Finite Group Databasebeta

← #85421

Subgroups — #85421

Order 384.

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

normal subgroups only

This group has 204 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#557⋊C3C211yesmaximal, characteristic, complemented
2#5395C211yesmaximal, characteristic, complemented
3#5395C211yesmaximal, characteristic, complemented
4#5394C211yesmaximal, characteristic, complemented
5C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
6C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3371—31nomaximal, Sylow
9C2×GL(2,𝔽3)C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#894—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15C4×SL(2,𝔽3)C2211yescharacteristic
16#859C2211yescharacteristic
17#559—31no—
18#557D311yescharacteristic, complemented
19#490—31no—
20#489—31no—
21#488—31no—
22#479—31no—
23#478—31no—
24#476—31no—
25#464—31no—
26#464—31no—
27#460—31no—
28#446—31no—
29#441—31no—
30#440—31no—
31C4×SD16—31no—
32C2×D12—41no—
33C2×C4×D3—41no—
34C2×SL(2,𝔽3)C2311yescharacteristic
35GL(2,𝔽3)—21no—
36GL(2,𝔽3)—21no—
37#226—21no—
38C3×C4⋊C4—41no—
39(C2×D3)⋊C4—41no—
40(C2×D3)⋊C4—41no—
41(C2×D3)⋊C4—41no—
42Dic3⋊C4—41no—
43C2×Q16—31no—
44C2×SD16—31no—
45C2×SD16—31no—
46C4⋊Q8—31no—
47#127—31no—
48#125—31no—
49#125—31no—
50#124—31no—
51#124—31no—
52#124—61no—
53#123—61no—
54#123—61no—
55C4×Q8—31no—
56C4×Q8—31no—
57C4×Q8C2×D311yescharacteristic
58C4×Q8C2×D311yescharacteristic
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62C4×D4—61no—
63#107—31no—
64#106—31no—
65#106—31no—
66C4⋊C8—31no—
67C4⋊C8—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#102—31no—
74#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78#102—31no—
79#97—31no—
80#97—31no—
81C4×C8—31no—
82C22×D3—41no—
83C22×D3—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89D12—82no—
90C4×D3—82no—
91SL(2,𝔽3)#4111yescharacteristic
92C2×Q8—31no—
93C2×Q8—31no—
94C2×Q8C22×D311yescharacteristic
95C2×D4—31no—
96C2×D4—31no—
97C2×D4—61no—
98C2×D4—61no—
99C22×C4—61no—
100C22×C4—61no—
101Q16—61no—
102Q16—61no—
103SD16—61no—
104SD16—61no—
105SD16—61no—
106SD16—61no—
107C2×C8—31no—
108C2×C8—31no—
109C2×C8—31no—
110C2×C8—31no—
111C4⋊C4—31no—
112C4⋊C4—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4S411yescharacteristic
120C22⋊C4—61no—
121C22⋊C4—61no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C42—31no—
127C42—31no—
128C42—31no—
129C42—61no—
130C22×C3—41no—
131C2×D3—42no—
132C2×D3—81no—
133C2×D3—81no—
134C2×D3—81no—
135C2×D3—81no—
136C2×D3—81no—
137C3×C4—81no—
138C3×C4—81no—
139C3×C4—81no—
140Dic3—81no—
141Dic3—81no—
142C23—61no—
143C23—61no—
144Q8—31no—
145Q8—61no—
146Q8—61no—
147Q8—121no—
148Q8#23911yescharacteristic
149D4—61no—
150D4—61no—
151D4—61no—
152D4—61no—
153D4—122no—
154D4—241no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—241no—
162C2×C4—241no—
163C2×C4C2×S411yescharacteristic
164C2×C4C2×S411yescharacteristic
165C2×C4C2×S411yescharacteristic
166C8—61no—
167C8—61no—
168C8—61no—
169C8—61no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—41no—
173D3—82no—
174D3—161no—
175C22—121no—
176C22—121no—
177C22—121no—
178C22—121no—
179C22—121no—
180C22—121no—
181C22C22×S411yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—31no—
186C4—31no—
187C4—61no—
188C4—61no—
189C4—61no—
190C4—121no—
191C4—121no—
192C3—41noSylow
193C2—241no—
194C2—241no—
195C2#591911yescharacteristic
196C2#592311yescharacteristic
197C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#557D3
prime radicalC4⋊GL(2,𝔽3)C2
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. #557⋊C3C2semidirect
2. #5395C2semidirect
3. #5395C2semidirect
4. #5394C2semidirect
6. C4×GL(2,𝔽3)C2semidirect
7. #5388C2semidirect
18. #557D3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
#53952
#53942
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
#53882
#33713
#8944