Finite Group Databasebeta

← C3×#32979 #10651586

Subgroups — C3×#32979 #10651586

Order 768.

1812 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#481C212yesmaximal, complemented
3C3×#2847C211yesmaximal, characteristic
4C3×#1874C214yesmaximal, complemented
5C3×#1853C214yesmaximal, complemented
6C3×#1824C212yesmaximal, complemented
7C3×#1779C211yesmaximal, characteristic, complemented
8#32979C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4—21no—
10C24×C3×C4C2211yescharacteristic
11C2×C3×#121C2212yes—
12C2×C3×#121C2211yescharacteristic
13C2×C3×#120—21no—
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4—22no—
16C22×C3×C4⋊C4—28no—
17C22×C3×C4⋊C4C2214yes—
18C22×C3×C4⋊C4C2212yes—
19C3×#481—28no—
20C2×C3×#106C2214yescomplemented
21C2×C3×#102C2214yescomplemented
22C2×C3×C22⋊C8C2212yes—
23C2×C3×C22⋊C8C2214yes—
24C3×#386—22no—
25C3×#386—21no—
26C3×#380C2211yescharacteristic
27C3×#380C2212yes—
28C3×#340C2218yescomplemented
29C23×C4⋊C4C2×C311yescharacteristic, complemented
30C2×#481C2×C312yescomplemented
31#2847C2×C311yescharacteristic
32#1874C2×C314yescomplemented
33#1853C2×C314yescomplemented
34#1824C2×C312yescomplemented
35#1779C2×C311yescharacteristic, complemented
36C25×C3D411yescharacteristic
37C22×C3×D4—21no—
38C22×C3×D4—22no—
39C22×C3×D4C2311yescharacteristic
40C23×C3×C4—24no—
41C23×C3×C4—28no—
42C23×C3×C4—28no—
43C23×C3×C4—24no—
44C23×C3×C4—22no—
45C23×C3×C4—48no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D412yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C22×C3×C8C2314yes—
51C3×#121—24no—
52C3×#121—28no—
53C3×#120—44no—
54C2×C3×C4⋊C4—28no—
55C2×C3×C4⋊C4—216no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—216no—
60C2×C3×C4⋊C4D418yescomplemented
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2314yes—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—22no—
68C3×#106—28no—
69C3×#102—28no—
70C3×C22⋊C8—24no—
71C3×C22⋊C8—28no—
72C3×#95—21no—
73C3×#95—22no—
74C24×C4—21no—
75C24×C4C22×C311yescharacteristic
76C2×#121C22×C311yescharacteristic
77C2×#121C22×C312yes—
78C2×#120—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4C22×C314yes—
83C22×C4⋊C4C22×C312yes—
84#481—28no—
85C2×#106C22×C314yescomplemented
86C2×#102C22×C314yescomplemented
87C2×C22⋊C8C22×C312yes—
88C2×C22⋊C8C22×C314yes—
89#386—22no—
90#386—21no—
91#380C22×C311yescharacteristic
92#380C22×C312yes—
93#340C22×C318yescomplemented
94C24×C3—22no—
95C24×C3—24no—
96C24×C3—24no—
97C24×C3—22no—
98C24×C3—21no—
99C24×C3SD1614yes—
100C24×C3C2×D411yescharacteristic
101C24×C3C2×D412yes—
102C2×C3×D4—22no—
103C2×C3×D4—24no—
104C2×C3×D4—42no—
105C2×C3×D4—44no—
106C2×C3×D4—42no—
107C2×C3×D4—44no—
108C2×C3×D4—44no—
109C2×C3×D4—42no—
110C22×C3×C4—232no—
111C22×C3×C4—22no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—216no—
120C22×C3×C4—28no—
121C22×C3×C4—24no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—432no—
125C22×C3×C4—48no—
126C22×C3×C4—48no—
127C22×C3×C4—48no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D414yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4#4118yes—
132C22×C3×C4C2411yescharacteristic
133C2×C3×C8—24no—
134C2×C3×C8—28no—
135C3×C4⋊C4—216no—
136C3×C4⋊C4—28no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4C2×D418yes—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—44no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—44no—
145C25C3×D411yescharacteristic
146C22×D4—21no—
147C22×D4—22no—
148C22×D4C23×C311yescharacteristic
149C23×C4—24no—
150C23×C4—28no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—48no—
155C23×C4C3×D411yescharacteristic
156C23×C4C3×D412yes—
157C23×C4C23×C311yescharacteristic
158C23×C4C23×C312yes—
159C22×C8C23×C314yes—
160#121—24no—
161#121—28no—
162#120—44no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—216no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—216no—
169C2×C4⋊C4C3×D418yescomplemented
170C2×C4⋊C4C23×C312yes—
171C2×C4⋊C4C23×C311yescharacteristic
172C2×C4⋊C4C23×C314yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177#106—28no—
178#102—28no—
179C22⋊C8—24no—
180C22⋊C8—28no—
181#95—21no—
182#95—22no—
183C23×C3—28no—
184C23×C3—28no—
185C23×C3—216no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—28no—
193C23×C3—216no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3#12012yes—
200C23×C3#12314yes—
201C23×C3C2×SD1612yes—
202C23×C3C2×SD1614yes—
203C23×C3AGL(1,ℤ/8)14yes—
204C23×C3#13712yes—
205C23×C3C22×D411yescharacteristic
206C3×D4—44no—
207C3×D4—84no—
208C3×D4—88no—
209C2×C3×C4—216no—
210C2×C3×C4—28no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—28no—
215C2×C3×C4—216no—
216C2×C3×C4—28no—
217C2×C3×C4—416no—
218C2×C3×C4—432no—
219C2×C3×C4—48no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4#12012yes—
227C2×C3×C4#12314yes—
228C2×C3×C4C22×D412yes—
229C2×C3×C4C2×#4114yes—
230C3×C8—44no—
231C24—22no—
232C24—24no—
233C24—24no—
234C24—22no—
235C24—21no—
236C24C3×SD1614yes—
237C24C2×C3×D411yescharacteristic
238C24C2×C3×D412yes—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—42no—
247C22×C4—232no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—216no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—432no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4C2×C3×D411yescharacteristic
266C22×C4C2×C3×D414yes—
267C22×C4C2×C3×D412yes—
268C22×C4C3×#4118yes—
269C22×C4C24×C311yescharacteristic
270C2×C8—24no—
271C2×C8—28no—
272C4⋊C4—216no—
273C4⋊C4—28no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4C2×C3×D418yes—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22×C3—232no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—24no—
286C22×C3—24no—
287C22×C3—24no—
288C22×C3—22no—
289C22×C3—24no—
290C22×C3—24no—
291C22×C3—22no—
292C22×C3—24no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3#44814yes—
299C22×C3#48118yescomplemented
300C22×C3C2×#12011yescharacteristic
301C22×C3C2×#12312yes—
302C22×C3C22×SD1611yescharacteristic
303C22×C3C2×AGL(1,ℤ/8)12yes—
304C22×C3C2×#13711yescharacteristic
305C3×C4—22no—
306C3×C4—24no—
307C3×C4—48no—
308C3×C4—82no—
309C3×C4—81no—
310C3×C4#52514yes—
311C23—28no—
312C23—28no—
313C23—216no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—28no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—216no—
322C23—41no—
323C23—42no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23C3×#12012yes—
328C23C3×#12314yes—
329C23C2×C3×SD1612yes—
330C23C2×C3×SD1614yes—
331C23C3×AGL(1,ℤ/8)14yes—
332C23C3×#13712yes—
333C23C22×C3×D411yescharacteristic
334D4—44no—
335D4—84no—
336D4—88no—
337C2×C4—216no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—28no—
343C2×C4—216no—
344C2×C4—28no—
345C2×C4—416no—
346C2×C4—432no—
347C2×C4—48no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4C3×#12012yes—
355C2×C4C3×#12314yes—
356C2×C4C22×C3×D412yes—
357C2×C4C2×C3×#4114yes—
358C8—44no—
359C2×C3—24no—
360C2×C3—28no—
361C2×C3—161no—
362C2×C3#284911yescharacteristic
363C2×C3C2×#44811yescharacteristic
364C2×C3#300311yescharacteristic
365C2×C3C2×#48112yes—
366C2×C3#309212yes—
367C22—232no—
368C22—24no—
369C22—22no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—22no—
377C22—24no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22—82no—
382C22—81no—
383C22C3×#44814yes—
384C22C3×#48118yescomplemented
385C22C2×C3×#12011yescharacteristic
386C22C2×C3×#12312yes—
387C22C22×C3×SD1611yescharacteristic
388C22C2×C3×AGL(1,ℤ/8)12yes—
389C22C2×C3×#13711yescharacteristic
390C4—22no—
391C4—24no—
392C4—48no—
393C4—81no—
394C4—82no—
395C4C3×#52514yes—
396C3#3297911yescharacteristic, direct product factor, Sylow
397C2—24no—
398C2—28no—
399C2—161no—
400C2C3×#284911yescharacteristic
401C2C2×C3×#44811yescharacteristic
402C2C3×#300311yescharacteristic
403C2C2×C3×#48112yes—
404C2C3×#309212yes—
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. C23×C3×C4⋊C4C2semidirect
2. C2×C3×#481C2semidirect
4. C3×#1874C2semidirect
5. C3×#1853C2semidirect
6. C3×#1824C2semidirect
7. C3×#1779C2semidirect
8. #32979C3direct (N × H)
20. C2×C3×#106C22semidirect
21. C2×C3×#102C22semidirect
28. C3×#340C22semidirect
29. C23×C4⋊C4C2×C3semidirect
30. C2×#481C2×C3semidirect
32. #1874C2×C3semidirect
33. #1853C2×C3semidirect
34. #1824C2×C3semidirect
35. #1779C2×C3semidirect
60. C2×C3×C4⋊C4D4semidirect
85. C2×#106C22×C3semidirect
86. C2×#102C22×C3semidirect
93. #340C22×C3semidirect
169. C2×C4⋊C4C3×D4semidirect
299. C22×C3#481semidirect
384. C22C3×#481semidirect
396. C3#32979direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#4812
C3×#28472
C3×#18742
C3×#18532
C3×#18242
C3×#17792
#329793