Finite Group Databasebeta

← #70995

Subgroups — #70995

Order 384.

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

normal subgroups only

This group has 224 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#5722C211yesmaximal, characteristic, complemented
2C3×#456C211yesmaximal, characteristic
3#5246C212yesmaximal
4#4548C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2205—31nomaximal, Sylow
7C3×2−1+4C2211yescharacteristic
8C3×#137—21no—
9C3×C4⋊Q8C2211yescharacteristic
10#951—22no—
11#944—21no—
12#944C2212yes—
13#888—22no—
14#887—22no—
15C4×Dic6—21no—
16C3×#104—21no—
17C3×#101C2211yescharacteristic
18#837—22no—
19#836C2212yes—
20Q82—31no—
21#456—32no—
22#456D311yescharacteristic
23#356—31no—
24#356—32no—
25C3×#41—22no—
26C3×#41—21no—
27C3×#41—41no—
28C2×C3×Q8—41no—
29C2×C3×Q8D411yescharacteristic
30C2×C3×Q8C2311yescharacteristic
31C2×Dic6—21no—
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C3×Q16—41no—
35C3×SD16—41no—
36C3×M16—21no—
37C3×C4⋊C4—22no—
38C3×C42D411yescharacteristic
39#216—42no—
40#214—42no—
41C4⋊Dic3—21no—
42Dic3⋊C4—22no—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3—22no—
46C4×Dic3—21no—
47C4×Dic3D412yes—
48#208—22no—
492−1+4C2×D311yescharacteristic
50#137—21no—
51#137—62no—
52C4⋊Q8—32no—
53C4⋊Q8—62no—
54C4⋊Q8—61no—
55C4⋊Q8C2×D311yescharacteristic
56C4×Q8—62no—
57C4×Q8—61no—
58#104—21no—
59#104—62no—
60#101—32no—
61#101C2×D311yescharacteristic
62C3×Q8—22no—
63C3×Q8—21no—
64C3×Q8—41no—
65C3×Q8—41no—
66C3×D4—22no—
67C3×D4—21no—
68C3×D4—41no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—42no—
72C2×C3×C4—41no—
73C2×C3×C4C2×D412yes—
74C2×C3×C4C2×D411yescharacteristic
75C2×Dic3—22no—
76C2×Dic3—22no—
77Dic6—24no—
78Dic6—42no—
79Dic6—81no—
80C3×C8—41no—
81C3⋊C8—42no—
82#41—22no—
83#41—21no—
84#41—41no—
85C2×Q8—32no—
86C2×Q8—41no—
87C2×Q8—61no—
88C2×Q8—61no—
89C2×Q8#6711yescharacteristic
90C2×Q8C22×D311yescharacteristic
91Q16—41no—
92Q16—122no—
93SD16—41no—
94SD16—122no—
95M16—21no—
96M16—62no—
97C4⋊C4—22no—
98C4⋊C4—62no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C4⋊C4—62no—
102C42—32no—
103C42—62no—
104C42—61no—
105C42#6711yescharacteristic
106C22×C3—41no—
107C22×C3#12011yescharacteristic
108C3×C4—22no—
109C3×C4—22no—
110C3×C4—21no—
111C3×C4—41no—
112C3×C4—41no—
113Dic3—24no—
114Dic3—42no—
115Q8—21no—
116Q8—22no—
117Q8—41no—
118Q8—41no—
119Q8—64no—
120Q8—122no—
121Q8—241no—
122D4—21no—
123D4—22no—
124D4—41no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4D3×D412yes—
132C2×C4C2×#6711yescharacteristic
133C8—41no—
134C8—122no—
135C2×C3—21no—
136C2×C3—81no—
137C2×C3B(4,𝔽2)11yescharacteristic
138C22—41no—
139C22#93711yescharacteristic
140C4—22no—
141C4—21no—
142C4—22no—
143C4—41no—
144C4—41no—
145C4—64no—
146C4—122no—
147C3#220511yescharacteristic, complemented, Sylow
148C2—21no—
149C2—81no—
150C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#456C2
socleC2×C3B(4,𝔽2)
prime radicalC3×2−1+4C22
hypercenter#456D3
hypocenterC3#2205
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. #5722C2semidirect
4. #4548C2semidirect
5. #4478C2semidirect
147. C3#2205semidirect

Maximal subgroups

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

subgroupindex
#57222
C3×#4562
#52462
#45482
#44782
#22053