Finite Group Databasebeta

← #68126

Subgroups — #68126

Order 384.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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×#892C211yesmaximal, characteristic
2#4719C212yesmaximal, complemented
3C3×#324C211yesmaximal, characteristic, complemented
4#4528C211yesmaximal, characteristic, complemented
5#4468C212yesmaximal
6#1464—31nomaximal, Sylow
7C22×C4×D3C411yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×(C2×D3)⋊C4C411yescharacteristic
10C2×(C2×D3)⋊C4C2211yescharacteristic
11C2×C4⋊Dic3C411yescharacteristic
12C2×C4×Dic3C411yescharacteristic
13C2×C4×Dic3C2211yescharacteristic
14#892—22no—
15#892—22no—
16C3×C22⋊C8C2212yes—
17#830C2212yes—
18#815—22no—
19Dic3⋊C8—22no—
20C2×#117—31no—
21#432—32no—
22#324—31no—
23#324D311yescharacteristic, complemented
24#323—32no—
25C23×D3C2×C411yescharacteristic
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C22×Dic3C2×C411yescharacteristic
30C22×Dic3C2×C411yescharacteristic
31C22×Dic3C2×C411yescharacteristic
32C2×C4×D3—22no—
33C2×C4×D3—22no—
34C2×C4×D3—21no—
35C2×C4×D3—21no—
36C2×C3×C8—22no—
37C3×C4⋊C4—22no—
38(C2×D3)⋊C4—22no—
39(C2×D3)⋊C4—22no—
40C4⋊Dic3—22no—
41C4×Dic3—21no—
42C4×Dic3—21no—
43C4×Dic3D412yes—
44C4×Dic3D412yes—
45C2×C3⋊C8—22no—
46C23×C4—31no—
47#117—62no—
48#117—62no—
49C2×C4⋊C4—31no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C22⋊C4—31no—
52C2×C22⋊C4—31no—
53C2×C42—31no—
54C2×C42—31no—
55C4⋊C8—62no—
56C22⋊C8—32no—
57C22⋊C8C2×D312yes—
58#97—62no—
59C23×C3C22×C411yescharacteristic
60C22×D3—21no—
61C22×D3—21no—
62C22×D3—41no—
63C22×D3—41no—
64C22×D3M1612yes—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4M1612yes—
71C2×C3×C4C2×D412yes—
72C2×Dic3—21no—
73C2×Dic3—21no—
74C2×Dic3—21no—
75C2×Dic3—41no—
76C2×Dic3—41no—
77C2×Dic3C22⋊C411yescharacteristic
78C2×Dic3C22⋊C411yescharacteristic
79C2×Dic3C22⋊C412yes—
80C4×D3—24no—
81C4×D3—24no—
82C3×C8—42no—
83C3⋊C8—42no—
84C24—31no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—31no—
88C22×C4—61no—
89C22×C4—61no—
90C22×C4—62no—
91C22×C4—62no—
92C22×C4C4×D311yescharacteristic
93C22×C4C4×D311yescharacteristic
94C22×C4C22×D311yescharacteristic
95C2×C8—22no—
96C2×C8—62no—
97C4⋊C4—22no—
98C4⋊C4—62no—
99C22⋊C4—62no—
100C22⋊C4—62no—
101C42—32no—
102C42—32no—
103C42—61no—
104C42—61no—
105C22×C3—21no—
106C22×C3—21no—
107C22×C3C2×C22⋊C411yescharacteristic
108C22×C3C2×M1611yescharacteristic
109C22×C3C2×M1611yescharacteristic
110C2×D3—22no—
111C2×D3—42no—
112C2×D3—41no—
113C2×D3—41no—
114C2×D3—42no—
115C3×C4—22no—
116C3×C4—22no—
117C3×C4—41no—
118Dic3—21no—
119Dic3—21no—
120Dic3—22no—
121Dic3—41no—
122C23—32no—
123C23—61no—
124C23—61no—
125C23—121no—
126C23—121no—
127C23C2×C4×D311yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—31no—
132C2×C4—31no—
133C2×C4—32no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—64no—
137C2×C4—64no—
138C2×C4—61no—
139C2×C4—61no—
140C2×C4—61no—
141C2×C4—121no—
142C2×C4—121no—
143C2×C4ΓL(1,𝔽25)12yes—
144C2×C4D3×D412yes—
145C8—42no—
146C8—122no—
147C2×C3—21no—
148C2×C3—21no—
149C2×C3#40711yescharacteristic
150C2×C3#41011yescharacteristic
151C2×C3#41311yescharacteristic
152D3—42no—
153C22—21no—
154C22—21no—
155C22—62no—
156C22—122no—
157C22—122no—
158C22—121no—
159C22—121no—
160C22D3×C22⋊C411yescharacteristic
161C22C2×ΓL(1,𝔽25)11yescharacteristic
162C22D3×M1611yescharacteristic
163C4—22no—
164C4—22no—
165C4—41no—
166C4—61no—
167C4—61no—
168C4—62no—
169C4—121no—
170C3#146411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—21no—
173C2—122no—
174C2#472611yescharacteristic
175C2#474011yescharacteristic
176C2#475111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3C2×C4
cosocleC22×C3×C4C23
hypercenter#324D3
hypocenterC3#1464
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #4719C2semidirect
3. C3×#324C2semidirect
4. #4528C2semidirect
23. #324D3semidirect
170. C3#1464semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
#47192
C3×#3242
#45282
#44682
#14643