Finite Group Databasebeta

← C3×#33009 #10651616

Subgroups — C3×#33009 #10651616

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic
4C3×#2855C211yesmaximal, characteristic, complemented
5C3×#1877C214yesmaximal, complemented
6C3×#1862C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9C3×#1786C211yesmaximal, characteristic
10#33009C311yesmaximal, 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×#484—24no—
23C3×#450—24no—
24C2×C3×#107C2212yescomplemented
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×#484C2×C311yescharacteristic, complemented
40C2×#450C2×C311yescharacteristic
41#2855C2×C311yescharacteristic, complemented
42#1877C2×C314yescomplemented
43#1862C2×C314yescomplemented
44#1824C2×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—48no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C2×C3×SD16—24no—
62C2×C3×SD16—28no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C4×D4—216no—
68C3×C4×D4—28no—
69C3×C4×D4—28no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D414yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—28no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—22no—
83C2×C3×C42—22no—
84C2×C3×C42D412yes—
85C2×C3×C42C2312yes—
86C3×#107—24no—
87C3×#103—24no—
88C3×#102—24no—
89C3×#102—28no—
90C3×C22⋊C8—22no—
91C3×C22⋊C8—22no—
92C3×#95—21no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C23×D4C22×C311yescharacteristic
97C24×C4—21no—
98C22×SD16C22×C312yes—
99C2×C22⋊Q8C22×C311yescharacteristic
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4C22×C314yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C42C22×C311yescharacteristic
107#484—24no—
108#450—24no—
109C2×#107C22×C312yescomplemented
110C2×#103C22×C312yescomplemented
111C2×#102C22×C312yes—
112C2×#102C22×C314yes—
113C2×C22⋊C8C22×C311yescharacteristic
114C2×C22⋊C8C22×C311yescharacteristic
115#391C22×C312yescomplemented
116#389C22×C312yescomplemented
117#387—22no—
118#385—21no—
119#381—21no—
120#380C22×C311yescharacteristic
121#340C22×C314yescomplemented
122#336C22×C314yescomplemented
123C24×C3—28no—
124C24×C3—24no—
125C24×C3—22no—
126C24×C3—44no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—24no—
133C2×C3×D4—28no—
134C2×C3×D4—216no—
135C2×C3×D4—216no—
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—24no—
139C2×C3×D4C2×D414yes—
140C2×C3×D4#4118yes—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—216no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—432no—
154C22×C3×C4—48no—
155C22×C3×C4—44no—
156C22×C3×C4—48no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C3×SD16—48no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—24no—
167C2×C3×C8—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4C2×D414yes—
176C3×C22⋊C4—216no—
177C3×C22⋊C4—42no—
178C3×C22⋊C4—42no—
179C3×C42—24no—
180C3×C42—22no—
181C3×C42—22no—
182C25—21no—
183C22×Q8C23×C311yescharacteristic
184C22×D4—28no—
185C22×D4—21no—
186C22×D4—22no—
187C22×D4C3×D414yes—
188C22×D4C23×C312yes—
189C23×C4—28no—
190C23×C4—21no—
191C23×C4—24no—
192C23×C4—48no—
193C23×C4C3×D411yescharacteristic
194C23×C4C3×D411yescharacteristic
195C23×C4C23×C311yescharacteristic
196C2×SD16—24no—
197C2×SD16—28no—
198C22×C8C23×C312yes—
199C22×C8C23×C312yes—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C4×D4—216no—
203C4×D4—28no—
204C4×D4—28no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C3×D414yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C3×D412yes—
220C2×C42C23×C312yes—
221#107—24no—
222#103—24no—
223#102—24no—
224#102—28no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—216no—
234C23×C3—24no—
235C23×C3—28no—
236C23×C3—48no—
237C23×C3—416no—
238C23×C3—416no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3—42no—
242C23×C3—48no—
243C23×C3#12012yes—
244C23×C3#13512yes—
245C23×C3#13712yes—
246C23×C3C22×D411yescharacteristic
247C3×Q8—42no—
248C3×Q8—42no—
249C3×D4—216no—
250C3×D4—24no—
251C3×D4—28no—
252C3×D4#12318yes—
253C2×C3×C4—28no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—432no—
264C2×C3×C4—48no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4#12012yes—
275C2×C3×C4AGL(1,ℤ/8)14yes—
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C2×#4114yes—
279C3×C8—42no—
280C3×C8—42no—
281C24—28no—
282C24—24no—
283C24—22no—
284C24—44no—
285C24—42no—
286C24—42no—
287C24C2×C3×D411yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—24no—
291C2×D4—28no—
292C2×D4—216no—
293C2×D4—216no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4C2×C3×D414yes—
298C2×D4C3×#4118yes—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—216no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—432no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—48no—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D412yes—
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D412yes—
320C22×C4C24×C311yescharacteristic
321SD16—48no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4C2×C3×D414yes—
334C22⋊C4—216no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C42—24no—
338C42—22no—
339C42—22no—
340C22×C3—22no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—22no—
344C22×C3—28no—
345C22×C3—432no—
346C22×C3—48no—
347C22×C3—48no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3#45214yes—
351C22×C3C2×#12011yescharacteristic
352C22×C3#52512yes—
353C22×C3C2×#13511yescharacteristic
354C22×C3C2×AGL(1,ℤ/8)12yes—
355C22×C3C2×#13711yescharacteristic
356C3×C4—22no—
357C3×C4—24no—
358C3×C4—42no—
359C3×C4—44no—
360C3×C4—81no—
361C3×C4—82no—
362C3×C4—81no—
363C3×C4C2×#12312yes—
364C3×C4#52512yes—
365C23—22no—
366C23—22no—
367C23—216no—
368C23—24no—
369C23—28no—
370C23—48no—
371C23—416no—
372C23—416no—
373C23—42no—
374C23—44no—
375C23—42no—
376C23—48no—
377C23C3×#12012yes—
378C23C3×#13512yes—
379C23C3×#13712yes—
380C23C22×C3×D411yescharacteristic
381Q8—42no—
382Q8—42no—
383D4—216no—
384D4—24no—
385D4—28no—
386D4C3×#12318yes—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—432no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4C3×#12012yes—
409C2×C4C3×AGL(1,ℤ/8)14yes—
410C2×C4C22×C3×D411yescharacteristic
411C2×C4C22×C3×D411yescharacteristic
412C2×C4C2×C3×#4114yes—
413C8—42no—
414C8—42no—
415C2×C3—24no—
416C2×C3—48no—
417C2×C3#284911yescharacteristic
418C2×C3C2×#45211yescharacteristic
419C2×C3#300311yescharacteristic
420C2×C3#310512yes—
421C2×C3#311012yes—
422C22—22no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—28no—
427C22—432no—
428C22—44no—
429C22—44no—
430C22—48no—
431C22—48no—
432C22C3×#45214yes—
433C22C2×C3×#12011yescharacteristic
434C22C3×#52512yes—
435C22C2×C3×#13511yescharacteristic
436C22C2×C3×AGL(1,ℤ/8)12yes—
437C22C2×C3×#13711yescharacteristic
438C4—24no—
439C4—22no—
440C4—42no—
441C4—44no—
442C4—81no—
443C4—81no—
444C4—82no—
445C4C2×C3×#12312yes—
446C4C3×#52512yes—
447C3#3300911yescharacteristic, direct product factor, Sylow
448C2—24no—
449C2—48no—
450C2C3×#284911yescharacteristic
451C2C2×C3×#45211yescharacteristic
452C2C3×#300311yescharacteristic
453C2C3×#310512yes—
454C2C3×#311012yes—
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×#484C2semidirect
4. C3×#2855C2semidirect
5. C3×#1877C2semidirect
6. C3×#1862C2semidirect
7. C3×#1824C2semidirect
8. C3×#1817C2semidirect
10. #33009C3direct (N × H)
24. C2×C3×#107C22semidirect
25. C2×C3×#103C22semidirect
30. C3×#391C22semidirect
31. C3×#389C22semidirect
36. C3×#340C22semidirect
37. C3×#336C22semidirect
39. C2×#484C2×C3semidirect
41. #2855C2×C3semidirect
42. #1877C2×C3semidirect
43. #1862C2×C3semidirect
44. #1824C2×C3semidirect
45. #1817C2×C3semidirect
109. C2×#107C22×C3semidirect
110. C2×#103C22×C3semidirect
115. #391C22×C3semidirect
116. #389C22×C3semidirect
121. #340C22×C3semidirect
122. #336C22×C3semidirect
447. C3#33009direct (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×#4842
C2×C3×#4502
C3×#28552
C3×#18772
C3×#18622
C3×#18242
C3×#18172
C3×#17862
#330093