Finite Group Databasebeta

← #85356

Subgroups — #85356

Order 384.

208 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 32 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#519⋊C3C211yesmaximal, characteristic, complemented
2#5421C211yesmaximal, characteristic
3#5419C211yesmaximal, characteristic, complemented
4#5395C211yesmaximal, characteristic, complemented
5C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5391C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3121—31nomaximal, Sylow
9C22×SL(2,𝔽3)C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#883—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#542—31no—
18#519D311yescharacteristic, complemented
19#485—31no—
20#482—31no—
21#477—31no—
22#475—31no—
23#464—31no—
24#464—31no—
25#451—31no—
26#448—31no—
27#441—31no—
28#440—31no—
29C4×Q16—31no—
30C4×SD16—31no—
31#433—31no—
32C2×#67—41no—
33C2×C4×D3—41no—
34C2×SL(2,𝔽3)—21no—
35C2×SL(2,𝔽3)C2311yescharacteristic
36GL(2,𝔽3)—21no—
37#226—21no—
38C3×C22⋊C4—41no—
39#217—41no—
40(C2×D3)⋊C4—41no—
41C4⋊Dic3—41no—
42Dic3⋊C4—41no—
43C22×Q8C2×D311yescharacteristic
44C2×Q16—31no—
45C2×Q16—31no—
46C2×SD16—31no—
47C2×SD16—31no—
48C4⋊Q8—31no—
49#126—61no—
50#124—31no—
51#124—61no—
52#123—61no—
53C22⋊Q8—31no—
54C22⋊Q8—61no—
55#121—31no—
56C4×Q8—31no—
57C4×Q8—31no—
58C4×Q8—31no—
59C4×Q8C2×D311yescharacteristic
60C4×Q8C2×D311yescharacteristic
61C4×D4—31no—
62#117—31no—
63#117—31no—
64#107—31no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69C4⋊C8—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#102—31no—
77#102—31no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#97—31no—
81C4×C8—31no—
82C23×C3—41no—
83C22×D3—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86#67—82no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89C2×Dic3—41no—
90C4×D3—82no—
91SL(2,𝔽3)#4111yescharacteristic
92C2×Q8—21no—
93C2×Q8—31no—
94C2×Q8—31no—
95C2×Q8—61no—
96C2×Q8—61no—
97C2×Q8C22×D311yescharacteristic
98C2×D4—31no—
99C2×D4—61no—
100C22×C4—31no—
101C22×C4—61no—
102Q16—61no—
103Q16—62no—
104Q16—61no—
105SD16—61no—
106SD16—62no—
107SD16—61no—
108C2×C8—31no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—31no—
112C4⋊C4—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C22⋊C4—31no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4S411yescharacteristic
128C42—31no—
129C42—31no—
130C42—61no—
131C42—61no—
132C22×C3—41no—
133C22×C3—81no—
134C22×C3—81no—
135C22×C3—81no—
136C2×D3—42no—
137C2×D3—81no—
138C2×D3—81no—
139C3×C4—81no—
140C3×C4—81no—
141Dic3—81no—
142Dic3—81no—
143Dic3—81no—
144C23—61no—
145C23C2×S411yescharacteristic
146Q8—31no—
147Q8—32no—
148Q8—61no—
149Q8—61no—
150Q8—61no—
151Q8—241no—
152Q8#23511yescharacteristic
153D4—61no—
154D4—61no—
155D4—241no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—241no—
166C2×C4C2×S411yescharacteristic
167C2×C4C2×S411yescharacteristic
168C8—61no—
169C8—61no—
170C8—61no—
171C8—61no—
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—161no—
176D3—82no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—121no—
181C22—121no—
182C22—121no—
183C22C22×S411yescharacteristic
184C4—21no—
185C4—21no—
186C4—31no—
187C4—31no—
188C4—61no—
189C4—61no—
190C4—61no—
191C4—121no—
192C4—121no—
193C4—121no—
194C3—41noSylow
195C2—41no—
196C2—241no—
197C2#591211yescharacteristic
198C2#592611yescharacteristic
199C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#519D3
prime radical#5421C2
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #519⋊C3C2semidirect
3. #5419C2semidirect
4. #5395C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
6. #5391C2semidirect
7. #5388C2semidirect
18. #519D3semidirect

Maximal subgroups

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

subgroupindex
#519⋊C32
#54212
#54192
#53952
C4×GL(2,𝔽3)2
#53912
#53882
#31213
#8834