Finite Group Databasebeta

← C3×#32983 #10651590

Subgroups — C3×#32983 #10651590

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#484C212yesmaximal, complemented
3C3×#2848C211yesmaximal, characteristic
4C3×#1872C214yesmaximal, complemented
5C3×#1852C214yesmaximal, complemented
6C3×#1824C212yesmaximal, complemented
7C3×#1779C211yesmaximal, characteristic
8#32983C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4—21no—
10C24×C3×C4C2211yescharacteristic
11C2×C3×C22⋊Q8C2212yes—
12C2×C3×C22⋊Q8C2211yescharacteristic
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4—22no—
15C22×C3×C4⋊C4—28no—
16C22×C3×C4⋊C4C2214yes—
17C22×C3×C4⋊C4C2212yes—
18C3×#484—28no—
19C2×C3×#107C2214yescomplemented
20C2×C3×#103C2214yescomplemented
21C2×C3×C22⋊C8C2212yes—
22C2×C3×C22⋊C8C2214yes—
23C3×#385—22no—
24C3×#385—21no—
25C3×#380C2211yescharacteristic
26C3×#380C2212yes—
27C3×#379—21no—
28C3×#340C2218yescomplemented
29C23×C4⋊C4C2×C311yescharacteristic
30C2×#484C2×C312yescomplemented
31#2848C2×C311yescharacteristic
32#1872C2×C314yescomplemented
33#1852C2×C314yescomplemented
34#1824C2×C312yescomplemented
35#1779C2×C311yescharacteristic
36C25×C3D411yescharacteristic
37C22×C3×Q8C2311yescharacteristic
38C23×C3×C4—24no—
39C23×C3×C4—28no—
40C23×C3×C4—28no—
41C23×C3×C4—24no—
42C23×C3×C4—22no—
43C23×C3×C4—48no—
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D412yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C22×C3×C8C2314yes—
49C3×C22⋊Q8—24no—
50C3×C22⋊Q8—28no—
51C2×C3×C4⋊C4—28no—
52C2×C3×C4⋊C4—216no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—216no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4D418yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2314yes—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C3×#107—28no—
68C3×#103—28no—
69C3×C22⋊C8—24no—
70C3×C22⋊C8—28no—
71C3×#95—21no—
72C3×#95—22no—
73C24×C4—21no—
74C24×C4C22×C311yescharacteristic
75C2×C22⋊Q8C22×C311yescharacteristic
76C2×C22⋊Q8C22×C312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4C22×C314yes—
81C22×C4⋊C4C22×C312yes—
82#484—28no—
83C2×#107C22×C314yescomplemented
84C2×#103C22×C314yescomplemented
85C2×C22⋊C8C22×C312yes—
86C2×C22⋊C8C22×C314yes—
87#385—22no—
88#385—21no—
89#380C22×C311yescharacteristic
90#380C22×C312yes—
91#379—21no—
92#340C22×C318yescomplemented
93C24×C3—22no—
94C24×C3—24no—
95C24×C3—24no—
96C24×C3—22no—
97C24×C3Q1614yes—
98C24×C3C2×D411yescharacteristic
99C24×C3C2×D412yes—
100C2×C3×Q8—22no—
101C2×C3×Q8—24no—
102C22×C3×C4—232no—
103C22×C3×C4—22no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—216no—
112C22×C3×C4—28no—
113C22×C3×C4—24no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—432no—
118C22×C3×C4—48no—
119C22×C3×C4—48no—
120C22×C3×C4—48no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D414yes—
123C22×C3×C4C2×D412yes—
124C22×C3×C4#4118yes—
125C22×C3×C4C2411yescharacteristic
126C2×C3×C8—24no—
127C2×C3×C8—28no—
128C3×C4⋊C4—216no—
129C3×C4⋊C4—28no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4—44no—
135C3×C4⋊C4C2×D418yes—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—44no—
138C3×C22⋊C4—44no—
139C3×C22⋊C4—42no—
140C25C3×D411yescharacteristic
141C22×Q8C23×C311yescharacteristic
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—28no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—48no—
148C23×C4C3×D411yescharacteristic
149C23×C4C3×D412yes—
150C23×C4C23×C311yescharacteristic
151C23×C4C23×C312yes—
152C22×C8C23×C314yes—
153C22⋊Q8—24no—
154C22⋊Q8—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C3×D418yes—
164C2×C4⋊C4C23×C312yes—
165C2×C4⋊C4C23×C311yescharacteristic
166C2×C4⋊C4C23×C314yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171#107—28no—
172#103—28no—
173C22⋊C8—24no—
174C22⋊C8—28no—
175#95—21no—
176#95—22no—
177C23×C3—28no—
178C23×C3—28no—
179C23×C3—216no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—28no—
187C23×C3—216no—
188C23×C3#12012yes—
189C23×C3#12314yes—
190C23×C3C2×Q1612yes—
191C23×C3C2×Q1614yes—
192C23×C3AGL(1,ℤ/8)14yes—
193C23×C3#13712yes—
194C23×C3C22×D411yescharacteristic
195C3×Q8—44no—
196C2×C3×C4—216no—
197C2×C3×C4—28no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—28no—
202C2×C3×C4—216no—
203C2×C3×C4—28no—
204C2×C3×C4—416no—
205C2×C3×C4—432no—
206C2×C3×C4—48no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4#12012yes—
215C2×C3×C4#12314yes—
216C2×C3×C4C22×D412yes—
217C2×C3×C4C2×#4114yes—
218C3×C8—44no—
219C24—22no—
220C24—24no—
221C24—24no—
222C24—22no—
223C24C3×Q1614yes—
224C24C2×C3×D411yescharacteristic
225C24C2×C3×D412yes—
226C2×Q8—22no—
227C2×Q8—24no—
228C22×C4—232no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—216no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—432no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4C2×C3×D411yescharacteristic
248C22×C4C2×C3×D414yes—
249C22×C4C2×C3×D412yes—
250C22×C4C3×#4118yes—
251C22×C4C24×C311yescharacteristic
252C2×C8—24no—
253C2×C8—28no—
254C4⋊C4—216no—
255C4⋊C4—28no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4C2×C3×D418yes—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22×C3—232no—
267C22×C3—24no—
268C22×C3—24no—
269C22×C3—24no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—22no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—22no—
276C22×C3—24no—
277C22×C3#45114yes—
278C22×C3#48418yescomplemented
279C22×C3C2×#12011yescharacteristic
280C22×C3C2×#12312yes—
281C22×C3C22×Q1611yescharacteristic
282C22×C3C2×AGL(1,ℤ/8)12yes—
283C22×C3C2×#13711yescharacteristic
284C3×C4—22no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—82no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4#52514yes—
291C23—28no—
292C23—28no—
293C23—216no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—28no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—216no—
302C23C3×#12012yes—
303C23C3×#12314yes—
304C23C2×C3×Q1612yes—
305C23C2×C3×Q1614yes—
306C23C3×AGL(1,ℤ/8)14yes—
307C23C3×#13712yes—
308C23C22×C3×D411yescharacteristic
309Q8—44no—
310C2×C4—216no—
311C2×C4—28no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—216no—
317C2×C4—28no—
318C2×C4—416no—
319C2×C4—432no—
320C2×C4—48no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4C3×#12012yes—
329C2×C4C3×#12314yes—
330C2×C4C22×C3×D412yes—
331C2×C4C2×C3×#4114yes—
332C8—44no—
333C2×C3—24no—
334C2×C3—28no—
335C2×C3#284911yescharacteristic
336C2×C3C2×#45111yescharacteristic
337C2×C3#300311yescharacteristic
338C2×C3C2×#48412yes—
339C2×C3#309212yes—
340C22—232no—
341C22—24no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—22no—
350C22—24no—
351C22C3×#45114yes—
352C22C3×#48418yescomplemented
353C22C2×C3×#12011yescharacteristic
354C22C2×C3×#12312yes—
355C22C22×C3×Q1611yescharacteristic
356C22C2×C3×AGL(1,ℤ/8)12yes—
357C22C2×C3×#13711yescharacteristic
358C4—22no—
359C4—24no—
360C4—48no—
361C4—81no—
362C4—82no—
363C4—81no—
364C4C3×#52514yes—
365C3#3298311yescharacteristic, direct product factor, Sylow
366C2—24no—
367C2—28no—
368C2C3×#284911yescharacteristic
369C2C2×C3×#45111yescharacteristic
370C2C3×#300311yescharacteristic
371C2C2×C3×#48412yes—
372C2C3×#309212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
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
2. C2×C3×#484C2semidirect
4. C3×#1872C2semidirect
5. C3×#1852C2semidirect
6. C3×#1824C2semidirect
8. #32983C3direct (N × H)
19. C2×C3×#107C22semidirect
20. C2×C3×#103C22semidirect
28. C3×#340C22semidirect
30. C2×#484C2×C3semidirect
32. #1872C2×C3semidirect
33. #1852C2×C3semidirect
34. #1824C2×C3semidirect
83. C2×#107C22×C3semidirect
84. C2×#103C22×C3semidirect
92. #340C22×C3semidirect
278. C22×C3#484semidirect
352. C22C3×#484semidirect
365. C3#32983direct (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×#4842
C3×#28482
C3×#18722
C3×#18522
C3×#18242
C3×#17792
#329833