Finite Group Databasebeta

← #69121

Subgroups — #69121

Order 384.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#5748C211yesmaximal, characteristic
2#5137C211yesmaximal, characteristic, complemented
3#4909C212yesmaximal, complemented
4C3×#349C211yesmaximal, characteristic, complemented
5#4518C212yesmaximal, complemented
6#2191—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×M16C2211yescharacteristic
9#924—21no—
10#909—22no—
11#908—22no—
12C2×Dic12C2211yescharacteristic, complemented
13#904—21no—
14#904C2212yes—
15C3×M32C2212yescomplemented
16#817C2211yescharacteristic, complemented
17#816—21no—
18Dic24—22no—
19#800—22no—
20#575—31no—
21#510—32no—
22#470—31no—
23#360—32no—
24#349D311yescharacteristic, complemented
25C22×C3×C4D411yescharacteristic
26C2×#67—41no—
27#235—22no—
28#235—22no—
29#235—41no—
30#235—41no—
31C2×D12—21no—
32C2×C4×D3—41no—
33C2×Dic6—21no—
34C2×Dic6—21no—
35C3×M16—22no—
36C2×C3×C8D411yescharacteristic
37C2×C3×C8C2311yescharacteristic
38#217—41no—
39C4⋊Dic3—21no—
40Dic3⋊C4—41no—
41Dic12—21no—
42Dic12—22no—
43Dic12D412yes—
44D24—21no—
45D24D412yes—
46#204—22no—
47#204—22no—
48C3×C16—22no—
49C2×#41—61no—
50#137—62no—
51AGL(1,ℤ/8)—62no—
52#135—32no—
53#135—61no—
54C2×Q16—31no—
55C2×M16C2×D311yescharacteristic
56C22⋊Q8—61no—
57Q32—62no—
58SD32—62no—
59M32C2×D312yescomplemented
60#106—31no—
61#103—61no—
62C23×C3D811yescharacteristic
63C22×D3—41no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4D811yescharacteristic
67C2×C3×C4C2×D411yescharacteristic
68#67—42no—
69#67—42no—
70C2×Dic3—41no—
71C2×Dic3—41no—
72C2×Dic3—41no—
73D12—22no—
74D12—41no—
75C4×D3—42no—
76C4×D3—42no—
77Dic6—22no—
78Dic6—41no—
79Dic6—41no—
80Dic6—41no—
81C3×C8—21no—
82C3×C8C2×D412yes—
83#41—62no—
84#41—62no—
85#41—121no—
86#41—121no—
87C2×Q8—61no—
88C2×Q8—61no—
89C2×D4—61no—
90C2×D4—121no—
91C22×C4—121no—
92C22×C4D1211yescharacteristic
93Q16—32no—
94Q16—61no—
95Q16—62no—
96SD16—62no—
97SD16—62no—
98D8—32no—
99D8—61no—
100M16—22no—
101C2×C8D1211yescharacteristic
102C2×C8C22×D311yescharacteristic
103C4⋊C4—61no—
104C4⋊C4—121no—
105C22⋊C4—121no—
106C16—22no—
107C22×C3—21no—
108C22×C3—41no—
109C22×C3C2×D811yescharacteristic
110C2×D3—42no—
111C2×D3—82no—
112C3×C4—21no—
113C3×C4#12011yescharacteristic
114C3×C4AGL(1,ℤ/8)11yescharacteristic
115Dic3—42no—
116Dic3—81no—
117Dic3—81no—
118C23—121no—
119C23D2411yescharacteristic
120Q8—62no—
121Q8—121no—
122Q8—121no—
123Q8—121no—
124D4—62no—
125D4—121no—
126D4—122no—
127D4—122no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—122no—
131C2×C4—121no—
132C2×C4—122no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4D2411yescharacteristic
136C2×C4C2×D1211yescharacteristic
137C8—21no—
138C8D3×D412yes—
139C2×C3—21no—
140C2×C3—41no—
141C2×C3#44711yescharacteristic
142D3—82no—
143C22—21no—
144C22—41no—
145C22—122no—
146C22—242no—
147C22C2×D2411yescharacteristic
148C4—21no—
149C4—122no—
150C4—241no—
151C4—241no—
152C4#88211yescharacteristic
153C4#90811yescharacteristic
154C3#219111yescharacteristic, complemented, Sylow
155C2—21no—
156C2—41no—
157C2—242no—
158C2#473211yescharacteristic
subgroupG/H
centerC2#4732
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#349C2
socleC2×C3#447
prime radical#5748C2
hypercenter#349D3
hypocenterC3#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #5137C2semidirect
3. #4909C2semidirect
4. C3×#349C2semidirect
5. #4518C2semidirect
12. C2×Dic12C22semidirect
15. C3×M32C22semidirect
16. #817C22semidirect
24. #349D3semidirect
59. M32C2×D3semidirect
154. C3#2191semidirect

Maximal subgroups

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

subgroupindex
#57482
#51372
#49092
C3×#3492
#45182
#21913