Finite Group Databasebeta

← #70994

Subgroups — #70994

Order 384.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#5734C211yesmaximal, characteristic, complemented
2C3×#456C211yesmaximal, characteristic, complemented
3#5245C212yesmaximal, complemented
4#4548C211yesmaximal, characteristic, complemented
5#4477C212yesmaximal, complemented
6#2204—31nomaximal, Sylow
7C3×2−1+4C2211yescharacteristic
8C3×#137—21no—
9C3×C4⋊Q8C2211yescharacteristic
10#949—22no—
11#947—21no—
12#947C2212yes—
13#895—22no—
14#892—22no—
15C4×D12—21no—
16C3×#104—21no—
17C3×#101C2211yescharacteristic
18#837—22no—
19#836C2212yes—
20#555—31no—
21#456D311yescharacteristic, complemented
22#454—32no—
23#356—31no—
24#355—32no—
25C3×#41—22no—
26C3×#41—21no—
27C3×#41—41no—
28C2×C3×Q8—41no—
29C2×C3×Q8D411yescharacteristic
30C2×C3×Q8C2311yescharacteristic
31C2×D12—21no—
32C2×D12—21no—
33C2×D12D412yes—
34C2×C4×D3—22no—
35C3×Q16—41no—
36C3×SD16—41no—
37C3×M16—21no—
38C3×C4⋊C4—22no—
39C3×C42D411yescharacteristic
40#215—42no—
41#213—42no—
42(C2×D3)⋊C4—22no—
43(C2×D3)⋊C4—22no—
44(C2×D3)⋊C4—22no—
45C4⋊Dic3—21no—
46C4×Dic3—21no—
47C4×Dic3D412yes—
48#208—22no—
492−1+4C2×D311yescharacteristic
50#137—21no—
51AGL(1,ℤ/8)—62no—
52C4⋊Q8C2×D311yescharacteristic
53#124—32no—
54#124—61no—
55#121—62no—
56C4×D4—61no—
57#117—62no—
58#104—21no—
59#104—62no—
60#101—32no—
61#101C2×D311yescharacteristic
62C22×D3—22no—
63C3×Q8—22no—
64C3×Q8—21no—
65C3×Q8—41no—
66C3×Q8—41no—
67C3×D4—22no—
68C3×D4—21no—
69C3×D4—41no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—42no—
73C2×C3×C4—41no—
74C2×C3×C4C2×D412yes—
75C2×C3×C4C2×D411yescharacteristic
76C2×Dic3—22no—
77D12—42no—
78D12—42no—
79D12—81no—
80C4×D3—44no—
81C3×C8—41no—
82C3⋊C8—42no—
83#41—22no—
84#41—21no—
85#41—41no—
86C2×Q8—41no—
87C2×Q8#6711yescharacteristic
88C2×Q8C22×D311yescharacteristic
89C2×D4—32no—
90C2×D4—61no—
91C2×D4—61no—
92C22×C4—62no—
93Q16—41no—
94SD16—41no—
95SD16—122no—
96D8—122no—
97M16—21no—
98M16—62no—
99C4⋊C4—22no—
100C4⋊C4—61no—
101C22⋊C4—62no—
102C22⋊C4—62no—
103C22⋊C4—62no—
104C42—32no—
105C42—61no—
106C42#6711yescharacteristic
107C22×C3—41no—
108C22×C3#12011yescharacteristic
109C2×D3—42no—
110C2×D3—42no—
111C2×D3—42no—
112C3×C4—22no—
113C3×C4—22no—
114C3×C4—21no—
115C3×C4—41no—
116C3×C4—41no—
117Dic3—42no—
118C23—62no—
119Q8—22no—
120Q8—21no—
121Q8—41no—
122Q8—41no—
123D4—22no—
124D4—21no—
125D4—41no—
126D4—122no—
127D4—122no—
128D4—241no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—62no—
134C2×C4—124no—
135C2×C4D3×D412yes—
136C2×C4C2×#6711yescharacteristic
137C8—41no—
138C8—122no—
139C2×C3—21no—
140C2×C3—81no—
141C2×C3B(4,𝔽2)11yescharacteristic
142D3—82no—
143C22—41no—
144C22—122no—
145C22—122no—
146C22—122no—
147C22#93711yescharacteristic
148C4—22no—
149C4—21no—
150C4—22no—
151C4—41no—
152C4—41no—
153C4—122no—
154C3#220411yescharacteristic, complemented, Sylow
155C2—21no—
156C2—81no—
157C2—242no—
158C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#456C2
socleC2×C3B(4,𝔽2)
hypercenter#456D3
hypocenterC3#2204
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. C3×#456C2semidirect
3. #5245C2semidirect
4. #4548C2semidirect
5. #4477C2semidirect
21. #456D3semidirect
154. C3#2204semidirect

Maximal subgroups

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

subgroupindex
#57342
C3×#4562
#52452
#45482
#44772
#22043