Finite Group Databasebeta

← #85397

Subgroups — #85397

Order 384.

277 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 42 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
1D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5421C212yesmaximal, complemented
3C2×#859C212yesmaximal, complemented
4C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5386C211yesmaximal, characteristic, complemented
6#3298—31nomaximal, Sylow
7C2×#237—41nomaximal
8C22×SL(2,𝔽3)C2212yes—
9C2×GL(2,𝔽3)C2211yescharacteristic
10C4×SL(2,𝔽3)C2211yescharacteristic
11#859C2212yes—
12#859C2211yescharacteristic
13D4×Q8D311yescharacteristic, complemented
14#547—31no—
15#487—31no—
16#483—32no—
17#475—31no—
18#465—32no—
19#461—31no—
20#448—32no—
21C4×SD16—31no—
22C8×D4—31no—
23C2×#103—32no—
24C2×C3×D4—41no—
25C2×#67—42no—
26C22×Dic3—42no—
27#237—48no—
28C2×C4×D3—41no—
29C2×Dic6—41no—
30C2×SL(2,𝔽3)—22no—
31C2×SL(2,𝔽3)C2311yescharacteristic
32GL(2,𝔽3)—21no—
33C2×#41—61no—
34C22×Q8C2×D312yes—
35C2×SD16—31no—
36C2×SD16—32no—
37C22×C8—32no—
38C4⋊Q8—31no—
39C4⋊Q8—31no—
40#123—62no—
41C22⋊Q8—32no—
42C22⋊Q8—61no—
43#121—32no—
44C4×Q8C2×D311yescharacteristic
45C4×D4—31no—
46C4×D4—31no—
47C2×C4⋊C4—32no—
48#106—31no—
49#106—32no—
50C4⋊C8—31no—
51#103—32no—
52#103—32no—
53#103—32no—
54#103—31no—
55#103—32no—
56#102—31no—
57C22⋊C8—32no—
58C4×C8—31no—
59C23×C3—42no—
60C22×D3—41no—
61C3×D4—44no—
62C2×C3×C4—41no—
63#67—48no—
64C2×Dic3—42no—
65C2×Dic3—41no—
66C2×Dic3—48no—
67C4×D3—44no—
68Dic6—44no—
69SL(2,𝔽3)#4111yescharacteristic, complemented
70#41—124no—
71C2×Q8—22no—
72C2×Q8—32no—
73C2×Q8—61no—
74C2×Q8—62no—
75C2×Q8C22×D311yescharacteristic
76C2×D4—31no—
77C2×D4—62no—
78C2×D4S411yescharacteristic
79C22×C4—32no—
80C22×C4—62no—
81C22×C4—61no—
82SD16—61no—
83SD16—61no—
84SD16—122no—
85C2×C8—31no—
86C2×C8—31no—
87C2×C8—32no—
88C2×C8—122no—
89C4⋊C4—31no—
90C4⋊C4—31no—
91C4⋊C4—31no—
92C4⋊C4—32no—
93C4⋊C4—61no—
94C4⋊C4—62no—
95C4⋊C4—61no—
96C22⋊C4—32no—
97C22⋊C4—61no—
98C22⋊C4—62no—
99C42—31no—
100C22×C3—44no—
101C22×C3—41no—
102C22×C3—82no—
103C22×C3—82no—
104C2×D3—42no—
105C2×D3—81no—
106C2×D3—81no—
107C3×C4—42no—
108Dic3—44no—
109Dic3—42no—
110C23—61no—
111C23C2×S412yes—
112Q8—31no—
113Q8—62no—
114Q8—62no—
115Q8—122no—
116Q8#23711yescharacteristic, complemented
117D4—61no—
118D4—61no—
119D4—124no—
120D4GL(2,𝔽3)14yescomplemented
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—32no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—61no—
127C2×C4—62no—
128C2×C4—124no—
129C2×C4—122no—
130C2×C4C2×S411yescharacteristic
131C8—61no—
132C8—62no—
133C8—61no—
134C2×C3—41no—
135C2×C3—41no—
136C2×C3—41no—
137C2×C3—84no—
138D3—82no—
139C22—22no—
140C22—22no—
141C22—121no—
142C22—121no—
143C22—121no—
144C22C2×GL(2,𝔽3)14yes—
145C22C22×S411yescharacteristic
146C4—31no—
147C4—31no—
148C4—62no—
149C4—61no—
150C4—121no—
151C4—122no—
152C4C2×GL(2,𝔽3)12yes—
153C3—41noSylow
154C2—24no—
155C2—241no—
156C2#591411yescharacteristic
157C2C22×GL(2,𝔽3)11yescharacteristic
158C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingD4×Q8D3
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×SL(2,𝔽3)C2semidirect
2. #5421C2semidirect
3. C2×#859C2semidirect
4. C4×GL(2,𝔽3)C2semidirect
5. #5386C2semidirect
13. D4×Q8D3semidirect
69. SL(2,𝔽3)#41semidirect
116. Q8#237semidirect
120. D4GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
D4×SL(2,𝔽3)2
#54212
C2×#8592
C4×GL(2,𝔽3)2
#53862
#32983
C2×#2374