Finite Group Databasebeta

← #85369

Subgroups — #85369

Order 384.

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

normal subgroups only

This group has 207 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#520⋊C3C211yesmaximal, characteristic, complemented
2#5427C211yesmaximal, characteristic
3#5425C211yesmaximal, characteristic, complemented
4#5395C211yesmaximal, characteristic, complemented
5C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5388C211yesmaximal, characteristic, complemented
7#5386C211yesmaximal, characteristic, complemented
8#3125—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#896—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#542—31no—
18#520D311yescharacteristic, complemented
19#483—31no—
20#481—31no—
21#476—31no—
22#475—31no—
23#464—31no—
24#463—31no—
25#452—31no—
26#449—31no—
27#442—31no—
28#441—31no—
29C4×SD16—31no—
30C4×SD16—31no—
31#433—31no—
32C2×C4×D3—41no—
33C2×Dic6—41no—
34#41⋊C3—21no—
35C2×SL(2,𝔽3)C2311yescharacteristic
36GL(2,𝔽3)—21no—
37#226—21no—
38C3×C4⋊C4—41no—
39(C2×D3)⋊C4—41no—
40(C2×D3)⋊C4—41no—
41Dic3⋊C4—41no—
42Dic3⋊C4—41no—
43C2×#41C2×D311yescharacteristic
44C2×Q16—31no—
45C2×SD16—31no—
46C2×SD16—31no—
47C2×D8—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×Q8C2×D311yescharacteristic
58C4×Q8C2×D311yescharacteristic
59C4×D4—31no—
60C4×D4—31no—
61C4×D4—31no—
62#117—31no—
63C2×C4⋊C4—31no—
64#106—31no—
65#106—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#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#97—31no—
81C4×C8—31no—
82C22×D3—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89C4×D3—82no—
90Dic6—82no—
91SL(2,𝔽3)#4111yescharacteristic
92#41—21no—
93#41—61no—
94C2×Q8—31no—
95C2×Q8—61no—
96C2×Q8C22×D311yescharacteristic
97C2×D4—31no—
98C2×D4—31no—
99C2×D4—61no—
100C22×C4—31no—
101C22×C4—61no—
102C22×C4—61no—
103Q16—61no—
104Q16—61no—
105SD16—62no—
106SD16—61no—
107SD16—61no—
108D8—62no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—31no—
112C2×C8—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—
121C4⋊C4S411yescharacteristic
122C22⋊C4—31no—
123C22⋊C4—31no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C42—31no—
130C42—31no—
131C42—61no—
132C22×C3—41no—
133C2×D3—42no—
134C2×D3—81no—
135C2×D3—81no—
136C3×C4—81no—
137C3×C4—81no—
138C3×C4—81no—
139Dic3—42no—
140Dic3—81no—
141Dic3—81no—
142C23—31no—
143C23—61no—
144Q8—31no—
145Q8—61no—
146Q8—61no—
147Q8—241no—
148Q8#23511yescharacteristic
149D4—32no—
150D4—61no—
151D4—61no—
152D4—61no—
153D4—241no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—241no—
163C2×C4—241no—
164C2×C4C2×S411yescharacteristic
165C2×C4C2×S411yescharacteristic
166C2×C4C2×S411yescharacteristic
167C8—61no—
168C8—61no—
169C8—61no—
170C8—61no—
171C2×C3—41no—
172C2×C3—41no—
173C2×C3—41no—
174D3—82no—
175C22—61no—
176C22—61no—
177C22—61no—
178C22—121no—
179C22—121no—
180C22—121no—
181C22C22×S411yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—31no—
186C4—31no—
187C4—61no—
188C4—61no—
189C4—121no—
190C4—121no—
191C4—121no—
192C3—41noSylow
193C2—121no—
194C2—241no—
195C2#591211yescharacteristic
196C2#592411yescharacteristic
197C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#520D3
prime radical#5427C2
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #520⋊C3C2semidirect
3. #5425C2semidirect
4. #5395C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
6. #5388C2semidirect
7. #5386C2semidirect
18. #520D3semidirect

Maximal subgroups

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

subgroupindex
#520⋊C32
#54272
#54252
#53952
C4×GL(2,𝔽3)2
#53882
#53862
#31253
#8964