Finite Group Databasebeta

← #76335

Subgroups — #76335

Order 384.

708 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#938C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#883C214yesmaximal, complemented
4C3×#390C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4975C211yesmaximal, characteristic, complemented
7#4949C214yesmaximal, complemented
8C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9#4935C211yesmaximal, characteristic, complemented
10#2501—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×C4×D3C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2214yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2214yes—
20#938—24no—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2214yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2214yes—
27C2×C4×Dic3C2211yescharacteristic
28D3×C4⋊C4—24no—
29#883—216no—
30C3⋊#95C2212yes—
31C2×#123—34no—
32C2×#121—31no—
33C22×C4⋊C4—31no—
34#399—34no—
35#390—31no—
36#390D311yescharacteristic, complemented
37#384—31no—
38#384—31no—
39C4×C22⋊C4—31no—
40C24×C3C2312yes—
41C23×D3C2311yescharacteristic
42C2×C3×D4—22no—
43C2×C3×D4—22no—
44C2×C3×D4—22no—
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—28no—
49C2×#67—24no—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2311yescharacteristic
53C22×Dic3C2312yes—
54C2×C4×D3—22no—
55C2×C4×D3—22no—
56C2×C4×D3—28no—
57C2×C4×D3—24no—
58C2×C4×D3D414yes—
59C3×C4⋊C4—22no—
60C3×C22⋊C4—28no—
61C3×C42—22no—
62#217—28no—
63#217—24no—
64(C2×D3)⋊C4D418yescomplemented
65C4⋊Dic3—28no—
66C4⋊Dic3—22no—
67C4⋊Dic3—22no—
68Dic3⋊C4—28no—
69Dic3⋊C4—24no—
70C4×Dic3—22no—
71C22×D4—32no—
72C22×D4C2×D311yescharacteristic
73C23×C4—31no—
74C23×C4—32no—
75#123—616no—
76#121—64no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—34no—
81C2×C4⋊C4—34no—
82C2×C4⋊C4—64no—
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—34no—
87C2×C22⋊C4C2×D314yes—
88C2×C42—31no—
89C2×C42C2×D311yescharacteristic
90#95—32no—
91C23×C3—24no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—24no—
96C23×C3C2411yescharacteristic
97C22×D3—21no—
98C22×D3—21no—
99C22×D3—22no—
100C22×D3C2×D414yes—
101C22×D3C2×D412yes—
102C3×D4—44no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—24no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D414yes—
108C2×C3×C4C2×D412yes—
109C2×C3×C4#4112yes—
110C2×C3×C4#4112yes—
111#67—48no—
112C2×Dic3—22no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—24no—
118C2×Dic3—24no—
119C2×Dic3—22no—
120C2×Dic3C2×D414yes—
121C2×Dic3C2×D412yes—
122C4×D3—28no—
123C4×D3—48no—
124C24—31no—
125C24C22×D312yes—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—64no—
130C2×D4—68no—
131C22×C4—31no—
132C22×C4—32no—
133C22×C4—34no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—64no—
137C22×C4—68no—
138C22×C4—62no—
139C22×C4—62no—
140C22×C4C22×D312yes—
141C22×C4C22×D312yes—
142C22×C4C22×D311yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—64no—
145C4⋊C4—68no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4—68no—
149C22⋊C4—28no—
150C22⋊C4—38no—
151C22⋊C4—64no—
152C22⋊C4—68no—
153C42—22no—
154C42—62no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3—44no—
159C22×C3—44no—
160C22×C3C22×D411yescharacteristic
161C22×C3C22×D412yes—
162C22×C3C2×#4111yescharacteristic
163C22×C3C2×#4111yescharacteristic
164C22×C32−1+412yes—
165C2×D3—22no—
166C2×D3—22no—
167C2×D3—24no—
168C2×D3—24no—
169C2×D3#12714yes—
170C3×C4—24no—
171C3×C4—42no—
172C3×C4#12314yescomplemented
173Dic3—22no—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—24no—
182C23—34no—
183C23—32no—
184C23—61no—
185C23—61no—
186C23—62no—
187C23C23×D311yescharacteristic
188D4—44no—
189D4—128no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—34no—
195C2×C4—32no—
196C2×C4—62no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—68no—
201C2×C4—64no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—64no—
205C2×C4—128no—
206C2×C4#23512yes—
207C2×C4D3×D414yes—
208C2×C4D3×D412yes—
209C2×C4#23712yes—
210C2×C3—82no—
211C2×C3C2×#12311yescharacteristic
212C2×C3C2×#12711yescharacteristic
213C2×C3#53611yescharacteristic
214C2×C3#54712yes—
215C2×C3#54712yes—
216D3—24no—
217C22—34no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—44no—
222C22—44no—
223C22—62no—
224C22—62no—
225C22—64no—
226C22—64no—
227C22C2×#23511yescharacteristic
228C22C2×D3×D411yescharacteristic
229C22C2×D3×D412yes—
230C22C2×#23711yescharacteristic
231C22#101012yes—
232C4—24no—
233C4—42no—
234C4—62no—
235C4—64no—
236C4—122no—
237C4—122no—
238C4#88314yescomplemented
239C3#250111yescharacteristic, complemented, Sylow
240C2—64no—
241C2—82no—
242C2C2×#88311yescharacteristic
243C2#555112yes—
244C2#561112yes—
245C2#567511yescharacteristic
246C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#938C2
hypercenter#390D3
hypocenterC3#2501
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
3. C2×#883C2semidirect
4. C3×#390C2semidirect
5. #5221C2semidirect
6. #4975C2semidirect
7. #4949C2semidirect
8. C4×(C2×D3)⋊C4C2semidirect
9. #4935C2semidirect
36. #390D3semidirect
64. (C2×D3)⋊C4D4semidirect
172. C3×C4#123semidirect
238. C4#883semidirect
239. C3#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×D3×C4⋊C42
C2×#8832
C3×#3902
#52212
#49752
#49492
C4×(C2×D3)⋊C42
#49352
#25013