Finite Group Databasebeta

← Dic3×#121 #82632

Subgroups — Dic3×#121 #82632

Order 384.

824 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×D4×Dic3C212yesmaximal, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C4×Dic3C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5220C212yesmaximal, complemented
7#5212C211yesmaximal, characteristic, complemented
8#4984C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4951C212yesmaximal, complemented
11Dic3×C22⋊C4C212yesmaximal, complemented
12C4×#121—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C3×#121C418yescomplemented
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26D4×Dic3—28no—
27D4×Dic3—24no—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4×Dic3—24no—
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2212yes—
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#121D311yescharacteristic
41C2×C4×D4—31no—
42C2×C4×D4—32no—
43C22×C42—31no—
44#390—31no—
45#389—31no—
46#388—32no—
47#386—32no—
48#380—31no—
49C4×C4⋊C4—31no—
50C4×C22⋊C4—32no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4C2×C418yescomplemented
58C2×C3×D4C2×C414yescomplemented
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4C2×C414yescomplemented
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×Dic3—22no—
66C22×Dic3—22no—
67C22×Dic3—22no—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C22×Dic3—24no—
71C22×Dic3—24no—
72C22×Dic3—24no—
73C22×Dic3D414yescomplemented
74C22×Dic3C2312yes—
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C3×C4⋊C4C2×C414yescomplemented
81C3×C22⋊C4C2×C418yescomplemented
82#217—24no—
83#217—24no—
84#217—24no—
85#217—24no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—24no—
89C4×Dic3—22no—
90C4×Dic3—22no—
91C4×Dic3—22no—
92C4×Dic3—24no—
93C4×Dic3D414yescomplemented
94C22×D4C2×D312yes—
95C22×D4C2×D311yescharacteristic
96C23×C4—31no—
97C23×C4—32no—
98C23×C4—31no—
99C23×C4C2×D311yescharacteristic
100#121Dic318yesdirect product factor
101C4×D4—64no—
102C4×D4—68no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4C2×D311yescharacteristic
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4C2×D312yes—
112C2×C42—31no—
113C2×C42—32no—
114C2×C42—31no—
115C2×C42—31no—
116C2×C42—31no—
117C2×C42—64no—
118#95—32no—
119#95—32no—
120#95—32no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3C22×C414yes—
132C23×C3C22×C412yes—
133C23×C3C2×D412yes—
134C23×C3#4112yes—
135C23×C3C2411yescharacteristic
136C3×D4—216no—
137C3×D4—28no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—28no—
143C2×C3×C4C22×C412yes—
144C2×C3×C4C22×C414yes—
145C2×C3×C4C22×C412yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4#4112yes—
148C2×Dic3—28no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—28no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—48no—
163C2×Dic3C2×D412yes—
164C2×Dic3C2×D412yes—
165C2×Dic3C2×D412yes—
166C2×Dic3C2×D412yes—
167C2×Dic3#4112yes—
168C2×Dic3#4112yes—
169C24C22×D312yes—
170C24C22×D311yescharacteristic
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4C2×Dic318yescomplemented
176C2×D4C2×Dic314yescomplemented
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—31no—
180C22×C4—34no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4—64no—
191C22×C4—64no—
192C22×C4—64no—
193C22×C4—62no—
194C22×C4C2×Dic314yescomplemented
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C22×C4C22×D311yescharacteristic
198C4⋊C4—64no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C4⋊C4C2×Dic314yescomplemented
202C22⋊C4—64no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C22⋊C4C2×Dic318yescomplemented
207C42—34no—
208C42—62no—
209C42—62no—
210C42—64no—
211C42—62no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3C4×D414yescomplemented
226C22×C3C23×C411yescharacteristic
227C22×C3C22×D411yescharacteristic
228C22×C3C22×D411yescharacteristic
229C22×C3C2×#4111yescharacteristic
230C22×C3C2×#4111yescharacteristic
231C22×C3C2×#4111yescharacteristic
232C22×C3C2×#4111yescharacteristic
233C3×C4—22no—
234C3×C4—24no—
235C3×C4C4×D414yescomplemented
236Dic3—22no—
237Dic3—22no—
238Dic3—22no—
239Dic3—42no—
240Dic3—42no—
241Dic3#12114yesdirect product factor
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23D3×D412yes—
253C23#23712yes—
254C23C22×Dic314yes—
255C23C22×Dic312yes—
256C23C23×D311yescharacteristic
257D4—216no—
258D4—28no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—28no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—32no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—68no—
271C2×C4—68no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—128no—
285C2×C4D3×D412yes—
286C2×C4#23712yes—
287C2×C4C22×Dic312yes—
288C2×C4C22×Dic314yes—
289C2×C4C22×Dic312yes—
290C2×C3—24no—
291C2×C3—44no—
292C2×C3C2×C4×D411yescharacteristic
293C2×C3C2×C4×D411yescharacteristic
294C2×C3C4×#4111yescharacteristic
295C2×C3C2×#12111yescharacteristic
296C2×C3#52511yescharacteristic
297C2×C3#52911yescharacteristic
298C2×C3#53211yescharacteristic
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22D4×Dic314yescomplemented
313C22C2×D3×D411yescharacteristic
314C22C2×D3×D411yescharacteristic
315C22C2×#23711yescharacteristic
316C22C2×#23711yescharacteristic
317C22D3×#4111yescharacteristic
318C22D3×#4111yescharacteristic
319C22C23×Dic311yescharacteristic
320C4—22no—
321C4—24no—
322C4—34no—
323C4—62no—
324C4—62no—
325C4—62no—
326C4—122no—
327C4—122no—
328C4D4×Dic314yescomplemented
329C3C4×#12111yescharacteristic, complemented, Sylow
330C2—24no—
331C2—44no—
332C2#559611yescharacteristic
333C2#560011yescharacteristic
334C2D3×#12111yescharacteristic
335C2#560611yescharacteristic
336C2C2×D4×Dic311yescharacteristic
337C2C2×D4×Dic311yescharacteristic
338C2Dic3×#4111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
Fitting, prime radicalC2×C3×#121C2
socle, cosocleC23×C3C24
hypercenterC2×#121D3
hypocenterC3C4×#121
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×D4×Dic3C2semidirect
3. C2×D4×Dic3C2semidirect
4. C22×C4×Dic3C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #5212C2semidirect
8. #4984C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4951C2semidirect
11. Dic3×C22⋊C4C2semidirect
19. C3×#121C4semidirect
57. C2×C3×D4C2×C4semidirect
58. C2×C3×D4C2×C4semidirect
61. C22×C3×C4C2×C4semidirect
73. C22×Dic3D4semidirect
80. C3×C4⋊C4C2×C4semidirect
81. C3×C22⋊C4C2×C4semidirect
93. C4×Dic3D4semidirect
100. #121Dic3direct (N × H)
175. C2×D4C2×Dic3semidirect
176. C2×D4C2×Dic3semidirect
194. C22×C4C2×Dic3semidirect
201. C4⋊C4C2×Dic3semidirect
206. C22⋊C4C2×Dic3semidirect
225. C22×C3C4×D4semidirect
235. C3×C4C4×D4semidirect
241. Dic3#121direct (N × H)
312. C22D4×Dic3semidirect
328. C4D4×Dic3semidirect
329. C3C4×#121semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×D4×Dic32
C22×C4×Dic32
#52212
#52202
#52122
#49842
Dic3×C4⋊C42
#49512
Dic3×C22⋊C42
C4×#1213