Finite Group Databasebeta

← C3×#32982 #10651589

Subgroups — C3×#32982 #10651589

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#483C212yesmaximal, complemented
3C3×#2848C211yesmaximal, characteristic
4C3×#1872C214yesmaximal, complemented
5C3×#1853C214yesmaximal, complemented
6C3×#1823C212yesmaximal, complemented
7C3×#1779C211yesmaximal, characteristic
8#32982C311yesmaximal, 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×#483—28no—
19C2×C3×#106C2214yescomplemented
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×#483C2×C312yescomplemented
31#2848C2×C311yescharacteristic
32#1872C2×C314yescomplemented
33#1853C2×C314yescomplemented
34#1823C2×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—44no—
44C23×C3×C4—44no—
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D412yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C22×C3×C8C2314yes—
50C3×C22⋊Q8—24no—
51C3×C22⋊Q8—28no—
52C2×C3×C4⋊C4—28no—
53C2×C3×C4⋊C4—216no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—216no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4D418yes—
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×#103—28no—
70C3×C22⋊C8—24no—
71C3×C22⋊C8—28no—
72C3×#95—21no—
73C3×#95—22no—
74C24×C4—21no—
75C24×C4C22×C311yescharacteristic
76C2×C22⋊Q8C22×C311yescharacteristic
77C2×C22⋊Q8C22×C312yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4C22×C314yes—
82C22×C4⋊C4C22×C312yes—
83#483—28no—
84C2×#106C22×C314yescomplemented
85C2×#103C22×C314yescomplemented
86C2×C22⋊C8C22×C312yes—
87C2×C22⋊C8C22×C314yes—
88#385—22no—
89#385—21no—
90#380C22×C311yescharacteristic
91#380C22×C312yes—
92#379—21no—
93#340C22×C318yescomplemented
94C24×C3—22no—
95C24×C3—24no—
96C24×C3—22no—
97C24×C3—24no—
98C24×C3SD1614yes—
99C24×C3C2×D411yescharacteristic
100C24×C3C2×D412yes—
101C2×C3×Q8—22no—
102C2×C3×Q8—24no—
103C22×C3×C4—232no—
104C22×C3×C4—22no—
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—24no—
112C22×C3×C4—216no—
113C22×C3×C4—28no—
114C22×C3×C4—24no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—48no—
119C22×C3×C4—432no—
120C22×C3×C4—48no—
121C22×C3×C4—44no—
122C22×C3×C4—44no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D414yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4#4118yes—
127C22×C3×C4C2411yescharacteristic
128C2×C3×C8—24no—
129C2×C3×C8—28no—
130C3×C4⋊C4—216no—
131C3×C4⋊C4—28no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—44no—
137C3×C4⋊C4C2×D418yes—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—44no—
140C3×C22⋊C4—42no—
141C3×C22⋊C4—44no—
142C25C3×D411yescharacteristic
143C22×Q8C23×C311yescharacteristic
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4C3×D411yescharacteristic
152C23×C4C3×D412yes—
153C23×C4C23×C311yescharacteristic
154C23×C4C23×C312yes—
155C22×C8C23×C314yes—
156C22⋊Q8—24no—
157C22⋊Q8—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C3×D418yes—
167C2×C4⋊C4C23×C312yes—
168C2×C4⋊C4C23×C311yescharacteristic
169C2×C4⋊C4C23×C314yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174#106—28no—
175#103—28no—
176C22⋊C8—24no—
177C22⋊C8—28no—
178#95—21no—
179#95—22no—
180C23×C3—28no—
181C23×C3—216no—
182C23×C3—28no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—28no—
190C23×C3—216no—
191C23×C3#12012yes—
192C23×C3#12314yes—
193C23×C3C2×SD1612yes—
194C23×C3C2×SD1614yes—
195C23×C3AGL(1,ℤ/8)12yes—
196C23×C3#13714yes—
197C23×C3C22×D411yescharacteristic
198C3×Q8—44no—
199C2×C3×C4—216no—
200C2×C3×C4—28no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—28no—
205C2×C3×C4—216no—
206C2×C3×C4—28no—
207C2×C3×C4—432no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—416no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4#12012yes—
219C2×C3×C4#12314yes—
220C2×C3×C4C22×D412yes—
221C2×C3×C4C2×#4114yes—
222C3×C8—44no—
223C24—22no—
224C24—24no—
225C24—22no—
226C24—24no—
227C24C3×SD1614yes—
228C24C2×C3×D411yescharacteristic
229C24C2×C3×D412yes—
230C2×Q8—22no—
231C2×Q8—24no—
232C22×C4—232no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—216no—
242C22×C4—28no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—48no—
248C22×C4—432no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4C2×C3×D411yescharacteristic
253C22×C4C2×C3×D414yes—
254C22×C4C2×C3×D412yes—
255C22×C4C3×#4118yes—
256C22×C4C24×C311yescharacteristic
257C2×C8—24no—
258C2×C8—28no—
259C4⋊C4—216no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4C2×C3×D418yes—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C22×C3—232no—
272C22×C3—24no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—22no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—22no—
281C22×C3—24no—
282C22×C3#45014yes—
283C22×C3#48318yescomplemented
284C22×C3C2×#12011yescharacteristic
285C22×C3C2×#12312yes—
286C22×C3C22×SD1611yescharacteristic
287C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C3C2×#13712yes—
289C3×C4—22no—
290C3×C4—24no—
291C3×C4—48no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4#52514yes—
296C23—28no—
297C23—28no—
298C23—216no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—28no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—216no—
307C23C3×#12012yes—
308C23C3×#12314yes—
309C23C2×C3×SD1612yes—
310C23C2×C3×SD1614yes—
311C23C3×AGL(1,ℤ/8)12yes—
312C23C3×#13714yes—
313C23C22×C3×D411yescharacteristic
314Q8—44no—
315C2×C4—216no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—216no—
322C2×C4—28no—
323C2×C4—432no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—416no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4C3×#12012yes—
335C2×C4C3×#12314yes—
336C2×C4C22×C3×D412yes—
337C2×C4C2×C3×#4114yes—
338C8—44no—
339C2×C3—24no—
340C2×C3—28no—
341C2×C3#284911yescharacteristic
342C2×C3C2×#45011yescharacteristic
343C2×C3#300411yescharacteristic
344C2×C3C2×#48312yes—
345C2×C3#309212yes—
346C22—232no—
347C22—24no—
348C22—22no—
349C22—24no—
350C22—24no—
351C22—24no—
352C22—24no—
353C22—24no—
354C22—24no—
355C22—22no—
356C22—24no—
357C22C3×#45014yes—
358C22C3×#48318yescomplemented
359C22C2×C3×#12011yescharacteristic
360C22C2×C3×#12312yes—
361C22C22×C3×SD1611yescharacteristic
362C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
363C22C2×C3×#13712yes—
364C4—22no—
365C4—24no—
366C4—48no—
367C4—81no—
368C4—82no—
369C4—81no—
370C4C3×#52514yes—
371C3#3298211yescharacteristic, direct product factor, Sylow
372C2—24no—
373C2—28no—
374C2C3×#284911yescharacteristic
375C2C2×C3×#45011yescharacteristic
376C2C3×#300411yescharacteristic
377C2C2×C3×#48312yes—
378C2C3×#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×#483C2semidirect
4. C3×#1872C2semidirect
5. C3×#1853C2semidirect
6. C3×#1823C2semidirect
8. #32982C3direct (N × H)
19. C2×C3×#106C22semidirect
20. C2×C3×#103C22semidirect
28. C3×#340C22semidirect
30. C2×#483C2×C3semidirect
32. #1872C2×C3semidirect
33. #1853C2×C3semidirect
34. #1823C2×C3semidirect
84. C2×#106C22×C3semidirect
85. C2×#103C22×C3semidirect
93. #340C22×C3semidirect
283. C22×C3#483semidirect
358. C22C3×#483semidirect
371. C3#32982direct (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×#4832
C3×#28482
C3×#18722
C3×#18532
C3×#18232
C3×#17792
#329823