Finite Group Databasebeta

← #72113

Subgroups — #72113

Order 384.

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

normal subgroups only

This group has 216 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
1C3×#575C211yesmaximal, characteristic
2#5195C212yesmaximal, complemented
3#5137C211yesmaximal, characteristic, complemented
4#4566C212yesmaximal, complemented
5#4553C211yesmaximal, characteristic, complemented
6#2191—31nomaximal, Sylow
7C2×C3×#41—21no—
8C3×#137—22no—
9C3×AGL(1,ℤ/8)—22no—
10C3×#135—21no—
11C3×#135C2212yes—
12C2×C3×M16C2211yescharacteristic
13#924—21no—
14C2×Dic12C2211yescharacteristic, complemented
15#829—22no—
16#827—22no—
17#817C2211yescharacteristic, complemented
18#816—21no—
19#812C2212yescomplemented
20#575D311yescharacteristic
21#510—32no—
22#470—31no—
23#360—32no—
24#349—31no—
25C3×#41—22no—
26C3×#41—22no—
27C3×#41—41no—
28C3×#41—41no—
29C2×C3×Q8—21no—
30C2×C3×D4—21no—
31C2×C3×D4—41no—
32C22×C3×C4—41no—
33C22×C3×C4D411yescharacteristic
34C2×Dic6—21no—
35C3×Q16—21no—
36C3×Q16D412yes—
37C3×SD16—22no—
38C3×SD16—22no—
39C3×D8—21no—
40C3×D8D412yes—
41C3×M16—22no—
42C2×C3×C8D411yescharacteristic
43C2×C3×C8C2311yescharacteristic
44#217—41no—
45C4⋊Dic3—21no—
46Dic3⋊C4—41no—
47Dic12—22no—
48C3⋊C16—22no—
49C2×#41—21no—
50#137—22no—
51AGL(1,ℤ/8)—22no—
52#135—21no—
53#135C2×D312yes—
54C2×Q16—31no—
55C2×M16C2×D311yescharacteristic
56C22⋊Q8—61no—
57Q32—62no—
58SD32—62no—
59M32—32no—
60#106—31no—
61#103—61no—
62C23×C3—41no—
63C23×C3D811yescharacteristic
64C3×Q8—22no—
65C3×Q8—41no—
66C3×D4—22no—
67C3×D4—41no—
68C3×D4—42no—
69C3×D4—42no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—42no—
73C2×C3×C4—41no—
74C2×C3×C4—42no—
75C2×C3×C4D811yescharacteristic
76C2×C3×C4C2×D411yescharacteristic
77C2×Dic3—41no—
78C2×Dic3—41no—
79Dic6—41no—
80Dic6—41no—
81C3×C8—21no—
82C3×C8C2×D412yes—
83#41—22no—
84#41—22no—
85#41—41no—
86#41—41no—
87C2×Q8—21no—
88C2×Q8—61no—
89C2×D4—21no—
90C2×D4—41no—
91C22×C4—41no—
92C22×C4#6711yescharacteristic
93Q16—21no—
94Q16—62no—
95Q16#6712yes—
96SD16—22no—
97SD16—22no—
98D8—21no—
99D8#6712yes—
100M16—22no—
101C2×C8#6711yescharacteristic
102C2×C8C22×D311yescharacteristic
103C4⋊C4—61no—
104C4⋊C4—121no—
105C22⋊C4—121no—
106C16—62no—
107C22×C3—21no—
108C22×C3—41no—
109C22×C3—42no—
110C22×C3—82no—
111C22×C3C2×D811yescharacteristic
112C3×C4—21no—
113C3×C4—42no—
114C3×C4#12011yescharacteristic
115C3×C4AGL(1,ℤ/8)11yescharacteristic
116Dic3—81no—
117Dic3—81no—
118C23—41no—
119C23#21311yescharacteristic
120Q8—22no—
121Q8—41no—
122Q8—121no—
123Q8—121no—
124D4—22no—
125D4—41no—
126D4—42no—
127D4—42no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4#21311yescharacteristic
136C2×C4C2×#6711yescharacteristic
137C8—21no—
138C8C2×#6712yes—
139C2×C3—21no—
140C2×C3—41no—
141C2×C3—82no—
142C2×C3#44711yescharacteristic
143C22—21no—
144C22—41no—
145C22—42no—
146C22—82no—
147C22C2×#21311yescharacteristic
148C4—21no—
149C4—42no—
150C4—241no—
151C4—241no—
152C4#93211yescharacteristic
153C4#95311yescharacteristic
154C3#219111yescharacteristic, complemented, Sylow
155C2—21no—
156C2—41no—
157C2—82no—
158C2#521711yescharacteristic
subgroupG/H
centerC2#5217
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
Fitting, prime radicalC3×#575C2
socleC2×C3#447
hypercenter#575D3
hypocenterC3#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
2. #5195C2semidirect
3. #5137C2semidirect
4. #4566C2semidirect
5. #4553C2semidirect
14. C2×Dic12C22semidirect
17. #817C22semidirect
19. #812C22semidirect
154. C3#2191semidirect

Maximal subgroups

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

subgroupindex
C3×#5752
#51952
#51372
#45662
#45532
#21913