Finite Group Databasebeta

← #71089

Subgroups — #71089

Order 384.

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

normal subgroups only

This group has 226 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, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5147C212yesmaximal, complemented
4#4553C211yesmaximal, characteristic, complemented
5#4488C212yesmaximal, complemented
6#2190—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×D8C2211yescharacteristic, complemented
9C2×C3×M16C2211yescharacteristic
10C3×#121—21no—
11#909—22no—
12#908—22no—
13#904—21no—
14#904C2212yes—
15C3×#106C2211yescharacteristic, complemented
16C3×#102—21no—
17#827—22no—
18#826—22no—
19#812C2212yescomplemented
20#575—31no—
21#509—32no—
22#469D311yescharacteristic, complemented
23#360—32no—
24#349—31no—
25C2×C3×D4—21no—
26C2×C3×D4—41no—
27C22×C3×C4D411yescharacteristic
28C2×#67—41no—
29#235—22no—
30#235—22no—
31#235—41no—
32#235—41no—
33C2×D12—21no—
34C2×C4×D3—41no—
35C2×Dic6—21no—
36C3×D8—22no—
37C3×M16—22no—
38C2×C3×C8D411yescharacteristic
39C2×C3×C8C2311yescharacteristic
40C3×C4⋊C4—21no—
41C3×C22⋊C4—41no—
42Dic12—21no—
43Dic12D412yescomplemented
44D24—21no—
45D24D412yescomplemented
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×D8C2×D311yescharacteristic, complemented
55C2×M16C2×D311yescharacteristic
56#121—21no—
57SD32—62no—
58D16—62no—
59M32—32no—
60#106C2×D311yescharacteristic, complemented
61#102—21no—
62C23×C3—41no—
63C23×C3D811yescharacteristic
64C22×D3—41no—
65C3×D4—41no—
66C3×D4—41no—
67C3×D4—82no—
68C2×C3×C4—21no—
69C2×C3×C4—21no—
70C2×C3×C4—41no—
71C2×C3×C4D811yescharacteristic
72C2×C3×C4C2×D411yescharacteristic
73#67—42no—
74#67—42no—
75C2×Dic3—41no—
76D12—22no—
77D12—41no—
78C4×D3—42no—
79C4×D3—42no—
80Dic6—22no—
81Dic6—41no—
82C3×C8—21no—
83C3×C8C2×D412yes—
84#41—62no—
85#41—62no—
86#41—121no—
87#41—121no—
88C2×Q8—61no—
89C2×D4—21no—
90C2×D4—41no—
91C2×D4—61no—
92C2×D4—121no—
93C22×C4—121no—
94C22×C4#6711yescharacteristic
95Q16—32no—
96Q16—61no—
97SD16—62no—
98SD16—62no—
99D8—22no—
100D8—32no—
101D8—61no—
102M16—22no—
103C2×C8#6711yescharacteristic
104C2×C8C22×D311yescharacteristic
105C4⋊C4—21no—
106C22⋊C4—41no—
107C16—62no—
108C22×C3—21no—
109C22×C3—41no—
110C22×C3—81no—
111C22×C3—81no—
112C22×C3—81no—
113C22×C3C2×D811yescharacteristic
114C2×D3—42no—
115C2×D3—82no—
116C3×C4—21no—
117C3×C4—81no—
118C3×C4#12011yescharacteristic
119C3×C4AGL(1,ℤ/8)11yescharacteristic
120Dic3—42no—
121C23—41no—
122C23—121no—
123C23#21311yescharacteristic
124Q8—62no—
125Q8—121no—
126D4—41no—
127D4—41no—
128D4—62no—
129D4—82no—
130D4—121no—
131D4—122no—
132D4—122no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—41no—
136C2×C4—122no—
137C2×C4—121no—
138C2×C4—122no—
139C2×C4#21311yescharacteristic
140C2×C4C2×#6711yescharacteristic
141C8—21no—
142C8D3×D412yes—
143C2×C3—21no—
144C2×C3—41no—
145C2×C3—161no—
146C2×C3#44711yescharacteristic
147D3—82no—
148C22—21no—
149C22—41no—
150C22—81no—
151C22—81no—
152C22—81no—
153C22—122no—
154C22—242no—
155C22C2×#21311yescharacteristic
156C4—21no—
157C4—81no—
158C4—122no—
159C4#93711yescharacteristic
160C4#94911yescharacteristic
161C3#219011yescharacteristic, complemented, Sylow
162C2—21no—
163C2—41no—
164C2—161no—
165C2—242no—
166C2#503611yescharacteristic
subgroupG/H
centerC2#5036
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#469C2
socleC2×C3#447
hypercenter#469D3
hypocenterC3#2190
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5748C2semidirect
2. C3×#469C2semidirect
3. #5147C2semidirect
4. #4553C2semidirect
5. #4488C2semidirect
8. C2×C3×D8C22semidirect
15. C3×#106C22semidirect
19. #812C22semidirect
22. #469D3semidirect
43. Dic12D4semidirect
45. D24D4semidirect
54. C2×D8C2×D3semidirect
60. #106C2×D3semidirect
161. C3#2190semidirect

Maximal subgroups

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

subgroupindex
#57482
C3×#4692
#51472
#45532
#44882
#21903