Finite Group Databasebeta

← C3×#33007 #10651614

Subgroups — C3×#33007 #10651614

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic
4C3×#2855C211yesmaximal, characteristic, complemented
5C3×#1877C214yesmaximal, complemented
6C3×#1864C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9C3×#1786C211yesmaximal, characteristic
10#33007C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×C3×SD16C2212yes—
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C4×D4—24no—
16C2×C3×C4×D4—24no—
17C2×C3×C4×D4—22no—
18C2×C3×C4×D4C2214yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×C3×C42C2211yescharacteristic
22C3×#483—24no—
23C3×#450—24no—
24C2×C3×#106C2212yescomplemented
25C2×C3×#103C2212yescomplemented
26C2×C3×#102C2214yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#391C2212yescomplemented
31C3×#389C2212yescomplemented
32C3×#387—22no—
33C3×#385—21no—
34C3×#381—21no—
35C3×#380C2211yescharacteristic
36C3×#340C2214yescomplemented
37C3×#336C2214yescomplemented
38C22×C4×D4C2×C311yescharacteristic
39C2×#483C2×C311yescharacteristic, complemented
40C2×#450C2×C311yescharacteristic
41#2855C2×C311yescharacteristic, complemented
42#1877C2×C314yescomplemented
43#1864C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1817C2×C311yescharacteristic, complemented
46#1786C2×C311yescharacteristic
47C25×C3—21no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—28no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—28no—
55C23×C3×C4—21no—
56C23×C3×C4—24no—
57C23×C3×C4—44no—
58C23×C3×C4—44no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×SD16—24no—
63C2×C3×SD16—28no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×C4×D4—216no—
69C3×C4×D4—28no—
70C3×C4×D4—28no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
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—21no—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42D412yes—
86C2×C3×C42C2312yes—
87C3×#106—24no—
88C3×#103—24no—
89C3×#102—24no—
90C3×#102—28no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—22no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C23×D4C22×C311yescharacteristic
98C24×C4—21no—
99C22×SD16C22×C312yes—
100C2×C22⋊Q8C22×C311yescharacteristic
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4C22×C314yes—
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C42C22×C311yescharacteristic
108#483—24no—
109#450—24no—
110C2×#106C22×C312yescomplemented
111C2×#103C22×C312yescomplemented
112C2×#102C22×C312yes—
113C2×#102C22×C314yes—
114C2×C22⋊C8C22×C311yescharacteristic
115C2×C22⋊C8C22×C311yescharacteristic
116#391C22×C312yescomplemented
117#389C22×C312yescomplemented
118#387—22no—
119#385—21no—
120#381—21no—
121#380C22×C311yescharacteristic
122#340C22×C314yescomplemented
123#336C22×C314yescomplemented
124C24×C3—28no—
125C24×C3—24no—
126C24×C3—22no—
127C24×C3—44no—
128C24×C3—42no—
129C24×C3—42no—
130C24×C3C2×D411yescharacteristic
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—24no—
134C2×C3×D4—28no—
135C2×C3×D4—216no—
136C2×C3×D4—216no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—24no—
140C2×C3×D4C2×D414yes—
141C2×C3×D4#4118yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—216no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4—48no—
154C22×C3×C4—432no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—48no—
158C22×C3×C4SD1614yescomplemented
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2411yescharacteristic
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—24no—
169C2×C3×C8—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×D414yes—
178C3×C22⋊C4—216no—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C42—24no—
182C3×C42—22no—
183C3×C42—22no—
184C25—21no—
185C22×Q8C23×C311yescharacteristic
186C22×D4—28no—
187C22×D4—21no—
188C22×D4—22no—
189C22×D4C3×D414yes—
190C22×D4C23×C312yes—
191C23×C4—28no—
192C23×C4—21no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C3×D411yescharacteristic
197C23×C4C3×D411yescharacteristic
198C23×C4C23×C311yescharacteristic
199C2×SD16—24no—
200C2×SD16—28no—
201C22×C8C23×C312yes—
202C22×C8C23×C312yes—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C4×D4—216no—
206C4×D4—28no—
207C4×D4—28no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C3×D414yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C312yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C3×D412yes—
223C2×C42C23×C312yes—
224#106—24no—
225#103—24no—
226#102—24no—
227#102—28no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—216no—
237C23×C3—24no—
238C23×C3—28no—
239C23×C3—416no—
240C23×C3—416no—
241C23×C3—48no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—44no—
245C23×C3—48no—
246C23×C3#12012yes—
247C23×C3C2×SD1612yes—
248C23×C3AGL(1,ℤ/8)12yes—
249C23×C3C22×D411yescharacteristic
250C3×Q8—42no—
251C3×Q8—42no—
252C3×D4—216no—
253C3×D4—24no—
254C3×D4—28no—
255C3×D4#12318yes—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—432no—
267C2×C3×C4—44no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—48no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4#12012yes—
279C2×C3×C4C2×SD1614yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4114yes—
283C3×C8—42no—
284C3×C8—42no—
285C24—28no—
286C24—24no—
287C24—22no—
288C24—44no—
289C24—42no—
290C24—42no—
291C24C2×C3×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—24no—
295C2×D4—28no—
296C2×D4—216no—
297C2×D4—216no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4C2×C3×D414yes—
302C2×D4C3×#4118yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—216no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—432no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4C3×SD1614yescomplemented
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D412yes—
324C22×C4C2×C3×D412yes—
325C22×C4C24×C311yescharacteristic
326SD16—48no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4C2×C3×D414yes—
339C22⋊C4—216no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C42—24no—
343C42—22no—
344C42—22no—
345C22×C3—24no—
346C22×C3—22no—
347C22×C3—24no—
348C22×C3—22no—
349C22×C3—28no—
350C22×C3—432no—
351C22×C3—48no—
352C22×C3—48no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3#45014yes—
356C22×C3C2×#12011yescharacteristic
357C22×C3#52512yes—
358C22×C3C22×SD1611yescharacteristic
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3#57512yes—
361C3×C4—22no—
362C3×C4—24no—
363C3×C4—44no—
364C3×C4—42no—
365C3×C4—81no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4C2×#12312yes—
369C3×C4#52512yes—
370C23—22no—
371C23—22no—
372C23—216no—
373C23—24no—
374C23—28no—
375C23—48no—
376C23—416no—
377C23—416no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—48no—
382C23C3×#12012yes—
383C23C2×C3×SD1612yes—
384C23C3×AGL(1,ℤ/8)12yes—
385C23C22×C3×D411yescharacteristic
386Q8—42no—
387Q8—42no—
388D4—216no—
389D4—24no—
390D4—28no—
391D4C3×#12318yes—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—432no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4C3×#12012yes—
415C2×C4C2×C3×SD1614yes—
416C2×C4C22×C3×D411yescharacteristic
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C2×C3×#4114yes—
419C8—42no—
420C8—42no—
421C2×C3—24no—
422C2×C3—48no—
423C2×C3#284911yescharacteristic
424C2×C3C2×#45011yescharacteristic
425C2×C3#300511yescharacteristic
426C2×C3#310112yes—
427C2×C3#311412yes—
428C22—22no—
429C22—24no—
430C22—22no—
431C22—24no—
432C22—28no—
433C22—432no—
434C22—44no—
435C22—44no—
436C22—48no—
437C22—48no—
438C22C3×#45014yes—
439C22C2×C3×#12011yescharacteristic
440C22C3×#52512yes—
441C22C22×C3×SD1611yescharacteristic
442C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
443C22C3×#57512yes—
444C4—24no—
445C4—22no—
446C4—44no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—82no—
451C4C2×C3×#12312yes—
452C4C3×#52512yes—
453C3#3300711yescharacteristic, direct product factor, Sylow
454C2—24no—
455C2—48no—
456C2C3×#284911yescharacteristic
457C2C2×C3×#45011yescharacteristic
458C2C3×#300511yescharacteristic
459C2C3×#310112yes—
460C2C3×#311412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
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
2. C2×C3×#483C2semidirect
4. C3×#2855C2semidirect
5. C3×#1877C2semidirect
6. C3×#1864C2semidirect
7. C3×#1823C2semidirect
8. C3×#1817C2semidirect
10. #33007C3direct (N × H)
24. C2×C3×#106C22semidirect
25. C2×C3×#103C22semidirect
30. C3×#391C22semidirect
31. C3×#389C22semidirect
36. C3×#340C22semidirect
37. C3×#336C22semidirect
39. C2×#483C2×C3semidirect
41. #2855C2×C3semidirect
42. #1877C2×C3semidirect
43. #1864C2×C3semidirect
44. #1823C2×C3semidirect
45. #1817C2×C3semidirect
110. C2×#106C22×C3semidirect
111. C2×#103C22×C3semidirect
116. #391C22×C3semidirect
117. #389C22×C3semidirect
122. #340C22×C3semidirect
123. #336C22×C3semidirect
158. C22×C3×C4SD16semidirect
319. C22×C4C3×SD16semidirect
453. C3#33007direct (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×#4832
C2×C3×#4502
C3×#28552
C3×#18772
C3×#18642
C3×#18232
C3×#18172
C3×#17862
#330073