Finite Group Databasebeta

← #71091

Subgroups — #71091

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
2C3×#470C211yesmaximal, characteristic, complemented
3#5179C212yesmaximal, complemented
4#4553C211yesmaximal, characteristic, complemented
5#4488C212yesmaximal, complemented
6#2191—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×Q16C2211yescharacteristic, complemented
9C2×C3×M16C2211yescharacteristic
10C3×C22⋊Q8—21no—
11#909—22no—
12#908—22no—
13#904—21no—
14#904C2212yes—
15C3×#106C2211yescharacteristic, complemented
16C3×#103—21no—
17#829—22no—
18#828—22no—
19#812C2212yescomplemented
20#575—31no—
21#510—32no—
22#470D311yescharacteristic, complemented
23#360—32no—
24#349—31no—
25C2×C3×Q8—21no—
26C22×C3×C4D411yescharacteristic
27C2×#67—41no—
28#235—22no—
29#235—22no—
30#235—41no—
31#235—41no—
32C2×D12—21no—
33C2×C4×D3—41no—
34C2×Dic6—21no—
35C3×Q16—22no—
36C3×M16—22no—
37C2×C3×C8D411yescharacteristic
38C2×C3×C8C2311yescharacteristic
39C3×C4⋊C4—21no—
40C3×C4⋊C4—41no—
41C3×C22⋊C4—41no—
42Dic12—21no—
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×Q16C2×D311yescharacteristic, complemented
55C2×M16C2×D311yescharacteristic
56C22⋊Q8—21no—
57Q32—62no—
58SD32—62no—
59M32—32no—
60#106C2×D311yescharacteristic, complemented
61#103—21no—
62C23×C3D811yescharacteristic
63C22×D3—41no—
64C3×Q8—41no—
65C3×Q8—41no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4D811yescharacteristic
71C2×C3×C4C2×D411yescharacteristic
72#67—42no—
73#67—42no—
74C2×Dic3—41no—
75D12—22no—
76D12—41no—
77C4×D3—42no—
78C4×D3—42no—
79Dic6—22no—
80Dic6—41no—
81C3×C8—21no—
82C3×C8C2×D412yes—
83#41—62no—
84#41—62no—
85#41—121no—
86#41—121no—
87C2×Q8—21no—
88C2×Q8—61no—
89C2×D4—61no—
90C2×D4—121no—
91C22×C4—121no—
92C22×C4#6711yescharacteristic
93Q16—22no—
94Q16—32no—
95Q16—61no—
96SD16—62no—
97SD16—62no—
98D8—32no—
99D8—61no—
100M16—22no—
101C2×C8#6711yescharacteristic
102C2×C8C22×D311yescharacteristic
103C4⋊C4—21no—
104C4⋊C4—41no—
105C22⋊C4—41no—
106C16—62no—
107C22×C3—21no—
108C22×C3—41no—
109C22×C3C2×D811yescharacteristic
110C2×D3—42no—
111C2×D3—82no—
112C3×C4—21no—
113C3×C4—81no—
114C3×C4—81no—
115C3×C4#12011yescharacteristic
116C3×C4AGL(1,ℤ/8)11yescharacteristic
117Dic3—42no—
118C23—121no—
119C23#21311yescharacteristic
120Q8—41no—
121Q8—41no—
122Q8—62no—
123Q8—121no—
124D4—62no—
125D4—121no—
126D4—122no—
127D4—122no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—122no—
133C2×C4—121no—
134C2×C4—122no—
135C2×C4#21311yescharacteristic
136C2×C4C2×#6711yescharacteristic
137C8—21no—
138C8D3×D412yes—
139C2×C3—21no—
140C2×C3—41no—
141C2×C3#44711yescharacteristic
142D3—82no—
143C22—21no—
144C22—41no—
145C22—122no—
146C22—242no—
147C22C2×#21311yescharacteristic
148C4—21no—
149C4—81no—
150C4—81no—
151C4—122no—
152C4#93711yescharacteristic
153C4#94911yescharacteristic
154C3#219111yescharacteristic, complemented, Sylow
155C2—21no—
156C2—41no—
157C2—242no—
158C2#503611yescharacteristic
subgroupG/H
centerC2#5036
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#470C2
socleC2×C3#447
prime radical#5748C2
hypercenter#470D3
hypocenterC3#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #470

Complements

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

NH ≅ G/Ntype
2. C3×#470C2semidirect
3. #5179C2semidirect
4. #4553C2semidirect
5. #4488C2semidirect
8. C2×C3×Q16C22semidirect
15. C3×#106C22semidirect
19. #812C22semidirect
22. #470D3semidirect
54. C2×Q16C2×D3semidirect
60. #106C2×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
C3×#4702
#51792
#45532
#44882
#21913