Finite Group Databasebeta

← #74194

Subgroups — #74194

Order 384.

488 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 157 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×#872C211yesmaximal, characteristic
2C2×#869C211yesmaximal, characteristic, complemented
3C3×#382C211yesmaximal, characteristic, complemented
4#4987C212yesmaximal
5#4984C212yesmaximal, complemented
6#4677C212yesmaximal
7#4673C212yesmaximal
8#4659C212yesmaximal, complemented
9#4656C212yesmaximal, complemented
10#2628—31nomaximal, Sylow
11C22×C4×D3C2211yescharacteristic
12C22×Dic6C2212yes—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C42C2211yescharacteristic
15C2×(C2×D3)⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C4C2214yes—
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2214yes—
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2211yescharacteristic
23C2×C4×Dic3C2211yescharacteristic
24#872—24no—
25#869—24no—
26C3×#95C2212yes—
27C3×#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×#117—31no—
32#400—32no—
33#399—32no—
34#395—32no—
35#393—32no—
36#389—32no—
37#382D311yescharacteristic, complemented
38#380—32no—
39C23×D3C2311yescharacteristic
40C22×C3×C4C2314yes—
41C22×C3×C4C2311yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×Dic3C2311yescharacteristic
45C22×Dic3C2314yes—
46C22×Dic3C2311yescharacteristic
47C22×Dic3C2311yescharacteristic
48C2×C4×D3—22no—
49C2×C4×D3—22no—
50C2×C4×D3Q814yescomplemented
51C2×Dic6—24no—
52C2×Dic6—44no—
53C3×C4⋊C4—24no—
54C3×C42D414yes—
55(C2×D3)⋊C4—22no—
56(C2×D3)⋊C4—22no—
57(C2×D3)⋊C4—44no—
58C4⋊Dic3—24no—
59C4⋊Dic3—24no—
60C4⋊Dic3—44no—
61Dic3⋊C4—22no—
62Dic3⋊C4—22no—
63C4×Dic3—22no—
64C22×Q8—32no—
65C23×C4—31no—
66C4⋊Q8—64no—
67#117—64no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—34no—
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—34no—
77C2×C42—31no—
78C2×C42C2×D311yescharacteristic
79#95—34no—
80#95—34no—
81#95C2×D312yes—
82#95C2×D312yes—
83C23×C3C2411yescharacteristic
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—22no—
87C22×D3—22no—
88C22×D3C2×Q812yes—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—24no—
92C2×C3×C4—24no—
93C2×C3×C4—24no—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×D412yes—
96C2×C3×C4C2×D412yes—
97C2×C3×C4C2×Q812yes—
98C2×C3×C4#4112yes—
99C2×Dic3—21no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—24no—
104C2×Dic3—22no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107C2×Dic3—22no—
108C2×Dic3C2×Q812yes—
109C4×D3—28no—
110Dic6—48no—
111C24—31no—
112C2×Q8—64no—
113C2×Q8—124no—
114C22×C4—31no—
115C22×C4—34no—
116C22×C4—31no—
117C22×C4—34no—
118C22×C4—31no—
119C22×C4—62no—
120C22×C4—62no—
121C22×C4C22×D314yes—
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—24no—
126C4⋊C4—62no—
127C4⋊C4—62no—
128C4⋊C4—64no—
129C4⋊C4—64no—
130C4⋊C4—124no—
131C22⋊C4—62no—
132C22⋊C4—62no—
133C22⋊C4—124no—
134C42—62no—
135C42D1214yes—
136C22×C3C22×D411yescharacteristic
137C22×C3C22×Q811yescharacteristic
138C22×C3C2×#4111yescharacteristic
139C22×C3H(5,𝔽2)12yes—
140C22×C32−1+412yes—
141C2×D3—22no—
142C2×D3—24no—
143C2×D3—41no—
144C2×D3—41no—
145C2×D3—42no—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—44no—
149C3×C4C22⋊Q814yes—
150Dic3—22no—
151Dic3—44no—
152Dic3—41no—
153Dic3—41no—
154C23—32no—
155C23—61no—
156C23—61no—
157C23—62no—
158C23—62no—
159C23C23×D311yescharacteristic
160Q8—128no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—32no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—68no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—64no—
173C2×C4—64no—
174C2×C4—62no—
175C2×C4—64no—
176C2×C4—62no—
177C2×C4C2×D1212yes—
178C2×C4C2×D1212yes—
179C2×C4C2×D1212yes—
180C2×C4#23712yes—
181C2×C4D3×Q812yes—
182C2×C3C2×C22⋊Q811yescharacteristic
183C2×C3#53511yescharacteristic
184C2×C3#53611yescharacteristic
185C2×C3#54212yes—
186C2×C3#54412yes—
187D3—42no—
188C22—64no—
189C22—62no—
190C22—122no—
191C22—121no—
192C22—121no—
193C22C22×D1211yescharacteristic
194C22C2×#23711yescharacteristic
195C22C2×D3×Q811yescharacteristic
196C22#100912yes—
197C22#101012yes—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—62no—
202C4—124no—
203C4—121no—
204C4—121no—
205C4#89714yes—
206C3#262811yescharacteristic, complemented, Sylow
207C2—122no—
208C2C2×#89711yescharacteristic
209C2#552511yescharacteristic
210C2#552611yescharacteristic
211C2#568212yes—
212C2#568912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC23×D3C23
hypercenter#382D3
hypocenterC3#2628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#869C2semidirect
3. C3×#382C2semidirect
5. #4984C2semidirect
8. #4659C2semidirect
9. #4656C2semidirect
37. #382D3semidirect
50. C2×C4×D3Q8semidirect
206. C3#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#8722
C2×#8692
C3×#3822
#49872
#49842
#46772
#46732
#46592
#46562
#26283