Finite Group Databasebeta

← #71875

Subgroups — #71875

Order 384.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C3×#537C211yesmaximal, characteristic
2#5034C212yesmaximal, complemented
3#4986C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4528C211yesmaximal, characteristic, complemented
6#1616—31nomaximal, Sylow
7C2×C3×#41—21no—
8C3×C22⋊Q8—22no—
9C3×#121—22no—
10C3×#121—21no—
11C3×#121C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic3C2211yescharacteristic
14#832—22no—
15C3⋊#95—21no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#830C2212yes—
19#808—22no—
20#537D311yescharacteristic
21#481—32no—
22#400—31no—
23#327—32no—
24#324—31no—
25C3×#41—42no—
26C3×#41—42no—
27C2×C3×Q8—21no—
28C2×C3×D4—21no—
29C2×C3×D4—22no—
30C2×C3×D4—21no—
31C2×C3×D4D412yes—
32C22×C3×C4—22no—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C22×Dic3—21no—
37C22×Dic3—21no—
38C3×C4⋊C4—21no—
39C3×C4⋊C4—22no—
40C3×C4⋊C4D412yes—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—22no—
43C4⋊Dic3—22no—
44C2×C3⋊C8—22no—
45C2×#41—21no—
46C22⋊Q8—22no—
47#121—22no—
48#121—21no—
49#121C2×D312yes—
50C2×C4⋊C4—31no—
51C2×C4⋊C4C2×D311yescharacteristic
52#106—62no—
53#102—62no—
54C22⋊C8—32no—
55#95—31no—
56#95—61no—
57#95—61no—
58C23×C3—22no—
59C23×C3C2×D411yescharacteristic
60C3×Q8—42no—
61C3×D4—42no—
62C3×D4—42no—
63C3×D4—42no—
64C3×D4—42no—
65C3×D4—81no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—21no—
69C2×C3×C4—22no—
70C2×C3×C4—41no—
71C2×C3×C4—44no—
72C2×C3×C4—41no—
73C2×C3×C4SD1612yes—
74C2×C3×C4C2×D412yes—
75C2×Dic3—41no—
76C2×Dic3—41no—
77C2×Dic3—42no—
78C2×Dic3—41no—
79C2×Dic3—41no—
80C3⋊C8—42no—
81#41—42no—
82#41—42no—
83C2×Q8—21no—
84C2×D4—21no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4#6712yes—
88C22×C4—22no—
89C22×C4—61no—
90C22×C4—61no—
91C22×C4#6711yescharacteristic
92C22×C4#6711yescharacteristic
93C22×C4C22×D311yescharacteristic
94C2×C8—62no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—62no—
98C4⋊C4#6712yes—
99C22⋊C4—22no—
100C22⋊C4—22no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3—42no—
104C22×C3—42no—
105C22×C3—42no—
106C22×C3#12011yescharacteristic
107C22×C3C2×SD1611yescharacteristic
108C22×C3AGL(1,ℤ/8)11yescharacteristic
109C3×C4—22no—
110C3×C4—22no—
111C3×C4—42no—
112C3×C4—41no—
113Dic3—81no—
114Dic3—81no—
115C23—22no—
116C23C2×#6711yescharacteristic
117Q8—42no—
118D4—42no—
119D4—42no—
120D4—42no—
121D4—42no—
122D4—81no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—44no—
129C2×C4—41no—
130C2×C4—121no—
131C2×C4—121no—
132C2×C4—122no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4#21412yes—
136C2×C4C2×#6712yes—
137C8—122no—
138C2×C3—21no—
139C2×C3—21no—
140C2×C3—82no—
141C2×C3#45011yescharacteristic
142C2×C3#45411yescharacteristic
143C2×C3#45811yescharacteristic
144C22—21no—
145C22—21no—
146C22—42no—
147C22—42no—
148C22—42no—
149C22#93211yescharacteristic
150C22C2×#21411yescharacteristic
151C22#95311yescharacteristic
152C4—22no—
153C4—22no—
154C4—42no—
155C4—41no—
156C4—241no—
157C4—241no—
158C3#161611yescharacteristic, complemented, Sylow
159C2—21no—
160C2—21no—
161C2—82no—
162C2#521811yescharacteristic
163C2#524311yescharacteristic
164C2#524511yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#537C2
socleC22×C3#120
hypercenter#537D3
hypocenterC3#1616
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
2. #5034C2semidirect
3. #4986C2semidirect
4. #4537C2semidirect
5. #4528C2semidirect
158. C3#1616semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
#50342
#49862
#45372
#45282
#16163