Finite Group Databasebeta

← #2897⋊C3 #73237

Subgroups — #2897⋊C3 #73237

Order 384.

193 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C2×#994C211yesmaximal, characteristic
2C2×#131⋊C3C211yesmaximal, characteristic, complemented
3#520⋊C3C211yesmaximal, characteristic, complemented
4#2897C311yesmaximal, characteristic, complemented, Sylow
5#994—21no—
6#994C412yescomplemented
7C2×#41⋊C3—21no—
8C2×#41⋊C3C2211yescharacteristic
9#131⋊C3—21no—
10C4×SL(2,𝔽3)—21no—
11C3×#103—41nomaximal
12C2×H(5,𝔽2)C2×C311yescharacteristic
13C2×#131C2×C311yescharacteristic
14#520C2×C311yescharacteristic
15#418—31no—
16#417—31no—
17#416—31no—
18C2×#102—31no—
19C2×C3×Q8—41no—
20#41⋊C3—22no—
21#41⋊C3C2×C411yescharacteristic
22#41⋊C3D411yescharacteristic
23C2×SL(2,𝔽3)D411yescharacteristic
24C2×C3×C8—41no—
25C3×C4⋊C4—41no—
26H(5,𝔽2)—21no—
27H(5,𝔽2)—32no—
28H(5,𝔽2)—61no—
29H(5,𝔽2)C3×C412yescomplemented
30C2×#41—21no—
31C2×#41—31no—
32C2×#41C22×C311yescharacteristic
33C22×D4—31no—
34C22×D4—61no—
35#131—21no—
36#131—61no—
37C2×M16—31no—
38C22×C8—31no—
39C4×Q8—21no—
40C4×D4—61no—
41#117—31no—
42C2×C4⋊C4—31no—
43#103—31no—
44#103A411yescharacteristic
45#102—32no—
46#102—31no—
47#102—31no—
48C3×Q8—42no—
49C3×Q8—81no—
50C2×C3×C4—41no—
51C2×C3×C4—81no—
52C2×C3×C4—81no—
53SL(2,𝔽3)C22⋊C411yescharacteristic
54C3×C8—81no—
55C24—61no—
56#41—22no—
57#41—31no—
58#41—31no—
59#41—32no—
60#41—32no—
61#41—61no—
62#41—61no—
63#41—62no—
64#41C2×C3×C411yescharacteristic
65#41C3×D411yescharacteristic
66C2×Q8C3×D411yescharacteristic
67C2×Q8C2×A411yescharacteristic
68C2×D4—31no—
69C2×D4—32no—
70C2×D4—32no—
71C2×D4—31no—
72C2×D4—61no—
73C2×D4—61no—
74C2×D4—61no—
75C2×D4—61no—
76C2×D4—64no—
77C2×D4—62no—
78C2×D4—62no—
79C22×C4—31no—
80C22×C4—61no—
81C22×C4—61no—
82M16—61no—
83M16—61no—
84C2×C8—31no—
85C2×C8—61no—
86C2×C8—61no—
87C2×C8C2×A411yescharacteristic
88C4⋊C4—31no—
89C4⋊C4—61no—
90C4⋊C4C2×A411yescharacteristic
91C22⋊C4—61no—
92C42—61no—
93C22×C3—41no—
94C3×C4—41no—
95C3×C4—41no—
96C3×C4—82no—
97C3×C4—161no—
98C23—31no—
99C23—62no—
100C23—62no—
101C23—61no—
102C23—241no—
103C23—241no—
104Q8—21no—
105Q8—31no—
106Q8C3×C22⋊C411yescharacteristic
107Q8C4×A412yes—
108D4—31no—
109D4—31no—
110D4—32no—
111D4—32no—
112D4—62no—
113D4—62no—
114D4—62no—
115D4—62no—
116D4—61no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—31no—
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—61no—
127C2×C4—241no—
128C2×C4C22×A411yescharacteristic
129C8—21no—
130C8—61no—
131C2×C3—41no—
132C2×C3—41no—
133C2×C3—41no—
134C22—31no—
135C22—31no—
136C22—61no—
137C22—61no—
138C22—62no—
139C22—121no—
140C22—121no—
141C22—242no—
142C22—242no—
143C22D4×A411yescharacteristic
144C4—22no—
145C4—31no—
146C4—31no—
147C4—41no—
148C4—121no—
149C4C2×C4×A411yescharacteristic
150C4D4×A411yescharacteristic
151C3—41noSylow
152C2—61no—
153C2—61no—
154C2—122no—
155C2A4×C22⋊C411yescharacteristic
156C2#576⋊C311yescharacteristic
157C2#577⋊C311yescharacteristic
subgroupG/H
center, socleC22D4×A4
commutator#41C2×C3×C4
FrattiniC2×C4C22×A4
Fitting#2897C3
prime radicalC2×#994C2
cosocleC2×#41C22×C3
hypercenter#103A4
hypocenterQ8C3×C22⋊C4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#131⋊C3C2semidirect
3. #520⋊C3C2semidirect
4. #2897C3semidirect
6. #994C4semidirect
29. H(5,𝔽2)C3×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#9942
C2×#131⋊C32
#520⋊C32
#28973
C3×#1034