Finite Group Databasebeta

← C3×#33095 #10651702

Subgroups — C3×#33095 #10651702

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2854C211yesmaximal, characteristic
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#1926C214yesmaximal, complemented
4C3×#1919C214yesmaximal, complemented
5C3×#1874C212yesmaximal
6C3×#1853C212yesmaximal, complemented
7C3×#1817C211yesmaximal, characteristic, complemented
8#33095C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C42—21no—
13C22×C3×C42C2211yescharacteristic
14C2×C3×#106C2214yes—
15C2×C3×#102C2214yes—
16C2×C3×C22⋊C8C2212yes—
17C3×#390C2212yes—
18C3×#389C2212yes—
19C3×#388—22no—
20C3×#386—21no—
21C3×#380C2211yescharacteristic
22C3×C4×C4⋊C4—22no—
23C3×C4×C4⋊C4C2214yes—
24C2×C3×#95—21no—
25C3×#340C2214yes—
26C3×#340C2214yes—
27C3×#336C2214yes—
28#2854C2×C311yescharacteristic
29C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
30#1926C2×C314yescomplemented
31#1919C2×C314yescomplemented
32#1874C2×C312yes—
33#1853C2×C312yescomplemented
34#1817C2×C311yescharacteristic, complemented
35C22×C3×D4—21no—
36C22×C3×D4C2311yescharacteristic
37C23×C3×C4—22no—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×C3×C8C2314yes—
42C3×#123—44no—
43C3×#121—24no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—28no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4C2312yes—
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C3×C4⋊C4C2314yes—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C42—24no—
55C2×C3×C42—24no—
56C2×C3×C42—44no—
57C2×C3×C42D412yes—
58C2×C3×C42C2312yes—
59C3×#106—28no—
60C3×#102—28no—
61C3×C22⋊C8—24no—
62C3×#95—24no—
63C3×#95—21no—
64C3×#95—22no—
65C2×#123—21no—
66C2×#121C22×C311yescharacteristic
67C22×C4⋊C4C22×C312yes—
68C22×C42—21no—
69C22×C42C22×C311yescharacteristic
70C2×#106C22×C314yes—
71C2×#102C22×C314yes—
72C2×C22⋊C8C22×C312yes—
73#390C22×C312yes—
74#389C22×C312yes—
75#388—22no—
76#386—21no—
77#380C22×C311yescharacteristic
78C4×C4⋊C4—22no—
79C4×C4⋊C4C22×C314yes—
80C2×#95—21no—
81#340C22×C314yes—
82#340C22×C314yes—
83#336C22×C314yes—
84C24×C3—21no—
85C24×C3C2×D411yescharacteristic
86C2×C3×D4—24no—
87C2×C3×D4—22no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—24no—
94C22×C3×C4—28no—
95C22×C3×C4—22no—
96C22×C3×C4—24no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—42no—
100C22×C3×C4—44no—
101C22×C3×C4—44no—
102C22×C3×C4SD1614yes—
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4#4114yes—
107C22×C3×C4C2411yescharacteristic
108C2×C3×C8—24no—
109C2×C3×C8—24no—
110C2×C3×C8—24no—
111C3×C4⋊C4—28no—
112C3×C4⋊C4—24no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—44no—
116C3×C4⋊C4#4118yes—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—44no—
121C3×C42—24no—
122C3×C42—24no—
123C3×C42—48no—
124C22×D4—21no—
125C22×D4C23×C311yescharacteristic
126C23×C4—22no—
127C23×C4C3×D411yescharacteristic
128C23×C4C3×D411yescharacteristic
129C23×C4C23×C311yescharacteristic
130C22×C8C23×C314yes—
131#123—44no—
132#121—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C23×C311yescharacteristic
137C2×C4⋊C4C23×C312yes—
138C2×C4⋊C4C23×C314yes—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—44no—
146C2×C42C3×D412yes—
147C2×C42C23×C312yes—
148#106—28no—
149#102—28no—
150C22⋊C8—24no—
151#95—24no—
152#95—21no—
153#95—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—41no—
157C23×C3—42no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—42no—
161C23×C3#12312yes—
162C23×C3C2×SD1612yes—
163C23×C3AGL(1,ℤ/8)12yes—
164C23×C3C22×D411yescharacteristic
165C3×D4—44no—
166C3×D4—84no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—28no—
171C2×C3×C4—28no—
172C2×C3×C4—22no—
173C2×C3×C4—28no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—44no—
177C2×C3×C4—416no—
178C2×C3×C4—44no—
179C2×C3×C4—44no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4#12312yes—
187C2×C3×C4C2×SD1614yes—
188C2×C3×C4C2×#4112yes—
189C2×C3×C4C2×#4114yes—
190C3×C8—44no—
191C24—21no—
192C24C2×C3×D411yescharacteristic
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4C3×SD1614yes—
210C22×C4C2×C3×D412yes—
211C22×C4C2×C3×D411yescharacteristic
212C22×C4C2×C3×D412yes—
213C22×C4C3×#4114yes—
214C22×C4C24×C311yescharacteristic
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C4⋊C4—28no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4C3×#4118yes—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C42—24no—
229C42—24no—
230C42—48no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—81no—
236C22×C3—81no—
237C22×C3—82no—
238C22×C3—81no—
239C22×C3—82no—
240C22×C3#48114yes—
241C22×C3C2×#12311yescharacteristic
242C22×C3#52512yes—
243C22×C3C22×SD1611yescharacteristic
244C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C3#57512yes—
246C3×C4—24no—
247C3×C4—22no—
248C3×C4—42no—
249C3×C4—48no—
250C3×C4—81no—
251C3×C4—82no—
252C3×C4#52914yes—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—41no—
259C23—42no—
260C23C3×#12312yes—
261C23C2×C3×SD1612yes—
262C23C3×AGL(1,ℤ/8)12yes—
263C23C22×C3×D411yescharacteristic
264D4—44no—
265D4—84no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—28no—
275C2×C4—44no—
276C2×C4—416no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4C3×#12312yes—
286C2×C4C2×C3×SD1614yes—
287C2×C4C2×C3×#4112yes—
288C2×C4C2×C3×#4114yes—
289C8—44no—
290C2×C3—24no—
291C2×C3—161no—
292C2×C3#285311yescharacteristic
293C2×C3C2×#48111yescharacteristic
294C2×C3#309311yescharacteristic
295C2×C3#310312yes—
296C2×C3#311412yes—
297C22—22no—
298C22—24no—
299C22—22no—
300C22—24no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22C3×#48114yes—
307C22C2×C3×#12311yescharacteristic
308C22C3×#52512yes—
309C22C22×C3×SD1611yescharacteristic
310C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
311C22C3×#57512yes—
312C4—24no—
313C4—22no—
314C4—42no—
315C4—48no—
316C4—81no—
317C4—82no—
318C4C3×#52914yes—
319C3#3309511yescharacteristic, direct product factor, Sylow
320C2—24no—
321C2—161no—
322C2C3×#285311yescharacteristic
323C2C2×C3×#48111yescharacteristic
324C2C3×#309311yescharacteristic
325C2C3×#310312yes—
326C2C3×#311412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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×C4×C4⋊C4C2semidirect
3. C3×#1926C2semidirect
4. C3×#1919C2semidirect
6. C3×#1853C2semidirect
7. C3×#1817C2semidirect
8. #33095C3direct (N × H)
29. C2×C4×C4⋊C4C2×C3semidirect
30. #1926C2×C3semidirect
31. #1919C2×C3semidirect
33. #1853C2×C3semidirect
34. #1817C2×C3semidirect
319. C3#33095direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28542
C2×C3×C4×C4⋊C42
C3×#19262
C3×#19192
C3×#18742
C3×#18532
C3×#18172
#330953