Finite Group Databasebeta

← #70992

Subgroups — #70992

Order 384.

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

normal subgroups only

This group has 226 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×#455C211yesmaximal, characteristic
3#5243C212yesmaximal
4#4541C211yesmaximal, characteristic, complemented
5#4476C212yesmaximal, complemented
6#2201—31nomaximal, Sylow
7C3×H(5,𝔽2)C2211yescharacteristic
8C3×#137—21no—
9C3×#124C2211yescharacteristic
10#944—21no—
11#935—22no—
12#935C2212yes—
13#879—22no—
14#878—22no—
15#870—21no—
16C3×#104—21no—
17C3×#100C2211yescharacteristic
18#834—22no—
19#834C2212yes—
20#563—31no—
21#458—32no—
22#455D311yescharacteristic
23#354—31no—
24#352—32no—
25C3×#41—21no—
26C3×#41—41no—
27C2×C3×Q8D411yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—41no—
30C2×C3×D4C2311yescharacteristic
31C22×Dic3D412yes—
32C2×Dic6—21no—
33C3×Q16—41no—
34C3×SD16—41no—
35C3×M16—21no—
36C3×C22⋊C4—22no—
37C3×C42D411yescharacteristic
38#217—22no—
39#217—22no—
40#217—42no—
41#217D412yes—
42C4⋊Dic3—21no—
43C4⋊Dic3—21no—
44Dic3⋊C4—22no—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47Dic3⋊C4—42no—
48C4×Dic3—21no—
49H(5,𝔽2)C2×D311yescharacteristic
50#137—21no—
51C4⋊Q8—61no—
52#126—62no—
53#125—61no—
54#124C2×D311yescharacteristic
55#123—32no—
56#123—62no—
57C22⋊Q8—62no—
58#104—21no—
59#100C2×D311yescharacteristic
60#99—32no—
61#99—62no—
62C23×C3—22no—
63C23×C3C2×D412yes—
64C3×Q8—21no—
65C3×Q8—41no—
66C3×D4—21no—
67C3×D4—41no—
68C3×D4—42no—
69C3×D4—41no—
70C2×C3×C4—22no—
71C2×C3×C4—21no—
72C2×C3×C4—21no—
73C2×C3×C4—41no—
74C2×C3×C4C2×D411yescharacteristic
75C2×Dic3—22no—
76C2×Dic3—22no—
77C2×Dic3—42no—
78C2×Dic3—42no—
79Dic6—81no—
80C3×C8—41no—
81#41—21no—
82#41—41no—
83C2×Q8—61no—
84C2×Q8#6711yescharacteristic
85C2×D4—22no—
86C2×D4—41no—
87C2×D4C22×D311yescharacteristic
88C22×C4—32no—
89Q16—41no—
90SD16—41no—
91M16—21no—
92C4⋊C4—61no—
93C4⋊C4—62no—
94C4⋊C4—62no—
95C4⋊C4—61no—
96C4⋊C4—62no—
97C4⋊C4—122no—
98C22⋊C4—22no—
99C22⋊C4—32no—
100C22⋊C4—62no—
101C22⋊C4—62no—
102C22⋊C4—122no—
103C42—61no—
104C42#6711yescharacteristic
105C22×C3—22no—
106C22×C3—42no—
107C22×C3—41no—
108C22×C3—82no—
109C22×C3#12011yescharacteristic
110C3×C4—21no—
111C3×C4—41no—
112C3×C4—41no—
113C3×C4—41no—
114Dic3—42no—
115Dic3—42no—
116Dic3—82no—
117C23—22no—
118C23D3×D412yes—
119Q8—21no—
120Q8—41no—
121Q8—241no—
122D4—21no—
123D4—41no—
124D4—42no—
125D4—41no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—62no—
131C2×C4—62no—
132C2×C4—122no—
133C2×C4—122no—
134C2×C4C2×#6711yescharacteristic
135C8—41no—
136C2×C3—21no—
137C2×C3—42no—
138C2×C3—81no—
139C2×C3B(4,𝔽2)11yescharacteristic
140C22—22no—
141C22—42no—
142C22—41no—
143C22—82no—
144C22#93711yescharacteristic
145C4—21no—
146C4—41no—
147C4—41no—
148C4—41no—
149C4—122no—
150C4—122no—
151C4—242no—
152C3#220111yescharacteristic, complemented, Sylow
153C2—21no—
154C2—42no—
155C2—81no—
156C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#455C2
socleC2×C3B(4,𝔽2)
prime radicalC3×H(5,𝔽2)C22
hypercenter#455D3
hypocenterC3#2201
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #5672C2semidirect
4. #4541C2semidirect
5. #4476C2semidirect
152. C3#2201semidirect

Maximal subgroups

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

subgroupindex
#56722
C3×#4552
#52432
#45412
#44762
#22013