Finite Group Databasebeta

← #68116

Subgroups — #68116

Order 384.

228 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#5483C211yesmaximal, characteristic
2#4953C211yesmaximal, characteristic
3#4721C212yesmaximal, complemented
4C3×#323C211yesmaximal, characteristic
5#4467C212yesmaximal, complemented
6#1608—31nomaximal, Sylow
7C2×C3×C22⋊C4C2211yescharacteristic
8C2×#217—21no—
9C2×#217—21no—
10C2×C4⋊Dic3C2211yescharacteristic
11#924—21no—
12#924C2212yes—
13#878—22no—
14#878—22no—
15C3×C22⋊C8C2212yes—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#817—22no—
19#816—22no—
20#543—31no—
21#483—32no—
22#397—31no—
23#328—32no—
24#323D311yescharacteristic
25C24×C3D411yescharacteristic
26C22×C3×C4D411yescharacteristic
27C22×C3×C4C2311yescharacteristic
28C22×Dic3—21no—
29C22×Dic3—21no—
30C22×Dic3—21no—
31C22×Dic3—21no—
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C2×C3×C8—22no—
35C3×C22⋊C4—22no—
36#217—22no—
37#217—22no—
38#217—42no—
39C4⋊Dic3—21no—
40C4⋊Dic3—22no—
41C4⋊Dic3D412yes—
42Dic3⋊C4—22no—
43Dic3⋊C4—22no—
44C22⋊Q8—32no—
45C22⋊Q8—62no—
46C22⋊Q8—62no—
47C22⋊Q8—61no—
48C2×C4⋊C4—31no—
49C2×C22⋊C4—61no—
50C2×C22⋊C4—61no—
51C2×C22⋊C4C2×D311yescharacteristic
52#106—62no—
53#103—62no—
54C22⋊C8C2×D312yes—
55#95—31no—
56#95—61no—
57C23×C3—21no—
58C23×C3—21no—
59C23×C3—41no—
60C23×C3—41no—
61C23×C3SD1612yes—
62C23×C3C2×D411yescharacteristic
63C2×C3×C4—21no—
64C2×C3×C4—41no—
65C2×C3×C4—41no—
66C2×C3×C4C2×D412yes—
67C2×Dic3—22no—
68C2×Dic3—22no—
69C2×Dic3—41no—
70C2×Dic3—41no—
71C2×Dic3—41no—
72C2×Dic3—41no—
73C2×Dic3—41no—
74C2×Dic3—41no—
75C2×Dic3—81no—
76C2×Dic3—81no—
77Dic6—42no—
78Dic6—81no—
79C3×C8—42no—
80C24D1211yescharacteristic
81C2×Q8—32no—
82C2×Q8—61no—
83C22×C4—61no—
84C22×C4—61no—
85C22×C4—61no—
86C22×C4—61no—
87C22×C4D1211yescharacteristic
88C22×C4C22×D311yescharacteristic
89C2×C8—22no—
90C4⋊C4—32no—
91C4⋊C4—62no—
92C4⋊C4—61no—
93C4⋊C4—62no—
94C4⋊C4—62no—
95C22⋊C4—22no—
96C22⋊C4—62no—
97C22⋊C4—62no—
98C22⋊C4—122no—
99C22×C3—21no—
100C22×C3—21no—
101C22×C3—22no—
102C22×C3—42no—
103C22×C3—41no—
104C22×C3—41no—
105C22×C3—81no—
106C22×C3#12011yescharacteristic
107C22×C3C2×SD1611yescharacteristic
108C22×C3#13711yescharacteristic
109C3×C4—22no—
110C3×C4—41no—
111Dic3—42no—
112Dic3—42no—
113Dic3—81no—
114Dic3—81no—
115C23—21no—
116C23—21no—
117C23—41no—
118C23—41no—
119C23#20412yes—
120C23C2×D1211yescharacteristic
121Q8—122no—
122Q8—241no—
123C2×C4—21no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—62no—
127C2×C4—62no—
128C2×C4—121no—
129C2×C4—121no—
130C2×C4—121no—
131C2×C4—121no—
132C2×C4—121no—
133C2×C4—121no—
134C2×C4—241no—
135C2×C4—241no—
136C2×C4D3×D412yes—
137C8—42no—
138C2×C3—21no—
139C2×C3—21no—
140C2×C3—42no—
141C2×C3#44811yescharacteristic
142C2×C3#45511yescharacteristic
143C2×C3#45811yescharacteristic
144C22—21no—
145C22—21no—
146C22—22no—
147C22—42no—
148C22—41no—
149C22—41no—
150C22—81no—
151C22#88211yescharacteristic
152C22C2×#20411yescharacteristic
153C22#90911yescharacteristic
154C4—22no—
155C4—41no—
156C4—122no—
157C4—122no—
158C4—241no—
159C4—241no—
160C3#160811yescharacteristic, complemented, Sylow
161C2—21no—
162C2—21no—
163C2—42no—
164C2#473811yescharacteristic
165C2#474211yescharacteristic
166C2#474611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3#120
prime radicalC24×C3D4
hypercenter#323D3
hypocenterC3#1608
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
3. #4721C2semidirect
5. #4467C2semidirect
160. C3#1608semidirect

Maximal subgroups

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

subgroupindex
#54832
#49532
#47212
C3×#3232
#44672
#16083