Finite Group Databasebeta

← C3×#20661 #10639268

Subgroups — C3×#20661 #10639268

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic
2C2×C3×#428C212yesmaximal, complemented
3C3×#2280C211yesmaximal, characteristic
4C2×C3×#380C211yesmaximal, characteristic
5C3×#1809C214yesmaximal, complemented
6C3×#1808C212yesmaximal, complemented
7C3×#1807C212yesmaximal, complemented
8C3×#1736C212yesmaximal, complemented
9#20661C311yesmaximal, characteristic, direct product factor, Sylow
10C24×C3×C4C2211yescharacteristic
11C22×C3×M16—28no—
12C22×C3×M16C418yescomplemented
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2212yes—
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2212yes—
17C2×C3×#117C2212yes—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C22⋊C4—21no—
20C3×#428—28no—
21C2×C3×#107C2214yescomplemented
22C2×C3×#106C2214yescomplemented
23C3×#381—21no—
24C3×#380—21no—
25C3×#380—21no—
26C3×#380C2211yescharacteristic
27C3×#380C2212yes—
28C3×#380C2211yescharacteristic
29C3×#380C2212yes—
30C3×#380C2212yes—
31C3×#379—21no—
32C3×C4×C22⋊C4—21no—
33C3×C4×C22⋊C4—21no—
34C2×C3×#95—21no—
35C3×#340C2214yescomplemented
36C3×#340C2214yescomplemented
37C23×M16C2×C311yescharacteristic
38C2×#428C2×C312yescomplemented
39#2280C2×C311yescharacteristic
40C2×#380C2×C311yescharacteristic
41#1809C2×C314yescomplemented
42#1808C2×C312yescomplemented
43#1807C2×C312yescomplemented
44#1736C2×C312yescomplemented
45C25×C3D411yescharacteristic
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—22no—
49C23×C3×C4C2×C414yes—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4Q814yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C2×C3×M16—24no—
58C2×C3×M16—232no—
59C2×C3×M16—28no—
60C2×C3×M16C2×C414yes—
61C2×C3×M16C2×C414yes—
62C2×C3×M16C2×C418yescomplemented
63C2×C3×M16D418yes—
64C22×C3×C8—28no—
65C22×C3×C8C2×C418yescomplemented
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×#117—24no—
69C3×#117—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2314yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—44no—
83C2×C3×C42—22no—
84C3×#107—28no—
85C3×#106—28no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—21no—
89C3×#95—21no—
90C3×#95—22no—
91C24×C4C22×C311yescharacteristic
92C22×M16—28no—
93C22×M16C3×C418yescomplemented
94C22×M16C22×C311yescharacteristic
95C22×M16C22×C312yes—
96C22×M16C22×C311yescharacteristic
97C23×C8C22×C312yes—
98C2×#117C22×C312yes—
99C22×C4⋊C4C22×C312yes—
100C22×C22⋊C4—21no—
101#428—28no—
102C2×#107C22×C314yescomplemented
103C2×#106C22×C314yescomplemented
104#381—21no—
105#380—21no—
106#380—21no—
107#380C22×C311yescharacteristic
108#380C22×C312yes—
109#380C22×C311yescharacteristic
110#380C22×C312yes—
111#380C22×C312yes—
112#379—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C2×#95—21no—
116#340C22×C314yescomplemented
117#340C22×C314yescomplemented
118C24×C3—24no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—42no—
122C24×C3C4⋊C414yes—
123C24×C3C2×D411yescharacteristic
124C24×C3C2×D411yescharacteristic
125C24×C3C2×D411yescharacteristic
126C22×C3×C4—24no—
127C22×C3×C4—28no—
128C22×C3×C4—28no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—216no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—24no—
139C22×C3×C4—44no—
140C22×C3×C4—44no—
141C22×C3×C4—44no—
142C22×C3×C4C4⋊C414yes—
143C22×C3×C4C4⋊C414yes—
144C22×C3×C4C4⋊C414yes—
145C22×C3×C4C22×C412yes—
146C22×C3×C4C22×C412yes—
147C22×C3×C4C22×C412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D414yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×Q812yes—
153C22×C3×C4C2×Q812yes—
154C22×C3×C4C2×Q812yes—
155C22×C3×C4#4114yes—
156C22×C3×C4C2411yescharacteristic
157C3×M16—28no—
158C3×M16—216no—
159C3×M16C22⋊C4116yescomplemented
160C2×C3×C8—24no—
161C2×C3×C8—216no—
162C2×C3×C8C22×C414yes—
163C2×C3×C8C22×C414yes—
164C2×C3×C8C2×D418yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—44no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—42no—
178C3×C42—44no—
179C25C3×D411yescharacteristic
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4C2×C3×C414yes—
184C23×C4C3×D411yescharacteristic
185C23×C4C3×D411yescharacteristic
186C23×C4C3×D411yescharacteristic
187C23×C4C3×Q814yes—
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C311yescharacteristic
191C2×M16—24no—
192C2×M16—232no—
193C2×M16—28no—
194C2×M16C2×C3×C414yes—
195C2×M16C2×C3×C414yes—
196C2×M16C2×C3×C418yescomplemented
197C2×M16C3×D418yes—
198C22×C8—28no—
199C22×C8C2×C3×C418yescomplemented
200C22×C8C23×C312yes—
201C22×C8C23×C312yes—
202#117—24no—
203#117—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C23×C314yes—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—44no—
217C2×C42—22no—
218#107—28no—
219#106—28no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225C23×C3—22no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—22no—
234C23×C3—28no—
235C23×C3—28no—
236C23×C3—44no—
237C23×C3—48no—
238C23×C3—48no—
239C23×C3C2×C4⋊C412yes—
240C23×C3C2×C4⋊C412yes—
241C23×C3C2×C4⋊C412yes—
242C23×C3#12114yes—
243C23×C3AGL(1,ℤ/8)12yes—
244C23×C3#13712yes—
245C23×C3C22×D411yescharacteristic
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—28no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—28no—
256C2×C3×C4—28no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—88no—
267C2×C3×C4C2×C22⋊C414yes—
268C2×C3×C4C2×C4⋊C412yes—
269C2×C3×C4C2×C4⋊C412yes—
270C2×C3×C4C2×C4⋊C412yes—
271C2×C3×C4#11714yes—
272C2×C3×C4#12114yes—
273C2×C3×C4C22⋊Q818yes—
274C2×C3×C4C23×C411yescharacteristic
275C2×C3×C4C22×D412yes—
276C2×C3×C4C22×Q811yescharacteristic
277C2×C3×C4C2×#4112yes—
278C3×C8—24no—
279C3×C8C2×C22⋊C418yescomplemented
280C24—24no—
281C24—22no—
282C24—22no—
283C24—42no—
284C24C3×C4⋊C414yes—
285C24C2×C3×D411yescharacteristic
286C24C2×C3×D411yescharacteristic
287C24C2×C3×D411yescharacteristic
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—216no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4C3×C4⋊C414yes—
305C22×C4C3×C4⋊C414yes—
306C22×C4C3×C4⋊C414yes—
307C22×C4C22×C3×C412yes—
308C22×C4C22×C3×C412yes—
309C22×C4C22×C3×C412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D414yes—
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×Q812yes—
315C22×C4C2×C3×Q812yes—
316C22×C4C2×C3×Q812yes—
317C22×C4C3×#4114yes—
318C22×C4C24×C311yescharacteristic
319M16—28no—
320M16—216no—
321M16C3×C22⋊C4116yescomplemented
322C2×C8—24no—
323C2×C8—216no—
324C2×C8C22×C3×C414yes—
325C2×C8C22×C3×C414yes—
326C2×C8C2×C3×D418yes—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C42—44no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—416no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3#38014yes—
356C22×C3#42814yescomplemented
357C22×C3C22×C4⋊C411yescharacteristic
358C22×C3C2×#12112yes—
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3C2×#13711yescharacteristic
361C22×C3#57512yes—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—82no—
365C3×C4—84no—
366C3×C4—82no—
367C3×C4#38014yes—
368C3×C4C22×C22⋊C411yescharacteristic
369C3×C4C2×#11711yescharacteristic
370C3×C4C2×C22⋊Q812yes—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—28no—
380C23—28no—
381C23—22no—
382C23—44no—
383C23—48no—
384C23—48no—
385C23C2×C3×C4⋊C412yes—
386C23C2×C3×C4⋊C412yes—
387C23C2×C3×C4⋊C412yes—
388C23C3×#12114yes—
389C23C3×AGL(1,ℤ/8)12yes—
390C23C3×#13712yes—
391C23C22×C3×D411yescharacteristic
392C2×C4—28no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—88no—
413C2×C4C2×C3×C22⋊C414yes—
414C2×C4C2×C3×C4⋊C412yes—
415C2×C4C2×C3×C4⋊C412yes—
416C2×C4C2×C3×C4⋊C412yes—
417C2×C4C3×#11714yes—
418C2×C4C3×#12114yes—
419C2×C4C3×C22⋊Q818yes—
420C2×C4C23×C3×C411yescharacteristic
421C2×C4C22×C3×D412yes—
422C2×C4C22×C3×Q811yescharacteristic
423C2×C4C2×C3×#4112yes—
424C8—24no—
425C8C2×C3×C22⋊C418yescomplemented
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—42no—
429C2×C3C2×#38011yescharacteristic
430C2×C3C2×#42811yescharacteristic
431C2×C3#291111yescharacteristic
432C2×C3#305512yes—
433C2×C3#305612yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—416no—
446C22—42no—
447C22—42no—
448C22C3×#38014yes—
449C22C3×#42814yescomplemented
450C22C22×C3×C4⋊C411yescharacteristic
451C22C2×C3×#12112yes—
452C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
453C22C2×C3×#13711yescharacteristic
454C22C3×#57512yes—
455C4—22no—
456C4—22no—
457C4—84no—
458C4—82no—
459C4—82no—
460C4C3×#38014yes—
461C4C22×C3×C22⋊C411yescharacteristic
462C4C2×C3×#11711yescharacteristic
463C4C2×C3×C22⋊Q812yes—
464C3#2066111yescharacteristic, direct product factor, Sylow
465C2—24no—
466C2—24no—
467C2—42no—
468C2C2×C3×#38011yescharacteristic
469C2C2×C3×#42811yescharacteristic
470C2C3×#291111yescharacteristic
471C2C3×#305512yes—
472C2C3×#305612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
5. C3×#1809C2semidirect
6. C3×#1808C2semidirect
7. C3×#1807C2semidirect
8. C3×#1736C2semidirect
9. #20661C3direct (N × H)
12. C22×C3×M16C4semidirect
21. C2×C3×#107C22semidirect
22. C2×C3×#106C22semidirect
35. C3×#340C22semidirect
36. C3×#340C22semidirect
38. C2×#428C2×C3semidirect
41. #1809C2×C3semidirect
42. #1808C2×C3semidirect
43. #1807C2×C3semidirect
44. #1736C2×C3semidirect
62. C2×C3×M16C2×C4semidirect
65. C22×C3×C8C2×C4semidirect
93. C22×M16C3×C4semidirect
102. C2×#107C22×C3semidirect
103. C2×#106C22×C3semidirect
116. #340C22×C3semidirect
117. #340C22×C3semidirect
159. C3×M16C22⋊C4semidirect
196. C2×M16C2×C3×C4semidirect
199. C22×C8C2×C3×C4semidirect
279. C3×C8C2×C22⋊C4semidirect
321. M16C3×C22⋊C4semidirect
356. C22×C3#428semidirect
425. C8C2×C3×C22⋊C4semidirect
449. C22C3×#428semidirect
464. C3#20661direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C2×C3×#4282
C3×#22802
C2×C3×#3802
C3×#18092
C3×#18082
C3×#18072
C3×#17362
#206613