Finite Group Databasebeta

← #70989

Subgroups — #70989

Order 384.

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

normal subgroups only

This group has 216 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#5672C211yesmaximal, characteristic, complemented
2C3×#454C211yesmaximal, characteristic
3#5246C212yesmaximal, complemented
4#4542C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2203—31nomaximal, Sylow
7C3×2−1+4C2211yescharacteristic
8C3×AGL(1,ℤ/8)—21no—
9C3×#124C2211yescharacteristic
10#951—22no—
11#944C2212yes—
12#935—21no—
13#879—22no—
14#878—22no—
15#870—21no—
16C3×#104—21no—
17C3×#101C2211yescharacteristic
18#837—22no—
19#836C2212yes—
20#563—31no—
21#456—32no—
22#454D311yescharacteristic
23#356—32no—
24#355—31no—
25C3×#41—22no—
26C3×#41—21no—
27C3×#41—41no—
28C2×C3×Q8—41no—
29C2×C3×Q8C2311yescharacteristic
30C2×C3×D4D411yescharacteristic
31C22×Dic3—21no—
32C2×Dic6D412yes—
33C3×SD16—41no—
34C3×D8—41no—
35C3×M16—21no—
36C3×C22⋊C4—22no—
37C3×C42D411yescharacteristic
38#217—22no—
39#217—21no—
40#216—42no—
41#214—42no—
42C4⋊Dic3—21no—
43C4⋊Dic3—21no—
44Dic3⋊C4—22no—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47C4×Dic3D412yes—
48#208—22no—
492−1+4C2×D311yescharacteristic
50#137—62no—
51AGL(1,ℤ/8)—21no—
52C4⋊Q8—32no—
53#126—62no—
54#125—61no—
55#124C2×D311yescharacteristic
56#123—61no—
57C22⋊Q8—62no—
58#104—21no—
59#104—62no—
60#101—32no—
61#101C2×D311yescharacteristic
62C23×C3—21no—
63C3×Q8—22no—
64C3×Q8—21no—
65C3×Q8—41no—
66C3×D4—22no—
67C3×D4—21no—
68C3×D4—41no—
69C3×D4—41no—
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—
77C2×Dic3—81no—
78Dic6—42no—
79C3×C8—41no—
80C3⋊C8—42no—
81#41—22no—
82#41—21no—
83#41—41no—
84C2×Q8—32no—
85C2×Q8—41no—
86C2×Q8C22×D311yescharacteristic
87C2×D4#6711yescharacteristic
88C22×C4—61no—
89Q16—122no—
90SD16—41no—
91SD16—122no—
92D8—41no—
93M16—21no—
94M16—62no—
95C4⋊C4—61no—
96C4⋊C4—62no—
97C4⋊C4—61no—
98C4⋊C4—62no—
99C4⋊C4—62no—
100C22⋊C4—22no—
101C22⋊C4—62no—
102C22⋊C4—61no—
103C42—32no—
104C42#6711yescharacteristic
105C22×C3—42no—
106C22×C3—41no—
107C22×C3—41no—
108C22×C3#12011yescharacteristic
109C3×C4—22no—
110C3×C4—21no—
111C3×C4—41no—
112C3×C4—41no—
113Dic3—42no—
114Dic3—42no—
115C23—21no—
116Q8—21no—
117Q8—22no—
118Q8—41no—
119Q8—122no—
120D4—21no—
121D4—22no—
122D4—41no—
123D4—41no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4—241no—
130C2×C4D3×D412yes—
131C2×C4C2×#6711yescharacteristic
132C8—41no—
133C8—122no—
134C2×C3—21no—
135C2×C3—81no—
136C2×C3—81no—
137C2×C3B(4,𝔽2)11yescharacteristic
138C22—42no—
139C22—41no—
140C22—41no—
141C22#93711yescharacteristic
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—122no—
147C4—122no—
148C3#220311yescharacteristic, complemented, Sylow
149C2—21no—
150C2—81no—
151C2—81no—
152C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
Fitting, prime radicalC3×#454C2
socleC2×C3B(4,𝔽2)
hypercenter#454D3
hypocenterC3#2203
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #5672C2semidirect
3. #5246C2semidirect
4. #4542C2semidirect
5. #4478C2semidirect
148. C3#2203semidirect

Maximal subgroups

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

subgroupindex
#56722
C3×#4542
#52462
#45422
#44782
#22033