Finite Group Databasebeta

← #3025⋊C3 #73259

Subgroups — #3025⋊C3 #73259

Order 384.

171 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 18 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#5950C211yesmaximal, characteristic, complemented
2#458⋊C3C211yesmaximal, characteristic, complemented
3B(4,𝔽2)⋊C3C211yesmaximal, characteristic, complemented
4#3025C311yesmaximal, characteristic, complemented, Sylow
5#997C2211yescharacteristic
6C2×C4×A4—41nomaximal
7#585C2×C311yescharacteristic, complemented
8#525—31no—
9#458—31no—
10#458C2×C311yescharacteristic, complemented
11B(4,𝔽2)—31no—
12B(4,𝔽2)C2×C311yescharacteristic
13#410—31no—
14C22×A4—41no—
15#41⋊C3—81no—
16C4×A4—41no—
17C4×A4—41no—
18C4×A4—81no—
192−1+4—61no—
20H(5,𝔽2)—31no—
21H(5,𝔽2)—61no—
22H(5,𝔽2)C22×C311yescharacteristic
23C2×#41—31no—
24C2×#41—61no—
25C2×#41—61no—
26C23×C4A411yescharacteristic, complemented
27#123—31no—
28#123—31no—
29C22⋊Q8—61no—
30#121—61no—
31#120—31no—
32#120—31no—
33C4×D4—61no—
34C4×D4—61no—
35#117—31no—
36#99—32no—
37#99—31no—
38#99—31no—
39C2×A4—42no—
40C2×A4—41no—
41C2×A4—81no—
42C2×C3×C4—161no—
43SL(2,𝔽3)—81no—
44C24C2×A411yescharacteristic
45#41—21no—
46#41—62no—
47#41—62no—
48#41—61no—
49#41—62no—
50#41—61no—
51#41—62no—
52#41—62no—
53#41—61no—
54C2×Q8—31no—
55C2×Q8—61no—
56C2×Q8—61no—
57C2×D4—31no—
58C2×D4—31no—
59C2×D4—31no—
60C2×D4—61no—
61C2×D4—62no—
62C2×D4—61no—
63C2×D4—61no—
64C2×D4—61no—
65C2×D4—61no—
66C2×D4—61no—
67C22×C4—21no—
68C22×C4—31no—
69C22×C4—31no—
70C22×C4—61no—
71C22×C4—61no—
72C22×C4—61no—
73C22×C4C2×A411yescharacteristic
74C22×C4C2×A411yescharacteristic
75C4⋊C4—61no—
76C4⋊C4—61no—
77C4⋊C4—61no—
78C22⋊C4—31no—
79C22⋊C4—31no—
80C22⋊C4—61no—
81C22⋊C4—61no—
82C22⋊C4—61no—
83C22⋊C4—61no—
84C42—61no—
85C22×C3—161no—
86A4—41no—
87A4—81no—
88C3×C4—161no—
89C3×C4—161no—
90C23—21no—
91C23—31no—
92C23—31no—
93C23—42no—
94C23—61no—
95C23—61no—
96C23C22×A411yescharacteristic
97Q8—21no—
98Q8—62no—
99Q8—61no—
100D4—62no—
101D4—62no—
102D4—62no—
103D4—61no—
104D4—61no—
105D4—62no—
106D4—122no—
107D4—122no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—31no—
112C2×C4—41no—
113C2×C4—61no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—62no—
117C2×C4—61no—
118C2×C4—61no—
119C2×C4—61no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—121no—
124C2×C3—162no—
125C2×C3—161no—
126C22—31no—
127C22—41no—
128C22—41no—
129C22—62no—
130C22—61no—
131C22—61no—
132C22—81no—
133C22—121no—
134C22—121no—
135C22—121no—
136C22—241no—
137C4—31no—
138C4—41no—
139C4—61no—
140C4—61no—
141C4—121no—
142C4—121no—
143C4#120⋊C311yescharacteristic
144C3—161noSylow
145C2—42no—
146C2—61no—
147C2—121no—
148C2—121no—
149C2C2×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC4#120⋊C3
commutator, cosocle, hypocenterH(5,𝔽2)C22×C3
FrattiniC23C22×A4
Fitting#3025C3
socleC2C2×#120⋊C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5950C2semidirect
2. #458⋊C3C2semidirect
3. B(4,𝔽2)⋊C3C2semidirect
4. #3025C3semidirect
7. #585C2×C3semidirect
10. #458C2×C3semidirect
26. C23×C4A4semidirect

Maximal subgroups

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

subgroupindex
#59502
#458⋊C32
B(4,𝔽2)⋊C32
#30253
C2×C4×A44