Finite Group Databasebeta

← C3×#33004 #10651611

Subgroups — C3×#33004 #10651611

Order 768.

1808 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2854C211yesmaximal, characteristic, complemented
5C3×#1879C214yesmaximal, complemented
6C3×#1864C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9C3×#1786C211yesmaximal, characteristic, complemented
10#33004C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×C3×D8C2212yes—
14C2×C3×#123—21no—
15C2×C3×#121C2211yescharacteristic
16C2×C3×C4×D4—24no—
17C2×C3×C4×D4—24no—
18C2×C3×C4×D4—22no—
19C2×C3×C4×D4C2214yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#481—24no—
24C3×#447—24no—
25C2×C3×#106C2212yescomplemented
26C2×C3×#102C2214yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#390C2212yescomplemented
32C3×#389C2212yescomplemented
33C3×#388—22no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×#340C2214yescomplemented
37C3×#336C2214yescomplemented
38C22×C4×D4C2×C311yescharacteristic, complemented
39C2×#481C2×C311yescharacteristic, complemented
40C2×#447C2×C311yescharacteristic
41#2854C2×C311yescharacteristic, complemented
42#1879C2×C314yescomplemented
43#1864C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1817C2×C311yescharacteristic, complemented
46#1786C2×C311yescharacteristic, complemented
47C25×C3—21no—
48C22×C3×D4—28no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C22×C3×D4C2311yescharacteristic
54C22×C3×D4C2312yes—
55C23×C3×C4—28no—
56C23×C3×C4—21no—
57C23×C3×C4—24no—
58C23×C3×C4—48no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×D8—24no—
63C2×C3×D8—28no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×#123—44no—
67C3×#121—22no—
68C3×#121—22no—
69C3×C4×D4—216no—
70C3×C4×D4—28no—
71C3×C4×D4—28no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—22no—
86C2×C3×C42—22no—
87C2×C3×C42D412yes—
88C2×C3×C42C2312yes—
89C3×#106—24no—
90C3×#102—24no—
91C3×#102—28no—
92C3×#102—24no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×#95—21no—
96C3×#95—22no—
97C23×D4C22×C311yescharacteristic
98C24×C4—21no—
99C22×D8C22×C312yes—
100C2×#123—21no—
101C2×#121C22×C311yescharacteristic
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—22no—
105C2×C4×D4C22×C314yes—
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C42C22×C311yescharacteristic
109#481—24no—
110#447—24no—
111C2×#106C22×C312yescomplemented
112C2×#102C22×C312yes—
113C2×#102C22×C314yes—
114C2×#102C22×C312yescomplemented
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#390C22×C312yescomplemented
118#389C22×C312yescomplemented
119#388—22no—
120#386—21no—
121#380C22×C311yescharacteristic
122#340C22×C314yescomplemented
123#336C22×C314yescomplemented
124C24×C3—28no—
125C24×C3—24no—
126C24×C3—22no—
127C24×C3—21no—
128C24×C3—44no—
129C24×C3—42no—
130C24×C3—42no—
131C24×C3C2×D411yescharacteristic
132C2×C3×D4—28no—
133C2×C3×D4—216no—
134C2×C3×D4—216no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4C2×D414yes—
145C2×C3×D4#4118yes—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—216no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—432no—
158C22×C3×C4—48no—
159C22×C3×C4—44no—
160C22×C3×C4—48no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2411yescharacteristic
167C3×D8—48no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—24no—
171C2×C3×C8—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—44no—
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—24no—
185C3×C42—22no—
186C3×C42—22no—
187C25—21no—
188C22×D4—28no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4C3×D414yes—
193C22×D4C23×C311yescharacteristic
194C22×D4C23×C312yes—
195C23×C4—28no—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—48no—
199C23×C4C3×D411yescharacteristic
200C23×C4C3×D411yescharacteristic
201C23×C4C23×C311yescharacteristic
202C2×D8—24no—
203C2×D8—28no—
204C22×C8C23×C312yes—
205C22×C8C23×C312yes—
206#123—44no—
207#121—22no—
208#121—22no—
209C4×D4—216no—
210C4×D4—28no—
211C4×D4—28no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C3×D414yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C3×D412yes—
228C2×C42C23×C312yes—
229#106—24no—
230#102—24no—
231#102—28no—
232#102—24no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235#95—21no—
236#95—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—216no—
240C23×C3—24no—
241C23×C3—28no—
242C23×C3—48no—
243C23×C3—416no—
244C23×C3—416no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—42no—
248C23×C3—48no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3#12012yes—
255C23×C3#13512yes—
256C23×C3AGL(1,ℤ/8)12yes—
257C23×C3C22×D411yescharacteristic
258C3×D4—216no—
259C3×D4—24no—
260C3×D4—28no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—84no—
264C3×D4#12318yescomplemented
265C2×C3×C4—28no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—432no—
276C2×C3×C4—48no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4#12012yes—
286C2×C3×C4AGL(1,ℤ/8)14yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4114yes—
290C3×C8—42no—
291C3×C8—42no—
292C24—28no—
293C24—24no—
294C24—22no—
295C24—21no—
296C24—44no—
297C24—42no—
298C24—42no—
299C24C2×C3×D411yescharacteristic
300C2×D4—28no—
301C2×D4—216no—
302C2×D4—216no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4C2×C3×D414yes—
313C2×D4C3×#4118yes—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—216no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—432no—
326C22×C4—48no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4C2×C3×D412yes—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D412yes—
333C22×C4C2×C3×D412yes—
334C22×C4C24×C311yescharacteristic
335D8—48no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—44no—
346C4⋊C4C2×C3×D414yes—
347C22⋊C4—216no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C42—24no—
353C42—22no—
354C42—22no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—22no—
359C22×C3—28no—
360C22×C3—432no—
361C22×C3—48no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—81no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3#44914yes—
371C22×C3C2×#12011yescharacteristic
372C22×C3#52512yes—
373C22×C3C2×#13511yescharacteristic
374C22×C3C2×AGL(1,ℤ/8)12yes—
375C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
376C3×C4—22no—
377C3×C4—24no—
378C3×C4—42no—
379C3×C4—44no—
380C3×C4—82no—
381C3×C4—81no—
382C3×C4C2×#12312yes—
383C3×C4#52512yes—
384C23—22no—
385C23—22no—
386C23—216no—
387C23—24no—
388C23—28no—
389C23—48no—
390C23—416no—
391C23—416no—
392C23—42no—
393C23—44no—
394C23—42no—
395C23—48no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23C3×#12012yes—
402C23C3×#13512yes—
403C23C3×AGL(1,ℤ/8)12yes—
404C23C22×C3×D411yescharacteristic
405D4—216no—
406D4—24no—
407D4—28no—
408D4—42no—
409D4—42no—
410D4—84no—
411D4C3×#12318yescomplemented
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—432no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4C3×#12012yes—
433C2×C4C3×AGL(1,ℤ/8)14yes—
434C2×C4C22×C3×D411yescharacteristic
435C2×C4C22×C3×D411yescharacteristic
436C2×C4C2×C3×#4114yes—
437C8—42no—
438C8—42no—
439C2×C3—24no—
440C2×C3—48no—
441C2×C3—161no—
442C2×C3#284911yescharacteristic
443C2×C3C2×#44911yescharacteristic
444C2×C3#300311yescharacteristic
445C2×C3#310512yes—
446C2×C3#310912yes—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—24no—
451C22—28no—
452C22—432no—
453C22—44no—
454C22—44no—
455C22—48no—
456C22—48no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—82no—
462C22C3×#44914yes—
463C22C2×C3×#12011yescharacteristic
464C22C3×#52512yes—
465C22C2×C3×#13511yescharacteristic
466C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
467C22C2×C3×AGL(1,ℤ/8)12yes—
468C4—24no—
469C4—22no—
470C4—42no—
471C4—44no—
472C4—81no—
473C4—82no—
474C4C2×C3×#12312yes—
475C4C3×#52512yes—
476C3#3300411yescharacteristic, direct product factor, Sylow
477C2—24no—
478C2—48no—
479C2—161no—
480C2C3×#284911yescharacteristic
481C2C2×C3×#44911yescharacteristic
482C2C3×#300311yescharacteristic
483C2C3×#310512yes—
484C2C3×#310912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
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
1. C22×C3×C4×D4C2semidirect
2. C2×C3×#481C2semidirect
4. C3×#2854C2semidirect
5. C3×#1879C2semidirect
6. C3×#1864C2semidirect
7. C3×#1824C2semidirect
8. C3×#1817C2semidirect
9. C3×#1786C2semidirect
10. #33004C3direct (N × H)
25. C2×C3×#106C22semidirect
28. C2×C3×#102C22semidirect
31. C3×#390C22semidirect
32. C3×#389C22semidirect
36. C3×#340C22semidirect
37. C3×#336C22semidirect
38. C22×C4×D4C2×C3semidirect
39. C2×#481C2×C3semidirect
41. #2854C2×C3semidirect
42. #1879C2×C3semidirect
43. #1864C2×C3semidirect
44. #1824C2×C3semidirect
45. #1817C2×C3semidirect
46. #1786C2×C3semidirect
111. C2×#106C22×C3semidirect
114. C2×#102C22×C3semidirect
117. #390C22×C3semidirect
118. #389C22×C3semidirect
122. #340C22×C3semidirect
123. #336C22×C3semidirect
264. C3×D4#123semidirect
411. D4C3×#123semidirect
476. C3#33004direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4812
C2×C3×#4472
C3×#28542
C3×#18792
C3×#18642
C3×#18242
C3×#18172
C3×#17862
#330043