Finite Group Databasebeta

← #73074

Subgroups — #73074

Order 384.

162 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 24 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×#983C211yesmaximal, characteristic
2#519⋊C3C211yesmaximal, characteristic, complemented
3#5420C211yesmaximal, characteristic, complemented
4#1905—31nomaximal, Sylow
5C22×SL(2,𝔽3)C2211yescharacteristic
6#983—21no—
7#983—21no—
8C2×GL(2,𝔽3)C411yescharacteristic, complemented
9C2×#226C411yescharacteristic, complemented
10C4×SL(2,𝔽3)—21no—
11#859—21no—
12#806—41nomaximal
13C2×#137—31no—
14#519D311yescharacteristic, complemented
15#421—31no—
16#419—31no—
17#410—31no—
18#344—31no—
19H(3,ℤ/4)—31no—
20C2×#67—41no—
21C2×SL(2,𝔽3)C2×C411yescharacteristic
22C2×SL(2,𝔽3)D411yescharacteristic
23C2×SL(2,𝔽3)D411yescharacteristic
24GL(2,𝔽3)—21no—
25#226—21no—
26C3×C22⋊C4—41no—
27#217—41no—
28C2×#41—31no—
29C22×Q8C2×D311yescharacteristic
30#137—61no—
31#137—61no—
32#137—61no—
33#137—61no—
34C2×Q16—31no—
35C2×Q16—31no—
36C2×SD16—31no—
37C2×SD16—31no—
38C2×M16—31no—
39C2×M16—31no—
40C4×Q8—21no—
41C4×Q8—61no—
42#117—31no—
43#117—31no—
44#117—31no—
45#104—62no—
46#103—61no—
47#103—61no—
48#99—61no—
49#99—61no—
50C23×C3—41no—
51C22×D3—41no—
52C2×C3×C4—81no—
53#67—81no—
54#67—81no—
55C2×Dic3—41no—
56C2×Dic3—81no—
57SL(2,𝔽3)C22⋊C411yescharacteristic
58#41—61no—
59#41—61no—
60#41—61no—
61#41—61no—
62C2×Q8—31no—
63C2×Q8—31no—
64C2×Q8—31no—
65C2×Q8—31no—
66C2×Q8C4×D311yescharacteristic
67C2×Q8D1211yescharacteristic
68C2×Q8#6711yescharacteristic
69C2×D4—31no—
70C2×D4—61no—
71C22×C4—31no—
72C22×C4—61no—
73Q16—61no—
74Q16—61no—
75Q16—61no—
76Q16—61no—
77SD16—61no—
78SD16—61no—
79SD16—61no—
80SD16—61no—
81M16—61no—
82M16—61no—
83M16—61no—
84M16—61no—
85C2×C8—31no—
86C2×C8—31no—
87C2×C8—61no—
88C4⋊C4—61no—
89C4⋊C4—61no—
90C4⋊C4—61no—
91C22⋊C4—31no—
92C22⋊C4—61no—
93C22⋊C4S411yescharacteristic
94C42—32no—
95C42—61no—
96C42—61no—
97C22×C3—41no—
98C22×C3—41no—
99C22×C3—41no—
100C22×C3—161no—
101C2×D3—81no—
102C2×D3—161no—
103C3×C4—161no—
104Dic3—81no—
105Dic3—161no—
106C23—61no—
107C23C2×S411yescharacteristic
108Q8—31no—
109Q8—31no—
110Q8—31no—
111Q8—61no—
112Q8—61no—
113Q8—61no—
114Q8(C2×D3)⋊C411yescharacteristic
115D4—61no—
116D4—61no—
117D4—121no—
118D4—121no—
119C2×C4—21no—
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—31no—
125C2×C4—31no—
126C2×C4—61no—
127C2×C4—61no—
128C2×C4—121no—
129C2×C4—121no—
130C2×C4—121no—
131C8—61no—
132C8—61no—
133C8—121no—
134C2×C3—41no—
135C2×C3—81no—
136C2×C3—81no—
137C2×C3—81no—
138D3—161no—
139C22—41no—
140C22—121no—
141C22—241no—
142C22C4×S411yescharacteristic
143C22C4⋊S411yescharacteristic
144C22GL(2,ℤ/4)11yescharacteristic
145C4—31no—
146C4—31no—
147C4—31no—
148C4—31no—
149C4—41no—
150C4—121no—
151C4—121no—
152C4—241no—
153C3—41noSylow
154C2—21no—
155C2—21no—
156C2—21no—
157C2—241no—
158C2(C2×S4)⋊C411yescharacteristic
subgroupG/H
center, socleC2(C2×S4)⋊C4
commutatorC2×SL(2,𝔽3)C2×C4
FrattiniC23C2×S4
Fitting#519D3
prime radicalC2×#983C2
cosocleC22×SL(2,𝔽3)C22
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #519⋊C3C2semidirect
3. #5420C2semidirect
8. C2×GL(2,𝔽3)C4semidirect
9. C2×#226C4semidirect
14. #519D3semidirect

Maximal subgroups

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

subgroupindex
C2×#9832
#519⋊C32
#54202
#19053
#8064