Finite Group Databasebeta

← C3×#27663 #10646270

Subgroups — C3×#27663 #10646270

Order 768.

860 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#435C211yesmaximal, characteristic
2C22×C3×#106C211yesmaximal, characteristic, complemented
3C3×#2487C211yesmaximal, characteristic
4C3×#2486C211yesmaximal, characteristic
5C3×#2088C212yesmaximal, complemented
6C3×#2077C212yesmaximal, complemented
7C3×#1943C212yesmaximal, complemented
8C3×#1824C211yesmaximal, characteristic, complemented
9C3×#1823C211yesmaximal, characteristic, complemented
10C3×#1809C212yesmaximal, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#27663C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#123—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#435—24no—
20C2×C3×#107C2212yes—
21C2×C3×#106—24no—
22C2×C3×#106C2212yes—
23C2×C3×#106C2212yes—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#400—21no—
29C3×#399—21no—
30C3×#398—21no—
31C3×#397—21no—
32C3×#389C2212yes—
33C3×#389C2212yes—
34C3×#386—21no—
35C3×#385—21no—
36C3×#381—21no—
37C3×#380C2211yescharacteristic
38C3×#380C2211yescharacteristic
39C3×#380C2212yes—
40C3×#380C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#435C2×C311yescharacteristic
47C22×#106C2×C311yescharacteristic, complemented
48#2487C2×C311yescharacteristic
49#2486C2×C311yescharacteristic
50#2088C2×C312yescomplemented
51#2077C2×C312yescomplemented
52#1943C2×C312yescomplemented
53#1824C2×C311yescharacteristic, complemented
54#1823C2×C311yescharacteristic, complemented
55#1809C2×C312yescomplemented
56#1808C2×C311yescharacteristic, complemented
57C22×C3×D4D411yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×M16—22no—
63C2×C3×M16—24no—
64C22×C3×C8—24no—
65C22×C3×C8D414yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#123—44no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×#107—24no—
94C3×#106—24no—
95C3×#106—24no—
96C3×C4⋊C8—22no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8—22no—
99C3×#97—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—22no—
105C3×#95—22no—
106C22×M16C22×C311yescharacteristic
107C23×C8C22×C311yescharacteristic
108C2×#123—21no—
109C2×C4×D4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112#435—24no—
113C2×#107C22×C312yes—
114C2×#106—24no—
115C2×#106C22×C312yes—
116C2×#106C22×C312yes—
117C2×C4⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C22⋊C8C22×C311yescharacteristic
120C2×#97C22×C311yescharacteristic
121#400—21no—
122#399—21no—
123#398—21no—
124#397—21no—
125#389C22×C312yes—
126#389C22×C312yes—
127#386—21no—
128#385—21no—
129#381—21no—
130#380C22×C311yescharacteristic
131#380C22×C311yescharacteristic
132#380C22×C312yes—
133#380C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137#340C22×C312yes—
138#340C22×C312yes—
139C24×C3C2×D411yescharacteristic
140C24×C3C2×D411yescharacteristic
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—82no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C3×M16—44no—
167C2×C3×C8—22no—
168C2×C3×C8—28no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8C2×D412yes—
174C2×C3×C8C2×D412yes—
175C2×C3×C8#4114yes—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—44no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—82no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—82no—
195C3×C42—21no—
196C3×C42—21no—
197C22×D4C3×D411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C311yescharacteristic
202C2×M16—22no—
203C2×M16—24no—
204C22×C8—24no—
205C22×C8C3×D414yes—
206C22×C8C23×C311yescharacteristic
207C22×C8C23×C311yescharacteristic
208C22×C8C23×C311yescharacteristic
209C22×C8C23×C311yescharacteristic
210#123—44no—
211C4×D4—22no—
212C4×D4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C3×D411yescharacteristic
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C312yes—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C312yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C3×D411yescharacteristic
231C2×C22⋊C4C3×D411yescharacteristic
232C2×C42C23×C311yescharacteristic
233#107—24no—
234#106—24no—
235#106—24no—
236C4⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#97—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—22no—
253C23×C3#12112yes—
254C23×C3AGL(1,ℤ/8)12yes—
255C23×C3#13712yes—
256C23×C3C22×D411yescharacteristic
257C3×D4—44no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—84no—
285C2×C3×C4—84no—
286C2×C3×C4#12112yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+412yes—
291C3×C8—22no—
292C3×C8—41no—
293C3×C8—41no—
294C3×C8#12314yescomplemented
295C24C2×C3×D411yescharacteristic
296C24C2×C3×D411yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D411yescharacteristic
319C22×C4C2×C3×D412yes—
320C22×C4C3×#4112yes—
321C22×C4C24×C311yescharacteristic
322M16—44no—
323C2×C8—22no—
324C2×C8—28no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8C2×C3×D412yes—
330C2×C8C2×C3×D412yes—
331C2×C8C3×#4114yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3#46814yes—
363C22×C3C2×#12111yescharacteristic
364C22×C3#54712yes—
365C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
366C22×C3C2×#13711yescharacteristic
367C22×C3#57712yes—
368C3×C4—22no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—82no—
373C3×C4—82no—
374C3×C4—82no—
375C3×C4—82no—
376C3×C4C2×#12311yescharacteristic
377C3×C4#53611yescharacteristic
378C3×C4#55212yes—
379C23—22no—
380C23—22no—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—21no—
385C23—22no—
386C23C3×#12112yes—
387C23C3×AGL(1,ℤ/8)12yes—
388C23C3×#13712yes—
389C23C22×C3×D411yescharacteristic
390D4—44no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4C3×#12112yes—
420C2×C4C22×C3×D411yescharacteristic
421C2×C4C2×C3×#4112yes—
422C2×C4C2×C3×#4111yescharacteristic
423C2×C4C3×2−1+412yes—
424C8—22no—
425C8—41no—
426C8—41no—
427C8C3×#12314yescomplemented
428C2×C3—24no—
429C2×C3—81no—
430C2×C3#248611yescharacteristic
431C2×C3C2×#46811yescharacteristic
432C2×C3#305711yescharacteristic
433C2×C3#334412yes—
434C2×C3#334512yes—
435C22—22no—
436C22—22no—
437C22—24no—
438C22—24no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22C3×#46814yes—
445C22C2×C3×#12111yescharacteristic
446C22C3×#54712yes—
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C2×C3×#13711yescharacteristic
449C22C3×#57712yes—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4C2×C3×#12311yescharacteristic
459C4C3×#53611yescharacteristic
460C4C3×#55212yes—
461C3#2766311yescharacteristic, direct product factor, Sylow
462C2—24no—
463C2—81no—
464C2C3×#248611yescharacteristic
465C2C2×C3×#46811yescharacteristic
466C2C3×#305711yescharacteristic
467C2C3×#334412yes—
468C2C3×#334512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#106C2semidirect
5. C3×#2088C2semidirect
6. C3×#2077C2semidirect
7. C3×#1943C2semidirect
8. C3×#1824C2semidirect
9. C3×#1823C2semidirect
10. C3×#1809C2semidirect
11. C3×#1808C2semidirect
12. #27663C3direct (N × H)
47. C22×#106C2×C3semidirect
50. #2088C2×C3semidirect
51. #2077C2×C3semidirect
52. #1943C2×C3semidirect
53. #1824C2×C3semidirect
54. #1823C2×C3semidirect
55. #1809C2×C3semidirect
56. #1808C2×C3semidirect
294. C3×C8#123semidirect
427. C8C3×#123semidirect
461. C3#27663direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4352
C22×C3×#1062
C3×#24872
C3×#24862
C3×#20882
C3×#20772
C3×#19432
C3×#18242
C3×#18232
C3×#18092
C3×#18082
#276633