Finite Group Databasebeta

← #68102

Subgroups — #68102

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×#877C211yesmaximal, characteristic
2#4719C212yesmaximal, complemented
3C3×#323C211yesmaximal, characteristic
4#4525C211yesmaximal, characteristic
5#4469C212yesmaximal, complemented
6#1464—31nomaximal, Sylow
7C23×Dic3C411yescharacteristic
8C2×C3×C22⋊C4C2211yescharacteristic
9C2×#217C411yescharacteristic
10C2×Dic3⋊C4C411yescharacteristic
11C2×Dic3⋊C4C2211yescharacteristic
12C2×C4×Dic3C411yescharacteristic
13C2×C4×Dic3C2211yescharacteristic
14#877—22no—
15#877—22no—
16C3×C22⋊C8C2212yes—
17#830C2212yes—
18#815—22no—
19Dic3⋊C8—22no—
20C2×#117—31no—
21#432—32no—
22#324—32no—
23#323—31no—
24#323D311yescharacteristic
25C24×C3C2×C411yescharacteristic
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4C2311yescharacteristic
28C22×Dic3—22no—
29C22×Dic3—21no—
30C22×Dic3—21no—
31C22×Dic3—22no—
32C22×Dic3C2×C411yescharacteristic
33C22×Dic3C2×C411yescharacteristic
34C22×Dic3C2×C411yescharacteristic
35C22×Dic3C2×C411yescharacteristic
36C2×C3×C8—22no—
37C3×C22⋊C4—22no—
38#217—22no—
39Dic3⋊C4—22no—
40Dic3⋊C4—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⋊C4—31no—
51C2×C22⋊C4—31no—
52C2×C22⋊C4C2×D311yescharacteristic
53C2×C42—31no—
54C2×C42—31no—
55C4⋊C8—62no—
56C22⋊C8—32no—
57C22⋊C8C2×D312yes—
58#97—62no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3M1612yes—
64C23×C3C22×C411yescharacteristic
65C2×C3×C4—21no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4C2×D412yes—
69C2×Dic3—24no—
70C2×Dic3—24no—
71C2×Dic3—21no—
72C2×Dic3—21no—
73C2×Dic3—21no—
74C2×Dic3—21no—
75C2×Dic3—21no—
76C2×Dic3—41no—
77C2×Dic3—41no—
78C2×Dic3C22⋊C411yescharacteristic
79C2×Dic3C22⋊C411yescharacteristic
80C2×Dic3C22⋊C412yes—
81C2×Dic3M1612yes—
82C3×C8—42no—
83C3⋊C8—42no—
84C24C4×D311yescharacteristic
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—31no—
88C22×C4—31no—
89C22×C4—61no—
90C22×C4—61no—
91C22×C4—62no—
92C22×C4—62no—
93C22×C4C4×D311yescharacteristic
94C22×C4C22×D311yescharacteristic
95C2×C8—22no—
96C2×C8—62no—
97C4⋊C4—62no—
98C4⋊C4—62no—
99C22⋊C4—22no—
100C22⋊C4—62no—
101C42—32no—
102C42—32no—
103C42—61no—
104C42—61no—
105C22×C3—21no—
106C22×C3—21no—
107C22×C3—22no—
108C22×C3—42no—
109C22×C3—42no—
110C22×C3—41no—
111C22×C3—41no—
112C22×C3C2×C22⋊C411yescharacteristic
113C22×C3C2×M1611yescharacteristic
114C22×C3C2×M1611yescharacteristic
115C3×C4—22no—
116C3×C4—41no—
117Dic3—21no—
118Dic3—21no—
119Dic3—22no—
120Dic3—22no—
121Dic3—41no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23ΓL(1,𝔽25)12yes—
127C23C2×C4×D311yescharacteristic
128C2×C4—21no—
129C2×C4—31no—
130C2×C4—31no—
131C2×C4—32no—
132C2×C4—32no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—64no—
136C2×C4—64no—
137C2×C4—61no—
138C2×C4—61no—
139C2×C4—61no—
140C2×C4—61no—
141C2×C4—61no—
142C2×C4—121no—
143C2×C4—121no—
144C2×C4D3×D412yes—
145C8—42no—
146C8—122no—
147C2×C3—21no—
148C2×C3—21no—
149C2×C3—42no—
150C2×C3#40711yescharacteristic
151C2×C3#41011yescharacteristic
152C2×C3#41311yescharacteristic
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—42no—
158C22—41no—
159C22—41no—
160C22D3×C22⋊C411yescharacteristic
161C22C2×ΓL(1,𝔽25)11yescharacteristic
162C22D3×M1611yescharacteristic
163C4—22no—
164C4—41no—
165C4—61no—
166C4—61no—
167C4—62no—
168C4—62no—
169C4—121no—
170C3#146411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—21no—
173C2—42no—
174C2#472611yescharacteristic
175C2#474011yescharacteristic
176C2#474511yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3C2×C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1464
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4719C2semidirect
5. #4469C2semidirect
170. C3#1464semidirect

Maximal subgroups

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

subgroupindex
C2×#8772
#47192
C3×#3232
#45252
#44692
#14643