Finite Group Databasebeta

← #2898⋊C3 #73238

Subgroups — #2898⋊C3 #73238

Order 384.

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

normal subgroups only

This group has 191 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
1C2×2−1+4⋊C3C211yesmaximal, characteristic
2C2×#131⋊C3C211yesmaximal, characteristic, complemented
3#520⋊C3C211yesmaximal, characteristic, complemented
4#2898C311yesmaximal, characteristic, complemented, Sylow
52−1+4⋊C3—21no—
62−1+4⋊C3C412yescomplemented
7C2×#41⋊C3C2211yescharacteristic
8C22×SL(2,𝔽3)—21no—
9#131⋊C3—21no—
10C4×SL(2,𝔽3)—21no—
11C3×#102—41nomaximal
12C2×2−1+4C2×C311yescharacteristic
13C2×#131C2×C311yescharacteristic, complemented
14#520C2×C311yescharacteristic, complemented
15#419—31no—
16#417—31no—
17#416—31no—
18C2×#103—31no—
19C2×C3×D4—41no—
20#41⋊C3C2×C411yescharacteristic
21#41⋊C3D411yescharacteristic
22C2×SL(2,𝔽3)—22no—
23C2×SL(2,𝔽3)D411yescharacteristic
24C2×C3×C8—41no—
25C3×C4⋊C4—41no—
262−1+4—21no—
272−1+4—32no—
282−1+4—61no—
292−1+4C3×C412yescomplemented
30C2×#41—31no—
31C2×#41—61no—
32C2×#41C22×C311yescharacteristic
33C22×Q8—21no—
34C22×Q8—31no—
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—32no—
44#103—31no—
45#103—31no—
46#102—31no—
47#102A411yescharacteristic
48C23×C3—81no—
49C3×D4—42no—
50C3×D4—81no—
51C2×C3×C4—41no—
52C2×C3×C4—81no—
53SL(2,𝔽3)C22⋊C411yescharacteristic
54C3×C8—81no—
55#41—31no—
56#41—31no—
57#41—32no—
58#41—32no—
59#41—61no—
60#41—61no—
61#41—64no—
62#41—62no—
63#41—62no—
64#41C2×C3×C411yescharacteristic
65#41C3×D411yescharacteristic
66C2×Q8—22no—
67C2×Q8—32no—
68C2×Q8—32no—
69C2×Q8—31no—
70C2×Q8—61no—
71C2×Q8—61no—
72C2×Q8—61no—
73C2×Q8—62no—
74C2×Q8C3×D411yescharacteristic
75C2×D4—31no—
76C2×D4—61no—
77C2×D4C2×A411yescharacteristic
78C22×C4—31no—
79C22×C4—61no—
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—
94C22×C3—82no—
95C22×C3—161no—
96C22×C3—161no—
97C3×C4—41no—
98C3×C4—41no—
99C3×C4—161no—
100C23—21no—
101C23—31no—
102Q8—31no—
103Q8—32no—
104Q8—62no—
105Q8—62no—
106Q8—61no—
107Q8C3×C22⋊C411yescharacteristic
108D4—21no—
109D4—31no—
110D4—31no—
111D4—32no—
112D4—62no—
113D4—62no—
114D4C4×A412yes—
115C2×C4—21no—
116C2×C4—31no—
117C2×C4—31no—
118C2×C4—31no—
119C2×C4—31no—
120C2×C4—31no—
121C2×C4—62no—
122C2×C4—62no—
123C2×C4—62no—
124C2×C4—61no—
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—
134C2×C3—162no—
135C22—22no—
136C22—31no—
137C22—31no—
138C22—41no—
139C22—41no—
140C22—61no—
141C22—61no—
142C22D4×A411yescharacteristic
143C4—31no—
144C4—31no—
145C4—41no—
146C4—62no—
147C4—121no—
148C4C2×C4×A411yescharacteristic
149C4D4×A411yescharacteristic
150C3—41noSylow
151C2—42no—
152C2—61no—
153C2—61no—
154C2A4×C22⋊C411yescharacteristic
155C2#577⋊C311yescharacteristic
156C2#578⋊C311yescharacteristic
subgroupG/H
center, socleC22D4×A4
commutator#41C2×C3×C4
FrattiniC2×C4C22×A4
Fitting#2898C3
prime radicalC2×2−1+4⋊C3C2
cosocleC2×#41C22×C3
hypercenter#102A4
hypocenterQ8C3×C22⋊C4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#131⋊C3C2semidirect
3. #520⋊C3C2semidirect
4. #2898C3semidirect
6. 2−1+4⋊C3C4semidirect
13. C2×#131C2×C3semidirect
14. #520C2×C3semidirect
29. 2−1+4C3×C4semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
C2×#131⋊C32
#520⋊C32
#28983
C3×#1024