Finite Group Databasebeta

← C3×#21388 #10639995

Subgroups — C3×#21388 #10639995

Order 768.

1290 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×#529C211yesmaximal, characteristic
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#1989C212yesmaximal, complemented
5C3×#1983C212yesmaximal, complemented
6C3×#1874C212yesmaximal
7C3×#1872C212yesmaximal, complemented
8C3×C4×#103C212yesmaximal, complemented
9C3×C4×#102C212yesmaximal, complemented
10#21388C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#529—24no—
12C3×#529C418yescomplemented
13C2×C3×#126—21no—
14C2×C3×#125C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#123—21no—
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C42—21no—
25C22×C3×C42C2211yescharacteristic
26C3×#431—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2214yes—
29C2×C3×#102C2214yes—
30C2×C3×C22⋊C8C2212yes—
31C2×C3×C4×C8C2212yes—
32C3×C4×C4⋊C4—21no—
33C3×C4×C4⋊C4—21no—
34C3×C4×C4⋊C4C2212yes—
35C3×C4×C4⋊C4C2212yes—
36C2×C3×#95—21no—
37C3×#340C2214yes—
38C3×#340C2214yes—
39C2×#529C2×C311yescharacteristic
40C2×#431C2×C311yescharacteristic, complemented
41C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
42#1989C2×C312yescomplemented
43#1983C2×C312yescomplemented
44#1874C2×C312yes—
45#1872C2×C312yescomplemented
46C4×#103C2×C312yescomplemented
47C4×#102C2×C312yescomplemented
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—21no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—22no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2314yes—
57C3×#126—28no—
58C3×#125—22no—
59C3×#125C2×C414yescomplemented
60C3×#124—22no—
61C3×#124C2×C414yescomplemented
62C3×#123—28no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8C2×C414yes—
65C3×#121—22no—
66C3×#121C2×C414yes—
67C3×C4×Q8—22no—
68C3×C4×Q8C2×C414yescomplemented
69C3×C4×D4—28no—
70C3×C4×D4—22no—
71C3×C4×D4C2×C414yescomplemented
72C3×#117—28no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—28no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42—44no—
91C2×C3×C42C2×C414yes—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D414yes—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—24no—
98C3×#103—28no—
99C3×#102—28no—
100C3×C22⋊C8—24no—
101C3×C4×C8—24no—
102C3×#95—22no—
103C3×#95—22no—
104#529—24no—
105#529C3×C418yescomplemented
106C2×#126—21no—
107C2×#125C22×C311yescharacteristic
108C2×#124C22×C311yescharacteristic
109C2×#123—21no—
110C2×C22⋊Q8C22×C311yescharacteristic
111C2×#121C22×C311yescharacteristic
112C2×C4×Q8C22×C311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4C22×C311yescharacteristic
115C2×#117—21no—
116C22×C4⋊C4C22×C312yes—
117C22×C42—21no—
118C22×C42C22×C311yescharacteristic
119#431—24no—
120C2×C4⋊C8C22×C312yes—
121C2×#103C22×C314yes—
122C2×#102C22×C314yes—
123C2×C22⋊C8C22×C312yes—
124C2×C4×C8C22×C312yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C22×C312yes—
128C4×C4⋊C4C22×C312yes—
129C2×#95—21no—
130#340C22×C314yes—
131#340C22×C314yes—
132C24×C3—21no—
133C24×C3C2×D411yescharacteristic
134C2×C3×Q8—21no—
135C2×C3×Q8—24no—
136C2×C3×Q8C22×C412yes—
137C2×C3×D4—24no—
138C2×C3×D4—21no—
139C2×C3×D4—24no—
140C2×C3×D4—44no—
141C2×C3×D4C22×C412yes—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—28no—
147C22×C3×C4—24no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4C22⋊C414yes—
157C22×C3×C4C22⋊C414yes—
158C22×C3×C4C22×C412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—24no—
167C2×C3×C8—24no—
168C2×C3×C8—24no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4C22×C412yes—
178C3×C4⋊C4C22×C412yes—
179C3×C4⋊C4#4118yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C42—24no—
186C3×C42—44no—
187C3×C42—44no—
188C3×C42C22⋊C414yes—
189C3×C42C22⋊C414yes—
190C3×C42C22×C412yes—
191C3×C42C22×C412yes—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194C22×Q8C23×C311yescharacteristic
195C22×D4—21no—
196C22×D4C23×C311yescharacteristic
197C23×C4—22no—
198C23×C4—21no—
199C23×C4C3×D411yescharacteristic
200C23×C4C3×D411yescharacteristic
201C23×C4C23×C311yescharacteristic
202C22×C8C23×C314yes—
203#126—28no—
204#125—22no—
205#125C2×C3×C414yescomplemented
206#124—22no—
207#124C2×C3×C414yescomplemented
208#123—28no—
209C22⋊Q8—22no—
210C22⋊Q8C2×C3×C414yes—
211#121—22no—
212#121C2×C3×C414yes—
213C4×Q8—22no—
214C4×Q8C2×C3×C414yescomplemented
215C4×D4—28no—
216C4×D4—22no—
217C4×D4C2×C3×C414yescomplemented
218#117—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C314yes—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—44no—
237C2×C42C2×C3×C414yes—
238C2×C42C3×D411yescharacteristic
239C2×C42C3×D411yescharacteristic
240C2×C42C3×D414yes—
241C2×C42C23×C311yescharacteristic
242C2×C42C23×C311yescharacteristic
243C4⋊C8—24no—
244#103—28no—
245#102—28no—
246C22⋊C8—24no—
247C4×C8—24no—
248#95—22no—
249#95—22no—
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—22no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3C2×C22⋊C412yes—
259C23×C3#13514yes—
260C23×C3C22×D411yescharacteristic
261C3×Q8—24no—
262C3×Q8—42no—
263C3×D4—24no—
264C3×D4—48no—
265C3×D4—42no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—28no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—28no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—28no—
279C2×C3×C4—416no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—44no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—48no—
287C2×C3×C4C2×C22⋊C412yes—
288C2×C3×C4C2×C22⋊C412yes—
289C2×C3×C4C2×C22⋊C412yes—
290C2×C3×C4C2×C22⋊C412yes—
291C2×C3×C4C2×C22⋊C412yes—
292C2×C3×C4AGL(1,ℤ/8)12yes—
293C2×C3×C4#13712yes—
294C2×C3×C4C23×C411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4114yes—
297C3×C8—44no—
298C24—21no—
299C24C2×C3×D411yescharacteristic
300C2×Q8—21no—
301C2×Q8—24no—
302C2×Q8C22×C3×C412yes—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—24no—
306C2×D4—44no—
307C2×D4C22×C3×C412yes—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C3×C22⋊C414yes—
323C22×C4C3×C22⋊C414yes—
324C22×C4C22×C3×C412yes—
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D412yes—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C24×C311yescharacteristic
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—24no—
343C4⋊C4C22×C3×C412yes—
344C4⋊C4C22×C3×C412yes—
345C4⋊C4C3×#4118yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C42—24no—
352C42—44no—
353C42—44no—
354C42C3×C22⋊C414yes—
355C42C3×C22⋊C414yes—
356C42C22×C3×C412yes—
357C42C22×C3×C412yes—
358C42C2×C3×D412yes—
359C42C2×C3×D412yes—
360C22×C3—24no—
361C22×C3—24no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3#41614yes—
370C22×C3C22×C22⋊C411yescharacteristic
371C22×C3#52512yes—
372C22×C3C2×#13512yes—
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C3C2×#13711yescharacteristic
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—48no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4#41714yes—
384C3×C4C4×#4112yes—
385C3×C4#53212yes—
386C23—21no—
387C23—21no—
388C23—22no—
389C23—22no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23C2×C3×C22⋊C412yes—
395C23C3×#13514yes—
396C23C22×C3×D411yescharacteristic
397Q8—24no—
398Q8—42no—
399D4—24no—
400D4—48no—
401D4—42no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—28no—
415C2×C4—416no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4C2×C3×C22⋊C412yes—
424C2×C4C2×C3×C22⋊C412yes—
425C2×C4C2×C3×C22⋊C412yes—
426C2×C4C2×C3×C22⋊C412yes—
427C2×C4C2×C3×C22⋊C412yes—
428C2×C4C3×AGL(1,ℤ/8)12yes—
429C2×C4C3×#13712yes—
430C2×C4C23×C3×C411yescharacteristic
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C2×C3×#4114yes—
433C8—44no—
434C2×C3—24no—
435C2×C3—82no—
436C2×C3#229711yescharacteristic
437C2×C3C2×#41611yescharacteristic
438C2×C3C2×#41711yescharacteristic
439C2×C3#310512yes—
440C2×C3#310812yes—
441C22—22no—
442C22—22no—
443C22—24no—
444C22—24no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—82no—
449C22—82no—
450C22C3×#41614yes—
451C22C22×C3×C22⋊C411yescharacteristic
452C22C3×#52512yes—
453C22C2×C3×#13512yes—
454C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
455C22C2×C3×#13711yescharacteristic
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—48no—
464C4C3×#41714yes—
465C4C3×C4×#4112yes—
466C4C3×#53212yes—
467C3#2138811yescharacteristic, direct product factor, Sylow
468C2—24no—
469C2—82no—
470C2C3×#229711yescharacteristic
471C2C2×C3×#41611yescharacteristic
472C2C2×C3×#41711yescharacteristic
473C2C3×#310512yes—
474C2C3×#310812yes—
subgroupG/H
centerC22×C3×C4C2×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#431C2semidirect
3. C2×C3×C4×C4⋊C4C2semidirect
4. C3×#1989C2semidirect
5. C3×#1983C2semidirect
7. C3×#1872C2semidirect
8. C3×C4×#103C2semidirect
9. C3×C4×#102C2semidirect
10. #21388C3direct (N × H)
12. C3×#529C4semidirect
40. C2×#431C2×C3semidirect
41. C2×C4×C4⋊C4C2×C3semidirect
42. #1989C2×C3semidirect
43. #1983C2×C3semidirect
45. #1872C2×C3semidirect
46. C4×#103C2×C3semidirect
47. C4×#102C2×C3semidirect
59. C3×#125C2×C4semidirect
61. C3×#124C2×C4semidirect
68. C3×C4×Q8C2×C4semidirect
71. C3×C4×D4C2×C4semidirect
105. #529C3×C4semidirect
205. #125C2×C3×C4semidirect
207. #124C2×C3×C4semidirect
214. C4×Q8C2×C3×C4semidirect
217. C4×D4C2×C3×C4semidirect
467. C3#21388direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4312
C2×C3×C4×C4⋊C42
C3×#19892
C3×#19832
C3×#18742
C3×#18722
C3×C4×#1032
C3×C4×#1022
#213883