yescharacteristic
422C2×C4C3×2−1+412yes—
423C8—44no—
424C2×C3—24no—
425C2×C3—82no—
426C2×C3#248611yescharacteristic
427C2×C3C2×#45911yescharacteristic
428C2×C3#303911yescharacteristic
429C2×C3C2×#48011yescharacteristic
430C2×C3#309211yescharacteristic
431C2×C3#322611yescharacteristic
432C2×C3#323811yescharacteristic
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22C3×#45914yes—
447C22C2×C3×#12111yescharacteristic
448C22C2×C3×#12311yescharacteristic
449C22C3×#53611yescharacteristic
450C22C22×C3×D811yescharacteristic
451C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
452C22C2×C3×#13711yescharacteristic
453C22C2×C3×#13711yescharacteristic
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—81no—
460C4—84no—
461C4—81no—
462C4C3×#48014yescomplemented
463C4C3×#54712yes—
464C4C3×#55212yes—
465C3#2764411yescharacteristic, direct product factor, Sylow
466C2—24no—
467C2—82no—
468C2C3×#248611yescharacteristic
469C2C2×C3×#45911yescharacteristic
470C2C3×#303911yescharacteristic
471C2C2×C3×#48011yescharacteristic
472C2C3×#309211yescharacteristic
473C2C3×#322611yescharacteristic
474C2C3×#323811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#384C2semidirect
4. C3×#2083C2semidirect
5. C3×#2071C2semidirect
6. C3×#1947C2semidirect
7. C3×#1823C2semidirect
8. C3×#1802C2semidirect
9. C3×#1797C2semidirect
10. #27644C3direct (N × H)
39. C2×#384C2×C3semidirect
40. #2083C2×C3semidirect
41. #2071C2×C3semidirect
42. #1947C2×C3semidirect
43. #1823C2×C3semidirect
44. #1802C2×C3semidirect
45. #1797C2×C3semidirect
365. C3×C4#480semidirect
462. C4C3×#480semidirect
465. C3#27644direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4802
C3×C4×#1212
C2×C3×#3842
C3×#20832
C3×#20712
C3×#19472
C3×#18232
C3×#18022
C3×#17972
#276443