Finite Group Databasebeta

← #75968

Subgroups — #75968

Order 384.

600 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#892C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic
3C2×#872C211yesmaximal, characteristic
4C3×#389C211yesmaximal, characteristic, complemented
5#4989C212yesmaximal
6#4984C211yesmaximal, characteristic, complemented
7#4975C212yesmaximal, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4668C212yesmaximal
11#4650C212yesmaximal, complemented
12#2317—31nomaximal, Sylow
13C22×C4×D3C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32#892C418yescomplemented
33D3×C4⋊C4—24no—
34#872—24no—
35C3×#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#117—31no—
40C2×#117—31no—
41C22×C4⋊C4—31no—
42#389—31no—
43#389—31no—
44#389—31no—
45#389D311yescharacteristic, complemented
46#385—32no—
47#385—32no—
48#384—32no—
49#382—32no—
50C23×D3C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C2×C4×D3—24no—
62C2×C4×D3—42no—
63C2×C4×D3—42no—
64C2×C4×D3C2×C414yes—
65C2×C4×D3D414yes—
66C2×C4×D3Q814yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69C3×C4⋊C4C2×C414yescomplemented
70C3×C42—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4C2×C418yes—
74C4⋊Dic3—22no—
75C4⋊Dic3—42no—
76C4⋊Dic3C2×C414yescomplemented
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4—24no—
80Dic3⋊C4—42no—
81Dic3⋊C4—42no—
82C4×Dic3—22no—
83C4×Dic3C2×C418yescomplemented
84C23×C4—31no—
85C23×C4—32no—
86#117—38no—
87#117—64no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—64no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C4⋊C4C2×D312yes—
99C2×C4⋊C4C2×D311yescharacteristic
100C2×C22⋊C4—32no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—31no—
103C2×C42—32no—
104C2×C42—31no—
105C2×C42C2×D311yescharacteristic
106#95—32no—
107#95—32no—
108#95—32no—
109#95C2×D312yes—
110C23×C3C2411yescharacteristic
111C22×D3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3C22×C412yes—
116C22×D3C2×D412yes—
117C22×D3C2×Q812yes—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4C22×C414yes—
130C2×C3×C4C22×C412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×Q812yes—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3C22×C414yes—
145C2×Dic3C22×C412yes—
146C2×Dic3C2×D412yes—
147C2×Dic3C2×Q812yes—
148C4×D3—48no—
149C4×D3C4⋊C4116yescomplemented
150C24—31no—
151C22×C4—31no—
152C22×C4—34no—
153C22×C4—34no—
154C22×C4—34no—
155C22×C4—32no—
156C22×C4—31no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—64no—
160C22×C4—122no—
161C22×C4—122no—
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D312yes—
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D312yes—
166C22×C4C22×D311yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—34no—
169C4⋊C4—42no—
170C4⋊C4—64no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C4⋊C4—122no—
177C4⋊C4C4×D314yescomplemented
178C22⋊C4—38no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C42—22no—
182C42—38no—
183C42—62no—
184C22×C3C23×C411yescharacteristic
185C22×C3C22×D411yescharacteristic
186C22×C3C22×Q811yescharacteristic
187C22×C3H(5,𝔽2)11yescharacteristic
188C22×C3H(5,𝔽2)11yescharacteristic
189C22×C32−1+411yescharacteristic
190C22×C32−1+411yescharacteristic
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—22no—
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—22no—
197C2×D3C2×C4⋊C414yes—
198C3×C4—24no—
199C3×C4—42no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4C2×C4⋊C414yes—
203Dic3—24no—
204Dic3—42no—
205Dic3—41no—
206Dic3—41no—
207Dic3C2×C4⋊C414yes—
208C23—32no—
209C23—32no—
210C23—32no—
211C23—61no—
212C23—61no—
213C23—61no—
214C23—61no—
215C23C23×D311yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—316no—
231C2×C4—34no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—61no—
243C2×C4—128no—
244C2×C4C2×C4×D314yes—
245C2×C4C2×C4×D312yes—
246C2×C4D3×D412yes—
247C2×C4D3×Q812yes—
248C2×C3C22×C4⋊C411yescharacteristic
249C2×C3#52011yescharacteristic
250C2×C3#52011yescharacteristic
251C2×C3#53511yescharacteristic
252C2×C3#53611yescharacteristic
253C2×C3#54411yescharacteristic
254C2×C3#54411yescharacteristic
255D3—24no—
256C22—34no—
257C22—62no—
258C22—62no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—62no—
263C22C22×C4×D311yescharacteristic
264C22C2×D3×D411yescharacteristic
265C22#100411yescharacteristic
266C22C2×D3×Q811yescharacteristic
267C22#100711yescharacteristic
268C22#100911yescharacteristic
269C22#101011yescharacteristic
270C4—24no—
271C4—34no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—64no—
276C4—122no—
277C4—121no—
278C4—121no—
279C4D3×C4⋊C414yes—
280C3#231711yescharacteristic, complemented, Sylow
281C2—64no—
282C2C2×D3×C4⋊C411yescharacteristic
283C2#554611yescharacteristic
284C2#557411yescharacteristic
285C2#560711yescharacteristic
286C2#562811yescharacteristic
287C2#568911yescharacteristic
288C2#572911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#389C2
socle, cosocleC23×C3C24
prime radicalC23×D3C23
hypercenter#389D3
hypocenterC3#2317
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
4. C3×#389C2semidirect
6. #4984C2semidirect
7. #4975C2semidirect
8. #4972C2semidirect
9. #4928C2semidirect
11. #4650C2semidirect
32. #892C4semidirect
45. #389D3semidirect
69. C3×C4⋊C4C2×C4semidirect
76. C4⋊Dic3C2×C4semidirect
83. C4×Dic3C2×C4semidirect
149. C4×D3C4⋊C4semidirect
177. C4⋊C4C4×D3semidirect
280. C3#2317semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×D3×C4⋊C42
C2×#8722
C3×#3892
#49892
#49842
#49752
#49722
#49282
#46682
#46502
#23173