Finite Group Databasebeta

← #69120

Subgroups — #69120

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
2#5135C211yesmaximal, characteristic, complemented
3#4908C212yesmaximal, complemented
4C3×#349C211yesmaximal, characteristic, complemented
5#4518C212yesmaximal, complemented
6#2190—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×M16C2211yescharacteristic
9#930—21no—
10#909—22no—
11#908—22no—
12#904—21no—
13#904C2212yes—
14C2×D24C2211yescharacteristic, complemented
15C3×M32C2212yescomplemented
16#821—21no—
17#817C2211yescharacteristic, complemented
18#800—22no—
19D48—22no—
20#575—31no—
21#509—32no—
22#469—31no—
23#360—32no—
24#349D311yescharacteristic, complemented
25C22×C3×C4D411yescharacteristic
26C2×#67—41no—
27C2×#67—41no—
28#235—22no—
29#235—22no—
30#235—41no—
31#235—41no—
32C2×D12—21no—
33C2×D12—21no—
34C2×C4×D3—41no—
35C2×Dic6—21no—
36C3×M16—22no—
37C2×C3×C8D411yescharacteristic
38C2×C3×C8C2311yescharacteristic
39(C2×D3)⋊C4—41no—
40C4⋊Dic3—21no—
41Dic12—21no—
42Dic12D412yescomplemented
43D24—21no—
44D24—22no—
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×D8—31no—
55C2×M16C2×D311yescharacteristic
56#121—61no—
57SD32—62no—
58D16—62no—
59M32C2×D312yescomplemented
60#106—31no—
61#102—61no—
62C23×C3D811yescharacteristic
63C22×D3—41no—
64C22×D3—41no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4D811yescharacteristic
68C2×C3×C4C2×D411yescharacteristic
69#67—42no—
70#67—42no—
71#67—82no—
72C2×Dic3—41no—
73C2×Dic3—41no—
74D12—22no—
75D12—41no—
76D12—41no—
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—61no—
90C2×D4—61no—
91C2×D4—121no—
92C2×D4—121no—
93C22×C4—121no—
94C22×C4D1211yescharacteristic
95Q16—32no—
96Q16—61no—
97SD16—62no—
98SD16—62no—
99D8—32no—
100D8—61no—
101D8—62no—
102M16—22no—
103C2×C8D1211yescharacteristic
104C2×C8C22×D311yescharacteristic
105C4⋊C4—61no—
106C22⋊C4—121no—
107C16—22no—
108C22×C3—21no—
109C22×C3—41no—
110C22×C3C2×D811yescharacteristic
111C2×D3—42no—
112C2×D3—82no—
113C2×D3—81no—
114C2×D3—81no—
115C2×D3—81no—
116C3×C4—21no—
117C3×C4#12011yescharacteristic
118C3×C4AGL(1,ℤ/8)11yescharacteristic
119Dic3—42no—
120Dic3—81no—
121C23—121no—
122C23—121no—
123C23D2411yescharacteristic
124Q8—62no—
125Q8—121no—
126D4—62no—
127D4—121no—
128D4—122no—
129D4—122no—
130D4—121no—
131D4—121no—
132D4—242no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—122no—
136C2×C4—121no—
137C2×C4—122no—
138C2×C4—121no—
139C2×C4D2411yescharacteristic
140C2×C4C2×D1211yescharacteristic
141C8—21no—
142C8D3×D412yes—
143C2×C3—21no—
144C2×C3—41no—
145C2×C3#44711yescharacteristic
146D3—82no—
147D3—161no—
148C22—21no—
149C22—41no—
150C22—122no—
151C22—241no—
152C22—241no—
153C22—242no—
154C22—241no—
155C22C2×D2411yescharacteristic
156C4—21no—
157C4—122no—
158C4—241no—
159C4#88211yescharacteristic
160C4#90811yescharacteristic
161C3#219011yescharacteristic, complemented, Sylow
162C2—21no—
163C2—41no—
164C2—242no—
165C2—481no—
166C2#473211yescharacteristic
subgroupG/H
centerC2#4732
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#349C2
socleC2×C3#447
hypercenter#349D3
hypocenterC3#2190
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
1. #5748C2semidirect
2. #5135C2semidirect
3. #4908C2semidirect
4. C3×#349C2semidirect
5. #4518C2semidirect
14. C2×D24C22semidirect
15. C3×M32C22semidirect
17. #817C22semidirect
24. #349D3semidirect
42. Dic12D4semidirect
45. D24D4semidirect
59. M32C2×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
#51352
#49082
C3×#3492
#45182
#21903