Finite Group Databasebeta

← #76664

Subgroups — #76664

Order 384.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#946C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3C2×#897C211yesmaximal, characteristic
4C2×C42×D3C211yesmaximal, characteristic
5C3×#391C211yesmaximal, characteristic, complemented
6#4975C211yesmaximal, characteristic, complemented
7#4929C211yesmaximal, characteristic, complemented
8#4667C214yesmaximal
9#4647C214yesmaximal, complemented
10#2857—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C4×D3C2211yescharacteristic
13C22×C4×D3C2212yes—
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17#946—24no—
18#944—24no—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2214yes—
25C2×Dic3⋊C4C2214yes—
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#897—24no—
29C42×D3—24no—
30C3×#95C2214yes—
31C3⋊#95C2214yes—
32C2×C4⋊Q8—31no—
33C2×C22⋊Q8—31no—
34C2×C22⋊Q8—31no—
35C22×C42—31no—
36#391—31no—
37#391D311yescharacteristic, complemented
38#389—34no—
39#384—31no—
40#380—34no—
41C23×D3C2311yescharacteristic
42C2×C3×Q8—22no—
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2311yescharacteristic
52C22×Dic3C2312yes—
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3—24no—
56C2×C4×D3—24no—
57C2×C4×D3D414yes—
58C2×C4×D3Q818yescomplemented
59C2×Dic6—22no—
60C2×Dic6—22no—
61C2×Dic6—22no—
62C3×C4⋊C4—22no—
63C3×C42—22no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66C4⋊Dic3—22no—
67C4⋊Dic3—24no—
68Dic3⋊C4—28no—
69Dic3⋊C4—24no—
70Dic3⋊C4—28no—
71C4×Dic3—24no—
72C4×Dic3D414yes—
73C22×Q8—31no—
74C22×Q8C2×D311yescharacteristic
75C23×C4—31no—
76C23×C4—32no—
77C4⋊Q8—64no—
78C22⋊Q8—64no—
79C22⋊Q8—64no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—34no—
84C2×C4⋊C4—34no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C42—31no—
89C2×C42—32no—
90C2×C42—64no—
91C2×C42C2×D311yescharacteristic
92#95—34no—
93#95C2×D314yes—
94C23×C3C2411yescharacteristic
95C22×D3—21no—
96C22×D3—21no—
97C22×D3—22no—
98C22×D3C2×D412yes—
99C22×D3C2×Q814yes—
100C3×Q8—44no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—24no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×D412yes—
110C2×C3×C4C2×Q814yes—
111C2×C3×C4#4112yes—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—24no—
118C2×Dic3—22no—
119C2×Dic3C2×D412yes—
120C2×Dic3C2×D414yes—
121C2×Dic3C2×Q814yes—
122C2×Dic3#4114yes—
123C4×D3—28no—
124C4×D3—216no—
125Dic6—44no—
126C24—31no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—62no—
131C2×Q8—62no—
132C2×Q8—62no—
133C22×C4—32no—
134C22×C4—31no—
135C22×C4—34no—
136C22×C4—38no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—64no—
140C22×C4—64no—
141C22×C4—62no—
142C22×C4—62no—
143C22×C4C22×D312yes—
144C22×C4C22×D312yes—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—64no—
149C4⋊C4—64no—
150C4⋊C4—68no—
151C4⋊C4—68no—
152C4⋊C4—62no—
153C22⋊C4—64no—
154C22⋊C4—64no—
155C42—22no—
156C42—34no—
157C42—64no—
158C22×C3C22×D411yescharacteristic
159C22×C3C22×D411yescharacteristic
160C22×C3C22×Q812yes—
161C22×C3C2×#4111yescharacteristic
162C22×C3C2×#4112yes—
163C2×D3—22no—
164C2×D3—24no—
165C2×D3—22no—
166C2×D3—24no—
167C2×D3C4⋊Q814yescomplemented
168C3×C4—24no—
169C3×C4—42no—
170C3×C4—42no—
171C3×C4#12114yes—
172Dic3—22no—
173Dic3—42no—
174Dic3—42no—
175Dic3C22⋊Q818yescomplemented
176C23—32no—
177C23—34no—
178C23—61no—
179C23—61no—
180C23—62no—
181C23C23×D311yescharacteristic
182Q8—44no—
183Q8—124no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—34no—
192C2×C4—34no—
193C2×C4—34no—
194C2×C4—32no—
195C2×C4—616no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—68no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—64no—
204C2×C4#23512yes—
205C2×C4D3×D412yes—
206C2×C4D3×D412yes—
207C2×C4D3×Q814yes—
208C2×C3C2×#12111yescharacteristic
209C2×C3C2×C22⋊Q812yes—
210C2×C3C2×C4⋊Q811yescharacteristic
211C2×C3#53211yescharacteristic
212C2×C3#53312yes—
213D3—24no—
214C22—34no—
215C22—62no—
216C22—62no—
217C22—64no—
218C22—64no—
219C22C2×#23511yescharacteristic
220C22C2×D3×D411yescharacteristic
221C22C2×D3×D411yescharacteristic
222C22C2×D3×Q812yes—
223C22D3×#4112yes—
224C4—24no—
225C4—38no—
226C4—42no—
227C4—42no—
228C4—62no—
229C4—122no—
230C4—122no—
231C4#88414yes—
232C3#285711yescharacteristic, complemented, Sylow
233C2—64no—
234C2C2×#88411yescharacteristic
235C2#557812yes—
236C2D3×C22⋊Q812yes—
237C2#566811yescharacteristic
238C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC23×D3C23
hypercenter#391D3
hypocenterC3#2857
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
5. C3×#391C2semidirect
6. #4975C2semidirect
7. #4929C2semidirect
9. #4647C2semidirect
37. #391D3semidirect
58. C2×C4×D3Q8semidirect
167. C2×D3C4⋊Q8semidirect
175. Dic3C22⋊Q8semidirect
232. C3#2857semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#9442
C2×#8972
C2×C42×D32
C3×#3912
#49752
#49292
#46672
#46472
#28573