Finite Group Databasebeta

← C3×#32980 #10651587

Subgroups — C3×#32980 #10651587

Order 768.

1508 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
1C22×C3×#117C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic, complemented
3C2×C3×#480C211yesmaximal, characteristic, complemented
4C3×#2847C211yesmaximal, characteristic
5C3×#1875C214yesmaximal, complemented
6C3×#1854C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1823C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#32980C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#121C2211yescharacteristic
14C2×C3×#120—21no—
15C2×C3×#117—22no—
16C2×C3×#117—24no—
17C2×C3×#117—24no—
18C2×C3×#117C2214yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42—21no—
23C3×#481—24no—
24C3×#480—24no—
25C2×C3×#107C2212yescomplemented
26C2×C3×#106C2212yescomplemented
27C2×C3×#102C2212yescomplemented
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2214yes—
32C3×#386—21no—
33C3×#386—22no—
34C3×#380C2211yescharacteristic
35C3×#380C2212yes—
36C3×#340C2214yescomplemented
37C3×#340C2214yescomplemented
38C22×#117C2×C311yescharacteristic, complemented
39C2×#481C2×C311yescharacteristic, complemented
40C2×#480C2×C311yescharacteristic, complemented
41#2847C2×C311yescharacteristic
42#1875C2×C314yescomplemented
43#1854C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1823C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic, complemented
47C25×C3D411yescharacteristic
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—22no—
52C23×C3×C4—22no—
53C23×C3×C4—28no—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#121—22no—
63C3×#121—22no—
64C3×#121—28no—
65C3×#120—44no—
66C3×#117—216no—
67C3×#117—28no—
68C3×#117—28no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D414yescomplemented
74C2×C3×C4⋊C4D414yescomplemented
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—28no—
86C2×C3×C42—24no—
87C3×#107—24no—
88C3×#106—24no—
89C3×#102—24no—
90C3×#102—24no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—28no—
94C3×#95—21no—
95C3×#95—22no—
96C24×C4C22×C311yescharacteristic
97C2×#121C22×C311yescharacteristic
98C2×#121C22×C312yes—
99C2×#120—21no—
100C2×#117—22no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117C22×C314yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C42—21no—
108#481—24no—
109#480—24no—
110C2×#107C22×C312yescomplemented
111C2×#106C22×C312yescomplemented
112C2×#102C22×C312yescomplemented
113C2×#102C22×C312yescomplemented
114C2×C22⋊C8C22×C311yescharacteristic
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C314yes—
117#386—22no—
118#386—21no—
119#380C22×C311yescharacteristic
120#380C22×C312yes—
121#340C22×C314yescomplemented
122#340C22×C314yescomplemented
123C24×C3—22no—
124C24×C3—24no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—21no—
128C24×C3—42no—
129C24×C3C2×D412yes—
130C24×C3C2×D411yescharacteristic
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—24no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4—42no—
137C2×C3×D4—44no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C22×C3×C4—216no—
141C22×C3×C4—216no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4118yes—
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—24no—
169C2×C3×C8—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4C2×D414yes—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—216no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—44no—
184C3×C42—216no—
185C25C3×D411yescharacteristic
186C22×D4—22no—
187C22×D4—21no—
188C22×D4C23×C311yescharacteristic
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4C3×D412yes—
195C23×C4C3×D411yescharacteristic
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C312yes—
198C22×C8C23×C312yes—
199C22×C8C23×C312yes—
200#121—22no—
201#121—22no—
202#121—28no—
203#120—44no—
204#117—216no—
205#117—28no—
206#117—28no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C3×D414yescomplemented
212C2×C4⋊C4C3×D414yescomplemented
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C312yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—28no—
224C2×C42—24no—
225#107—24no—
226#106—24no—
227#102—24no—
228#102—24no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—28no—
232#95—21no—
233#95—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—28no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—48no—
244C23×C3—48no—
245C23×C3—44no—
246C23×C3—44no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3#12012yes—
253C23×C3#12314yes—
254C23×C3#13512yes—
255C23×C3AGL(1,ℤ/8)12yes—
256C23×C3C22×D411yescharacteristic
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—88no—
260C3×D4—84no—
261C2×C3×C4—216no—
262C2×C3×C4—28no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—28no—
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—48no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—48no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4#12012yes—
284C2×C3×C4#12314yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4114yes—
288C3×C8—42no—
289C3×C8—42no—
290C24—22no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—42no—
296C24C2×C3×D412yes—
297C24C2×C3×D411yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—24no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—216no—
308C22×C4—216no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D412yes—
330C22×C4C2×C3×D412yes—
331C22×C4C3×#4118yes—
332C22×C4C24×C311yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—24no—
344C4⋊C4C2×C3×D414yes—
345C4⋊C4C2×C3×D414yes—
346C22⋊C4—216no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C42—216no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—24no—
359C22×C3—416no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—81no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3#44914yes—
370C22×C3C2×#12011yescharacteristic
371C22×C3C2×#12312yes—
372C22×C3C2×#13511yescharacteristic
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C3#57512yes—
375C3×C4—22no—
376C3×C4—24no—
377C3×C4—44no—
378C3×C4—44no—
379C3×C4—82no—
380C3×C4—81no—
381C3×C4#52512yes—
382C3×C4#52512yes—
383C23—22no—
384C23—22no—
385C23—28no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—48no—
393C23—48no—
394C23—44no—
395C23—44no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23C3×#12012yes—
402C23C3×#12314yes—
403C23C3×#13512yes—
404C23C3×AGL(1,ℤ/8)12yes—
405C23C22×C3×D411yescharacteristic
406D4—42no—
407D4—42no—
408D4—88no—
409D4—84no—
410C2×C4—216no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4C3×#12012yes—
433C2×C4C3×#12314yes—
434C2×C4C22×C3×D411yescharacteristic
435C2×C4C22×C3×D411yescharacteristic
436C2×C4C2×C3×#4114yes—
437C8—42no—
438C8—42no—
439C2×C3—24no—
440C2×C3—44no—
441C2×C3—161no—
442C2×C3#284911yescharacteristic
443C2×C3C2×#44911yescharacteristic
444C2×C3#300511yescharacteristic
445C2×C3#309112yes—
446C2×C3#309412yes—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—416no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—81no—
460C22—81no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22C3×#44914yes—
465C22C2×C3×#12011yescharacteristic
466C22C2×C3×#12312yes—
467C22C2×C3×#13511yescharacteristic
468C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
469C22C3×#57512yes—
470C4—22no—
471C4—24no—
472C4—44no—
473C4—44no—
474C4—81no—
475C4—82no—
476C4C3×#52512yes—
477C4C3×#52512yes—
478C3#3298011yescharacteristic, direct product factor, Sylow
479C2—24no—
480C2—44no—
481C2—161no—
482C2C3×#284911yescharacteristic
483C2C2×C3×#44911yescharacteristic
484C2C3×#300511yescharacteristic
485C2C3×#309112yes—
486C2C3×#309412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#117C2semidirect
2. C2×C3×#481C2semidirect
3. C2×C3×#480C2semidirect
5. C3×#1875C2semidirect
6. C3×#1854C2semidirect
7. C3×#1824C2semidirect
8. C3×#1823C2semidirect
9. C3×#1779C2semidirect
10. #32980C3direct (N × H)
25. C2×C3×#107C22semidirect
26. C2×C3×#106C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
36. C3×#340C22semidirect
37. C3×#340C22semidirect
38. C22×#117C2×C3semidirect
39. C2×#481C2×C3semidirect
40. C2×#480C2×C3semidirect
42. #1875C2×C3semidirect
43. #1854C2×C3semidirect
44. #1824C2×C3semidirect
45. #1823C2×C3semidirect
46. #1779C2×C3semidirect
73. C2×C3×C4⋊C4D4semidirect
74. C2×C3×C4⋊C4D4semidirect
110. C2×#107C22×C3semidirect
111. C2×#106C22×C3semidirect
112. C2×#102C22×C3semidirect
113. C2×#102C22×C3semidirect
121. #340C22×C3semidirect
122. #340C22×C3semidirect
211. C2×C4⋊C4C3×D4semidirect
212. C2×C4⋊C4C3×D4semidirect
478. C3#32980direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C2×C3×#4812
C2×C3×#4802
C3×#28472
C3×#18752
C3×#18542
C3×#18242
C3×#18232
C3×#17792
#329803