Finite Group Databasebeta

← #82633

Subgroups — #82633

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×#926C211yesmaximal, characteristic, complemented
4#5221C211yesmaximal, characteristic, complemented
5#5221C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#5212C211yesmaximal, characteristic, complemented
8#4984C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4951C212yesmaximal, complemented
11#4950C212yesmaximal, complemented
12#2324—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C3×#121C418yescomplemented
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26D4×Dic3—24no—
27#926—24no—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#121D311yescharacteristic
40C2×C4×D4—31no—
41C2×#117—31no—
42#390—31no—
43#390—32no—
44#389—31no—
45#388—32no—
46#386—32no—
47#385—32no—
48#380—31no—
49C4×C4⋊C4—31no—
50C24×C3C2311yescharacteristic
51C24×C3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4C2×C414yes—
57C2×C3×D4C2×C418yescomplemented
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4C2×C414yescomplemented
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×Dic3—24no—
65C22×Dic3—42no—
66C22×Dic3—42no—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C3×C4⋊C4C2×C414yescomplemented
74C3×C22⋊C4C2×C418yescomplemented
75#217—22no—
76#217—22no—
77#217—24no—
78#217—42no—
79#217—42no—
80#217—42no—
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—22no—
85C4⋊Dic3—42no—
86C4×Dic3—22no—
87C4×Dic3—42no—
88C4×Dic3D414yescomplemented
89C22×D4C2×D311yescharacteristic
90C22×D4C2×D312yes—
91C23×C4—32no—
92C23×C4C2×D311yescharacteristic
93#121Dic318yescomplemented
94C4×D4—64no—
95#117—64no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊C4—31no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D312yes—
109C2×C42—31no—
110C2×C42—31no—
111C2×C42—32no—
112#95—32no—
113#95—32no—
114#95—32no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3C22×C414yes—
128C23×C3C22×C412yes—
129C23×C3C2411yescharacteristic
130C3×D4—28no—
131C3×D4—48no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—28no—
137C2×C3×C4C22×C412yes—
138C2×C3×C4C22×C414yes—
139C2×C3×C4C22×C412yes—
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yes—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—48no—
159C2×Dic3C2×D412yes—
160C2×Dic3C2×D412yes—
161C24C22×D311yescharacteristic
162C24C22×D312yes—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4C2×Dic314yes—
168C2×D4C2×Dic318yescomplemented
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—64no—
178C22×C4—122no—
179C22×C4—122no—
180C22×C4C2×Dic314yescomplemented
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—122no—
189C4⋊C4C2×Dic314yescomplemented
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—64no—
193C22⋊C4—122no—
194C22⋊C4—122no—
195C22⋊C4—122no—
196C22⋊C4C2×Dic318yescomplemented
197C42—34no—
198C42—62no—
199C42—122no—
200C22×C3—22no—
201C22×C3—24no—
202C22×C3—22no—
203C22×C3—24no—
204C22×C3—22no—
205C22×C3—24no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3C23×C411yescharacteristic
215C22×C3C22×D411yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C3H(5,𝔽2)11yescharacteristic
219C22×C3H(5,𝔽2)11yescharacteristic
220C22×C32−1+411yescharacteristic
221C3×C4—22no—
222C3×C4—24no—
223C3×C4C4×D414yes—
224Dic3—22no—
225Dic3—22no—
226Dic3—41no—
227Dic3—42no—
228Dic3—41no—
229Dic3—42no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23C22×Dic314yes—
243C23C22×Dic312yes—
244C23C23×D311yescharacteristic
245D4—28no—
246D4—48no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—128no—
271C2×C4D3×D412yes—
272C2×C4#23712yes—
273C2×C4C22×Dic312yes—
274C2×C4C22×Dic314yes—
275C2×C4C22×Dic312yes—
276C2×C3—42no—
277C2×C3—44no—
278C2×C3C2×C4×D411yescharacteristic
279C2×C3#51811yescharacteristic
280C2×C3#52011yescharacteristic
281C2×C3#53511yescharacteristic
282C2×C3#53711yescharacteristic
283C2×C3#54011yescharacteristic
284C2×C3#54211yescharacteristic
285C22—22no—
286C22—24no—
287C22—22no—
288C22—24no—
289C22—22no—
290C22—24no—
291C22—41no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22C2×D3×D411yescharacteristic
300C22C2×#23711yescharacteristic
301C22#100411yescharacteristic
302C22#100411yescharacteristic
303C22#100911yescharacteristic
304C22#101011yescharacteristic
305C22C23×Dic311yescharacteristic
306C4—22no—
307C4—24no—
308C4—62no—
309C4—62no—
310C4—122no—
311C4—121no—
312C4—121no—
313C4—122no—
314C4D4×Dic314yes—
315C3#232411yescharacteristic, complemented, Sylow
316C2—42no—
317C2—44no—
318C2#560311yescharacteristic
319C2#560711yescharacteristic
320C2#560811yescharacteristic
321C2#561911yescharacteristic
322C2C2×D4×Dic311yescharacteristic
323C2#579811yescharacteristic
324C2#582711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
Fitting, prime radicalC2×C3×#121C2
socle, cosocleC23×C3C24
hypercenterC2×#121D3
hypocenterC3#2324
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×#926C2semidirect
4. #5221C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #5212C2semidirect
8. #4984C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4951C2semidirect
11. #4950C2semidirect
17. C3×#121C4semidirect
57. C2×C3×D4C2×C4semidirect
60. C22×C3×C4C2×C4semidirect
73. C3×C4⋊C4C2×C4semidirect
74. C3×C22⋊C4C2×C4semidirect
88. C4×Dic3D4semidirect
93. #121Dic3semidirect
168. C2×D4C2×Dic3semidirect
180. C22×C4C2×Dic3semidirect
189. C4⋊C4C2×Dic3semidirect
196. C22⋊C4C2×Dic3semidirect
315. C3#2324semidirect

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
C2×#9262
#52212
#52202
#52122
#49842
Dic3×C4⋊C42
#49512
#49502
#23243