Finite Group Databasebeta

← #77004

Subgroups — #77004

Order 384.

568 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 141 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×#935C212yesmaximal
2C2×#878C212yesmaximal, complemented
3C3×#394C211yesmaximal, characteristic
4#4979C211yesmaximal, characteristic, complemented
5#4975C211yesmaximal, characteristic, complemented
6#4952C212yesmaximal, complemented
7#4950C212yesmaximal, complemented
8#4944C212yesmaximal, complemented
9#4657C212yesmaximal, complemented
10#2680—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20#935—28no—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#878—28no—
29C3×#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C2×#123—32no—
35C2×C22⋊Q8—32no—
36#398—32no—
37#397—32no—
38#395—31no—
39#394D311yescharacteristic
40#388—32no—
41#385—32no—
42#384—31no—
43C24×C3C2312yes—
44C2×C3×D4—24no—
45C2×C3×D4—41no—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×Dic3—24no—
51C22×Dic3—24no—
52C22×Dic3—42no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C2×Dic6—24no—
58C2×Dic6—41no—
59C3×C4⋊C4—41no—
60C3×C22⋊C4—24no—
61C3×C22⋊C4—24no—
62#217—24no—
63#217—24no—
64#217D418yes—
65C4⋊Dic3—24no—
66C4⋊Dic3—41no—
67Dic3⋊C4—24no—
68Dic3⋊C4—24no—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71C4×Dic3—22no—
72C22×Q8—31no—
73C22×D4C2×D311yescharacteristic
74C23×C4—32no—
75#123—68no—
76C22⋊Q8—68no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4C2×D312yes—
88C2×C22⋊C4C2×D312yes—
89C2×C42—31no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95—31no—
94#95C2×D311yescharacteristic
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3C2×D414yes—
102C23×C3C2411yescharacteristic
103C3×D4—44no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4#4112yes—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—48no—
125C2×Dic3C2×D414yes—
126C2×Dic3C2×D414yes—
127Dic6—44no—
128C24C22×D312yes—
129C2×Q8—64no—
130C2×Q8—121no—
131C2×D4—24no—
132C2×D4—41no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—64no—
138C22×C4—64no—
139C22×C4—122no—
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D311yescharacteristic
144C4⋊C4—41no—
145C4⋊C4—64no—
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—64no—
150C4⋊C4—121no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—38no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C42—62no—
157C22×C3—24no—
158C22×C3—24no—
159C22×C3—24no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3C22×D412yes—
165C22×C3C2×#4111yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C32−1+411yescharacteristic
168C22×C32−1+412yes—
169C3×C4—22no—
170C3×C4—41no—
171C3×C4—41no—
172C3×C4—42no—
173Dic3—24no—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23D3×D414yes—
184C23C23×D311yescharacteristic
185Q8—124no—
186D4—44no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—34no—
197C2×C4—34no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—128no—
209C2×C4#23712yes—
210C2×C3—44no—
211C2×C3#53612yes—
212C2×C3#53911yescharacteristic
213C2×C3D4211yescharacteristic
214C2×C3#54712yes—
215C2×C3#56311yescharacteristic
216C22—24no—
217C22—24no—
218C22—24no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22C2×D3×D412yes—
224C22C2×#23711yescharacteristic
225C22#100411yescharacteristic
226C22#101011yescharacteristic
227C22#101012yes—
228C4—22no—
229C4—41no—
230C4—41no—
231C4—42no—
232C4—64no—
233C4—64no—
234C4—122no—
235C4—122no—
236C3#268011yescharacteristic, complemented, Sylow
237C2—44no—
238C2#559011yescharacteristic
239C2#560211yescharacteristic
240C2#561112yes—
241C2#564812yes—
242C2#567211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2680
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#878C2semidirect
4. #4979C2semidirect
5. #4975C2semidirect
6. #4952C2semidirect
7. #4950C2semidirect
8. #4944C2semidirect
9. #4657C2semidirect
236. C3#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#9352
C2×#8782
C3×#3942
#49792
#49752
#49522
#49502
#49442
#46572
#26803